VOCATIONAL
MATHEMATICS
FOR GIRLS
BY
WILLIAM H. DOOLEY
AUTHOR OF "VOCATIONAL MATHEMATICS
"TEXTILES," ETC.
D. C. HEATH & CO., PUBLISHERS
BOSTON NEW YORK CHICAGO
COPYRIGHT, 1917,
BY D. C. HEATH & Co.
IB?
PREFACE
THE author has had, during the last ten years, considerable
experience in organizing and conducting intermediate and sec-
ondary technical schools for boys and girls. During this time
he has noticed the inability of the average teacher in mathe-
matics to give pupils practical applications of the subject.
Many teachers are not familiar with the commercial and rule
of thumb methods of solving mathematical problems of every-
day life. Too often a girl graduates from the course in mathe-
matics without being able to " commercialize " or apply her
mathematical knowledge in such a way as to meet the needs
of trade, commerce, and home life.
It is to overcome this difficulty that the author has prepared
this book on vocational mathematics for girls. He does not
believe in omitting the regular secondary school course in
mathematics, but offers vocational mathematics as an introduc-
tion to the regular course.
The problems have been used by the author during the past
few years with girls of high school age. . The method of teach-
ing has consisted in arousing an interest in mathematics by
showing its value in daily life. Important facts, based upon
actual experience and observation, are recalled to the pupil's
mind before she attempts to solve the problems.
A discussion of each division of the subject usually precedes
the problems. This information is provided for the regular
teacher in mathematics who may not be familiar with the
subject or the terms used. The book contains samples of
iii
A 1
IV PREFACE
problems from all occupations that women are likely to enter,
from the textile mill to the home.
The author has received valuable suggestions from his for-
mer teachers and from the following : Miss Lilian Baylies
Green, Editor Ladies' Home Journal, Philadelphia, Pa. ; Miss
Bessie Kingman, Brockton High School, Brockton, Mass. ; Mrs.
Ellen B. McGowan, Teachers College, New York City ; Miss
Susan Watson, Instructor at Peter Bent Brigham Hospital,
Boston ; Mr. Prank F. Murdock, Principal Normal School,
North Adams, Mass. ; Mr. Frank Rollins, Principal Bushwick
High School, Brooklyn ; Mr. George M. Lattimer, Mechanics
Institute, Rochester, N. Y. ; Mr. J. J. Eaton, Director of In-
dustrial Arts, Yonkers, N. Y. ; Dr. Mabel Belt, Baltimore, Md. ;
Mr. Curtis J. Lewis, Philadelphia, Pa. ; Mrs. F. H. Consalus,
Washington Irving High School, New York City ; Miss Griselda
Ellis, Girls' Industrial School, Newark, N. J. ; Mr. J. C. Dono-
hue, Technical High School, Syracuse, N. Y. ; Mr. W. E. Weaf er,
Hutchinson-Central High School, Buffalo, N. Y. ; The Bur-
roughs Adding Machine Company ; The Women's Educational
and Industrial Union ; the Department of Agriculture, Wash-
ington, D. C. ; and Reports of Conference of New York State
Vocational Teachers.
This preface would not be complete without reference to
the author's wife, Mrs. Ellen V. Dooley, who has offered many
valuable suggestions and corrected both the manuscript and
the proof.
The author will be pleased to receive any suggestions or
corrections from any teacher.
CONTENTS
PART I— REVIEW OF ARITHMETIC
CHAPTER PAGE
I. ESSENTIALS OF ARITHMETIC . ... . . 1
Fundamental Processes ; Fractions ; Decimals ; Com-
pound Numbers ; Percentage ; Ratio and Proportion ;
Involution ; Evolution.
II. MENSURATION . . . 64
Circles ; Triangles ; Quadrilaterals ; Polygons ; Ellipses ;
Pyramid ; Cone ; Sphere ; Similar Figures.
III. INTERPRETATION OF RESULTS ...... 80
Reading of Blue Print ; Plans of a Home ; Drawing to
Scale ; Estimating Distances and Weight ; Methods of
Solving Examples.
PART II — PROBLEMS IN HOMEMAKING
IV. THE DISTRIBUTION OF INCOME 89
Incomes of American Families ; Division of Income ;
Expense Account Book.
V. FOOD ,100
Different Kinds of Food ; Kitchen Weights and Meas-
ures ; Cost of Meals ; Recipes ; Economical Marketing.
VI. PROBLEMS ON THE CONSTRUCTION OF A HOUSE . . 128
Advantages of Different Types of Houses ; Building
Materials ; Taxes ; United States Revenue.
VII. COST OF FURNISHING A HOUSE 146
Different Kinds of Furniture ; Hall ; Floor Coverings ;
Linen ; Living Room ; Bedroom ; Dining Room ; Value
of Coal ; How to Read Gas Meters ; How to Read Elec-
tric Meters ; Heating.
VI CONTENTS
CHAPTER
VIII. THRIFT AND INVESTMENT 178
Different Institutions of Savings ; Bonds ; Stocks ; Ex-
change ; Insurance.
PART III — DRESSMAKING AND MILLINERY
IX. PROBLEMS IN DRESSMAKING 198
Fractions of a Yard ; Tucks ; Hem ; Ruffles ; Cost of
Finished Garments ; Millinery Problems.
X. CLOTHING . . .217
Parts of Cloth ; Materials of Yarn ; Kinds ; Weight.
PART IV— THE OFFICE AND THE STORE
XL ARITHMETIC FOR OFFICE ASSISTANTS .... 233
Rapid Calculations ; Invoices ; Profit and Loss ; Time
Sheets and Pay Rolls.
XII. ARITHMETIC FOR SALESGIRLS AND CASHIERS . . 260
Saleslips ; Extensions ; Making Change.
XIII. CIVIL SERVICE 268
PART V — ARITHMETIC FOR NURSES
XIV. ARITHMETIC FOR NURSES 276
Apothecary's Weights and Measures ; Household Meas-
ures ; Approximate Equivalents of Metric and English
Weights and Measures ; Doses ; Strength of Solutions ;
Prescription Reading.
PART VI — PROBLEMS ON THE FARM
XV. PROBLEMS ON THE FARM 304
APPENDIX 317
Metric System ; Graphs ; Formulas ; Useful Mechanical
Information.
INDEX 365
VOCATIONAL MATHEMATICS
FOR GIRLS
PART I — REVIEW OF ARITHMETIC
CHAPTER I
Notation and Numeration
A unit is one thing ; as, one book, one pencil, one inch.
A number is made up of units and tells how many units are
taken.
An integer is a whole number.
A single figure expresses a certain number of units and is said to be in
the units column. For example, 5 or 8 is a single figure in the units
column ; 53 is a number of two figures and has the figure 3 in the units
column and the figure 5 in the tens column, for the second figure
represents a certain number of tens. Each column has its own name,
as shown below.
Sp3 00 r- d
•9s ? o ~ * " m
| I | | 1 | I | | |
JjlJjsJjJJjl
138, 695, 4O 7, 125
Reading Numbers. — For convenience in reading and writing
numbers they are separated into groups of three figures each
by commas, beginning at the right :
138,695,407,125.
The first group is 125 units.
The second group is 407 thousands.
The third group is 695 millions.
The fourth group is 138 billions.
2 A\K7^d<TION;A^.MA*fPSEMATICS FOR GIRLS
The preceding number is read one hundred thirty-eight
billion, six hundred ninety-five million, four hundred seven
thousand, one hundred twenty-five ; or 138 billion, 695 million,
407 thousand, 125.
Roman Numerals
A knowledge of Roman numerals is very important. Dates
in buildings and amounts on prescriptions are usually expressed
in Roman numerals. They are also used for numbering
chapters and dials. The following capital letters, seven in
all, are used to express Roman numerals :
IIIVXLCD M
One Two Five Ten Fifty 100 500 1000
All other numbers are expressed by combining the letters
according to the following principles :
1. When a letter is repeated, the value is repeated. Thus,
II represents 2 ; XXX, 30.
2. When a letter of less value is placed after one of greater
value, the lesser is added to the greater. Thus, VII, 7 — two
added to five.
3. When a letter of less value is placed before one of greater
value, the lesser is taken from the greater. Thus, IX, 9 — ten
less one.
Read the following Roman numerals according to the above
rules :
1.
Ill
9.
XIX
17.
LXVI
2.
XXX
10.
LXXVII
18.
MDC
3.
ccc
11.
DCCCVII
19.
LXXII
4.
MMM
12.
XL
20.
CCLI
5.
VII
13.
XC
21.
DCLXVI
6.
LXXX
14.
IX
22.
DCXIV
7.
XXII
15.
XD
23.
MDXLVI
8.
XVIII
16.
XM
24.
MDCCXXIX
REVIEW OF ARITHMETIC 3
Express the following numbers in Roman numerals :
1. 14 4. 81 7. 281 10. 314 13. 1837
2. 42 5. 73 8. 509 11. 573 14. 1789
3. 69 6. 67 9. 812 12. 874 15. 80,003
Standard Mathematics Sheet. — To avoid errors in solving problems
the work should be done in such a way as to show each step and to make
it easy to check the answer when found. Paper of standard size, 8£ in.
by 11 in., should be used. Rule each sheet as in the following diagram,
set down each example with its proper number in the margin, and clearly
show the different steps required for the solution. To show that the
answer obtained is correct, the proof should follow the example.
STANDARD MATHEMATICS SHEET
1.
Mary Smith — 100
Hp Vocational Mathematics
\ 10-2-12 No. 10
1,203 2^
2,672 2?
31,118 23
480 10
39
19,883
'55,395 Ans.
2.
Proof:
3.
The pupil should write or print her name and class, the date when the
problem is finished, and the number of the problem on the Standard
4 VOCATIONAL MATHEMATICS FOR GIRLS
Mathematics Sheet. If the question contains several divisions or prob-
lems, they should be tabulated — (a), (/>), etc. — at the left of the prob-
lems inside the margin line. A line should be drawn between problems
to separate them.
Addition
Addition is the process of finding the sum of two or more
numbers. The result obtained by this process is called the
sum or amount.
The sign of addition is an upright cross, +- , called plus. The
sign is placed between the two numbers to be added.
Thus, 9 inches + 7 inches (read nine inches plus seven inches).
The sign of equality is two short horizontal parallel lines, = ,
and means equals or is equal to.
Thus, the statement that 8 feet + 6 feet = 14 feet, means that six feet
added to eight feet (or 8 feet plus 6 feet) equals fourteen feet.
To find the sum or amount of two or more numbers.
EXAMPLE. — An agent for a flour mill sold the following num-
ber of barrels of flour during the day : 1203, 2672, 31,118, 480,
39, and 19,883 bbl. How many barrels did he sell during the
day?
[The abbreviation for barrels is bbl.]
1,203 2$ The sum of the units column is 3 + 9 + 0
2 672 20 4-8 + 2 + 3 = 25 units, or 20 and 5 more ;
^1 1 1 k 9£ ^ ^s tens, so leave the 5 under the units
" column and add the 2 tens in the tens column.
*r The sum of the tens column is 2 + 8 + 3+8
39 +1+7 + 0 = 29 tens. 29 tens equal 2 hun-
19,883 dreds and 9 tens. Place the 9 tens under
Sum 55,395 bbl. the tens column and add the 2 hundreds
to the hundreds column. 2+8 + 4 + 1+6
+ 2 = 23 hundreds ; 23 hundreds are equal to 2 thousands and 3 hundreds.
Place the 3 hundreds under the hundreds column and add the 2 thousands
to the next column. 2 + 9 + 1 + 2+ 1 = 15 thousands, or 1 ten-thousand
and 5 thousands. Add the 1 ten-thousand to the ten-thousands column
REVIEW OF ARITHMETIC 5
and the sum is 1 + 1 + 3 = 5. Write the 5 in the ten-thousands column.
Hence, the sum is 55,395 bbl.
TEST. — Repeat the process, beginning at the top of the right-hand
column.
Exactness is very important in arithmetic. There is only
one correct answer. Therefore it is necessary to be accurate
in performing the numerical calculations. A check of some
kind should be made on the work. The simplest check is to
estimate the answer before solving the problem. If there is
a great discrepancy between the estimated answer and the
answer in the solution, the work is probably wrong. It is
also necessary to be exact in reading the problem.
EXAMPLES
1. Write the following numbers as figures and add them :
Seventy-five thousand three hundred eight ; seven million two
hundred five thousand eight hundred forty-nine.
2. In a certain year the total output of copper from the
mines was worth $ 58,638,277.86. Express this amount in words.
3. Solve the following :
386 + 5289 + 53666 + 3001 + 291 -f 38 = ?
4. The cost of the Panama Canal was estimated in 1912 to
be $ 375,000,000. Express this amount in words.
5. A farmer's wife received the following number of eggs
in four successive weeks : 692, 712, 684, and 705 eggs. How
many eggs were received during the four weeks ?
6. A woman buys a two-family house for $6511.00. She
makes the following repairs : mason-work, $ 112.00 ; plumb-
ing, $ 146.00 ; carpenter work, $ 208.00 ; painting and decora-
ting, $ 319.00. How much does the house cost her ?
7. Add the following numbers, left to right :
a. 108, 219, 374, 876, 763, 489, 531, 681, 104 ;
b. 3846, 5811, 6014, 8911, 7900, 3842, 5879.
6 VOCATIONAL MATHEMATICS FOR GIRLS
8. According to the census of 1910 the population of the
United States, exclusive of the outlying possessions, consisted
of 47,332,277 males and 44,639,989 females. What was the
total population?
9. Wire for electric lights was run around four sides of
three rooms. If the first room was 13 ft. long and 9 ft. wide ;
the second 18 ft. long and 18 ft. wide; and the third 12 ft.
long and 7 ft. wide, what was the total length of wire re-
quired? Remember that electric lights require two wires.
10. Find the sum :
46 Ib. + 135 Ib. + 72 Ib. + 39 Ib. + 427 Ib. + 64 Ib. + 139 Ib.
Subtraction
Subtraction is the process of finding the difference between
two numbers, or of finding what number must be added to a
given number to equal a given sum. The minuend is the num-
ber from which we subtract ; the subtrahend is the number
subtracted ; and the difference or remainder is the result of the
subtraction.
The sign of subtraction is a short horizontal line, — , called
minus, and is placed before the number to be subtracted.
Thus, 12 — 8 = 4 is read twelve minus (or less) eight equals four.
To find the difference of two numbers.
EXAMPLE. — A house was purchased for $ 8074.00 twenty-
five years ago. It was recently sold at auction for $ 4869.00.
What was the loss ?
Write the smaller number under the
Minuend $8074.00 greater, with units of the same order in
Subtrahend $4869.00 the same vertical line. 9 cannot be taken
Remainder $3205.00 from 4' so chan§e l *f *° unif 7Tbe 1
ten that was changed from the 7 tens
makes 10 units, which added to the 4 units makes 14 units. Take 9
from the 14 units and 5 units remain. Write the 5 under the unit col-
umn. Since 1 ten was changed from 7 tens, there are 6 tens left, and 6
from 6 leaves 0. Write 0 under the tens column. Next, 8 hundred can-
REVIEW OF ARITHMETIC 7
not be taken from 0 hundred, so 1 thousand (ten hundred) is changed
from the thousands column. 8 hundred from 10 hundred leave 2 hun-
dred. Write the 2 under the hundreds column. Since 1 thousand has
been taken from the 8 thousand, there are left 7 thousand to subtract the
4 thousand from, which leaves 3 thousand. Write 3 under the thousand
column. The whole remainder is 83205.00.
PROOF. — If the sum of the subtrahend and the remainder equals the
minuend, the answer is correct.
EXAMPLES
1. Subtract 1001 from 79,999.
2. A box contained one gross (144) of wood screws. If 48
screws were used on a job, how many screws were left in the
box?
3. What number must be added to 3001 to produce a sum
of 98,322?
4. Barrels are usually marked with the gross weight and tare
(weight of empty barrel). If a barrel of sugar is marked 329
Ib. gross weight and 19 Ib. tare, find the net weight of sugar.
5. A box contains a gross (144) of pencils. If 109 are
removed, how many remain?
6. A farmer received 1247 quarts of milk in October and
1189 quarts in November. What was the difference ?
7. A housewife purchases a $ 800.00 baby grand piano for
$ 719.00. How much does she save ?
8. No. 1 cotton yam contains 840 yards to the pound,
while No. 1 worsted yarn contains 560 yards to the pound.
What is the difference in length?
9. A young lady saved $453.00 during five years. She
spent $ 189.00 on a sea trip. How much remained ?
10. 69,221 - 3008 = ?
11. The population of New York City in 1900 was 3,437,202
and in 1910 was 4,766,883. What was the increase from 1900
to 1910?
8 VOCATIONAL MATHEMATICS FOR GIRLS
12. If there are 374,819 wage-earning women in a certain
city having a total population of 3,366,416 persons, how many of
the residents are not wage-earning women ?
13. In the year 1820 only 8385 immigrants arrived in the
United States. In 1842, 104,565 immigrants arrived. How
many more arrived in 1842 than in 1820 ?
14. The first great shoemaker settled in Lynn, Mass., in
1636. How many years is it since he arrived in Lynn ?
Multiplication
Multiplication is the process of rinding the product of two
numbers.
Thus, 8x3 may be read 8 multiplied by 3, or 8 times 3, and means
8 added to itself 3 times, or 8 + 8 + 8 = 24 and 8x3 = 24.
The numbers multiplied together are called factors. The
multiplicand is the number multiplied ; the multiplier is the
number multiplied by ; and the result is called the product.
The sign of multiplication is an oblique cross, x , which
means multiplied by or times.
Thus, 7x4 may be. read 7 multiplied by 4, or 7 times 4.
To find the product of two numbers.
EXAMPLE. — A certain set of books weighs 24 Ib. What is
the weight of 17 sets ?
Write the multiplier under the multipli-
Mvltiplicand 24 Ib. cand, units under units, tens under tens,
Multiplier 17 etc. 7 times 4 units equal 28 units, which
1(58 are 2 tens and 8 units. Place the 8 under
24 the units column. The 2 tens are to be
p , T7)« 11 added to the tens product. 7 times 2 tens
are 14 tens + the 2 tens are 16 tens, or 1
hundred and 6 tens. Place the 6 tens in the tens column and the 1 hun-
dred in the hundreds column. 168 is a partial product. To multiply by
the 1, proceed as before, but as 1 is a ten, write the first number, which
is 4 of this partial product, under the tens column, and the next number
under the hundreds column, and so on. Add the partial products, and
their sum is the whole product, or 408 Ib.
REVIEW OF ARITHMETIC 9
EXAMPLES
1. A milliner ordered 58 spools of wire, each spool contain-
ing 100 yards. How many yards did she order in all ?
2. Each shoe box contains 12 pairs of shoes. How many
pairs in 423 boxes ?
3. Multiply 839 by 291.
4. A mechanic sent in the following order for bolts : 12
bolts, 6 Ib. each; 9 bolts, 7 Ib. each; 11 bolts, 3 Ib. each; 6
bolts, 2 Ib. each; and 20 bolts, 3 Ib. each. What was the
total weight of the order ?
5. Find the product of 1683 and 809.
To multiply by 10, 100, 1000, etc., annex as many ciphers to
the multiplicand as there are ciphers in the multiplier.
EXAMPLE. — 864 x 100 = 86,400.
EXAMPLES
Multiply and read the answers to the following :
1. 869 x 10 8. 100 x 500
2. 1011 x 100 9. 1000 x 900
3. 10,389 x 1000 10. 10,000 x 500
4. 11,298 X 30,000 11. 10,000 x 6000
5. 58,999 x 400 12. 1,000,000 x 6000
6. 681,719 x 10 13. 1,891,717 x 400
7. 801,369 x 100 14. 10,000,059 x 78,911
Division
Division is the process of finding how many times one num-
ber is contained in another. The dividend is the number to be
divided ; the divisor is the number by which the dividend is
divided ; the quotient is the result of the division. When a
number is not contained an equal number of times in another
number, what is left over is called a remainder.
10 VOCATIONAL MATHEMATICS FOR GIRLS
The sign of division is -j-, and when placed between two
numbers signifies that the first is to be divided by the second.
Thus, 56 -s- 8 is read 56 divided by 8.
Division is also indicated by writing the dividend above the
divisor with a line between.
Thus, 5/ ; this is read 56 divided by 8.
In division we are given a product and one of the factors to
find the other factor.
To find how many times one number is contained in another.
EXAMPLE. — A manufacturer desires to distribute a surplus
of $ 8035.00 among his employees so that each one will re-
ceive $ 3.00 How many employees will receive $ 3.00 ? How
much is left over ?
Write the numbers in the manner
2678 Employees indicated at the left. 8 thousand is in
Divisor 3)8035 Dividend the thousands column. The nearest 8
g thousand can be divided into groups
2Q of 3 is 2 (thousand) times, which gives
.„ 6 thousand. Write 2 as the first
figure in the quotient over 8 in the
dividend. Place the 6 (thousand)
21 under the 8 thousand and subtract ;
25 the remainder is 2 thousand, or 20
24 hundred. 3 is contained in 20 hun-
Remainder ~T dred 6 hundred times, or 18 hundred
and 2 hundred remainder. Write 6
as the next figure in the quotient. Add the 3 tens in the dividend to the
2 hundred, or 20 tens, and 23 tens is the next dividend to be divided. 3 is
contained in 23 tens 7 times, or 21 tens with a remainder of 2 tens. Write
7 as the next figure in the quotient. 2 tens, or 20 units, plus the 5 units
from the quotient make 25 units. 3 is contained in 25, 8 times. Write 8
as the next figure in the quotient. 24 units subtracted from 25 units
leave a remainder of 1 unit. Then the answer is 2678 employees and
1 dollar left over.
PROOF. — Find the product of the divisor and quotient, add the re-
mainder, if any, and if the sum equals the dividend, the answer is correct.
REVIEW OF ARITHMETIC 11
EXAMPLES
1. A strip of sheeting measures 81" in width. How many
pieces 6" wide can be cut from it? Would there be a re-
mainder ?
2. How many pieces 6" long can be cut from a piece
of velvet 62" long, if no allowance is made for waste in
cutting ?
3. If the cost of constructing 162 miles of railway was
$ 4,561,200, what was the cost per mile ?
4. If a job which took 379 hours was divided equally
among 25 women, how many even hours would each woman
work, and how much overtime would one of the number have
to put in to complete the job?
5. The "over-all" dimension on a drawing was 18' 9".
The distance was to be spaced off into 14-inch lengths, begin-
ning at one end. How many such lengths could be spaced ?
How many inches would be left at the other end ?
6. If a locomotive consumed 18 gallons of fuel oil per mile
of freight service, how far could it run with 2036 gallons
of oil?
7. If 6 eggs weigh one pound, how many cases each
containing 36 eggs could be filled from a stock of 48 Ib. of
eggs?
8. The American people spend three hundred million dollars
every year on shoes, and average three pairs a person. What
is the average (wholesale) cost per pair, assuming that there
are 91,972,266 people in the United States ?
9. The enlisted strength of the army of the United States
in 1914 was 91,402 with an upkeep charge of $ 92,076,145.51.
What did it cost the United States per man to maintain its
standing army that year ?
10. Divide 38,910 by 3896.
12 VOCATIONAL MATHEMATICS FOR GIRLS
REVIEW EXAMPLES
1. A farmer's daughter raised on the farm 5 loads of pota-
toes containing 38 bu., 29 bu., 43 bu., 39 bu., and 29 bu.
respectively. She sold 12 bu. to each of three families, and
34 bu. to each of four families. How many bushels were
left?
2. Five pieces of cloth are placed end to end. If each
piece contains 38 yards, what is the total length ?
3. I bought a chair for $ 3, a mat for $ 1, a table for
$4, and gave in payment a $20 bill. What change did I
receive ?
4. A teacupful contains 4 fluid ounces. How many teacup-
fuls in 64 fluid ounces ?
5. No. 30 cotton yarn contains 25,200 yards to a pound.
How many pounds of yarn in 630,000 yards ?
6. The consumption of water in a city during the month
of December was 116,891,213 gallons and during January
115,819,729 gallons. How much was the decrease in con-
sumption ?
7. An order to a machine shop called for 598 sewing
machines each weighing 75 pounds. What was the total weight ?
8. If a strip of carpet weighs 4 Ib. per foot of length, find
the weight of one measuring 16' 9" in length.
9. Multiply 641 and 225.
10. Divide 24,566 by 319.
11. An order was given for ties for a railroad 847 miles
long. If each" mile required 3017 ties, how many ties would be
needed ?
12. How many gallons of milk are used every day by
two hospitals, if one uses 25 gallons per day and the other 6
gallons less ?
REVIEW. OF ARITHMETIC 13
Factors
The factors of a number are the integers which when multi-
plied together produce that number.
Thus, 21 is the product of 3 and 7 ; hence, 3 and 7 are the factors of 21.
Separating a number into its factors is called factoring.
A number that has no factors but itself and 1 is a prime
number.
The prime numbers up to 25 are 2, 3, 5, 7, 11, 13, 17, 19 and 23.
A prime number used as a factor is & prime factor.
Thus, 3 and 5 are prime factors of 15.
Every prime number except 2 and 5 ends with 1, 3, 7, or 9.
To find the prime factors of a number.
EXAMPLE. — Find the prime factors of 84.
2)84 The prime number 2 divides 84 evenly, leaving the quotient
2)42 ^' which 2 divides evenly. The next quotient is 21 which 3
ov>^ divides, giving a quotient 7. 7 divided by 7 gives the last
^— quotient 1 which is indivisible. The several divisors are the
1L prime factors. So 2, 2, 3, and 7 are the prime factors
1 of 84.
PROOF. — The product of the prime factors gives the number.
EXAMPLES
Find the prime factors :
1. 63 4. 636 7. 1155
2. 60 5. 1572 8. 7007
3. 250 6. 2800 9. 13104
Cancellation
To reject a factor from a number divides the number by that
factor ; to reject the same factors from both dividend and divisor
does not affect the quotient. This process is called cancellation.
This method can be used to advantage in many everyday cal-
culations.
EXAMPLE. — Divide 12 x 18 x 30 by 6 x 9 x 4.
14 VOCATIONAL MATHEMATICS FOR GIRLS
1
2 Jl 15 By this method it is not
Dividend }$ X J8 X £0 necessary to multiply be-
Divisor 0X9X4 =3° °UOft6n<' foredividing- locate
lj ^ ' the division by writing
the divisor under the divi-
1 dend with a line between.
Since 6 is a factor of 6
and 12, and 9 of 9 and 18, respectively, they may be cancelled from both
divisor and dividend. Since 2 in the dividend is a factor of 4 in the
divisor it may be cancelled from both, leaving 2 in the divisor. Then the
2 being a factor of 30 in the dividend, is cancelled from both, leaving 15.
The product of the uncancelled factors is 30. Therefore, the quotient
is 30.
PROOF. — If the product of the divisor and the quotient equal the
dividend, the answer is correct.
EXAMPLES
Indicate and find quotients by cancellation :
1. Divide 36 x 27 x 49 x 38 x 50 by 70 x 18 x 15.
2. What is the quotient of 36 x 48 X 16 divided by 27 x 24
X8?
3. How many pounds of tea at 50 cents a pound must be
given in exchange for 15 pounds of butter at 40 cents a
pound ?
4. There are 16 ounces in a pound ; 30 pounds of steel will
produce how many horseshoes, if each weighs 6 ounces ?
5. Divide the product of 10, 75, 9, and 96 by the product of
5, 12, 15, and 9.
6. I sold 16 dozen eggs at 30 cents a dozen and took my
pay in butter at 40 cents a pound : how many pounds did I
receive ?
7. A dealer bought 16 cords of wood at $ 4 a cord and sold
them for $ 96 ; find the gain per cord.
REVIEW OF ARITHMETIC 15
Greatest Common Divisor
The greatest common divisor of two or more numbers is the
greatest number that will exactly divide each of the numbers.
To find the greatest common divisor of two or more numbers.
EXAMPLE. — Find the greatest common divisor of 90 and
150.
90 = 2x3x5x3 2)90 150 First Method
150 = 2x3x5x5 5)45 75 The prime factors com-
Ans. 30 = 2 x 3 x 5 3)9 15 mon to both 90 aild 15°
~Q H are 2, 3, and 5. Since
2 x 3 x 5 = 30 Ans. the ^eatest common di~
visor of two or more num-
90)150(1 hers is the product of
go their common factors, 30
7^\\QA/-i is- tne greatest common
divisor of 90 and 150.
60
Greatest Common Divisor 30)60(2 Second Method
gQ To find the greatest
common divisor when
the numbers cannot be readily factored, divide the larger by the smaller,
then the last divisor by the last remainder until there is no remainder.
The last divisor will be the greatest common divisor. If the greatest com-
mon divisor is to be found of more than two numbers, find the greatest
common divisor of two of them, then of this divisor and the third num-
ber, and so on. The last divisor will be the greatest common divisor of
all of them.
EXAMPLES
Find the greatest common divisor :
1. 270, 810. 3. 504, 560. 5. 72, 153, 315, 2187.
2. 264,312. 4. 288,432,1152.
Least Common Multiple
The product of two or more numbers is called a multiple of
each of them ; 4, 6, 8, 12 are multiples of 2. The common
16 VOCATIONAL MATHEMATICS FOR GIRLS
multiple of two or more numbers is a number that is divisible
by each of the numbers without a remainder ; 60 is a common
multiple of 4, 5, 6.
The least common multiple of two or more numbers is the
smallest common multiple of the number; 30 is the least
common multiple of 3, 5, 6.
To find the least common multiple of two or more numbers.
EXAMPLE. — Find the least common multiple of 21, 28,
First Method
21 = 3 X 7 Take all the factors of the first number, all of
28 = 2 X 2 X J the second not already represented in the first, etc.
30 = 2 X £ X 5 Tnus>
3 x 7 x 2 x 2 x 5 = 420 L. C. M.
Second Method
2)21 28 30
3)21 14 15
7)7 14 5
125
2 x 3 x 7 x 1 X 2 x 5 = 420 L. C. M.
Divide any two or more numbers by a prime factor contained in them,
like 2 in 28 and 30. Write 21 which is not divided by the 2 for the next
quotient together with the 14 and 15. 3 is a prime factor of 21 and 15
which gives a quotient of 7 and 5 with 14 written in the quotient undi-
vided. 7 is a prime factor of 7 and 14 which gives a remainder of 1, 2 ;
and 5 undivided is written down as before. The product 420 of all these
divisors and the last quotients is the least common multiple of 21, 28,
and 30.
EXAMPLES
Find the least common multiple :
1. 18, 27, 30. 2. 15, 60, 140, 210. 3. 24, 42, 54, 360.
4. 25,20,35,40. 5. 24,48,96,192.
6. What is the shortest length of rope that can be cut into
pieces 32', 36', and 44' long?
REVIEW OF ARITHMETIC 17
Fractions
A fraction is one or more equal parts of a unit. If an apple
be divided into two equal parts, each part is one-half of the
apple, and is expressed by placing the number 1 above the
number 2 with a short line between : |-. A fraction always
indicates division. In 1, 1 is the dividend and 2 the divisor ;
1 is called the numerator and 2 is called the denominator.
A common fraction is one which is expressed by a numerator
written above a line and a denominator below. The nu-
merator and denominator are called the terms of the fraction.
A proper fraction is a fraction whose value is less than 1 ; its
numerator is less than its denominator, as f, £, f, |^-. An
improper fraction is a fraction whose value is 1 or more than 1;
its numerator is equal to or greater than its denominator, as f,
-}-§-. A number made up of an integer and a fraction is a
mixed number. Read with the word and between the whole
number and the fraction : 4T9g-, 3-J-, etc.
The value of a fraction is the quotient of the numerator
divided by the denominator. •
EXERCISE
Read the following :
1. f 3. 121 5. 51 7. 9^ 9. J
2. « 4. 81 6. 6J 8. 12A
Reduction of Fractions
Reduction of fractions is the process of changing their form
without changing their value.
To reduce a fraction to higher terms.
Multiplying the denominator and the numerator of the given
fraction by the same number does not change the value of the
fraction.
18 VOCATIONAL MATHEMATICS FOR GIRLS
EXAMPLE. — Reduce •§ to thirty-seconds.
The denominator must be multiplied by 4 to
_ X _ — _ Ans. obtain 32 ; so the numerator must be multiplied
8 4 32 by the same number in order that the value of
the fraction may not be changed.
EXAMPLES
Change the following :
1. | to 27ths. 6. /-j to 75ths.
2. -i-l to 60ths. 7. £§- to 144ths.
3. I to 40ths. 8. fj to 168ths.
4. | to 56ths. 9. || to 522ds.
5. T9Q to 50ths. 10. £ff to 9375ths.
A fraction is said to be in its lowest terms when the numera-
tor and the denominator are prime to each other.
To reduce a fraction to its lowest terms.
Dividing the numerator and the denominator of a fraction
by the same number does not change the value of the fraction.
The process of dividing the numerator and denominator of a
fraction by a number common to both may be continued until
the terms are prime to each other.
EXAMPLE. — Reduce |-| to fourths.
The denominator must be divided by 4 to give
12 3 j the new denominator 4 ; then the numerator must be
16 4 divided by the same number so as not to change the
value of the fraction.
If the terms of a fraction are large numbers, find their
greatest common divisor and divide both terms by that.
EXAMPLE. — Reduce f £f£ to fourths.
(1) 2166)2888(1 (2) 2166 = 3 ,
2166 2888 4 "
G. C. D. 722)2166(3
2166
REVIEW OF ARITHMETIC 19
EXAMPLES
Reduce to lowest terms :
1- A 3- Ml 5- H 7- W 9- ttt
2- »» 4- it 6. TW 8. HI 10.
To reduce an integer to an improper fraction.
EXAMPLE. — Reduce 25 to fifths.
25 times | = if* Ans. , *" l t5
25 times f , or if £.
To reduce a mixed number to an improper fraction.
EXAMPLE. — Reduce 16^ to an improper fraction.
_ I sevenths Since jn j there are ^ in 16 there must
112 be 16 times £, or i|a.
__4 sevenths H^ + I = 1**-
116 sevenths, = 1^.
EXAMPLES
Reduce to improper fractions :
1. 3£ 3. 17J 5. 13J- 7. 359T%
2. 16& 4. 121 6. 27^ 8. 482i|
9. 25^ 10. Reduce 250 to 16ths.
11. Change 156 to a fraction whose denominator shall be 12.
12. In $ 730 how many fourths of a dollar ?
13. Change 12f to 16ths. 14. Change 24| to 18ths.
To reduce an improper fraction to an integer or mixed number
divide the numerator by the denominator.
EXAMPLE: — Reduce -3T8g-5- to an integer or mixed number.
24
16)385
^2 Smce T! e(lual 1* ¥<r wil1 eQual as many
— — times 1 as 16 is contained in 385, or 24^
bo Z4:^Q Ans. timeSt
64
1
20 VOCATIONAL MATHEMATICS FOR GIRLS
EXAMPLES
Reduce to integers or mixed numbers :
1- tt * Aff± 7. 8J)f 1Q.
2. \v9- 5- -3¥<r6JL 8- W 11-
3. *«* 6. A™ 9. aflftn 12. ||
When fractions have the same denominator their denomi-
nator is called a common denominator.
Thus, |f, T%, y's have a common denominator.
The smallest common denominator of two or more fractions is
their least common denominator.
Thus, |§, T\, T2s become f, f , £ when changed to their least common
denominator.
The common denominator of two or more fractions is a
common multiple of their denominators.
The least common denominator of two or more fractions is
the least common multiple of their denominators.
EXAMPLE. — Reduce f and f to fractions having a common
denominator.
3. x 6. _ i_8. The common denominator must be a
* 4 _ 2 o common multiple of the denominators 4
= ^ 20 and 6' and Since 24 is the Product of the
¥ == 1TT anc*- ¥ = "2T denominators, it is a common multiple of
them. Therefore, 24 is a common denominator of f and f .
To reduce fractions to fractions having the least common denominator.
EXAMPLE. — Reduce -|, ^, and -£% to fractions having the
least common denominator.
2"> S 6 12 ^e least common de*
* nominator must be the
^)^ ^ ^ least common multiple of
112 the denominators 3, 6, 12,
2 x 3 x 2 = 12 L.C.M. which is 12.
I = J^. ; I = |f . _?_ _ _L. ^ws< Divide the least common
multiple 12 by the denom-
inator of each fraction, and multiply both terms by the quotient. If the
REVIEW OF ARITHMETIC 21
denominators should be prime to each other, their product would be their
least common denominator.
EXAMPLES
Reduce to fractions having a common denominator :
I- i, f 5. f , f , f
2 |,| 6. |, f,|-
3. f , i 7. 1 f , f, f
4- f T4> * 8- i> A. *, i
Reduce to fractions having least common denominator:
1- t, 1, -h * *, f > A, 4
2- i i A 6. f , f , J, |
3- AJ 2T> f 7. Which fraction is larger,
Addition of Fractions
Only fractions with a common denominator can be added.
If the fractions have not the same denominator, reduce them
to a common denominator, add their numerators, and place
their sum over the common denominator. The result should be
reduced to its lowest terms. If the result is an improper
fraction, it should be reduced to an integer or mixed number.
EXAMPLE. — Add J, ^, and T9g.
a ^a The least
1. 2)4 6 16 common multi-
2)2 3 8 pie of the de-
13 4 48 L. C. M. nominators is
nominator of each fraction and multiplying both terms by the quotient
give ff, |f, || . The fractions are now like fractions, and are added by
adding their numerators and placing the sum over the common denomi-
nator. Hence, the sum is -W/-, or 2?7^.
22 VOCATIONAL MATHEMATICS FOR GIRLS
EXAMPLE. — Add 5f , 7T^, and 6T7^.
First find the sum of the fractions,
which is f§> or iff- Add this to the
sum of the integers, 18. 18 + If § =
= 19|f . ^4rcs. 19M-
EXAMPLES
1. Find the " over-all " dimension of a drawing if the
separate parts measure T%", f ", |", and f", respectively.
2. Find the sum of |, }, -J-, |£, and f 1.
3. Find the sum of 3f , 4f , and 2TV
4. A seam T3g of an inch wide is made on both sides of a
piece of cloth 27 inches wide. What is the width after the
seams are made ?
5. I bought cotton cloth valued at $ 6 J, silk at $ 13J, hand-
kerchiefs for $2J, and hose for $2J. What was the whole
cost?
6. A ribbon was cut into two pieces, one 8f" and the other
5-fa" long. If Jg-" was allowed for waste in cutting, what was
the length of the ribbon ?
7. Three pieces of cloth contain 38^, 12-J-, and 53|- yards re-
spectively. What is their total length in yards ?
8. Add: 101 7f 11, if.
9. Add : 1361, 184 j, 416J, 125|.
Subtraction of Fractions
Only fractions with a common denominator can be sub-
tracted. If the fractions have not the same denominator,
reduce them to a common denominator and write the differ-
ence of their numerators over the common denominator. The
result should be reduced to its lowest terms.
REVIEW OF ARITHMETIC 23
EXAMPLE. — Subtract f from J.
The least common denominator of f
5 _ I = | _ |- = J. Ans. and f is 6. f = f , and f = f . Their
difference is £.
EXAMPLE. — From 11| subtract 5f .
•MI _ jQ_g When the fractions are changed to
® their least common denominator, they are
4B~ == 4TT n| _ 45 _ s cannot be subtracted from f,
6f = 6J- ^ns- hence 1 is taken from 11 units, changed to
sixths, and added to the f, which makes f . lOf — 4| = 6| = 6^.
EXAMPLES
1. From eleven yards of cloth, If yards were cut for a
jacket and 3J yards for a coat. How many yards were left ?
2. From a firkin of butter containing 271 lb. there were
sold 3| lb. and 11^ lb. How many pounds remained?
3. The sum of two fractions is f . One of the fractions is
^. Find the other.
4. Laura had $ 1\ and gave away $ 2^ and $ 3J. How
much remained ?
5. The sum of 2 numbers is 371 and one of the numbers is
28f . Find the other number.
o
6. By selling goods for $ 431, I lost $ 271. What was the
cost?
7. A man sells 9| yards from a piece of cloth containing
34 yds. How many yards remain ?
8. Mr. Brown sold goods for $ 56y3¥, gaining $ 12. What
did they cost ?
9. A dealer had 208 tons of coal and sold 92| tons. How
much remained ?
10. If I buy a ton of coal for $ 6J and sell for $ 71, how
much do I gain ?
24 VOCATIONAL MATHEMATICS FOR GIRLS
14. There were 48 J gallons in the tank. First 41 gallons
were used, then 5^ gallons, and last 2f gallons. How many
gallons were left in the tank ?
15. What is the difference between T9T and if ?
16. What is the difference between 32 J and 3£J ?
17. A piece of dress goods contains 60 yd. If four cuts
of 12 L, 9 1, 18f, and 101 yd. respectively are made, what
remains ?
Multiplication of Fractions
To multiply fractions, multiply the numerators together for the
neiv numerator and multiply the denominators together for the
new denominator.
Cancel when possible. The word of between two fractions
is equivalent to the sign of multiplication.
To multiply a mixed number by an integer, multiply the whole
number and the fraction separately by the integer then add the
products.
To multiply two mixed numbers, change each to an improper
fraction and multiply.
EXAMPLE. — Multiply | by f .
\ multiplied by f is the same as | of f . 3 and 5 are prime to each other
so that answer is f . This method of solution is the same as multiplying
the numerators together for a new numerator and the denominators for
a new denominator. Cancellation shortens the process.
EXAMPLE. — Find the product of 124 J and 5.
124f
~ If the fraction and integer are mul-
~^T * v s _ is _ QS tiplied separately by 5, the result is 5
6t ? times f = -V = 3f, and 5 times 124 =
620 620. 620 + 3f =623f .
623f Ans.
REVIEW OF ARITHMETIC 25
EXAMPLES
1. William earns 831 cents a day. How much will he
earn in five weeks ?
2. One bag of flour costs 75 cents. How much will three
barrels cost ? A barrel holds 8 bags.
3. From a barrel of flour containing 196 lb., 241- lb. were
taken. At another time \ of the remainder was taken. How
many pounds were left ?
4. Multiply J of f by f of f .
5. Multiply 26f by 91.
6. Find the cost of 19| yd. of cloth at 161 cents a yard.
7. At $ 121 each, how many tables can be bought for
$280?
8. I paid $ 6 1 for a barrel of flour and sold it for $ T97 more.
How much did I sell it for ?
9. What is the cost of 18 yards of cloth at 15J cents a yard ?
10. If coal cost $7-J- a ton, how much will 8J tons cost ?
11. Multiply : 32| by 8 j.
Division of Fractions
To divide one fraction by another, invert the divisor and
proceed as in multiplication of fractions. Change integers and
mixed numbers to improper fractions.
EXAMPLE. — Divide f x f by f x f .
A 3 /? H 3 The divisor f x f is inverted and the
^X-X^X^ = -. Ans. result obtained by the process of cancel-
5 0 ?> 5 lation.
26 VOCATIONAL MATHEMATICS FOR GIRLS
EXAMPLE. — Divide 3156f by 5.
Ans.
* When the integer of a mixed
30 number is large, it may be
15 divided as follows : 5 in 3156f ,
15 1 J = |- 631 times, with a remainder of
(^ If. This remainder divided by
K 6 gives 270, which is placed at
K the right of the quotient.
If
EXAMPLE. — Divide 3682 by 5J.
When the dividend is a large number and
5 1) 3682 the divisor a mixed number, it is useful to re-
2 2 member that multiplying both dividend and di-
TT \-oflyi ' visor by the same number does not change the
quotient. In this example we can multiply
boy Yj Ans. footh dividend and divisor by 2 and then divide
as with whole numbers. The quotient is 669T5T.
A fraction having a fraction for one or both of its terms is
called a complex fraction.
To reduce a complex fraction to a simple fraction.
EXAMPLE. — Reduce _I to a simple fraction.
6
Change 4f and 7f to improper fractions, J/ and 4/, respectively. Per-
form the division indicated with the aid of cancellation and the result will
be |f.
EXAMPLES
1. Divide^ by f 7. 296-=-10i = ?
2. Divide TV by f. 8. 28,769 ^7|=?
3. Divide |f by i. 7i_?
4. Divide $ by \. ' ft
5. Dividef by f. iofi
6. 384| -- 5 = ? 1 X
REVIEW OF ARITHMETIC 27
REVIEW PROBLEMS IN FRACTIONS
1. Two and one half yards of cloth cost $ 2.75. What is
the price per yard ?
2. An 8i-qt. can of milk is bought from a farmer for. 60
cents. What is the cost per quart ?
3. I paid 56 cents for f of a yard of lace. What was the
price per yard ?
4. A farmer's daughter sold a weekly supply of eggs for
$ 5.70. If she received 28^ cents a dozen, how many dozen
did she sell ?
5. If a narrow piece of goods, 6J yd. long, is cut into pieces
6} inches long, how many pieces can be cut? How much
remains ? Allow -J- in. for waste.
6. What is the cost of 18^ pounds of crackers at 17^ cents
a pound ?
7. A gallon (U. S. Standard capacity) contains 231 cubic
inches.
a. Give number of cubic inches in J gallon.
b. Give number of cubic inches in 1 quart.
c. Give number of cubic inches in 1 pint.
d. Give number of cubic inches in 1 pint.
8. A woman earns $ 2% a day. If she spends $ If, how
much does she save ? How many weeks (six full working
days) will it take to save $ 90 ?
9. I paid 56 cents for f of a yard of lace. What was the
price per yard ?
10. A furniture dealer sold a table for $ 141, a couch for
$ 45f , a desk for $ llf , and some chairs for $ 27T%. Find the
amount of his sales.
11. A woman had $ 200. She lost £ of it, gave away J the
remainder, and spent $ 20J. How much had she left ?
12. I gave $ 16J for 33 yards of cloth. How much did one
yard cost ?
28 VOCATIONAL MATHEMATICS FOR GIRLS
REVIEW OF ARITHMETIC 29
22.
i
-A
_ 9
33.
f -
i =
9
44.
i
"A
__ 9
23.
i
-A
__ 9
34.
t
A =
9
45.
|
-A
= ?
24.
i
-A
_ 9
35.
t
1 .
3T -
9
46.
•H-
- A
= ?
25.
i
-A
__ 9
36.
t -
A =
?
47.
T2
- A
_ 9
26.
&
-A
_ 9
37.
tt-
A =
9
48.
A
-A
_ 9
27.
T¥S
-A
_ 9
38.
i -
sV =
9
49.
I
-i
_ 9
28.
A
. JL
1 6
_ 9
39.
ti-
A =
9
50.
I
-i
_ 9
29.
A
-A
_ 9
40.
H-
A =
9
51.
i
"~ 8"
_ 9
30.
A
- A
r>
41.
A-
T2 =
9
52.
i
-TV
= ?
31.
i
i
~ "2
= ?
42.
it-
A =
'?
53.
1
-1T2
= ?
32.
f
-i
_ 9
43.
i -
if =
= ?
54.
i
-A
_ 9
Multiplication
1.
ix
1 =
9
19.
i x
2- =
9
37.
A
xi
_ 9
2.
ix
1
9
20.
t x
i =
9
38.
A
xi
= 9
3.
ix
1
8 '
9
21.
i x
i =
9
39.
A
xi
= 9
4.
ix
T6 =
9
22.
1 vx
1 _
1 6
9
40.
A
xA
= 9
5.
ix
A =
9
23.
i x
sV =
9
41.
A
x A
_ 9
6.
ix
A =
9
24,
i x
A =
9
42.
A
x A
_ 9
7.
ix
i =
9
25.
A x
i =
9
43.
A
xi
= 9
8.
ix
i =
9
26.
AX
i =
9
44.
A
xi
_ 9
9.
ix
i =
9
27.
AX
i
9
45.
6V
xi
= 9
10.
ix
A =
9
28.
AX
A =
9
46.
6\
xA
_ 9
11.
ix
A =
9
29.
AX
A =
9
47.
A'
x A
= 9
12.
ix
A =
9
30.
AX
i
6 4
9
48.
A
xA
_ 9
13.
fx
i =
9
31.
3 v
4 X
i =
9
49.
I
xi
= 9
14.
— X
4 =
9
32.
3 V
4 A
i =
9
50.
J
xi
_ 9
15.
f x
1
8
9
33.
3 x
i =
9
51.
i
xi
_ 9
16.
fx
A =
9
34.
f x
A =
9
52.
1
xA
_ 9
17.
f x
A =
9
35.
3 V
4 A
A =
9
53.
1
x A
= ?
18.
f x
A =
9
36.
f x
\
Q
54.
f
x A
_ 9
30 VOCATIONAL MATHEMATICS FOR GIRLS
Division
1.
i-*-i
_ 9
19.
i
"^ \
— 9
37.
3T
-5-J =?
2.
i . i
"2" ~ ¥
_ 9
20.
i
+4
= 9
38.
3V
. i o
3.
i + i
_ 9
21.
i
+ 4
9
39.
A
-7- 1 = ?
4.
I^-T6
_ 9
22.
i
8"
^-y1^
. — 9
40.
A
•*• T6 = ?
5.
i-s-A
= 9
23.
i
"^A
. = 9
41.
aV
^A=?
6.
i^-6¥
9
24.
i
-r- ^
.= 9
42.
sV
-t. 1 = ?
7.
1 . 1
¥ ~ 2
_ 9
25.
T6
. 1
~ 2"
_ 9
43.
A
H-J -?
8.
1 . 1
¥ ' ¥
= 9
26.
T6
+4
_ 9
44.
A
-^-i =?
9.
i-*-i
_ 9
27.
A
-i
_ 9
45.
A
_:_ 1 _ 9
10.
4 + A
_ 9
28.
T6
-5- T\
r— 9
46.
-h
-«- T 6 = ?
11.
i-i-A
— 9
29.
TV
-i- g\
, = ?
47.
A
••• -fa = ?
12.
i-^s-V
= 9
30.
iV
-*-A
. = 9
48.
6T
•*• A = ?
13.
4 + i
_ 9
31.
8
¥
. 1
~ 2"
_ 9
49.
1
-v-1 =?
14.
4 -"4
= ?
32.
I
_._ 1
= 9
50.
1
^-¥ =?
15.
4 + 4
= 9
33.
3
4
+ t
_ 9
51.
1
+ i =?
16.
i-iV
_ 9
34.
8
¥
^-Tl
r=?
52.
i
-J-iV = ?
17.
•5 _:_ _1_
_ 9
35.
f
^-^
. _ 9
53.
J
^A=?
18.
"8 ~*~~6¥
= 9
36.
i
-7-g-1^
.= 9
54.
1
•*- 6¥ = ?
Decimal Fractions
A power is the product of equal factors, as 10 x 10 = 100.
10 x 10 x 10 = 1000. 100 is the second power of 10. 1000 is
the third power of 10.
A decimal fraction or decimal is a fraction whose denominator-
is 10 or a power of 10. A common fraction may have any
number for its denominator, but a decimal fraction must always
have for its denominator 10, or a power of 10. A decimal is
written at the right of a period (.), called the decimal point.
A figure at the right of a decimal point is called a decimal
figure.
^ = .5 ; TVo = -25 ; T^ = .07 ; T^ = .016.
REVIEW OF ARITHMETIC 31
A mixed decimal is an integer and a decimal ; as, 16.04.
To read a decimal, read the decimal as an integer, and give
it the denomination of the right-hand figure. To write, a deci-
mal, write the numerator, prefixing ciphers when necessary to
express the denominator, and place the point at the left.
There must be as many decimal places in the decimal as there
are ciphers in the denominator.
EXAMPLES
Read the following numbers :
1.
.7
7.
.4375
13.
.0000054
19.
9.999999
2.
.07
8.
.03125
14.
35.18006
20.
.10016
3.
.007
9.
.21875
15.
.0005
21.
.000155
4.
.700
10.
.90625
16.
100.000104
22.
.26
5.
.125
11.
.203125
17.
9.1632002
23.
.1
6.
.0625
12.
.234375!
18.
30.3303303
24.
.80062
Express decimally :
1. Four tenths.
2. Three hundred twenty-five thousandths.
3. Seventeen thousand two hundred eleven hundred-thou-
sandths.
4. Seventeen hundredths. 6. Five hundredths.
5. Fifteen thousandths. 7. Six ten-thousandths.
8. Eighteen and two hundred sixteen hundred-thousandths.
9. One hundred twelve hundred-thousandths.
10. 10 millionths. 11. 824 ten-thousandths.
12. Twenty-nine hundredths.
13. 324 and one hundred twenty-six millionths.
14. 7846 hundred-millionths.
32 VOCATIONAL MATHEMATICS FOR GIRLS
1C 563 1 2123 3 2 86 5 4
16- -nnnnnnnrj TO"O> nnnnr> TTF> r^nnnnnr-
17. One and one tenth.
18. One and one hundred-thousandth.
19. One thousand four and twenty-nine hundred ths.
Reduction of Decimals
Ciphers annexed to a decimal do not change the value .of
the decimal; these ciphers are called decimal ciphers. For
each cipher prefixed to a decimal, the value is diminished ten-
fold. The denominator of a decimal — when expressed — is
always 1 with as many ciphers as there are decimal places in
the decimal.
To reduce a decimal to a common fraction.
Write the numerator of the decimal omitting the point for the
numerator of the fraction. For the denominator write 1 with as
many ciphers annexed as there are decimal places in the decimal.
Tfien reduce to lowest terms.
EXAMPLE. — Reduce .25 and .125 to common fractions.
1 Write 25 for the numerator and
0~ 25 2$ 1 A 1 for the denominator with two O's,
~ 100 ~~ ^00 ~~ 4 " which makes ^ ; T^ reduced to
4 lowest terms is \.
1
H OK _ 125 _ fflfi _ 1 * .125 is reduced to a common frac-
= 1000 ~~ ) ~~ 8 ' tion in the same way.
EXAMPLE. — Reduce .371 to a common fraction.
37 has for its denominator 1
= x- = Ans
100 100 2 ^[00 8 " This is a complex fraction
4 which reduced to lowest terms
REVIEW OF ARITHMETIC
EXAMPLES
Reduce to common fractions :
1. .09375
6. 2.25
11. .16|
16. .87J
2. .15625
7. 16.144
12. .331
17. .66 1
3. .015625
8. 25.0000100
13. .061
18. .36J
4. .609375
9. 1084.0025
14. .140625
19. .83^
5. .578125
10. .121
15. .984375
20. .621
To reduce a common fraction to a decimal.
Annex decimal ciphers to the numerator and divide by the de-
nominator. Point off from the right of the quotient as many
places as there are ciphers annexed. If there are not figures
enough in the quotient, prefix ciphers.
The division will not always be exact, i.e. -f = .142f or .142+.
EXAMPLE. — Reduce J to a decimal.
.75
4)3.00
28
20
I = .75
EXAMPLES
Reduce to decimals :
1. JQ 6. -I 11. ¥L ™> fi 21. ^
2 irhr 7 ti 12 2To 17- 16i 22- 25.12^
3. ^ 8. If 13. ^ 18. 66| 23. 331
4. 1 9. A 14- 12i 19- if 24- A
5. | 10. T^ 15. T6T 20. | 25.
Addition of Decimals
To add decimals, write them so that their decimal points are in
a column. Add as in integers, and place the point in the sum
directly under the points above it.
34 VOCATIONAL MATHEMATICS FOR GIRLS
EXAMPLE. — Find the sum of 3,87,2.0983, 5.00831, .029,
.831.
3.87
2 Q983 Place these numbers, one under the other, with
' , decimal points in a column, and add as in addition
of integers. The sum of these numbers should
have the decimal point in the same column as the
.831 numbers that were added.
11.83661 Ans.
EXAMPLES
Find the sum :
1. 5.83, 7.016, 15.0081, and 18.3184.
2. 12.031, 0.0894, 12.0084, and 13.984.
3. .0765, .002478, .004967, .0007862, .17896.
4. 24.36, 1.358, .004, and 1632.1.
5. .175, 1.75, 17.5, 175., 1750.
6. 1., .1, .01, .001, 100, 10., 10.1, 100.001.
7. Add 5 tenths; 8063 millionths; 25 hundred-thousandths ;
48 thousandths; 17 millionths; 95 ten-millionths ; 5, and 5
hundred-thousandths ; 17 ten-thousandths.
8. Add 24f , 171 .0058, 71, 9Ty
9. 32.58, 28963.1, 287.531, 76398.9341.
10. 145., 14.5, 1.45, .145, .0145.
Subtraction of Decimals
To subtract decimals, write, the smaller number under the
larger tvith the decimal point of the subtrahend directly under the
decimal point of the minuend. Subtract as in integers, and place
the point directly under the points above.
EXAMPLE. — Subtract 2.17857 from 4.3257.
Write the lesser number under the greater,
4.32570 Minuend with the decimal points under each other.
2.17857 Subtrahend Add a 0 to the minuend, 4.3257, to give it the
2.14713 Remainder same, denominator as the subtrahend. Then
subtract as in subtraction of integers. Write
the remainder with decimal point under the other two points.
REVIEW OF ARITHMETIC 35
EXAMPLES
Subtract :
1. 59.0364-30.8691 = ? 3. .0625 - .03125 = ?
2. 48.7209-12.0039 = ? 4. .00011 - .000011 = ?
5. 10 -.1 + . 0001 = ?
6. From one thousand take five thousandths.
7. Take 17 hundred-thousandths from 1.2.
8. From 17.371 take 14.161.
9. Prove that 1 and .500 are equal.
10. Find the difference between y3^4^ and ^-fl^ff.
Multiplication of Decimals
To multiply decimals proceed as in integers, and give to the
product as many decimal figures as there are in both multiplier
and multiplicand. When there are not figures enough in the
product, prefix ciphers.
EXAMPLE. — Find the product of 6.8 and .63.
6.8 Multiplicand
63 Multiplier 6<^ *s tne multiplicand and .63 the multiplier.
~nr\4 Their product is 4.284 with three decimal figures,
the number of decimal figures in the multiplier
and multiplicand.
4.284 Product
EXAMPLE. — Find the product of .05 and .3.
.05 Multiplicand The product of .05 and .3 is .015 with a cipher
.3 Multiplier prefixed to make the three decimal figures re-
.015 Product quired in the product.
EXAMPLES
Find the products :
1. 46.25 x. 125 3. .015 x. 05
2. 8.0625 X .1875 4. 25.863 x 44-
36 VOCATIONAL MATHEMATICS FOR GIRLS
5. 11.11x100 8. .325xl2|
6. .5625 x 6.28125 9. .001542 x .0052
7. .326 x 2.78 10. 1.001 x 1.01
To multiply by 10, 100, 1000, etc., remove the point one place
to the right for each cipher in the multiplier.
This can be performed without writing the multiplier.
EXAMPLE.— Multiply 1.625 by 100.
1.625 x 100 = 162.5
To multiply by 200, remove the point to the right and multiply
by 2.
EXAMPLE. — Multiply 86.44 by 200.
86.44.
2
17,288
EXAMPLES
Find the product of :
1. 1 thousand by one thousandth.
2. 1 million by one millionth.
3. 700 thousands by 7 hundred-thousandths.
4. 3.894 x 3000 5. 1.892 x 2000.
Division of Decimals
To divide decimals proceed as in integers, and give to the quo-
tient as many decimal figures as the number in the dividend ex-
ceeds those in the divisor.
EXAMPLE. — Divide 12.685 by .5.
The number of decimal figures in
Divisor .5)12.685 Dividend the quotient, 12.685, exceeds the num-
25.37 Quotient her of decimal figures in the divisor, .5,
by two. So there must be two deci-
mal figures in the quotient.
REVIEW OF ARITHMETIC
37
EXAMPLE. — Divide 399.552 by 192.
When the divisor is an integer,
2.081 Quotient tjie point jn the quotient should be
placed directly over the point in
the dividend, and the division per-
formed as in integers. This may
be proved by multiplying divisor
Divisor 192)399.552 Dividend
384
1555
1536
192
192
by quotient, which would give the
dividend.
Divisor 1.25.)28.78.884 Dividend
250
EXAMPLE. — Divide 28.78884 by 1.25.
When the divisor contains
23.031+ Quotient decimal figures, move the point
in both divisor and dividend as
many places to the right as
there are decimal places in the
divisor, which is equivalent to
multiplying both divisor and
dividend by the same number
and does not change the quo-
tient. Then place the point in
the quotient as if the divisor
were an integer. In this ex-
ample, the multiplier of both
378
375
388
375
134
125
9 Remainder
dividend and divisor is 100.
EXAMPLES
Find the quotients :
1. .0625 — .125 5. 1000 - .001
2. 315.432 - .132 6. 2.496 -.136
3. .75 -.0125 7. 28000-16.8
4. 125-^121
8. 1.225-4.9
9. 3.1416-27
10. 8.33-5
To divide by 10, 100, 1000, etc., remove the point one place to
the left for each cipher in the divisor.
To divide by 200, remove the point two places to the left, and
divide by 2.
38 VOCATIONAL MATHEMATICS FOR GIRLS
EXAMPLES
Find the quotients :
1. 38.64 — 10 6. 865.45-=- 5000
2. 398.42-1000 7. 38.28-400
3. 1684.32-1000 8. 2.5-500
4. 1.155-100 9. .5-10
5. 386.54-2000 10. .001-1000
REVIEW EXAMPLES
1. Add 28.03, .1674, .08309, 7.00091, .1895.
2. Subtract 1.00894 from 13.0194.
3. Multiply 83.74 x 3.1416.
4. Divide 3.1416 by 8.5.
5. Perform the following calculations : .7854 x 35 x 7.5.
6. Perform the following calculations :
65.3 x 3.1416 x .7854
600 x 3.5 x 8.3
7. Change the following fractions to decimals :
00 2V (&) A, 00 eV , W yiu> 00 TV, (/) A, (?) A-
8. Change the following decimals to common fractions :
(a) .331 (6) .25, (c) .125, (d) .375, (e) .437J, (/) .875.
Parts of 100 or 1000
1. What part of 100 is 12£ ? 25 ? 33| ?
2. What part of 1000 is 125? 250? 333|?
3. How much is i of 100? Of 1000?
4. How much is l of 100 ? Of 1000 ?
5. What is J of 100 ? Of 1000?
EXAMPLE. — How much is 25 times 24 ?
100 times 24 = 2400.
25 times 24 = 1 as much as 100 times 24 = 600. Ans.
REVIEW OF ARITHMETIC 39
Short Method of Multiplication
To multiply by
25, multiply by 100 and divide by 4 ;
331, multiply by 100 and divide by 3 ;
16}, multiply by 100 and divide by 6 ;
121, multiply by 100 and divide by 8 ;
9, multiply by 10 and subtract the multiplicand ;
11, if more than two figures, multiply by 10 and add the
multiplicand to the product ;
11, if two figures, place the figure that is their sum between
them.
63 x 11 = 693 74 x 11 = 814
Note that when the sum of the two figures exceeds nine, the one in the
tens place is carried to the figure at the left.
EXAMPLES
Multiply by the short process :
1. 81 by 11 = ? 10. 68 by 16f = ?
2. 75 by 331 = ? 11. 112 by 11 = ?
3. 128 by 12J = ? 12. 37 by 11 = ?
4. 87 by 11 = ? 13. 4183 by 11 = ?
5. 19 by 9 = ? 14. 364 by 33i = ?
6. 846 by 11 = ? 15. 8712 by 121 = ?
7. 88 by 11 = ? 16. 984 by 16} = ?
8. 19 by 11 = ? 17. 36 by 25 = ?
9. 846 by 16} = ? 18. 30 by 3331 = ?
Aliquot Parts of $1.00
The aliquot parts of a number are the numbers that are
exactly contained in it. The aliquot parts of 100 are 5, 20,
121, 16}, 331, etc.
The monetary unit of the United States is the dollar, con-
taining one hundred cents, which are written decimally.
40 VOCATIONAL MATHEMATICS FOR GIRLS
6 J cents = $ -^ 25 cents == $ 1 = quarter dollar
81 cents = $ T^ 33 J cents = $ 1
12| cents = $ -J- 50 cents = $ 1 = half dollar
16 J cents = $ 1
10 mills = 1 cent, ct. = $ .01 or $ 0.01
5 cents = 1 " nickel " = $ .05
10 cents = 1 dime, d. = $ .10
10 dimes = 1 dollar, $ = $ 1.00
10 dollars = 1 eagle, E. = $ 10.00
EXAMPLE. — What will 69 pairs of stockings cost at 16 J
cents a pair ?
69 pairs will cost 69 x 16f cts., or 69 x $ \ = -\9- = $ llf = $ 11.50.
EXAMPLE. — At 25^ a peck, how many pecks of potatoes
can be bought for $ 8.00 ?
8-5-^ = 8x^ = 32 pecks. Ans.
Review of Decimals
1. For work on a job one woman receives $ 13.75, a second
woman $ 12.45, a third woman $ 14.21, and a fourth woman
$ 21.85. What is the total amount paid for the work ?
2. A pipe has an inside diameter of 3.067 inches and an
outside diameter of 3.428 inches. What is the thickness of
the metal of the pipe ?
3. At 4| cts. a pound, what will be the cost of 108 boxes of
salt each weighing 29 Ib. ?
4. A dressmaker receives $ 121.50 for doing a piece of
work. She gives $ 12.25 to one of her helpers and $ 10.50
to another. She also pays $ 75.75 for material. How much
does she make on the job ?
5. An automobile runs at the rate of 91 miles an hour.
How long will it take it to go from Lowell to Boston, a dis-
tance of 26.51 miles ?
REVIEW OF ARITHMETIC 41
6. A man uses a gallon of gasoline in traveling 16 miles.
If a gallon costs 23 cents, what is the cost of fuel per mile ?
7. Which is cheaper, and how much, to have a 13J cents
an hour woman take 13^ hours on a piece of work, or hire a
17| cents an hour woman who can do it in 9^ hours ?
8. On Monday 1725.25 Ib. of coal are used, on Tuesday
2134.43 Ib., on Wednesday 1651.21 Ib., on Thursday 1821.42
Ib., on Friday 1958.82 Ib., and on Saturday 658.32 Ib. How
many pounds of coal are used during the week ?
9. If, in the example above, there were 10,433.91 Ib. of
coal on hand at the beginning of the week, how much was left
at the end of the week ?
10. The distance traveled in an automobile is measured by an
instrument called a speedometer. A man travels in a week the
following distances: 87.5 mi., 49.75 mi., 112.60 mi., 89.7 mi.,
119.3 mi., and 93.75 mi. What is the total distance traveled ?
11. An English piece of currency corresponding to our five-
dollar bill is called a pound sterling and is worth $4.866|.
How much more is a five-dollar bill than a pound ?
12. An alloy is made of copper and zinc. If .66 is copper
and .34 is zinc, how many pounds of zinc and how many
pounds of copper will there be in a casting of the alloy
weighing 98 Ib. ?
13. A train leaves New York at 2.10 P.M. and arrives in
Philadelphia at 4.15 P.M. The distance is 90 miles. What is
the average rate per hour of the train ?
14. The weight of a foot of Ty steel bar is 1.08 Ib. Find
the weight of a 21-foot bar.
15. A steam pump pumps 3.38 gallons of water to each
stroke and the pump makes 51.1 strokes per minute. How
many gallons of water will it pump in an hour ?
16. At 121 cents per hour, what will be the pay for 23^ days
if the days are 10 hours each ?
42 VOCATIONAL MATHEMATICS FOR GIRLS
Compound Numbers
A number composed of different kinds of concrete units that
are related is a compound number : as, 3 bu. 2 pk. 1 qt.
A denomination is a name given to a unit of measure or of
weight. A number having one or more denominations is also
called a denominate number.
Reduction is the process of changing a number from one
denomination to another without changing its value.
Changing to a lower denomination is called reduction descend-
ing : as, 2 bu. 3 pk. = 88 qt. Changing to a higher denomi-
nation is called reduction ascending ; as, 88 qt. = 2 bu. 3 pk.
Linear Measure is used in measuring lines or distance
Table
12 inches (in.) = 1 foot, ft.
3 feet = 1 yard, yd.
5| yards, or 161 feet = 1 rod, rd.
320 rods, or 5280 feet = 1 mile, mi.
1 mi. = 320 rd. = 1760 yd. = 5280 ft. = 03,360 in.
Square Measure is used in measuring surfaces.
Table
144 square inches = 1 square foot, sq. ft.
9 square feet = 1 square yard, sq. yd.
30^ square yards j = l e rod pd
272£ square feet J
160 square rods = 1 acre, A.
640 acres = 1 square mile, sq. mi.
1 sq. mi. = 640 A. = 102,400 sq. rd. = 3,097,600 sq. yd.
Cubic Measure is used in measuring volumes or solids.
Table
1728 cubic inches = 1 cubic foot, cu. ft.
27 cubic feet = 1 cubic yard, cu. yd.
16 cubic feet = 1 cord foot, cd. ft.
8 cord feet, or 128 cu. ft. = 1 cord, cd.
1 cu. yd. = 27 cu. ft. = 46,656 cu. in.
REVIEW OF ARITHMETIC 43
Liquid Measure is used in measuring liquids.
Table
4 gills (gi.) = 1 pint, pt.
2 pints = 1 quart, qt.
4 quarts = 1 gallon, gal.
1 gal. = 4 qt. = 8 pt. = 32 gi.
A gallon contains 231 cubic inches.
The standard barrel is 31£ gal., and the hogshead 63 gal.
Dry Measure is used in measuring roots, grain, vegetables,
etc.
Table
2 pints = 1 quart, qt.
8 quarts = 1 peck, pk.
4 pecks = 1 bushel, bu.
1 bu. = 4 pk. = 32 qt. = 64 pints.
The bushel contains 2150.42 cubic inches; 1 dry quart contains
67.2 cu. in. A cubic foot is ff of a bushel.
Avoirdupois Weight is used in weighing all common articles ;
as, coal, groceries, hay, etc.
Table
16 ounces (oz.) = 1 pound, Ib.
100 pounds = 1 hundredweight, cwt. ;
or cental, ctl.
20 cwt., or 2000 Ib. = 1 ton, T.
1 T. = 20 cwt. = 2000 Ib. = 32,000 oz.
The long ton of 2240 pounds is used at the United States Custom
House and in weighing coal at the mines.
Measure of Time.
Table
60 seconds (sec.) = 1 minute, min.
60 minutes = 1 hour, hr.
24 hours = 1 day, da.
7 days = 1 week, wk.
365 days = 1 year, yr.
366 days = 1 leap year.
100 years = 1 century.
44 VOCATIONAL MATHEMATICS FOR GIRLS
Counting.
Table
12 things = 1 dozen, doz.
12 dozen = 1 gross, gr.
12 gross = 1 great gross, G. gr.
Paper Measure.
Table
24 sheets = 1 quire 2 reams = 1 bundle
20 quires = 1 ream 5 bundles = 1 bale
Reduction Descending
EXAMPLE. — Reduce 17 yd. 2 ft. 9 in. to inches.
1 yd. = 3 ft.
17 yd. = 17 x 3 = 51 ft.
51 + 2 = 53 ft.
1 ft. = 12 in.
53 ft. = 53 x 12 = 636 in.
636 + 9 = 645 in. Am.
EXAMPLES
Reduce to lower denominations :
1. 46 rd. 4 yd. 2 ft. to feet.
2. 4 A. 15 sq. rd. 4 sq. ft. to square inches.
3. 16 cu. yd. 25 cu. ft. 900 cu. in. to cubic inches.
4. 15 gal. 3 qt. 1 pt. to pints.
5. 27 da. 18 hr. 49 min. to seconds.
Reduction Ascending
EXAMPLE. — Reduce 1306 gills to higher denominations.
4)1306 gi. Since in 1 pt. there are 4 gi., in 1306 gi.
2)326 pt. + 2 gi. there are as many pints as 4 gi. are contained
4)163 qt. times in 1306 gi., or 326 pt. and 2 gi. remainder.
40 gal. + 3 qt. In the same way the quarts and gallons are
40 gal. 3 qt. 2 gi. Ans. found. So there are in 1306 gi., 40 gal. 3 qt.
2gi.
REVIEW OF ARITHMETIC 45
EXAMPLES
Reduce to higher denominations :
1. Reduce 225,932 in. to miles, etc.
2. Change 1384 dry pints to higher denominations.
3. In 139,843 sq. in. how many square miles, rods, etc. ?
4. How many cords of wood in 3692 cu. ft. ?
5. How many bales in 24,000 sheets of paper ?
A denominate fraction is a fraction of a unit of weight or
measure.
To reduce denominate fractions to integers of lower denominations.
Change the fraction to the next lower denomination. Treat
the fractional part of the product in the same way, and so pro-
ceed to the required denomination.
EXAMPLE. — Reduce f of a mile to rods, yards, feet, etc.
f of 320 rd. = -i-6^-0- rd. = 228f rd.
f of V yd. = # yd. = 3^ yd.
\ of 3 ft. = Of ft.
f of 12 in. = -3/ in. = 5} in.
f of a mile = 228 rd. 3 yd. 0 ft. 5| in.
The same process applies to denominate decimals.
To reduce denominate decimals to denominate numbers.
EXAMPLE. — Reduce .87 bu. to pecks, quarts, etc.
.87 bu. .84 qt.
4 2
Change the decimal fraction to
3.48 pk. 1.68 pt. the next iower denomination. Treat
.48 pk. the decimal part of the product in the
g same way, and so proceed to the re-
o OA OJ. quired denomination.
3 pk. 3 qt. 1.68 pt. Ans.
46 VOCATIONAL MATHEMATICS FOR GIRLS
EXAMPLES
Reduce to integers of lower denominations :
1. f of an acre. 3. ^ of a ton.
2. .3125 of a gallon. 4. .51625 of a mile.
5. Change f of a year to months and days.
6. .2364 of a ton.
7. What is the value of | of 1^ of a mile ?
8. Reduce -|^ bu. to integers of lower denominations.
9. .375 of a month.
10. T9? acre are equal to how many square rods, etc. ?
Addition of Compound Numbers
EXAMPLE. — Find the sum of 7 hr. 30 min. 45 sec., 12 hr.
25 min. 30 sec., 20 hr. 15 inin. 33 sec., 10 hr. 27 mm. 46 sec.
hr. min. sec.
7 30 45 The sum of the seconds = 154 sec. =
12 25 30 2 min. 34 sec. Write the 34 sec. under
20 15 33 the sec. column and add the 2 min. to
10 27 46 the min. column. Add the other columns
50 39 34 in the same way.
50 hr. 39 min. 34 sec. Ans.
Subtraction of Compound Numbers
EXAMPLE. — From 39 gal. 2 qt. 2 pt. 1 gi. take 16 gal. 2 qt.
3 pt. 3 gi.
. As 3 gi. cannot be taken from 1 gi., 4 gi.
or 1 pt. are borrowed from the pt. column
and added to the 1 gi. Subtract 3 gi. from
— — the 5 gi. and the remainder is 2 gi. Continue
in the same way until all are subtracted.
22 gal. 6 qt. 2 gi. Ans. 22 gal 3 qt Q pt 2 gi>
REVIEW OF ARITHMETIC 47
Multiplication of Compound Numbers
EXAMPLE. — Multiply 4 yd. 2 ft. 8 in. by 8.
yd. ft. in. 8 times 8 in. = 64 in. = 5 ft. 4 in. Place the
428 4 in. under the in. column, and add the 5 ft. to
8_ the product of 2 ft. by 8, which equals 21 ft. = 7 yd.
39 0 4 Add 7 yd. to the product of 4 yd. by 8 = 39 yd.
39 yd. 4 in. Ans.
Division of Compound Numbers
EXAMPLE. — Find ^ of 42 rd. 4 yd. 2 ft. 8 in.
rd. yd. ft. in.
35)42 4 2 8(1 rd.
35
7
6J -fa of 42 rd. = 1 rd. ; re-
3£ 35)24|(0ft. mainder, 7 rd. = 38| yd.;
35 12 add 4 yd. = 42£ yd. ^ of
38£ 294 42J yd. = 1 yd. ; remainder,
+ 4 +8 7£ yd., = 22| ft. = 24£ ft.
35JI2| yd. (1 yd. 35)3T)2(8f§ in. ^ of 24£ ft. = 0 ft. 24£ ft.
35_ 280 =294 in. ; add 8 in. =302 in.
~7£ ~22 -s\ of 302 in. = 8|| in.
3
22£ ft. 1 rd. 1 yd. 8f| in. Ans.
12
Difference between Dates
EXAMPLE. — Find the time from Jan. 25, 1842, to July 4,
1896.
1896 74 It is customary to consider 30 days
1842 1 25 to a month. July 4, 1896, is the 1896th
54 yr. 6 mo. 9 da. Ans. yr., 7th mo., 4th da., and Jan. 25, 1842,
is the 1842d yr., 1st. mo., 25th da.
Subtract, taking 30 da. for a month.
48 VOCATIONAL MATHEMATICS FOR GIRLS
EXAMPLE. — What is the exact number of days between
Dec. 16, 1895, and March 12, 1896 ?
Dec. 15 Do not count the first day mentioned. There
Jan. 31 are 15 days in December, after the 16th. Jan-
Feb. 29 uary has 31 days, February 29 (leap year),
Mar. 12 and 12 days in March ; making 87 days.
87 days. Ans.
EXAMPLES
1. How much time elapsed from the landing of the Pil-
grims, Dec. 11, 1620, to the Declaration of Independence,
July 4, 1776?
2. Washington was born Feb. 22, 1732, and died Dec. 14,
1799. How long did he live?
3. Mr. Smith gave a note dated Feb. 25, 1896, and paid it
July 12, 1896. Find the exact number of days between its date
and the time of payment.
4. A carpenter earning $ 2.50 per day commenced Wednes-
day morning, April 1, 1896, and continued working every week
day until June 6. How much did he earn ?
5. Find the exact number of days between Jan. 10, 1896,
and May 5, 1896.
6. John goes to bed at 9.15 P.M. and gets up at 7.10 A.M.
How many minutes does he spend in bed ?
To multiply or divide a compound number by a fraction.
To multiply by a fraction, multiply by the numerator, and
divide the product by the denominator.
To divide by a fraction, multiply by the denominator, and divide
the product by the numerator.
When the multiplier or divisor is a mixed number, reduce to
an improper fraction, and proceed as above.
REVIEW OF ARITHMETIC 49
EXAMPLES
1. How much is f of 16 hr. 17 min. 14 sec. ?
2. A field contains 10 A. 12 sq. rd. of land, which is f
of the whole farm. Find the size of the farm.
3. If a train runs 60 mi. 35 rd. 16 ft. in one hour, how far
will it run in 12f hr. at the same rate of speed ?
4. Divide 14 bu. 3 pk. 6 qt. 1 pt. by }.
5. Divide 5 yr. 1 mo. 1 wk. 1 da. 1 hr. 1 min. 1 sec. by 3f .
REVIEW EXAMPLES
1. A time card on a piece of work states that 2 hours and
15 minutes were spent on a skirt, 1 hour and 12 minutes on a
waist, 2 hours and 45 minutes on a petticoat, and 1 hour and
30 minutes on a jacket. What was the number of hours spent
on all the work ?
2. How many parts of a sewing machine, each weighing 14
oz., can be obtained from 860 Ib. of metal if nothing is allowed
for waste ?
3. How many feet long must a dry goods store be to hold
a counter 8' 6", a bench 14' 4", a desk 4' 2", and a counter
7' 5", placed side by side, if 3' 3" are allowed between the
pieces of furniture and between the walls and the counters ?
4. How many gross in a lot of 968 buttons ?
5. Find the sum of 7 hr. 30 min. 45 sec., 12 hr. 25 min.
30 sec., 20 hr. 15 min. 33 sec., 10 hr. 27 min. 46 sec.
6. If a train is run for 8 hr. at the average rate of 50
mi. 30 rd. 10 ft. per hour, how great is the distance covered ?
7. A telephone pole is 31 ft. long. If 4 ft. 7 in. are under
ground, how high (in inches) is the top of the pole above the
street ?
8. If 100 bars of iron, each 2f long, weigh 70 Ib., what is
the total weight of 2300 bars ?
50 VOCATIONAL MATHEMATICS FOR GIRLS
9. If a cubic foot of water weighs 62^ lb., how many
ounces does it weigh ?
10. A farmer's wife made 9 pounds 7 ounces of butter and
sold it at 41 cents a pound. How much did she receive ?
11. A peck is what part of a bushel ?
12. A quart is what part of a bushel ? of a peck ?
13. I have 84 lb. 14 oz. of salt which I wish to put into
packages of 2 lb. 6 oz. each. How many packages will
there be ?
14. If one bottle holds 1 pt. 3 gi., how many dozen bottles
will be required to hold 65 gal. 2 qt. 1 pt. ?
15. How many pieces 51" long can be cut from a rod 16' 8"
long, if 5" are allowed for waste ?
16. What is the entire length of a .railway consisting of five
different lines measuring respectively 160 mi. 185 rd. 2 yd.,
97 mi. 63 rd. 4 yd., 126 mi. 272 rd. 3 yd., 67 mi. 199 rd. 5 yd.,
and 48 ini. 266 rd. 5 yd. ?
Percentage
Percentage is a process of solving questions of relation by
means of hundredths or per cent (%).
Every question in percentage involves three elements : the
rate per cent, the base, and the percentage.
The rate per cent is the number of hundredths taken.
The base is the number of which the hundredths are taken.
The percentage is the result obtained by taking a certain per
cent of a number.
Since the percentage is the result obtained by taking a cer-
tain per cent of a number, it follows that the percentage is the
product of the base and the rate. The rate and base are always
factors, the percentage is the product.
EXAMPLE. — How much is 8 % of $ 200 ?
8 % of $200 = 200 x .08 = $ 16. (1)
REVIEW OF ARITHMETIC 51
In (1) we have the three elements: 8% is the rate, $200 is the base,
and $ 16 is the percentage.
Since $ 200 x .08 = $ 16, the percentage ;
$ 16 -=- .08 = $ 200, the base ;
and $ 16 -^ $ 200 = .08, the rate.
If any two of these elements are given, the other may be
found :
Base x Rate = Percentage
Percentage -5- Rate = Base
Percentage -5- Base = Rate
Per cent is commonly used in the decimal form, but many
operations may be much shortened by using the common frac-
tion form.
1 % = .01 = T^ i % = .001 or .005
10%= .10 = A 33|%=.33i = |
100 % = LOO = 1 81 % = -081 = .0825
121 % = .12J or .125 = 1 1 % = .00 J- = .00125
There are certain per cents that are used so frequently that
we should memorize their equivalent fractions.
10=
20%= I
25% =}
37*%= I
40% =|
50% = J
60%= f
75 % = }
80%=*
s4%=I
EXAMPLES
1. Find 75 % of $ 368.
2. Find 15 % of $ 412.
3. 840 is 331 % of what number ?
4. 615 is 15 % of what number ?
5. What per cent of 12 is 8 ?
52 VOCATIONAL MATHEMATICS FOR GIRLS
6. What per cent of a foot is 8 inches ? 11 inches ? 4 inches ?
7. A technical high school contains 896 pupils ; 476 of the
pupils are girls. What per cent of the school is girls ?
8. Out of a gross of bottles of mucilage 9 were broken.
What was the per cent broken ?
Trade Discount
Merchants and jobbers have a price list. From this list
they give special discounts according to the credit of the cus-
tomer and the amount of supplies purchased, etc. If they
give more than one discount, it is understood that the first
means the discount from the list price, while the second denotes
the discount from the remainder.
EXAMPLES
1. What is the price of 200 spools of cotton at $ 36.68 per
M. at 40 % off ?
2. Supplies from a dry goods store amounted to $ 58.75. If
121 % Were allowed for discount, what was the amount paid ?
3. A dealer received a bill amounting to $ 212.75. Suc-
cessive discounts of 15%, 10%, and 5% were allowed.
What was the amount to be paid ?
4. 2 % is usually discounted on bills paid within 30 days.
If the following are to be paid within 30 days, what will be
the amounts due ?
a. $ 30.19 c. $399.16 e. $1369.99
b. 2816.49 d. 489.01 /. 918.69
5. Millinery supplies amounted to $ 127.79 with a discount
of 40 % and 15 %. What was the net price ?
6. What single discount is equivalent to a discount of 45 %
and 10 % ?
7. What single discount is equivalent to 20 %, and 10 % ?
REVIEW OF ARITHMETIC 53
Simple Interest
Money that is paid for the use of money is called interest.
The money for the use of which interest is paid is called the
principal, and the sum of the principal and interest is called
the amount.
Interest at 6 % means 6 % of the principal for 1 year ; 12
months of 30 days each are usually regarded as a year in com-
puting interest. There are several methods of computing
interest.
EXAMPLE. — What is the interest on $ 100 for 3 years at 6 % ?
$100
.06
$ 6.00 interest for one year. Or, ^fa x ^ x f = $18. Ans.
3
$ 18.00 interest for 3 years. Ans.
$ 100 + $ 18 = $ 118, amount.
Principal x Rate x Time = Interest.
EXAMPLE. — What is the interest on $ 297.62 for 5 yr. 3 mo.
at 6 % ?
$297.62 3
.06 nr 0 w $ 297.62 ^ 21 _ $ 18750.06 _ ^ 09 7*
yjL » TTT^ * ~~ ~+ * ~V — ^7 — '1? vo> ' o>
$17.8572 100 1 £ 200
5j 2
4.4643
89.2860 NOTE. —Final results should not include
$93.7503 $93.75. Ans. mills. Mills are disregarded if less than 5,
and called another cent if 5 or more.
EXAMPLES
1. What is the interest on $ 586.24 for 3 months at 6 % ?
2. What is the interest on $ 816.01 for 9 months at 5 % ?
3. What is the interest 011 $ 314.72 for 1 year at 4 % ?
4. What is the interest on $ 876.79 for 2 yr. 3 mo. at 4£ % ?
5. What is the interest on $ 2119.70 for 6 yr. -2 mo. 13 da.
at 51 % ?
54 VOCATIONAL MATHEMATICS FOR GIRLS
The Six Per Cent Method
By the 6 % method it is convenient to find first the interest
of $ 1, then multiply it by the principal.
EXAMPLE, —r What is the interest on $ 50.24 at 6 % for 2 yr.
8 mo. 18 da. ?
Interest on $ 1 for 2 yr. =2 x $ .06 = $ . 12
Interest on $ 1 for 8 mo. = 8 x $ .OOJ = .04
Interest on $ 1 for 18 da. = 18 x $ .000* - .003
Interest on $ 1 for 2 yr. 8 mo. 18 da. $ .163
Interest on $ 50.24 is 50.24 times $ .163 = $ 8.19. Ans.
Second Method. — Interest on any sum for 60 days at 6 % is
•j-J-g- of that sum and may be expressed by moving the decimal point
two places to the left. The interest for 6 days may be expressed
by moving the decimal three places to the left.
EXAMPLE. — What is the interest on $ 394.50 for 96 days at
6%?
$3.9450, interest on $394.50 for 60 days at 6 %.
1.9725, interest on $394.50 for 30 days at 6 °Jo.
.3945, interest on $ 394.50 for 6 days at 6 °Jo.
$6.3120, interest on $394.50 for 96 days at 6 %. Ans. $ 6.31.
EXAMPLE. — What is the interest on $ 529.70 for 78 days at
8%?
$5.297, interest on $529.70 for 60 days at 6 %.
1.589, interest on $ 529.70 for 18 days (6 days x 3) .
$6.886, interest on $ 529.70 for 78 days at 6 %.
8 % =6 % +£ of 6%.
$6.886 + $2.295 = $9.181. Ans. $9.18.
EXAMPLES
Find the interest and amount of the following :
1. $ 2350 for 1 yr. 3 mo. 6 da. at 6 %.
2. $ 125.75 for 2 yr. 5 mo. 17 da. at 7 %.
3. $ 950.63 for 3 yr. 7 mo. 21 da. at 5 %.
4. $ 625.57 for 2 yr. 8 mo. 28 da. at 8 %.
REVIEW OF ARITHMETIC 55
Exact Interest
When the time includes days, interest computed by the 6%
method is not strictly exact, by reason of using only 30 days
for a month, which makes the year only 360 days. The day is
therefore reckoned as -^ of a year, whereas it is -^ of a year.
To compute exact interest, find the exact time in days, and con-
sider 1 day's interest as ^-^ of 1 year's interest.
EXAMPLE. — Find the exact interest of $ 358 for 74 days at
7%.
$358 x .07 = $25.06, 1 year's interest.
74 days' interest is -/^ of 1 year's interest.
^ of $ 25.06 = $ 5.08. Ans.
Qr $358 _7_ J74 _ ,
1 X100 365~
EXAMPLES
Find the exact interest of :
1. $324 for 15 da. at 5 %.
2. $253 for 98 da. at 4%.
3. $624 for 117 da. at 7 %.
4. $ 620 from Aug. 15 to Nov. 12 at 6 %.
5. $ 153.26 for 256 da. at 5| % •
6. $ 540.25 from June 12 to Sept. 14 at 8 %.
Rules for Computing Interest
The following will be found to be excellent rules for finding the inter-
est on any principal for any number of days.
Divide the principal by 100 and proceed as follows:
2 % — Multiply by number of days to run, and divide by 180.
21 % — Multiply by number of days, and divide by 144.
3 % — Multiply by number of days, and divide by 120.
3* °l° — Multiply by number of days, and divide by 102.86.
56
VOCATIONAL MATHEMATICS FOR GIRLS
4 % — Multiply by number of days, and divide by 90.
5 % — Multiply by number of days, and divide by 72.
6 % — Multiply by number of days, and divide by 60.
7 % — Multiply by number of days, and divide by 51.43.
8 % — Multiply by number of days, and divide by 45.
Savings Bank Compound Interest Table
Showing the amount of § 1, from 1 year to 15 years, with compound
interest added semiannually, at different rates.
PER CENT
3
4
5
6
7
8
9
iyear
1 01
102
102
1 03
03
1 04
104
1 year
1 03
104
1 05
1 06
07
1 08
109
1^ years
104
1 06
107
109
10
112
1 14
2 years
106
108
1 10
1 12
14
116
1 19
2| years
1 07
1 10
1 13
1 15
18
1 21
1 24
3 years
1 09
1 12
1 15
1 19
22
1 26
130
3| years
1 10
1 14
1 18
1 22
27
1 31
136
4 years
1 12
1 17
1 21
1 26
131
1 36
1 42
4^ years
1 14
1 19
124
1 30
1 36
1 42
1 48
5 years
1 16
1 21
128
1 34
41
148
1 55
5J years
1 17
1 24
131
138
45
153
1 62
6 years
1 19
1 26
1 34
142
51
1 60
169
Q\ years
1 21
1 29
1 37
146
56
1 66
1 77
7 years
123
1 31
1 41
1 51
61
1 73
185
7| years
1 24
1 34
144
1 55
67
1 80
1 93
8 years
1 26
1 37
148
1 60
73
1 87
202
8| years
128
139
1 52
1 65
79
1 94
2 11
9 years
1 30
142
1 55
170
85
202
220
9| years
132
1 45
1 59
175
92
2 10
230
10 years
1 34
1 48
163
1 80
98
2 19
241
11 years
1 38
1 54
1 72
1 91
2 13
236
263
12 years
1 42
1 60
1 80
203
228
256
287
13 years
1 47
167
190
2 15
2 44
277
314
14 years
1 51
1 73
199
228
2 62
299
342
15 years
1 56
1 80
209
242
280
324
374
REVIEW OF ARITHMETIC 57
EXAMPLES
Solve the following problems by using the tables on page 56 :
1. What is the compound interest of $1 at the end of
81 years at 6 % ?
2. What is the compound interest of $ 1 at the end of 11
years at 6 °/0 ?
3. How long will it take $ 400 to double itself at 5 % ,
compound interest?
4. How long will it take $ 580 to double itself at 5£ % ,
compound interest ?
5. How long will it take $615 to double itself at 8 %,
simple interest?
6. How long will it take $784 to double itself at 7%,
simple interest ?
7. Find the interest of $ 684 for 94 days at 3 %.
8. Find the interest of $ 1217 for 37 days at 4 %.
9. Find the interest of $681.14 for 74 days at 4|- %.
10. Find the interest of $414.50 for 65 days at 5 %.
11. Find the interest of $384.79 for 115 days at 6 %.
Ratio and Proportion
Ratio is the relation between two numbers. It is found
by dividing one by the other. The ratio of 4 to 8 is 4 ^- 8 = i.
The terms of the ratio are the two numbers compared. The
first term of a ratio is the antecedent, and the second the con-
sequent. The sign of the ratio is (:). (It is the division sign
with the line omitted.) Ratio may also be expressed fraction-
ally, as i£ or 16 : 4 ; or T3T or 3 : 17.
A ratio formed by dividing the consequent by the antece-
dent is an inverse ratio : 12 : 6 is the inverse ratio of 6 : 12.
The two terms of the ratio taken together form a couplet.
58 VOCATIONAL MATHEMATICS FOR GIRLS
Two or more couplets taken together form a compound ratio.
Thus, 2:5 6:11
A compound ratio may be changed to a simple ratio by
taking the product of the antecedents for a new antecedent,
and the product of the consequents for a new consequent ; as,
6x2:11x5, or 12:55.
Antecedent -+- Consequent = Eatio
Antecedent -+- Ratio = Consequent
Ratio x Consequent = Antecedent
To multiply or divide both terms of a ratio by the same
number does not change the ratio.
Thus 12 : 6 = 2
3x12:3x6 = 2
EXAMPLES
Find the ratio of
1. 20 : 300 Fractions with a common de-
2. 3 bu. : 3 pk. nominator have the same
3 21-16 ratio as their numerators.
«• 12:i »• A:*f «:*»**'**
5- i'* & f:|,f:|,|:|
6. 16: (?)=!•
Proportion
An equality of ratios is a proportion.
A proportion is usually expressed thus : 4 : 2 : : 12 : 6, and is
read 4 is to 2 as 12 is to 6.
A proportion has four terms, of which the first and third are
antecedents and the second and fourth are consequents. The
first and fourth terms are called extremes, and the second and
third terms are called means.
The product of the extremes equals the product of the
means.
REVIEW OF ARITHMETIC 59
To find an extreme, divide the product of the means by the given
extreme.
To find a mean, divide the product of the extremes by the given
mean.
EXAMPLES
Supply the missing term :
1. 1 : 836 : : 25 : ( ) 4. 10 yd. : 50 yd. : : $ 20 : ($ )
2. 6:24::( ) : 40 5. $f :$3f ::( ):5
3. ( ) : 15 : : 60 : 6
Simple Proportion
An equality of two simple ratios is a simple proportion.
EXAMPLE. — If 12 bushels of charcoal cost $ 4, what will 60
bushels cost ?
There is the same relation between the cost
of 12 bu. and the cost of 60 bu. as there is be-
tween the 12 bu. and the 60 bu. $4 is the
third term. The answer is the fourth term.
It must form a ratio of 12 and 60 that shall equal the ratio of $ 4 to the
answer. Since the third term is less than the required answer, the first
must be less than the second, and 12 : 60 is the first ratio. The product
of the means divided by the given extreme gives the other extreme, or $ 20.
EXAMPLES
Solve by proportion :
1. If 150 yd. of edging cost $ 6, how much will 1200 yd. cost ?
2. If 250 pounds of lead pipe cost $ 15, how much will 1200
pounds cost ?
3. If 5 men can dig a ditch in 3 days, how long will it take
2 men?
4. If 4 men can shingle a shed in 2 days, how long will it
take 3 men ?
5. The ratio of Simon's pay to Matthew's is -f. Simon
earns $ 18 per week. What does Matthew earn ?
60 VOCATIONAL MATHEMATICS FOR GIRLS
6. What will 11 1 yards of cambric cost if 50 yards cost
$6.75?
7. If it takes 7-J- yards of cloth, 1 yard wide, to make a
suit, how many yards of cloth, 44 inches wide, will it take to
make the same suit ?
8. If 21 yards of silk cost $ 52.50, what will 35 yards cost ?
9. A farm valued at $5700 is taxed for $38.19. What
should be the tax on property valued at $ 28,500 ?
10. If there are 7680 minims in a pint of water, how many
pints are there in 16,843 minims ?
11. There are approximately 15 grains in a gram. How
many grams in 641 grains ?
12. In a velocity diagram a line '3J in. long represents
45 ft. What would be the length of a line representing 30 ft.
velocity ?
13. When a post 11.5 ft. high casts a shadow on level ground
20.6 ft. long, a telephone pole nearby casts a shadow 59.2 ft.
long. How high is the pole ?
14. If 10 grams of silver nitrate dissolved in 100 cubic cen-
timeters of water will form a 10 % solution, how much silver
nitrate should be used in 1560 cubic centimeters of water ?
15. A ditch is dug in 14 days of 8 hours each. How many
days of 10 hours each would it have taken ?
16. If in a drawing a tree 38 ft. high is represented by 1^",
what on the same scale will represent the height of a house
47ft. high?
17. What will be the cost of 21 motors if 15 motors cost
$887?
18. If goods are bought at a discount of 25 % and are sold
at the list price, what per cent is gained ? (Assume $ 1 as
the list price.)
REVIEW OF ARITHMETIC 61
18. If a sewing machine sews 26 inches per minute on heavy
goods, how many yards will it sew in an hour ?
19. If a girl spends 28 cents a week for confectionery, how
much does she spend for it in three months ?
20. If a pole 8 ft. high casts a shadow 4J ft. long, how high
is a tree which casts a shadow 48 ft. long ?
Involution
The product of equal factors is a power.
The process of finding powers is involution.
The product of two equal factors is the second power, or
square, of the equal factor.
The product of three equal factors is the third power, or cube,
of the factor.
42 = 4 x 4 is 4 to the second power, or the square of 4.
23 = 2 x 2 x 2 is 2 to the third power, or the cube of 2.
34 =3x3x3x3 is 3 to the fourth power, or the fourth power of 4.
EXAMPLES
Find the powers :
1. 53 3. I4 5. (2i)2 7. 93
2. 1.1s 4. 252 6. 24 8. .152
Evolution
One of the equal factors of a power is a root.
One of two equal factors of a number is the square root.
One of three equal factors of a number is the cube root of it.
The square root of 16 = 4. The cube root of 27 = 3.
The radical sign (^/) placed before a number indicates that
its root is to be found. The radical sign alone before a number
indicates the square root.
Thus, \/9 = 3 is read, the square root of 9 = 3.
62 VOCATIONAL MATHEMATICS FOR GIRLS
A small figure placed in the opening of the radical sign is
called the index of the root, and shows what root is to be
taken.
Thus, \/8 = 2 is read, the cube root of 8 is 2.
Square Root
The square of a number composed of tens and units is equal
to the square of the tens, plus twice the product of the tens by
the units, plus the square of the units.
tens'2 + 2 x tens X units + units2
EXAMPLE. — What is the square root of 1225?
12'25(30 + 5 = 35 Separating
Tens2, 302 = 900 into periods of
2xtens = 2x3Q = 60[~325 two figures
2 x tens + units = 2 x 30 + 5 = 65 1 325 each , by a
checkmark ('),
beginning at units, we have 12'25. Since there are two periods in the
power, there must be two figures in the root, tens and units.
The greatest square of even tens contained in 1225 is 900, and its
square root is 30 (3 tens). Subtracting the square of the tens, 900, the
remainder consists of 2 x (tens x units) + units.
325, therefore, is composed of two factors, units being one of them,
and 2 x tens — units being the other. But the greater part of this factor
is 2 x tens (2 x 30 = 60). By trial we divide 325 by 60 to find the other
factor (units), which is 5, if correct. Completing the factor, we have
2 x tens + units = 65, which, multiplied by the other factor, 5, gives 325.
Therefore the square root is 30 + 5 = 35.
The area of every square surface is the product of two equal
factors, length, and width.
Finding the square root of a number, therefore, is equivalent
to finding the length of one side of a square surface, its area
being given.
1. Length x Width = Area
2. Area -r- Length = Width
3. Area -r- Width = Length
REVIEW OF ARITHMETIC 63
SHORT METHOD
EXAMPLE. — Find the square root of 1306.0996.
13'06. 09'96 (36. 14 Beginning at the decimal point, separate the
9 number into periods of two figures each, point-
66) 406 ing whole numbers to the left and decimals to
396 the right. Find the greatest square in the left-
721)1009 hand period, and write its root at the right.
721 Subtract the square from the left-hand period,
7224)28896 and bring down the next period for a dividend.
28896 Divide the dividend, with its right-hand
figure omitted, by twice the root already found,
and annex the quotient to the root, and to the divisor. Multiply this
complete divisor by the last root figure, and bring down the next period
for a dividend, as before.
Proceed in this manner till all the periods are exhausted.
When 0 occurs in the root, annex 0 to the trial divisor, bring down
the next period, and divide as before.
If there is a remainder after all the periods are exhausted, annex deci-
mal periods.
If, after multiplying by any root figure, the product is larger than the
dividend, the root figure is too large and must be diminished. Also the
last figure in the complete divisor must be diminished.
For every decimal period in the power, there must be a decimal figure
in the root. If the last decimal period does not contain two figures,
supply the deficiency by annexing a cipher.
EXAMPLES
Find the square root of :
1. 8836 5. \7xn 9. V3.532-6.28
2. 370881 6. 72.5 10. V625 + 1296
3. 29.0521 7. .009^ 11. _LX —
4. 46656 8. 1684.298431 12.
13. What.is the length of one side of a square field that has
an area equal to a field 75 rd. long and 45 rd. wide ?
CHAPTER II
MENSURATION
The Circle
A circle is a plane figure bounded by a curved line, called
the circumference, every point of which is equidistant from the
center.
The diameter is a straight line drawn,
from one point of the circumference
to another and passing through the
center.
The ratio of the circumference to
the diameter of any circle is always a
constant number, 3.1416+, approxi-
mately 3|, which is represented by
the Greek letter TT (pi).
C = Circumference
D = Diameter
The radius is a straight line drawn from the center to the
circumference.
Any portion of the circumference is an arc.
By drawing a number of radii a circle may be cut into a
series of figures, each one of which is called a sector. The area
of each sector is equal to one half the product of the arc and
radius. Therefore the area of the circle is equal to one half of
the product of the circumference and radius.
1 See Appendix for explanation and directions concerning the use of formulas.
64
MENSURATION 65
R X =
In this formula A equals area, TT = 3.1416, and R1 = the
radius squared.
^ = iz>x|<7
In this formula D equals the diameter and C the circum-
ference,
A=._V = 3.1416 g=.7864ly
4 4
EXAMPLE. — What is the area of a circle whose radius is
3ft.?
. ft
EXAMPLE. — What is the area of a circle whose circumfer-
ence is 10 ft. ?
X^X 10 = -^— = 7.1 sq.ft.
2 3.1416 2 3.1416
Area of a Ring. — On examining a flat iron ring it is clear that
the area of one side of the ring may be found by subtracting
the area of the inside circle from the area of the outside circle.
Let D = outside diameter
d = inside diameter
A = area of outside circle
a = area of inside circle
(1) A
66
VOCATIONAL MATHEMATICS FOR GIRLS
(2)
(3) A-a = -
4 4
Let B = area of circular ring = A — a
= — -c = ^Dz-d = .7854 D2 -
EXAMPLE. — If the outside diameter of a flat ring is 9" and
the inside diameter 7", what is the area of one side of the
ring?
#=.7854 (D2 -<f2)
B = .7854 (81 - 49) = .7854 x 32 = 25.1328 sq. in. Ans.
Angles
We make two common uses of angles : (1) to measure a cir-
cular movement, and (2) to measure a difference in direction.
A circle contains 360°, and the angles at the center of the
circle contain as many degrees as their corresponding arcs on
the circumference.
Angle POE has as many degrees as arc PE.
A right angle is measured by a quarter
of the circumference of the circle, which
is 90°.
The angle AOG is a right angle.
The angle AC, made with half the cir-
cumference of the circle, is a straight angle, and the two right
angles, AOG and GOC, which it contains, are supplementary
to each other. When the sum of two angles is equal to 90°,
they are said to be complementary angles, and one is the com-
plement of the other. When the sum of two angles equals 180°,
they are supplementary angles, and one is said to be the supple-
ment of the other.
MENSURATION 67
The number of degrees in an angle may be measured by a
protractor. The distance around a semicircular protractor is
PROTRACTOR — Semicircular, having 180°.
divided into 180 parts, each division measuring a degree. It
-is used by placing the center of the protractor on the vertex
and the base of the protractor on one side of the angle to be
measured. Where the other side of the angle cuts the circular
piece of the protractor, the size of the angle may be read in
degrees.
EXAMPLES
1. What is the area of a circular piece of velvet 8" in
diameter ?
2. What is the distance around the edge of a hat 6" in
diameter ?
3. Name the complements of angles of 30°, 45°, 65°, 70°,
85°.
4. Name the supplements of angles of 55°, 140°, 69°, 98° 44',
81° 19'.
5. What is the diameter of a wheel that is 12' 6" in circum-
ference ?
68 VOCATIONAL MATHEMATICS FOR GIRLS
6. What is the area of one side of a flat iron ring 14" inside
diameter and 18" outside diameter ?
7. The wheel of a child's carriage is 30" in diameter.
What is the length of the rubber tire that fits it ?
8. How much ribbon is needed to bind the edge of a circu-
lar cloth that exactly covers the top of a center table 28" in
diameter ?
9. A straw hat measures 30" around the rim. What is
the diameter of the hat ?
10. If a circular dining room table measures 12' 6" in cir-
cumference, what is the greatest distance across the table ?
Triangles
A triangle is a plane figure bounded by three straight lines.
Triangles are classified according to the relative lengths of
their sides and the size of their angles.
A triangle having equal sides is called equilateral. One
having two sides equal is isosceles. A triangle having no
sides equal is called scalene.
If the angles of a triangle are equal, the triangle is equi-
angular.
If one of the angles of a triangle is a right angle, the tri-
angle is a right triangle. In a right triangle the side opposite
the right angle is called the hypotenuse and is the longest side.
The other two sides of the right triangle are the legs, and are
at right angles to each other,
EQUILATERAL ISOSCELES SCALENE EIGHT
MENSURATION
69
KINDS OF TRIANGLES
Right Triangles
In a right triangle the
square of the hypotenuse
equals the sum of the
squares of the other two
sides or legs.
If the length of the hy-
potenuse and one leg of a
right triangle is known,
the other side may be
found by squaring the
hypotenuse and squaring
the leg, and extracting the
square root of their dif-
ference.
EXAMPLE. — If the hypotenuse of a right angle triangle is
30" and the base is 18", what is the altitude?
302 = 30 x 30 = 900
182 = 18 x 18 = 324
900 — 324 = 576
V57U = 24". Ans.
J8"
Areas of Triangles
The area of a triangle may be found when the length of the
three sides is given by adding the three sides together, divid-
ing by 2, and subtracting from this sum each side separately.
Multiply the four results together and find the square root of
their product.
70 VOCATIONAL MATHEMATICS FOR GIRLS
EXAMPLE. — What is the area of a triangle whose sides
measure 15, 16, and 17 inches, respectively ?
15
16
17 V24 x 9 x8 x7 = x/12096
2)48 V12096 = 109.98 sq. in. Am.
24-15=9
24 - 16 = 8
24 - 17 = 7
Area of a Triangle = J 5ase X Altitude
EXAMPLE. — What is the area of a triangle whose base is
17" and altitude 10"?
A = ~ x 17 x ;0 = 85 sq. in. Ans.
EXAMPLES
1. A ladder 17 ft. long standing on level ground reached to
a window 12 ft. from the ground. If it is assumed that the
wall is perpendicular, how far is the foot of the ladder from
the base of the wall ?
2. Find the area of a triangular piece of cloth having the
base 81" and the height measured from the opposite angle 56".
3. Find the length of the hypotenuse of a right triangle
with equal legs and having an area of 280 sq. in.
4. Find the length of a side of a right triangle with equal
legs and an area of 72 sq. in.
5. Find the hypotenuse of a right triangle with a base of
8" and the altitude of 7".
6. What is the area of a triangle whose sides measure 12,
19, and 21 inches ?
7. What is the altitude of an isosceles triangle having sides
8 ft. long and a base 6 ft. long ?
MENSURATION 71
Quadrilaterals
Four-sided plane figures are called quadrilaterals. Among
them are the trapezoid, trapezium, rectangle, rhombus, and rhom-
boid.
SQUARE RECTANGLE RHOMBOID RHOMBUS
TRAPEZIUM TRAPEZOID PARALLELOGRAM
KINDS OF QUADRILATERALS
A rectangle is a quadrilateral which has its opposite sides
parallel and its angles right angles. Its area equals the prod-
uct of its base and altitude.
A=ba
A trapezoid is a quadrilateral having only two sides parallel.
Its area is equal to the product of the altitude by one half the
sum of the bases.
In this formula c = length of longest side I
b = length of shortest side
a = altitude
A trapezium is a four-sided figure with no two sides parallel.
The area of a trapezium is found by dividing the trapezium
into triangles by means of a diagonal. Then the area may be
found if the diagonal and perpendicular heights of the triangles
are known.
72 VOCATIONAL MATHEMATICS FOR GIRLS
EXAMPLE. — In the trapezium ABCD if the diagonal is 43'
and the perpendiculars 11' and 17', respectively, what is the
area of the trapezium ?
43 X V- =^p = 236i sq. ft., area of ABC
43 X -V- = ^MH-=365| sq. ft., area of ADC
602 sq. ft., total area
Ans.
To find the areas of irregular figures,
draw the longest diagonal and upon this
diagonal drop perpendiculars from the ver-
tices of the figure. These perpendiculars will form trapezoids
and right triangles whose areas may be determined by the pre-
ceding rules. The sum of the areas of the separate figures will
give the area of the whole irregular figure.
Polygons
A plane figure bounded by straight lines is a polygon. A
polygon which has equal sides and equal angles is a regular
polygon.
The apothem of a regular polygon is the line drawn from the
center of the polygon perpen-
dicular to one of the sides.
A five-sided polygon is a
pentagon.
A six-sided polygon is a
PENTAGON HEXAGON
hexagon.
An eight-sided polygon is an octagon.
The shortest distance between the opposite sides of a regu-
lar hexagon is the perpendicular distance between them, and
is equal to the diameter of the inscribed circle.
The diameter of the circumscribed circle is the long diame-
ter of a regular hexagon.
The perimeter of a polygon is the sum of all its sides.
MENSURATION 73
The area of a regular polygon equals one half the product of
the apothem and the perimeter.
Formula ^ — \ aP
In this formula P = perimeter
a = apothem
Ellipse
Only the approximate circumference of an ellipse can be ob-
tained.
The circumference of an ellipse equals one half the product of
the sum of two diameters and TT.
If ^ = major diameter
d2 = minor diameter
C = circumference
then C- = ^±4,
The area of an ellipse is equal to one fourth the product of
the major and minor diameters by TT.
If A = area
dl = major diameter
d2 = minor diameter
then A = w^
4
EXAMPLES
1. Find the area of a trapezium if the diagonal is 93' and
the perpendiculars are 19' and 33'.
2. What is the area of a trapezoid whose parallel sides are
18 ft. and 12 ft., and the altitude 8 ft. ?
3. What is the distance around an ellipse whose major
diameter is 14" and minor diameter 8" ?
74 VOCATIONAL MATHEMATICS FOR GIRLS
4. In the map of a country a district is found to have two
of its boundaries approximately parallel and equal to 276 and
216 miles. If the breadth is 100 miles, what is its area ?
5. If the greater and lesser diameters of an elliptical man-
hole door are 2' 9" and 2' 6", what is its area ?
6. Find the area of a trapezium if the diagonal is 78" and
the perpendiculars 18" and 27".
7. The greater diameter of an elliptical funnel is 4 ft. 6 in.,
and the lesser diameter is 4 ft. What is its area ?
8. Find the perimeter of a hexagon having each side 15"
long.
9. What is the area of a pentagon whose apothem is 4y
and whose side is 5" ?
Volumes
The volume of a rectangular-shaped bar is found by multi-
plying the area of the base by the length. If the area is in
square inches, the length must be in inches.
The volume of a cube is equal to the cube of an edge.
The contents or volume of a cylindrical solid is equal to the
product of the area of the base by the height.
If S = contents or capacity of cylinder
R = radius of base
H = height of cylinder
TT = 3.1416+ or 2f. (approx.)
8 =
EXAMPLE. — Find the contents of a cylindrical tank whose
inside diameter is 14" and height 6'.
H= 6' = 72"
8 = ^ x 7 x 7 x 72 = 11,088 cu. in.
MENSURATION
75
The Pyramid
The volume of a pyramid equals one
third of the product of the area of the base
and the altitude.
The volume of a frustum of a pyramid
equals the product of one third the alti-
tude and the sum of the two bases and the
square root of the product of the bases.
The surface of a regular pyramid is equal to the product of
the perimeter of the bases and one half the slant height.
S = P X i 8/1
The Cone
A cone is a solid generated by a right triangle revolving on
one of its legs as an axis.
The altitude of the cone is the perpendicular distance from
the base to the apex.
The volume of a cone equals the product of the area of the
base and one third of the altitude.
F
or
V= .2618
EXAMPLE. — What is the volume of a cone 1-J-" $
in diameter and 4" high ?
Area of base = . 7854 x f
7.0686
= 1.7671 sq. in.
= .2618 x | x 4 = 2.3562 cu. in. Ans.
The lateral surface of a cone equals one half the product of
the perimeter of the base by the slant height.
76
VOCATIONAL MATHEMATICS FOR GIRLS
EXAMPLE. — What is the surface of a cone having a slant
height of 36 in., and a diameter of 14 in. ?
C = irD = 14 X
44 x 36
— . 4.4 tf
= 792 sq. in. Ans.
Frustum of a Cone
The frustum of a cone is the part of a cone included between
the base and a plane or upper base which is parallel to the
lower base.
The volume of a frustum of a cone equals the product of one
third of the altitude and the sum of the two bases and the
square root of their product.
When H = altitude
Bl == upper base
R = lower base
V=±H(B + 1? + VB&)
The lateral surface of a frustum of a cone equals one half the
product of the slant height and the sum of the perimeters
of the bases.
The Sphere
The volume of a sphere is equal to
where R is the radius.
The surface of a sphere is equal to
The Barrel
To find the cubical contents of a barrel, (1) multiply the
square of the largest diameter by 2, (2) add to this product
MENSURATION 77
the square of the head diameter, and (3) multiply this sum by
the length of the barrel and that product by .2618.
EXAMPLE. — Find the cubical contents of a barrel whose
largest diameter is 21" and head diameter 18", and whose
length is 33".
212 = 441 x 2 = 882 V= [(Z>2 x 2) + d2] x L x .2618
182 = 324 324 39798
120(3 .2618
33 10419-11 cu. in.
= 45.10 gal. Ans.
3618
3618 10419.11
39798 231
Similar Figures
Similar figures are figures that have exactly the same shape.
The areas of similar figures have the same ratio as the
squares of their corresponding dimensions.
EXAMPLE. — If two boilers are 15' and 20' in length, what is
the ratio of their surfaces ?
|jj. = f, ratio of lengths
§! = JL ratio of surfaces
4'2 16
One boiler is T9^ as large as the other. Ans.
The volumes of similar figures are to each other as the cubes
of their corresponding dimensions.
EXAMPLE. — If tw° iron balls have 8" and 12" diameters,
respectively, what is the ratio of their volumes ?
r8^ = |, ratio of diameters
= 28T, ratio of their volumes. Ans.
One ball weighs $7 as much as the other.
78 VOCATIONAL MATHEMATICS FOR GIRLS
EXAMPLES
1. Find the volume of a rectangular box with the following
inside dimensions : 8" by 10" and 4' long.
2. The radius of the small end of a bucket is 4 in. Water
stands in the bucket to a depth of 9 in., and the radius of the
surface of the water is 6 in. (1) Find the volume of the water
in cubic inches. (2) Find the volume of the water in gallons
if a cubic foot contains 7.48 gal.
3. What is the volume of a steel cone 2^" in diameter and
6" high?
4. Find the contents of a barrel whose largest diameter is
22", head diameter IS", and height 35".
5. What is the volume of a sphere 8" in diameter ?
6. What is the volume of a pyramid with a square base,
4" on a side and 11" high ?
7. What is the surface of a wooden cone with a 6" diameter
and 14" slant height ?
8. Find the surface of a pyramid with a perimeter of 18"
and a slant height of 11".
9. Find the volume of a cask whose height is 3^' and the
greatest radius 16", and the least radius 12", respectively.
10. How many gallons of water will a round tank hold which
is 4 ft. in diameter at the top, 5 ft. in diameter at the bottom,
and 8 ft. deep ? (231 cu. in. = 1 gal.)
11. What is the volume of a cylindrical ring having an
outside diameter of 6-J-", an inside diameter of 5Ty, and a
height of 3f " ? What is its outside area ?
12. If 9 tons of wild hay occupy a cube 7' x 1' X 7', how
many cubic feet in one ton of hay ?
13. A sphere has a circumference of 8.2467". (a) What is
its area ? (6) What is its volume ?
MENSURATION 79
14. If it is desired to make a conical can with a base 3.5" in
diameter to contain 1 pint, what must the height be ?
15. What is the area of one side of a flat ring if the inside
diameter is 2£' ' and the outside diameter 4£" ?
16. There are two balls of the same material with diameters
4" and V, respectively. If the smaller one weighs 3 lb., how
much does the larger one weigh ?
17. If the inside diameter of a ring is 5 in. , what must the
outside diameter be if the area of the ring is 6.9 sq. in. ?
18. How much less paint will it take to paint a wooden ball
4" in diameter than one 10" in diameter ?
19. What is the weight of a brass ball 3£" in diameter if
brass weighs .303 lb. per cubic inch ?
20. A cube is 19" 011 its edge, (a) Find its total area.
(&) Its volume.
21. If a barrel of water contains about 4 cu. ft., what is
the approximate weight of the barrel of water? (1 cu. ft.
of water weighs 62.5 lb.)
22. A conical funnel has an inside diameter of 19.25" at the
base and is 43" high inside, (a) Find its total area, (b) Find
its cubical contents.
23. A pointed heap of corn is in the shape of a cone. How
many bushels in a heap 10' high, with a base 20' in diameter ?
A bushel contains 2150.42 cu. in.
24. Find the capacity of a rectangular bin 6 ft. wide, 5 ft.
6 in. deep, and 8 ft. 3 in. long.
25. Find the capacity of a berry box with sloping sides 5.1"
by 5.1" on top, 4.3" by 4.3" at the bottom, and 2.9" in depth.
26. Find the capacity of a cylindrical measure 13" in
diameter and 6" deep.
27. How many tons of nut coal are in a bin 5 ft. wide and
8 ft. long if filled evenly to a depth of 4 ft. ? Average nut coal
weighs 52 lb. to a cubic foot.
CHAPTER III
INTERPRETATION OF RESULTS
Reading a Blue Print. — Everyone should know how to read
a blue print, which is the name given to working plans and
drawings with white lines upon a blue background. The blue
print is the language which the architect uses to the builder,
the machinist to the pattern maker, the engineer to the foreman
EXTERIOR VIEW OF COMPLETED HOUSE
of construction, and the designer to the workman. Through
following the directions of the blue print the carpenter, metal
worker, and mechanic are able to produce the object wanted
by the employer and his designer or draftsman.
Blue Print of a House. — An architect, in drawing the plans
of a house, usually represents the following views : the ex-
terior views to show the appearance of the house when it is
finished j views of each floor, including the basement, to show
80
INTERPRETATION OF RESULTS
81
WEST ELEVATION OF HOUSE
j. i i : : -___
EAST ELEVATION
the location of rooms, windows, doors, and stairs. Detailed
plans of sections are drawn for the contractors to show the
method of construction.
82 VOCATIONAL MATHEMATICS FOR GIRLS
NORTH ELEVATION
! — ' '
•*'—-*[ — — —-.--—— - — — — — — . , _;_
A— -——' — — — — -— £a rt ft fro rn
SOUTH ELEVATION
Pupils should be able to form a mental picture of the appear-
ance of a building constructed from any blue print plan set
before them. They should have practice in reading the plans
of the house and in computing the size of the rooms directly
from the blue print.
INTERPRETATION OF RESULTS
-FfagsloncCap
83
1. What is the height of the rooms on the first floor ?
2. What is the height of the rooms on the second floor ?
3. What is the height of the cellar, first, and second floors ?
EC.
GROUND FLOOR PLAN
84
VOCATIONAL MATHEMATICS FOR GIRLS
1. What is the frontage of the house ?
2. What is the depth of the house ?
3. What is the length and width of the front porch ?
4. What is the length and width of the living room ? the
dining room ? the kitchen ?
SECOND FLOOR PLAN
1. What is the size of each of the bedrooms ? (Compute
with aid of ground floor plan.)
2. What are the dimensions of the bathroom ?
3. How large is the storage room ?
Two views are usually necessary in every working drawing,
one the plan or top view obtained by looking down upon the
object, and the other the elevation or front view. When an
INTERPRETATION OF RESULTS 85
object is very complicated, a third view, called an end or
profile view is shown.
All the information, such as dimensions, etc., necessary to construct
whatever is represented by the blue print must be supplied on the draw-
ing. If the blue print represents a machine, it is necessary to show all
the parts of the machine put together in their proper places. This is
called an assembly drawing. Then there must be a drawing for each
part of the building or the machine, giving information as to the size,
shape, and number of the pieces. Then if there are interior sections,
these must be represented in section drawings.
Drawing to Scale. — As it is impossible to draw most objects
full size on paper, it is necessary to make the drawings pro-
portionately smaller. This is done by making all the dimen-
sions of the drawing a certain fraction of the true dimensions
of the object. A drawing made in this way is said to be drawn
to scale.
TRIANGULAR SCALE
The dimensions on the drawing designate the actual size of
the object — not of the drawing. If a drawing were made of
an iron door 25 inches long, it would be inconvenient to repre-
sent the actual size of the door, and the drawing might be made
half or quarter the size of the door, but on the drawing the
length would read 25 inches.
In making a drawing " to scale," it becomes very tedious to
be obliged to calculate all the small dimensions. In order to
obviate this work a triangular scale is used. It is a rule with
the different scales marked on it. By practice the student will
be able to use the scale with as much ease as the ordinary
rule.
QUESTIONS AND EXAMPLES
1. Tell what is the scale and the length of the drawing of
each of the following :
86 VOCATIONAL MATHEMATICS FOR GIRLS
a. An object 14" long drawn half size.
b. An object 2.6" long drawn quarter size.
c. An object 34" long drawn one third size.
d. An object 41" long drawn one twelfth size.
2. If a drawing made to the scale of f " = 1 ft. is reduced
i in size, what will the new scale be ?
3. A drawing is made -^ size. If the scale is doubled, how
many inches to the foot will the new scale be ?
4. On the TJg" scale, how many feet are there in 18 inches ?
5. On. the y scale, how many feet are there in 26 inches ?
6. On the \" scale, how many feet are there in 27 inches ?
7. If the drawing of a door is made ^ size and the length of
the drawing is 8^", what will it measure if made to scale 3"
= 1 f t. ?
8. What will be the dimensions of the drawing of a banquet
hall 582' by 195' if it is made to a scale of Ty ' = 1 ft. ?
Estimating Distances. — Everyone meets occasions in daily
life when it is of utmost importance that distance or weight
should be correctly estimated.
Few people have a clear conception of even our common standards of
measurements. This is due to the fact that the average person has never
given the proper attention to them. Improvement will be noticed after a
small amount of drill. To illustrate : if the distances of one inch, one
foot, one yard, six feet, and ten feet are measured off in a classroom so
that an actual view of standard distances is obtained, and then pupils are
asked to estimate other and unknown distances, they will estimate with a
greater degree of accuracy. Pupils should be able to estimate within
^ inch any distance up to a yard.
The power of estimating longer distances, such as the distance between
buildings, across streets, or between streets, may be developed by laying
off on a straight road one hundred feet, three hundred feet, and five hun-
dred feet sections, with the proper distance marked on each.
The same plan applies to heights of buildings, etc. Standards of alti-
tude may thus be established.
Pupils should measure in their homes pieces of furniture and wall
openings so that they may develop an eye for estimating distances.
INTERPRETATION OF RESULTS 87
1. Estimate the length and width of the schoolroom. Verify
this estimate by actual measurement and express the accuracy
of your estimate in per cent.
2. Estimate the height and width of the school door. Verify
this estimate by actual measurement and express the accuracy
of the estimate in per cent.
3. Estimate the width and length of the window panes ;
the width and length of the window sill.
Estimating Weights. — What is true concerning the advan-
tage of being able to estimate distances applies equally well to
weights.
In this, guesswork may be largely eliminated. A little mental figur-
ing on the part of the pupil will usually produce clear results. Weight
depends not only on volume but also on the density of the material.
Regular blocks of wood are excellent to begin with, and later small
spheres and rectangular blocks of different metals afford good material.
1. Select blocks of wood, coal, iron, lead, tin, or copper, and
estimate their respective weights.
2. Estimate the weight of a chair.
3. Estimate the weight of different persons.
Methods of Solving Examples. — Every commercial, household,
or mechanical problem or operation has two distinct sides : the
collecting of data, and the solving of the problem.
The first part, the collecting of data, demands a knowledge
of the materials and conditions under which the problem is
given, and calls for the exercise of judgment as to the neces-
sary accuracy of the work.
There are three ways by which a problem may be solved :
1. Exact method.
2. Rule of thumb method, by the use of a formula or a rule
committed to memory.
3. By means of tables.
The exact method of solving a problem in arithmetic is the
one usually taught in school and is the method obtained by
88 VOCATIONAL MATHEMATICS FOR GIRLS
analysis. Everyone should be able to solve a problem by the
exact method.
The Rule of Thumb Method. — Many of the problems that
arise in home, office, and industrial life have been met before,
and very careful judgment has been exercised in solving them.
As the result of this experience and the tendency to abbreviate
and devise shorter methods that give ' sufficiently accurate re-
sults, we find many rule of thumb methods used by the house-
wife, the storekeeper, the nurse, etc. The exact method would
require considerable time and the use of pencil and paper,
whereas in cases that are not too complicated the estimates,
based on experience or rule, give a quick and accurate result.
In solving problems involving the addition and subtraction
of fractions, use the yardstick or tape to carry on the compu-
tation. To illustrate : if we desire to add 1 and J- of a yard,
place the thumb over 1 of a yard divisions, then slide (move) the
thumb along the divisions corresponding to J- of a yard, and
then read the number of divisions passed over by the thumb.
In this case the result is 21 inches.
The Use of Tables. — In the commercial world the tendency
is to do everything in the quickest and the most economical
way. To illustrate : hand labor is more costly than machine
work, so, whenever possible, machine work is substituted for
hand labor. The same tendency applies to calculations in the
dressmaking shop or the office. The exact methods of doing
examples are not the quickest, nor are they more easily under-
stood and performed by the ordinary girl than the shorter
methods. Since a great many of the problems in calculation
that arise in the daily experiences of the office assistant, the
housewife, the dressmaker, the nurse, etc. are about ordinary
things and repeat themselves often, it is not necessary to work
them anew each time, if, when they are once solved, results are
kept on file in the form of tables.
See pages 220, 222, and 254 for tables used in this book.
PART II — PROBLEMS IN HOMEMAKING
CHAPTER IV
THE DISTRIBUTION OF INCOME
THE economic standing of every person in the community
depends upon three things : (1) earning capacity, (2) spend-
ing ability, and (3) the saving habit. The first regulates the
amount of income ; the second determines the purchasing
power after the amount is earned ; the third paves the way
to independence.
The welfare of every person, whether single or married,
depends upon the systematic and careful regulation of each
of these three items. No matter how large or small his wages
or salary, if he does not spend his money wisely and carefully,
or save each week or month a certain per cent of his earnings,
a young man or woman is not likely to make a success of life.
A young woman usually has more to do with the spending of money
than a young man. The wife is really the spender and the husband the
earner in the ordinary home. Therefore, it becomes necessary for every
young woman to know how to get one hundred cents out of a dollar. In
order to do this, she must know how to distribute the income over such
items as rent, food, clothing, incidental expenses due to sickness, pleas-
ure, or self-improvement. The proportion spent for each item should be
carefully regulated.
Incomes of American Families
The average family income of both foreign and native born heads is
about $ 725 a year ; that of families with native born heads alone is
about $ 800. Not more than one-fourth have incomes exceeding $ 1000.
The daily wages of adult men range from $ 1.50 to $ 5.00. This amounts
on the average from .$450 to $1500 a year.
The family, the head of which earns only a few hundred dollars a
year, must either be contented with comparatively low standards of liv-
89
90
VOCATIONAL MATHEMATICS FOR GIRLS
ing or obtain additional income, either through the labor of children or
from boarders or lodgers. The foreign-born workers resort to the labor
of children and mothers more than do the native Americans. The second
course is quite often adopted so that the average income of workingmen's
families is considerably greater than the average earnings of the heads of
the families.
INCOMES
EXPENDITURES
Based on Statistics of Twenty-five Thousand Families with an Average
Yearly Income of Seven Hundred and Fifty Dollars
EXAMPLES
1. The average workingman's family spends at least two-
fifths of its income for food. What per cent is spent for food ?
2. If the income of a workingman's family is $ 800, and the
amount spent for food is $ 350, what per cent is spent for food ?
3. One-fifth of the expenditure of workingmen's families is
for rent. What per cent ?
4. A family with an income of $ 800 spends $ 12.50 a
month for rent. What per cent of the income is spent for
rent? Is this too much?
5. A family's income is $ 760. The father contributes $ 601.
What per cent of the income is contributed by the father ?
6. A family of six has an income of $ 840. The father
contributes $ 592, mother $ 112, and one child the balance.
What per cent is contributed by the mother and child?
THE DISTRIBUTION OF INCOME 91
7. A man and wife have an income of $ 971. The husband
earns $ 514, the wife keeps boarders and lodgers, and provides
the rest of the income. What per cent of the income is con-
tributed by the boarders and lodgers ?
Cost of Subsistence
Shelter, warmth, and food demand from two-thirds to three-
fourths of the income of most workingmen's families. This
leaves for everything else — clothing, furniture, sickness, death,
insurance, religion, education, amusements, savings — only one-
third or one-fourth of the income. Between $ 200 and $ 250 a
year may be considered the usual outlay of workingmen's fami-
lies for all these purposes combined. It is in these respects
that the greatest difference appears between the families of
the comparatively poor and the families of the well-to-do. - The
well-to-do spend not only more in absolute amount, but also a
larger proportion of their incomes on these, in general, less
absolutely necessary things.
Clothes. — On the average, approximately one-eighth of the
income in workingmen's families goes for clothes. To those
who keep abreast of the fashions and who dress with some
elegance, it may seem quite preposterous that a family of five
should spend only $ 100 or less a year for clothing, but multi-
tudes of working-class families are clad with warmth and with
decency on such an expenditure.
EXAMPLES
1. If two-thirds of the average workingman's income is
spent for shelter, warmth, and food, what per cent is used ?
2. A family, receiving an income of $ 847, spends $ 579 for
shelter, warmth, and food. What per cent is used ?
3. If one-eighth of the income of the average workingmen's
family is spent for clothes, what is the per cent ?
4. A family receives an income of $ 768, and $ 94 is spent
for clothes. What per cent is spent for clothes ?
92 VOCATIONAL MATHEMATICS FOR GIRLS
The High Cost of Living
The average cost of living represents the amount that must
be expended during a given period by the average family
depending on an average income. The maximum or minimum
cost, however, is another phase of the problem. It no longer
involves the amount of dollars and cents necessary to buy
and pay for life's necessaries, but involves questions of home
management and housekeeping skill, which cannot be stand-
ardized. About 1907 food and other necessities of life began
to increase in cost — and this has continued to the present day.
EXAMPLES
1. In 1906 a ton of stove coal cost $ 5.75, and in 1915 $ 8.75.
What was the per cent of increase in the cost of coal ?
2. In 1907 a suit of clothes cost $15. The same suit in
1912 cost $ 19.75. What was the per cent of increase ?
3. In 1908 a barrel of flour cost $ 6.10. The same barrel of
flour cost in 1914 $ 8.25. What was the per cent of increase ?
Division of Income
A girl should always consider her income for the entire year
and divide it with some idea of time and relative proportion.
If she earns a good salary for only ten months of, the year,
she must save enough during those months to tide her over the
other two. For instance, if a teacher earns $ 60 a month for
10 months of the year, her actual monthly income is $ 50.
The milliner, the trained nurse, the actress, and sometimes even
the girl working in the mill have the same problem to confront.
No girl has a right to spend nearly all she earns on clothing,
neither should she spend too much for amusement. We find
from investigations that have been made that girls earning $ 8
or $ 10 a week usually spend about half their income on board
and laundry. Girls earning a larger income may pay more
for board, but not quite so great a fractional part. In these
THE DISTRIBUTION OF INCOME 93
days, when the cost of living is so' high, a girl should consider
carefully a position that includes her board and laundry, for in
such a position she will be better off financially at the end
of the year than her higher salaried sister, who has to pay
for the cost of her own living. The housegirl can save about
twice as much as the average stenographer.
We find that the average girl needs to spend about one-fifth
of her income for clothing. A poor manager will often spend
as much as one-third and not be very well dressed at that,
because she buys cheap materials, that have to be frequently
replaced, and follows every passing fad and style. Choose
medium, styles and good materials and you will look more
richly dressed. Keep the shoes shined, straight at the heel,
and the strings fresh. Keep gloves mended, and as clean as
possible. If you spend more on clothing than the allotted one-
fifth, you will have to go without something else. It may be
spending money, or it may be gift or charity money, and quite
often it is the bank account that suffers.
Every person should save some part of his income. One
never knows when sickness, lack of employment, or ill health
may come. Saving money is a habit and one that should be
acquired the very first year that a person earns his own living.
EXAMPLES
1. A girl earns $ 12 a week for 42 weeks, and in this time
spends $ 144 for clothing. Is she living within the per cent
of her income that should be spent for clothing ?
2. A salesgirl earns $ 8 a week. She spends $ 98 a year for
clothes. Is she living within her income ?
3. A girl earns $ 5 a week and pays half of it to her home.
She has two car fares and a 14-cent lunch each day. How much
should she spend on clothing each year ? How much has she
for spending money each week ? Should she save any money ?
4. Which girl is the better off financially, one earning $ 6
a week as a housemaid or one earning $ 7 a week in a store ?
94 VOCATIONAL MATHEMATICS FOR GIRLS
Buying Christmas Gifts
Let the gift be something useful. Do not be tempted by the
display of fancy Christmas articles, for it is on these that the
merchant makes his profit for extra decorations and light.
Think of the person for whom you are buying. She may not
have the same tastes as you have, so give something that she
will like rather than something she ought to like. For in-
stance, a certain girl may be very fond of light hair-ribbons
when you know that dark ones would be much more sensible,
but at Christmas give the light ones.
The stores always show an extra supply of fancy neckwear.
A collar cannot be worn more than three days without becom-
ing soiled, so even 25 cents is too much to pay for something
that cannot be cleansed. Over-trimmed Dutch collars and
jabots easily rip apart. Choose the plain ordinary ones that
you would be glad to wear any day. You see whole counters
of handkerchiefs displayed with embroidery, lace, and ruffles.
A linen handkerchief, even of very coarse texture, is more
suitable.
Be careful also about bright colors, for everything about the
store is so gay that ordinary things appear dull, but when
you get them out against the white snow, they will be bright
enough.
EXAMPLES
1. Shortly before Christmas I purchased -J- doz. handker-
chiefs for $ 1.50. One month later I purchased the same kind
of handkerchiefs at 16| cents each or 6 for $ 1.00. What per
cent did I save on the second purchase?
2. I also bought a chiffon scarf for which I paid $2.25.
Early in the fall I saw similar scarfs selling for $ 1.50. How
much did I lose by not making my purchase at that time?
What per cent did I lose ?
3. I bought at Christmas two pairs of silk stockings at
$ 1.50 per pair. If I had purchased the stockings in October
THE DISTRIBUTION OF INCOME
95
they would have cost me $ 1.121 per pair. How much would
I have saved ? What per cent would I have saved ?
An Expense Account Book
Every person and every family should keep an expense
account showing each year's record of receipts and expendi-
tures. A sample form is shown in page 96. Rule sheets in a
similar manner for the solution of the problems that follow.
At the end of the year a summary should be made of receipts and dis-
bursements in some such form as the following :
YEARLY SUMMARY
Receipts
Disbursements
RECEIPTS
Cash on hand January 1
Salary, etc.
Other Income
DISBURSEMENTS
Savings and Insurance
Rent
Food
Clothing
Laundry
Car fares
Stamps and Stationery
Health
Recreation
Education
Gifts, Church, Charity
Incidentals
Balance on Hand December 31
Totals
.
Rule similar sheets for the solution of the following problems.
96
VOCATIONAL MATHEMATICS FOR GIRLS
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THE DISTRIBUTION OF INCOME 97
EXAMPLES
1. A man and wife have an income of $ 1000 a year. The
disbursements for the month of October are as follows :
Rent, $ 15 Tooth paste, $.18
Telephone, 1.45 Provisions, 5.85
Repair on coat, 5.80 Life insurance, 7.40
Gas, .75 Coal, 7.50
Dinner, 1.60 Outlook (1 year), 3.00
Stationery, 2.51 Assistance, .60.
Fares, .85 Shampoo, .50
Groceries, 10.35 Fares, .60
Fruit, .30 Rubbers, .90
Theater, .50 Soap, .10
Papers, .06 Church, .25
Church, .25 Milk, .56
Milk, .71 Ice, .40
Ice, .45 Papers, .11
Electricity, 1.00 Vase for D., .75
Laundry, .50 Peroxide, .25
Enter the above disbursements in the expense account book.
Find the total amount.
What per cent was spent for food ? house ? clothing ?
housekeeping ? luxuries ? savings ?
2. The items of expense for the month of January, 1915, are :
rent and water, $ 15 ; operating expense : light and heat, $ 11 ;
food : meats, groceries, $ 30 ; labor or services, $ 16.65 ; cloth-
ing, $ 15 ; physician, $ 1 ; carfares, $ 2.85 ; books, $ 1.00 ;
amusements, $ 4 ; cigars, $ 1.00 ; gifts, $ 1.00 ; sundry ex-
penses, $ 1.50. Treat as in Ex. 1.
3. A family of two receives an income of $ 1200 a year.
They spend per week $ 6.93 for food, $ 3.51 for rent, $ 3.49
for operating expenses, $ 5.81 for contingency. What is the
amount for each item per month (4 weeks) ? per year
(52 weeks) ? What is the per cent of each item ?
98 VOCATIONAL MATHEMATICS FOR GIRLS
4. A young married woman has an income of $ 1200 a year
— $ 100 a month. The following represents the way she
spends her income a month :
Savings bank, $5.00 Ice (set aside in the winter months
Kent, 25.00 for the summer), $0.23
Insurance, 5.00 Necessary carfares (the house is
Groceries, 4.70 located in the country), 2.70
Meat and fish, 11.15 Recreation, 3.00
Milk, 2.79 Avocation, 3.00
Clothing, 12.00 Literature, .50
Heat and light, 5.00 Church and charity, .80
Laundry and supplies, 2.00 Remembrances, .75
Help, 4.00 Fire insurance, .09
Repairs and replenishing, .50
Enter the above in the expense account book.
How much was left toward the next month's expense ? Can
you improve on this? What is the per cent for food?
Clothing ?
5. A girl 14 years of age has cost her parents an amount
equal to the following items :
Food, $ 597. 16 Carfare to school, $ 11 .00
Clothing, 339.66 Doctor, 70.00
Furniture, 88.65 Dentist, 10.00
What is the per cent for each item ?
6. A family of seven — three grown people and four chil-
dren — lived in a southern city on $ 600 — $ 50 per month.
The expenses each month were as follows :
House rent, $ 12.00 Bread, $2.50
Groceries, 12.00 Beef, 2.50
Washing, 5.00 Vegetables, 2.00
What is the balance for clothing and fuel ? What is the
per cent of income spent for food ? Clothing ? Rent ? Suggest
points of saving.
THE DISTRIBUTION OF INCOME 99
7. A girl in New York City lives on $ 10 a week. The ex-
penses are as follows :
Board and washing, $ 300.00 Clothing and vacation, $ 95.00
Luncheons and carfare, 100.00 Church and charity, 10.00
How much can she save ? What is the per cent for cloth-
ing ? Incidentals ?
Can you suggest any improvements in the distribution of her
income ?
8. A family of four lives on $ 750 a year. The expenses
are as follows :
Rent, $180.00 Groceries, including vegetables,
Fuel, 52.00 butter, eggs, milk, kerosene,
Meat, oysters, cheese, 95.00 soap, etc., $241.00
Clothing, about 145.00
How much is left for the savings bank ? What per cent for
rent ? Food ? Clothing ? Operating expenses ?
CHAPTER V
FOOD
SINCE half of the income of the average family is spent for
food, it is important that this expenditure should be made
intelligently. Experts of the United States Department of
Agriculture estimate that the food waste in a great many of
the American homes is as high as 20 % . The causes of waste
may be classified as follows : Needlessly expensive material,
providing little nutrition ; failure to select according to season ;
great amounts thrown away ; poor preparation ; badly con-
structed ovens.
If this waste were checked, it would afford an increase in
the purchasing power of the income which would appreciably
lift the family to a higher plane of efficiency. The efficiency
of every person depends upon the energy and constant repair
of the body. A woman should know the cost of food and real-
ize what food value she is receiving for her money. It may
seem strange, but it is true, that "a Roman feast, a Lenten
fast, a Delmonico dinner, and the lunch of the wayside beggar
contain the same few elements of nutrition."
The art of cooking can transform the common but nutritious foods
into the most appetizing dishes. Further than this, the freshness and
attractiveness of the food, the way in which it is served, the sur-
roundings — all affect the appetite and the power to digest. Cost,
which must always be considered in the limited income in relation to
the nutritive value of food, is influenced by an equally important factor
— waste.
The problem of feeding our bodies is primarily a question of supply
and demand. Of course, the element of pleasure in eating is a properly
100
FOOD
normal one, for enjoyment aids digestion. We must, however, eat to live
rather than live to eat.
Every motion of the body and every thought in the brain destroy cer-
tain tissues. This material must be replaced from the food we eat. The
two objects of eating are tissue repair in the adult and tissue repair
plus growth in the child. As soon as we realize that these two pur-
poses should determine the character of the food we eat, then we
shall know the importance of intelligent instead of haphazard choice of
food.
To repair the body means to supply the elements needed to renew the
tissues that are worn or destroyed. We can separate the elements of the
human body broadly into water, protein, carbohydrates, fats, and ash.
Water composes 60 per cent of a normal man's body. In other words,
a 200-pound man is composed of only 80 pounds of solids, of which 18
per cent is composed of protein, 15 per cent of fat, 1 per cent of carbo-
hydrates, and 6 per cent of ash.
Water aids digestion and is necessary in the economy of life. Protein
is the basis of bone, muscle and other tissues, and essential to the body
structure. Fat is an important source of energy in the form of heat
and muscular power. Carbohydrates are transformed into fat and are
important constituents, though in small proportions in the human body.
Ash is composed of potash, soda, and phosphates of lime, and is necessary
in forming bone. The diet best fitted to supply all the needs of the healthy
human organism must contain a correct proportion of these elements ; it is
called the balanced ration.
Nutritive Ingredients (or Nutrients) of Food
What has thus far been said about the ingredients of food and the ways
in which they are used in the body may be briefly summarized in the
following manner :
Food as purchased
contains
Edible portion . . .
e.g. flesh of meat, yolk
and white of eggs, wheat,
flour, etc.
Refuse.
e.g. bones, entrails, shells,
bran, etc.
Water
Nutrients
Protein
Fats
Carbohydrates
Mineral mat-
ters.
.; OCATIONAL MATHEMATICS FOR GIRLS
Repairs tissues
All serve as
fuel to yield
I energy in the
forms of heat
and muscular
power.
in digestion, etc.
Uses of Nutrients in the Body
Protein Forms tissue
e g. white (albumen)
of eggs, curd (casein)
of milk, lean meat,
gluten of wheat, etc.
Fats Are stored in the body as fat
e.g. fat of meat, butter,
olive oil, oils of corn
and wheat, etc.
Carbohydrates .... Are transformed into fat .
e.g. sugar, starch, etc.
Mineral matters (ash) . Share in forming bone, assist
e.g. phosphates of lime,
potash, soda, etc.
The views thus presented lead to the following definitions : (1) Food
is that which, taken into the body, builds tissue or yields energy.
(2) The most healthful food is that which best fills the needs of the man.
(3) The cheapest food is that which furnishes the largest amount of
nutriment at the least cost. (4) The best food is that which is at the same
time most healthful and cheapest.
EXAMPLES
Carbohydrates are present in large proportions in all the cereals, bread,
and potatoes, and are almost 100 % pure in sugar. There is a widespread
notion that starch, which is a fat-producing element, is mostly contained
in potatoes, and many people who wish to reduce flesh omit potatoes
and substitute rice or larger quantities of bread or cereals. The fact is
that the white potato contains only 15 % carbohydrates and the sweet
potato 27 %, while rice has 77 % and the breads range from whole wheat
bread at 49 % to soda crackers at 73 % and the cereals from oats at 69 %
to rye, 78 %.
1. How many ounces of carbohydrates in J Ib. of white
potatoes ?
<2. Find the number of pounds of carbohydrates in 4 Ib. of
rice.
3. Find the number of ounces of carbohydrates in a loaf of
whole wheat bread (J Ib.)
FOOD 103
4. How many ounces of carbohydrates in a 2 Ib. package of
Quaker Oats ?
5. How many ounces of carbohydrates in 4 oz. of soda
crackers ?
EXAMPLES
The proportion of ash in foods is small, and as the body requires 6 per
cent, we must be sure to supply it in the food. Salt cod has the largest
proportion, 25 per cent, and we find it in good quantities in butter, dried
beef, smoked herring, and bacon.
Of the proteins, lean meat is the one most easily digested and assim-
ilated. Curiously enough, dried beef has the largest proportion of pro-
tein of any flesh meat, 30 per cent, while next in range are leg of lamb,
beef steak, roast beef, and fowl, with about 18 per cent.
Let us note the protein value of fish. Smoked herring, despised by
many, contains 36 per cent of protein, salt cod and canned salmon 21 per
cent, and perch, halibut, mackerel, and fresh cod average 18 per cent,
equal in food value per pound to the best beef and fowl. The peanut has
27 per cent of its weight protein, and peanut butter 29 per cent.
Fat is found in large proportion in nuts, especially in walnuts, which
contain 63 per cent, and cocoanuts 57 per cent. Bacon contains 67 per
cent of fat, and butter 85 per cent.
1. If a man weighs 189 pounds, how many pounds of water
in his weight ? How many pounds of solids ? How many
pounds of fat ? protein ? carbohydrates ?
2. How many ounces of protein in a pound and a half dried
beef ?
3. Give the number of ounces of protein in a pound fowl.
4. Give the number of ounces of protein in 1^ Ib. of herring.
5. How many ounces of fat in li Ib. of shelled walnuts ?
Kitchen Weights and Measures
Correct measurements are absolutely necessary to insure
successful results in cooking.
Sift flour, meal, powdered sugar and soda before measuring.
Many articles settle hard or in lumps and should be stirred
and pulverized before measuring.
104 VOCATIONAL MATHEMATICS FOR GIRLS
Measure all materials level full, leveling with knife. Do
not pack powdered articles. Butter, lard, etc., should be
packed in measure and leveled.
A half spoonful should be taken lengthwise and not crosswise.
A quarter spoonful should be taken by dividing a half crosswise.
A third spoonful is obtained by dividing twice crosswise.
EQUIVALENTS To MAKE ONE POUND
3 teaspoons equal 1 tablespoon. 4 cups flour.
4 tablespoons equal \ cup. 2£ cups corn meal.
2 cups equal 1 pint. 2| cups oatmeal.
2 pints equal 1 quart. 6 cups rolled oats.
4 quarts equal 1 gallon. 4^ cups rye meal.
4 cups flour equal 1 Ib. 2 cups rice.
2 cups sugar equal 1 Ib. 2 cups granulated sugar.
16 tablespoons dry ingredients 2f cups brown sugar.
equal 1 cup. 2f cups powdered sugar.
12 tablespoons liquid equal 1 cup. 3J cups confectioner's sugar.
2 cups butter.
2 cups chopped meat. .
4| cups ground coffee.
t. is the abbreviation for teaspoonful; and £&., for tablespoonful.
EXAMPLES
1. How many teaspoons in 4 tablespoons ?
2. How many tablespoons in j cup ?
3. How many cups are equivalent to 5 pints ?
4. Give the number of tablespoons in a pint.
5. Give the number of teaspoons in 3 quarts and 1 pint.
6. A cup of flour weighs how many ounces ? What part
of a pound ?
7. How many cups will 56 tablespoons of baking soda fill ?
8. A pint of liquid contains how many tablespoons ?
9. A cup of sugar weighs how many ounces ?
10. Two cups of corn meal is what part of a pound ?
FOOD 105
11. Give the weight in ounces and fractions of a pound of
the following quantities : —
(a) 1 cup of butter. (/) 1 ,cup of powdered sugar.
(b) 1 cup of rice. (g) 1 cup of brown sugar.
(c) 3 cups chopped meat. (7i) 1 cup of rye meal.
(d) 1 cup of coffee. (i) 1 cup of oatmeal.
(e) 1 cup of conf. sugar.
12. What is the cost of each of the following :
(a) 1 cup of oatmeal at 5 cts. a pound ?
(b) 2 cups of sugar at 5 Ibs. for 33 cts.?
(c) 2i cups of rice at 5 cts. a pound ?
(d) i cup of milk at 8 cts. a quart ?
(e) J cup of butter at 35 cts. a pound ?
(/) 2 eggs at 38 cts. a dozen ?
(g) | peck of potatoes at 72 cts. a bushel ?
(h) 31 level teaspoons of sugar at 8 cts. a pound ?
(i) i cup of vinegar at 9 cts. a quart.
Cost of Food
In order to calculate the cost of food it is necessary to know
price per pound, price per cupful, and even price per teaspoon-
ful. The price should be calculated to three decimal places
and the results tabulated as follows :
Cost per pound or quart.
Number of cupfuls in pound or quart.
Cost per cupful.
Cost per tablespoonful.
Cost per teaspoonful.
When it is once calculated the data may be used from day
to day in calculating the cost of food.
EXAMPLES
1. What is the cost per teaspoonful of cocoa at 38 cents a
pound ? (4 cups in a pound.)
106 VOCATIONAL MATHEMATICS FOR GIRLS
2. What is the cost of a third of a cup of powdered sugar
at 10 cents a pound ? (2 J cups in a pound.)
3. What is the cost of a tablespoonful of cream at 23 cents
a pint ?
4. What is the cost of 2 teaspoonfuls of sugar at 6-J- cents
a pound ? (2 cups in a pound.)
5. What is the cost of 6 nuts at 20 cents a pound ? (29
walnuts in a pound.)
6. What is the cost of 6 tablespoons of coffee at 35 cents a
pound ? (41 cups of coffee in a pound.)
7. What is the cost of 3 slices of toast at 5 cents a loaf ?
(10 slices in a loaf.)
8. What is the cost of a pat of butter at 38 cents a pound ?
(16 pats of butter in a pound.)
9. What is the cost of an ordinary serving of cornflakes
at 10 cents a pound ? (15 servings in a pound.)
10. What is the cost of a serving of cream (f of cup) at
24 cents a pint ?
11. What is the cost of an ordinary serving of macaroni at
12 cents a pound ? (11 servings to the pound.)
12. What is the cost of a serving of cheese at 20 cents a
pound ? (9 servings in a pound.)
13. What is the cost of a serving of cabbage salad if cab-
bage is 3 cents a pound and two servings in a pound? (A
tablespoonful of salad dressing of equal parts of oil and vinegar 0
Oil is 25 cents a half pint. Vinegar is 10 cents a quart.)
14. What is the cost of a serving of stewed apricots at 18
cents a pound ? (A half pound of sugar at 7 cents is added
to the apricots. Nine servings in a pound.)
FOOD 107
15. What is . the cost of an ordinary serving of mashed
potatoes at 25 cents a half peck ? (A teaspoonful of milk at
8 cents a quart to each serving. A half pat of butter at 37
cents a pound, sixteen pats in a pound. Twenty-one servings
in a half peck.)
16. What is the cost of a serving of grape jelly (1 oz.) at
13 cents a pound ?
Girls should know how to make out a tabulation of stand-
ard food materials, the current price for such material at the
local stores, and the cost of quantities commonly used in cook-
ing, as one cup or one tablespoonful. They should, in addi-
tion, know how to take common recipes that are used in cook-
ing classes 'and reckon the cost. This will aid in reckoning
the cost of meals and arranging them economically. In a like
manner, the cost of meals for one day and for one week may
be calculated to see how near they are living within their
income.
EXAMPLES
1. A supper consisting of the following is served for 14
people : codfish in tomato sauce, cereal muffins, cookies, and
tea. What is the cost per person if the codfish costs 30 cents,
the muffins 24 cents, the cookies 34 cents, the tea 10 cents,
and fuel 5 cents ?
2. What is the cost per person for the following meal
when 14 are served ? The meal consists of milk toast, stewed
prunes with lemon, chocolate layer cake, and tea. The milk
toast costs 25 cents, the prunes 25 cents, the cake 50 cents,
the tea 10 cents, and fuel 10 cents.
3. In a pound of rolled oats, costing 8 cents, there are 4
cups. What is the cost of a serving (J cup) of rolled oats ?
4. In a package of cream of wheat costing 14 cents there
are 41 cups. One eighth of a cup is necessary for a serving.
What is the cost of a serving ?
108 VOCATIONAL MATHEMATICS FOR GIRLS
5. Compute the cost of a cup of white sauce from the fol-
lowing recipe :
1 cup milk
4£ t. flour
4£ t. butter
t. salt
milk, 9c a quart
flour, 5c a pound
butter, 39c a pound
salt Ic a cup
6. A dinner consisting of mashed potatoes, peas, rib roast,
rolls, jelly, fruit salad, krummel torte, coffee, cream and sugar
is served for six. What is the cost per person? Estimate
portions from amounts given.
DISHES AMOUNTS
Mashed potatoes £ pk. potatoes at 45c per pk.
1^ cups milk at 2c per cup
3 tablespoons butter at 40c per Ib.
Peas 6 tablespoons or £ can at 13c per can
Rib roast f of 4-lb. roast at 28c per Ib.
Gravy 3 cups flour at . l|c per cup
Rolls ^ cup milk at 2c per cup
3 tablespoons sugar, 2 tablespoons butter, f yeast cake
at2c
Jelly £ glass at lOc per glass
Apple and 3 apples at 8c per doz.
grape salad \ Ib. grapes at lOc per Ib.
Saratoga flakes \ package at 15c per package
Salad dressing \ cup vinegar at 8c per qt.
1 egg at 30c per doz.
3 tablespoons sugar at 7c per Ib.
1 tablespoon butter at 40c per Ib.
Krummel Torte \ package dates at lOc per package
f cup nuts at 70c per Ib., 3 cups per Ib.
\\ eggs at 30c per doz.
\ pt. whipping cream for torte as well as for coffee at 15c
Coffee 6 tablespoons at 34c per Ib.
Sugar 3 teaspoons
Sugar total 7 tablespoons = TV cup at 7c per Ib.
Butter total 6 tablespoons - f cup at 40c per Ib.
FOOD 109
7. The following breakfast and luncheon were served for
six. What was the cost of each meal per person ?
DISHES AMOUNT PER PERSON
Orange 1 medium-sized, 30c a doz.
Toast 2 thin slices, ic a slice
Butter 1 ordinary pat, £c
Egg 1 medium-sized, 3c
Corn flakes 1 ordinary serving, .2c
Cream f cup, 15c \ pt.
Sugar 3| level teaspoons, 7c Ib.
Coffee 2£ tb. at 34c per Ib
Luncheon
Macaroni Ordinary serving, 4c
and cheese f cu. in., .02c
Cabbage salad Large serving, .02c
Cooked dressing 1 tablespoon, .OO^c
Stewed apricots 1 serving, 2c
Doughnut 1 large, Ic
Milk 1 cup, 3c
8. The following dinner was served for six. What is the
cost per person ?
DISHES AMOUNT PER PERSON
Rib roasfc 1 Fairly large serving, 9c
Brown gravy J
Butter Ordinary pat, £c
Mashed potatoes Ordinary serving, £c
Peas Very small, £c
Jelly Very small, Ic
Buns 2 buns, Ic apiece
Apple and £ apple, Jc
grape salad \ oz. of grapes, Ic
Cooked dressing 1 tablespoon, ^c
Saratoga flakes 3 portions, lOc a pkg. (12 portions)
Krummel Torte
Dates 6 dates, lOc a Ib. (30 dates in a Ib.)
Nuts 6 nuts, 18c a Ib. (22 nuts in a Ib.)
Whipping cream 1 tablespoon, 15c \ pt.
Sugar 2 teaspoons, 7c a Ib.
Cream 1 tablespoon, 25c a pt.
110 VOCATIONAL MATHEMATICS FOR GIRLS
9. The recipe for potato soup for a family of man, wife, and
two children is :
3 large potatoes 2 tb. flour
1 qt. milk 1| t. salt
2 slices onion dash pepper
3 tb. butter 1 t. chopped parsley
a. What is the recipe for five men, two women, and three
children (consider a child's diet one-half a man's diet, and a
woman's equal to a man's)? b. What is the recipe for one
person (adult) ?
10. The recipe for a vegetable soup for a family of husband,
wife, and two children is :
Beef, 1 Ib. 1^ - qt. water
Bones, 1 Ib. 5 tb. butter
£ cup carrot 1 tb. finely chopped parsley
\ cup turnip salt
\\ cups potato pepper
\ onion
a. What is the recipe for a family of three men, two
women, and a child ? b. What is the recipe for a child ?
11. The recipe for sour-milk biscuits for a family of hus-
band, wife, and two children is :
2£ c. flour 1 tb. fat (lard or butter)
1 t. salt 1 c. sour milk, or \ c. sour milk
£ t. soda £ c water
a. What is the recipe for a man, wife, two boarders, and
five children ? 6. What is the recipe for one adult ?
Food Values
The heating value of food is measured by the amount of heat
given off when burned. The food taken into the human system
is oxidized slowly in order to give us the ability to do work ;
FOOD 111
the number of heat units that food is capable of giving a body
represents the quantity of energy the food will provide.
There are two units for measuring heat : the English and metric unit.
The English unit is called a Calorie, and it represents the quantity of heat
necessary to raise a pound of water four degrees on the Fahrenheit scale.
The metric unit is also called a calorie and is the amount of heat neces-
sary to raise a gram of water one degree on the Centigrade scale. The
English unit is called a large Calorie and is represented by the large letter
C while the metric unit is called a small calorie and is represented by the
small letter c.
All scientific experiments are conducted in the metric system, while
our measurements in daily life are in the English system. It is only nec-
essary to know the English unit, which is used in this book.
The United States Department of Agriculture
The Department Bulletin No. 28 gives the fuel value of foods. It may
be well to know how the fuel value is determined. To illustrate : What
is the fuel value of flour ? Careful experiments by the Department of
Agriculture show that flour is composed of 10.6 % protein, 1.1 °/o fat, and
76.5 % carbohydrates. Other experiments have shown that :
1 gram of protein yields 4100 Calories (C)
1 gram of fat yields 9300 Calories (C)
1 gram of carbohydrates yields 4100 Calories (C)
or 1 ounce of protein yields 113 Calories (C)
1 ounce of fat yields 255 Calories (C)
1 ounce of carbohydrates yields 113 Calories (C)
Then each ounce of flour contains
0.106 ounce protein
0.011 ounce fat
0.763 ounce carbohydrates
Since each ounce of protein yields 113 C, 0.106oz. will yield 113 x 0.106,
or 11.98 C. 0.011 oz. of fat will yield 255 x 0.011, or 2.81 C. 0.763 oz.
of carbohydrates will yield 0.763 x 113, or 86.22 C.
EXAMPLES
1. Rice contains 6 % protein, 79.5 % carbohydrates, and
0.7 °/0 fat. What is the fuel value of 3 oz. rice ?
112 VOCATIONAL MATHEMATICS FOR GIRLS
2. Milk contains 4 % protein, 5 % carbohydrates, and 4 %
fat. What is the fuel value of 8 oz. milk ?
3. Beans contain 23.1 °/0 protein, 57 % carbohydrates, and
2 % fat. What is the fuel value of 5 oz. beans ?
4. Chicken contains 24.4 % protein and 2 % fat. What is
the fuel value of 7 oz. chicken ?
5. Pork (shoulder) contains 16 % protein and 32.8 % fat.
What is the fuel value of 2 Ib. pork ?
6. Butter contains 0.6 °/0 protein, 0.5 % carbohydrates, and
85 % fat. Wrhat is the fuel value of 1 Ib. butter ?
7. Eggs contain 14.9 % protein and 10.5 % fat. What is the
fuel value of 7 oz. eggs ?
8. Cornmeal contains 9.2 % protein, 70.6 % carbohydrates,
and 3.8 % fat. What is the fuel value of 3 Ib. cornmeal ?
9. English walnuts contain 16.6 °/0 protein, 63.4 % carbo-
hydrates, and 16.1% fat. What is the fuel value of f Ib.
nuts ?
It is clear that a balanced ration need not be an expensive one. The
amount of heat required by the body varies from 2000 to 3500 calories
approximately, dependent upon age, occupation, and sex. A family of a
working man, wife, and three children under sixteen years of age requires
12,000 total calories, 1200 to 1800 of protein, or from 10 to 25% of the
total amount required. The quantity of food taken at each meal may
vary, provided the total quantities each day reach the standard required.
Some authorities suggest about four-twelfths for breakfast, three-twelfths
for luncheon, and five-twelfths for dinner.
There are two defects in American diet. First, we fail to have a bal-
anced ration and, second, we think that the richer the food the more
nourishing it is, and that its goodness is in proportion to the hours spent
in its preparation.
The protein is the most valuable and expensive part of the food supply
and it is wise to have a list of proteins so that one can substitute the lesser
for the more expensive. Protein, of which we need 18 %, is found more
generally in fish than in meat, and the inexpensive peanut is an appe-
tizing substitute; fat, of which we need 15%, can be had from the
FOOD 113
fat of all meats, and carbohydrates are better obtained from potatoes
than from rich cakes, confectionery, and jellies. We are indebted to
modern inventions for a wide list of cooked and partially cooked foods
which have economized the time of the busy housewife and which have
enriched our breakfast table beyond that of other nations. The breakfast
cereals and the grains from which they are made, white bread, potatoes,
sugar, butter, and other fats may be classed as carbohydrates, while meat,
fish, eggs, milk, cheese, peas, beans, and cabbage are some of the repre-
sentatives of the protein group. These carbohydrates and nitrogenous
substances are not wholly such, but are more or less a mixture of other
things.
Making Up Menus
In making up menus it is necessary to have them balance
evenly. One should not have too much fat one day, too much
starch the next, etc. The menus for each day should hold
part of each kind of food, one meat (fish or eggs), one fat, one
starch, one tonic vegetable, and one laxative vegetable or
fruit.
The summer menus must be compiled most carefully, for
too much fat or too much meat tends to heat the body at an
excessive rate and should therefore be avoided.
Of the different food materials which are palatable, nutritious, and
otherwise suited for nourishment, we should select those that furnish the
largest amounts of available nutrients at the lowest cost. To do this it
is necessary to take into account not only the price per pound, quart, or
bushel of the different materials, but also the kinds and amounts of the
actual nutrients they contain and their fitness to meet the demands of
the body for nourishment. The cheapest food is that which supplies the
most nutriment for the least money. The most economical food is that
which is cheapest and at the same time best adapted to the needs of the
user.
There are various ways of comparing food materials with respect to the
relative cheapness or expensiveness of their nutritive ingredients. The best
way of estimating the relative pecuniary economy of different food mate-
rials is found in a comparison of the quantities of nutrients and energy
which can be obtained for a given sum, say 10 cents, at current prices.
This is illustrated in the table which follows :
114 VOCATIONAL MATHEMATICS FOR GIRLS
COMPARATIVE COST or DIGESTIBLE NUTRIENTS AND ENERGY IN
DIFFERENT FOOD MATERIALS AT AVERAGE PRICES
[It is estimated that a man at light to moderate muscular work requires
about 0.23 pounds of protein and 3050 Calories of energy per day.]
KIND OF FOOD MATERIAL
«
i_!
M
H
P*
1
3
p*
COST OP 1 LB. PRO-
TEIN l
ICosT OF 1,000 CAL-
ORIES ENERGY (a)
AMOUNT FOR 10 CENTS
Total weight of
food material
Protein
1
Carbohydrates
>,
SP
CD
C
W
Beef, sirloin ....
Do ....
Cents
25
20
15
16
14
12
12
9
5
25
16
20
16
12
22
18
12
10
18
7
10
12
25
18
Dollars
1.60
1.28
.96
.87
.76
.65
.75
.57
.35
.98
1.22
1.37
1.10
.92
1.60
1.30
6.67
.93
1.22
.45
.74
.57
4.30
3.10
Cents
25
20
15
18
16
13
17
13
7
32
11
22
18
10
13
11
3
46
38
22
9
13
111
80
Lbs.
0.40
.50
.67
.63
.71
.83
.83
1.11
2
.40
.63
.50
.63
.83
.45
.56
.83
1
.56
1.43
1
.83
.40
.56
Lbs.
0.06
.08
.10
.11
.13
.15
.13
.18
.29
.10
.08
.07
.09
.11
.06
.08
.02
.11
.08
.22
.13
.18
.02
.03
Lbs.
0.06
.08
.11
.08
.09
.10
.08
.10
.23
.03
.17
.07
.09
.19
.14
.18
.68
.02
.01
.20
.10
.01
Lbs.
.01
.02
Calories
410
515
685
560
630
740
595
795
1,530
315
890
445
560
1,035
735
915
2,950
220
265
465
1,135
760
90
125
Do ....
Beef, round ....
Do . .
Do . .
Beef, shoulder clod .
Do .
Beef, stew meat . . .
Beef, dried, chipped . .
Mutton chops, loin .
Mutton, leg ....
Do
Roast pork, loin . .
Pork, smoked ham .
Do
Pork, fat salt ....
Codfish, dressed, fresh .
Halibut, fresh ....
Cod, salt
Mackerel, salt, dressed .
Salmon, canned . . .
Oysters, solids, 50 cents
per quart
Oysters, solids, 35 cents
per quart .
1 The cost of 1 pound of protein means the cost of enough of the given ma-
terial to furnish 1 pound of protein, without regard to the amounts of the other
nutrients present. Likewise the cost of energy means the cost of enough ma-
terial to furnish 1000 Calories, without reference to the kinds and proportions
of nutrients in which the energy is supplied. These estimates of the cost of
protein and energy are thus incorrect in that neither gives credit for the value
of the other.
FOOD
115
COMPARATIVE COST OF DIGESTIBLE NUTRIENTS AND ENERGY IN
DIFFERENT FOOD MATERIALS AT AVERAGE PRICES — (Continued}
KIND OF FOOD MATERIAL
ed
t-1
a
W
PH
1
£
1
3*
:•
o
1
u
AMOUNT FOB 10 CENTS
^ >.
§g
;!
o
H W
Ji
2*3
||
11
31
£*
e
1
PH
1
1
T3
>,
1
£
a/
C
W
Lobster, canned . . .
Butter . .
Cents
18
20
25
30
24
16
8
16
!J
3
'4
2^
F>
8
6
5
4
5
5
2|
5
10
I1
1
I*
6
7
6
Dollars
1.02
20.00
25.00
30.00
2.09
1.39
.70
.64
1.09
.94
.31
.26
.32
.73
.53
.29
1.18
.77
.64
.51
.65
.29
2.08
6.65
4.21
1.00
.67
.60
1.33
5.00
10.00
12.00
8.75
Cents
46
6
7
9
39
26
13
8
11
10
2
2
2
4
4
2
5
5
4
3
4
3
22
77
23
5
3
3
8
8
27
40
47
3
Lbs.
.56
.50
.40
.33
.42
.63
1.25
.63
2.85
3.33
3.33
4
4
1.33
1.33
2.50
1.25
1.67
2
2.50
2
2
4
2
1
6.67
10
13.33
10
6.67
1.43
1.67
1.43
1.67
Lbs.
.10
.01
.05
.07
.14
.16
.09
.11
.32
.39
.31
.13
.19
.34
.08
.13
.16
.20
.15
.35
.05
.02
.02
.1
.15
.20
.08
.02
.01
.01
.01
Lbs.
.01
.40
.32
.27
.04
.06
.11
.20
.11
.13
.03
.04
.07
.02
.09
.16
.02
.02
.03
.01
.03
.01
.01
.01
.01
.01
.01
.02
.01
.01
Lbs.
.02
.14
.17
2.45
2.94
2.96
.98
.86
1.66
.97
.87
1.04
1.30
1.04
1.16
.18
.05
.18
.93
1.40
1.87
.54
.65
.18
.13
.09
1.67
Calories
225
1,705
1,365
1,125
260
385
770
1,185
885
1,030
5,440
6,540
6,540
2,235
2,395
4,500
2,025
2,000
2,400
3,000
2,340
3,040
460
130
430
1,970
2,950
3,935
1,200
1,270
370
250
215
2,920
Do
Do
Eggs, 36 cents per doz. .
Eggs, 24 cents per doz. .
Eggs, 12 cents per doz. .
Cheese
Milk, 7 cents per quart .
Milk, 6 cents per quart .
Wheat flour ....
Do
Corn meal, granular . .
Wheat breakfast food .
Oat breakfast food .
Oatmeal
Rice . . .
Wheat bread ....
Do .
Do
Rye bread
Beans, white, dried . .
Cabbage .
Celery . . .
Corn, canned ....
Potatoes, 90 cents per bu.
Potatoes, 60 cents per bu.
Potatoes, 45 cents per bu.
Turnips
Apples
Bananas .
Oranges
Strawberries ....
Su°"ar ...
116 VOCATIONAL MATHEMATICS FOR GIRLS
EXAMPLES
1. What is the most economical part of beef for a soup ?
2. What is the most economical part of mutton for boiling ?
3. What is the most economical part of pork for a roast ?
4. Is fresh or salt codfish more economical ?
5. What is the fuel value of 3 oz. oatmeal ?
6. What is the fuel value of 3 oz. rice ?
7. What is the fuel value of 4 oz. strawberries ?
8. What is the fuel value of 6 oz. milk ?
EXAMPLES
Since several hundred Calories are required each day for a person's
diet, it is most convenient in computing meals to think of our foods in
100-Calorie portions. Therefore it is desirable to know how to compute
this portion and tabulate it for future reference.
1. 42 qt. of milk give 36,841 Calories. What is the weight
of a 100-C portion ?
2. 3J Ib. of flour give 1610.5 Calories. What is the weight
of a 100-C portion ?
3. i Ib. of dates give 393.75 Calories. What is the weight
of a 100-C portion ?
4. If J of a cup of flaked breakfast food gives approximately
100 C, what is the food value of 1 Ib.?
5. If ^ of a cup of skimmed milk gives approximately 100
C, what is the food value of 1 qt.?
6. A teaspoonful of fat gives 100 C. What is the food
value of 1 Ib. lard ?
7. If J- of a medium-sized egg gives a food value of 100 C,
what is the food value of an egg ?
8. 4 thin slices of bacon (1 oz.) give a food value of 100 C.
What is the food value of 9 Ib. of bacon ?
FOOD
117
9. If f oz. of sweet chocolate has a food value of 100 C,
what is the food value of \ lb.?
10. Ten large pears have the value of 100 C, which is the
same as for 2 doz. raisins. What is the food value of a single
raisin ?
11. Find the individual cost of feeding the following families
per week and per day. Find the number of Calories per indi-
vidual per day. (Arrange results in a column as suggested.)
FAMILY No. IN FAMILY TOTAL COST TOTAL CALORIES
A 5
B 7
C 3
D 3
E 7
F 6
G 7
H 4
I 4
J 6
K 8
L 6
M 7
N 14
O 6
13.60
86224
15.06
99928.64
11.21
101966.75
6.68
33744.14
15.01
130557.04
12.89
93456.34
17.77
11063.91
11.86
90891.3
10.23
50490
16.47
69385.9
10.37
112197.3
16.08
930262
30.89
86006.8
32.91
141517
12.31
85582.8
Economical Use of Meat
It is important to reduce waste by using as much as possible
of the bone, fat, and trimmings, not usually served with the
meat. If nothing better can be done with them, the bones and
trimmings are profitably used in the soup kettle, and the fat
can be saved for cooking to be used in place of more expensive
butter and lard. The bits of meat not served with the main
dish, or remaining after the first serving, may be seasoned and
recooked in many palatable ways. Or they can be combined
with vegetables, pie crust, or other materials, thus extending
the meat flavor over a large quantity of less expensive food.
118 VOCATIONAL MATHEMATICS FOR GIRLS
Different kinds and cuts vary considerably in price. Sometimes the
cheaper cuts contain a larger proportion of refuse than the more expen-
sive, and the apparent cost is really more than the actual cost of the
more edible portion. Aside from this advantage, that of the more ex-
pensive cuts lies in the tenderness and flavor, rather than in the nutritive
value. Tenderness depends on the character of the muscle fibers arid
connective tissues of which the meat is composed. Flavor depends
partly on the fat present in the tissues, but mainly on nitrogenous bodies
known as extractives, which are usually more abundant or of more
agreeable flavor in the more tender parts of the animal. The heat of
cooking dissolves the connective tissues of tough meat and in a measure
makes it more tender, but heat above the boiling point or even a little
lower tends to change the texture of muscle fibers. Hence tough meats
must be carefully cooked at low heat long applied in order to soften the
connective tissue. For this purpose the fireless cooker may be used to
great advantage.
Steers and Beef
Steers are bought from the farmer by the hundredweight
(cwt.). They are inspected and then weighed. After they
are killed and dressed, they are washed several times and sent
to the cooler. The carcass must be left in the cooler several
days before it can be cut. It is then divided into eight
standard cuts and each piece weighed separately.
Sixty per cent of the meat used in this country is produced
in the Federally inspected slaughtering and packing houses, of
which there are nearly 900, located in 240 cities.
EXAMPLES
1. A steer weighing 1093 Ib. was purchased for $ 7.42 per
cwt. What was paid for him ?
2. The live weight of a steer is 1099 Ib. ; the dressed weight
641 Ib. What is the difference ? What is the percentage of
beef in the animal ?
3. A steer with a dressed weight of 677 Ib. was cut into the
following parts : two ribs weighing 61 Ib. each, 2 loins 103 Ib.,
2 rounds 154 Ib., and suet 21 Ib. What was the percentage of
each part to the total amount ?
FOOD 119
4. A steer with a dressed weight of 644 Ib. was sold at
$ 10.51 per cwt. What was paid ?
5. If the value of ribs is 18^, loins 18 j£ rounds 9f£
what is the value of cuts in problem 3 ?
6. A housewife buys 8J Ib. of meat every Monday, 9^ Ib.
on Wednesday, and 10J Ib. on Saturday. What is the total
amount of meat purchased in a week ?
7. The live weight of a low-grade steer was 947 Ib. and
dressed weight 475 Ib. What is the per cent of dressed to
live weight ? What did the steer sell for at 6^ cts. live
weight ? What was the selling price per cwt. ?
8. A high-grade steer weighed live weight 1314 Ib. and
dressed weight 897 Ib. What is the per cent of "dressed to
live weight? What did the steer sell for at 9 cts. a pound
live weight ? What was the selling price per cwt. ? Note
the difference in the price between low- and high-grade steers
due to the fact that the latter have a greater proportion of the
higher priced cuts.
9. A steer was killed weighing 632 Ib. and sold for
$ 10.38 cwt. a. What was the selling price ? b. What was
the average price per pound ? c. What was the percentage
of each cut to total value ? d. What was the total value of
each cut ?
CUTS WEIGHT PRICE PER POUND (Wholesale)
2 Ribs 58 Ib. $ .17
2 Loins 100 .18£
2 Rounds 150 .09f
2 Chucks 160 .08
2 Flanks 30 .05|
2 Shanks 26 .05
2 Briskets 32 .08£
Navel End 46 .05
Neck Piece 8 .Olf
2 Kidneys 2 .05
Suet 20 .08
632 Ib.
120 VOCATIONAL MATHEMATICS FOR GIRLS
Cuts of Beef
The cuts of beef differ with the locality and the packing
house. The general method of cutting up a side of beef is
illustrated in the following figure.
STANDARD BEEF CUTS — CHICAGO STYLE
1 — Round
Rump Roast
Round Steak
Corned Beef
Hamburger Steak
Dried Beef
Shank — Soup Bone
2 — Loin
Sirloin Steak
Porterhouse Steak
Club Steak
Beef Tenderloin
3— Flank
Flank Steak
Hamburger Steak
Corned Beef
4 — Ribs
Rib Roasts
5 — Navel End
Short Ribs
Corned Beef
Soup Meat
6 — Brisket
Corned Beef
Soup Meat
Pot Roast
7 — Fore Shank
Soup Bone
8 — Chuck
Shoulder Steak
Shoulder Roast
Pot Roast
Stews
FOOD
121
STANDARD PORK CUTS — CHICAGO STYLE
1 — Short-cut Ham
Ham
2 — Picnic Ham
or California Ham
3 — Boston Butt
Pickled Pork
Pork Shoulder
Pork Steak
4 — Clear Plate
Dry Salt or Barrel Pork
5 — Belly
Bacon
Spare Ribs
Brisket Bacon
Salt Pork
6— Loin
Pork Roast
Pork Chops
Pork Tenderloin
I — Fat Back
Paprika Bacon
Dry Salt Fat Backs
Barrel Pork
EXAMPLES
Hogs are usually killed when nine or ten months old. The weight
is 75 % to 80 % of live weight. The method of cutting up a side of pork
differs considerably from that employed with other meats. A large por-
tion of the carcass of a dressed pig consists of almost clear fat. This fur-
nishes the cuts which are used for salt pork and bacon.
1. A hog weighed at the end of 9 months 249 Ib. When he
was killed and dressed, he weighed 203 Ib. What was the
per cent of dressed to live weight ?
2. A hog weighing 251 Ib. was sold for 81 cents live weight.
When he was dressed, he weighed 204 Ib. What should he
sell for per cwt. (dressed) in order to cover the price of
purchase ?
122 VOCATIONAL MATHEMATICS FOR G'IRLS
3. Sugar-cured hams and bacons are made by rubbing salt
into the pieces and placing a brine solution of the following
proportions over them in a barrel, before smoking them:
8 Ib. salt, 21 Ib. brown sugar, 2 oz. saltpeter in four gallons of
water for every 100 Ib. of meat. What percentage of the solu-
tion is salt ? Sugar ? (Consider a pint of water equal to a
pound.)
4. Sausages are made by mixing pork trimmings from the
ham with fat and spices, and placing in casings. If 3 Ib. of
ham are added to 1 Ib. fat pork, what is the percentage of lean
pork?
STANDARD MUTTON CUTS — CHICAGO STYLE
I —Leg
Leg of Mutton
Mutton Chops
2 — Loin
Loiu Roast
Mutton Chops
3 — Hotel Mack
Rib Chops
Crown Roast
4 — Breast
Mutton Stew
' 5— Chuck
Shoulder Roast
Stew
Shoulder Chops
EXAMPLES
1. A butcher buys 169 sheep at $5.75 a head. He sells
them so as to receive on the average $ 6.12i for each. What
does he gain ?
FOOD 123
2. A market man bought 19 dressed sheep for $81.75.
What was the average price ?
3. A sheep weighed 138 Ib. live weight and 72 Ib. dressed.
What was the per cent of dressed to live weight ?
4. A dressed sheep when cut weighed as follows :
Leg 23.1 Ib, each Neck 3.4 Ib. Breast 8.2 Ib.
Loin 18.4 Ib. each Shoulder 5.1 Ib. each Shank 5.3 Ib. each
Ribs 15.3 Ib. each
What was the total dressed weight ? What was the percent-
age of each cut to the dressed weight ?
Length of Time Required to Cook Mutton
Boiling
Mutton, per pound 15 minutes
Baking
Mutton, leg, rare, per pound ... 10 minutes
Mutton, leg, well done, per pound . 15 minutes
Mutton, loin, rare, per pound . . 8 minutes
Mutton, shoulder, stuffed, per pound 15 minutes
Mutton, saddle, rare, per pound . . 9 minutes
Lamb, well done, per pound ... 15 minutes
Broiling
Mutton chops, French 8 minutes
Mutton chops, English 10 minutes
EXAMPLES
Give the fraction of an hour required
(a) To boil mutton (2 Ib.).
(b) To bake leg of mutton (3 Ib.).
(c) To bake loin of mutton (4 Ib.).
(d) To broil mutton chops (French).
(e) To broil mutton chops (English).
(/) To bake shoulder of mutton (5 Ib.).
124 VOCATIONAL MATHEMATICS FOR GIRLS
Fish is a very economical kind of food. It can be obtained
fresh at a reasonable figure in seacoast towns.
1. During the year 1913, 170,000,000 Ib. of fish were brought
into Boston, and sold for $ 7,000,000. What was the average
price per pound ?
2. If 528,000,000 Ib. of fish were caught in the waters of
New England during the year 1913, it would represent one-
quarter of the catch of the entire country. What is the catch
of the entire country ?
3. A pound of smoked ham at 24 cents contains 16 % protein,
while a pound of haddock at 7 cents contains 18 % protein.
How much more protein in a pound of haddock than in a pound
of ham ? (In ounces.)
4. For the same value, how much more protein can you pur-
chase in the haddock than in the ham ?
5. A pound of pork chops at 25 cents contains 17 % protein ;
a pound of herring at 8 cents contains 19 %. How much more
protein is there in the pound of herring than in the pork chops ?
6. For the same value, how much more protein can be pur-
chased in the pound of herring than in pork chops ?
7. A pound of sirloin steak at 30 cents gives the same amount
of protein as the pork chops in example 6. For the same value
how much more protein can be obtained from haddock than
from the steak ? What per cent of protein per pound in had-
dock ? Use data in Example 3.
8. If fish can be purchased at any time at not over 12 cents
per pound, and meats at not less than 20 cents per pound, what
is the per cent of saving by buying fish ?
9. If 5.3 % of the total expenses for foodstuffs is for fish,
and 22 % of the family earnings goes for food, what is the
amount spent for each ? Family income $ 894.
FOOD 125
Economical Marketing
The most economical way to purchase food is to buy in bulk. Fancy
packages with elaborate labels must be paid for by the consumer. All
realize the convenience of package goods, the saving in cost of preparation
and cooking and the ease with which they are kept clean and wholesome,
but the additional expense is enormous, in sonre instances as high as 300 °/o .
EXAMPLES
1. If the retail price of dried beef is 50 cents a pound,
how much more per pound do I pay for dried beef, when I
purchase a package weighing 3|- oz. for 18 cents ? What
per cent more do I pay ?
2. Wheat costs the farmer or producer 11 cents per pound.
I purchase a package of wheat preparation weighing 5 oz. for
10 cents. How much more do I pay for wheat per pound than
it costs to produce it ? What per cent more do I pay ?
3. Good apples cost $ 2.75 per barrel. If I purchase a peck
for 50 cents, at what rate am I paying for apples per barrel ?
(A standard apple barrel contains 2-J- bushels.) How much
would I save a peck, if a few families in the neighborhood
joined me in purchasing a barrel ?
4. Codfish retails at 17 cents a pound. A group of families
sent one of their members to the wharf and she purchased for
60 cents a codfish weighing 6 Ib. How much was saved per
pound ? What per cent ?
5. Print butter is molded by placing a quantity of tub
butter in a mold. If the tub butter costs 34 cts. a pound and
the print butter 42 cts. a pound, how much cheaper (per cent)
is the tub butter than print butter? Does it afford the same
nourishment?
6. A pint can of evaporated milk costs 10 cents and con-
tains the food element of 2 J quarts of fresh milk at 8 cents a
quart. What is the saving per quart of milk ?
126 VOCATIONAL MATHEMATICS FOR GIRLS
Every housewife should possess the following articles in the
kitchen so as to be able to verify everything she buys :
1 good 20-lb. scale 1 dry quart measure
1 peck measure 1 liquid quart measure
1 half-peck measure 00-inch steel tape
1 quarter peck measure - 8-oz. graduate
The above should be tested and " sealed " by the Super-
intendent of Weights and Measures. Check the goods bought
and see if weight and volume agree with what was ordered.
EXAMPLES
1. If a gallon contains 231 cu. in., how many cubic inches
are there in a quart ?
2. If a bushel contains 2150.42 cu. in., how many cubic
inches are there in a dry quart ?
3. If a half-bushel basket or box, heaping measure, must
contain five-eighths bushel, stricken l measure, how many cubic
inches does the basket contain ?
4. A box 12 by 14 by 16 inches when stricken full will
hold a heaping bushel. How many cubic inches in the box ?
5. A dealer often sells dry commodities by liquid measure.
If a quart of string beans were sold by liquid measure for 15
cts., how much would the customer lose ? What is the differ-
ence in per cent between liquid and dry quart measure ?
6. A grocer sold a peck of apples to a housewife. As he
was about to place the apples in the basket, the woman called
his attention to the fact that the measure was not " heaping."
He placed four more apples in the measure. When she
reached home she counted 24 apples. What would have been
the per cent loss if she had not called his attention to the
measure ?
1 Stricken measure is measure that is not heaped, but even full.
FOOD 127
7. A " five-pound " pail of lard was found to weigh 4 Ib.
11 oz. What per cent was lost to the customer ?
8. A package (supposed to be a pound) sold for 12 cents
and was found to weigh 141 ounces. How much did the
consumer lose ?
9. A quart of ice cream was bought for 40 cents. The box
was found to be 121 cjc short. How much did the consumer
lose?
10. A girl bought a quart of berries for ten cents. '• The box
was found to contain 54.5 cu. in. How much was lost ?
11. A pound of print butter cost 39 cents and was found to
weigh 14 J ounces. How much did the consumer lose ?
CHAPTER VI
PROBLEMS ON THE CONSTRUCTION OF A HOUSE
MOST people live either in a flat or a house. Each has its
advantages and its disadvantages. The work of a flat is all on
one floor ; there are no stairs, halls, cellars, furnaces, and side-
walks to care for, and when the building is heated by steam,
there is only the kitchen fire or a gas range to look after.
These are the advantages and they reduce the work of the
home to very simple proportions.
Then, too, it is possible to find comfortable flats at a moderate price in a
neighborhood where it would be impossible to build a small house. How-
ever, in these flats some of the rooms are not well lighted and ventilated,
and one is dependent upon the janitor for many services which are not
always pleasantly performed, though fees are constantly expected. The
long flights of stairs are a great drawback, because people will not go out as
much as they should, on account of the exhausting climb on their return.
The small house, in country or city, brings more healthful
mental and physical surroundings than the flat. Perfect venti-
lation, light, sunshine, and freedom from all petty restrictions
give a more vigorous tone to body and mind. If the house is
in the suburbs and there is some land with it, where a few
vegetables and flowers can be cultivated, it has an added charm
and blessing in the form of healthful outdoor work : furnace,
cellar, and grounds for the husband's . share ; house, from
garret to cellar, for the wife's share. In a flat a man can
escape nearly all duties about the house, but in the little house
he must bear his share.
If one lives in the suburbs, the time and money spent in going to and
from the city is quite an item, but the cheaper rent usually more than
balances the traveling expense. A person should not pay over 25 % of
income for rent. In case a person receives an income of $ 1500 or over,
and has a savings bank deposit of about $ 1500, it is usually better to
128
CONSTRUCTION OF A HOUSE 129
purchase a house than to rent. Money may be borrowed from either the
cooperative bank or the savings bank.
The total rent of a house a year should be at least 10% of the value of
the house and land : 6 % represents interest on the investment, and 4 %
covers taxes and depreciation. In a flat the middle floor should cost
approximately 10% more than the first floor, and the top 10% less than
the first floor.
EXAMPLES
1. A single house and land cost $ 2800. What should be
considered the rent per year ?
2. A two-family house cost $ 5600. (a) What should be
the rent per month ? (6) What should be the rent of each flat ?
3. A three-family house costs $ 6500. What should be the
rent of each floor ?
4. A family desires to build a cottage-style, garnbrel roof
house containing seven rooms, bath, reception hall, cemented
cellar, and small storage attic. It is finished inside with North
Carolina pine and has hard-pine floors, fir doors, open plumb-
ing, two coats of plaster, furnace heat, and electric light. The
first floor has three rooms and a reception hall. The second
floor has three chambers, bath, and sewing room over the hall.
The architect finds that the cost of materials in the summer
and late fall varies as follows :
AMOUNT SAVED
ITEM SUMMER BY BUILDING
IN THE FALL
Mason work $200 $1(5.00
Brick and cement 90 7.20
Lumber 500 60.00
Finish 125 12.50
Plumbing 225 22.50
Heat (furnace) 100 10.00
Paint and paper 200 20.00
Plastering 200 16.00
Electric wiring 40 3.20
Electric light fixtures 40 4.00
Labor (carpenters) 450
Profit to contractor 213 27.52
130 VOCATIONAL MATHEMATICS FOR GIRLS
(a) What is the total cost in each case? (6) What is the
difference in per cent ? What is the per cent difference in each
item?
Economy of Space
Many persons who build houses, barns, and other buildings
do not understand the fundamental fact that there is more
space in a square building than in a long one, and that the
further they depart from the square form the more their build-
ing will cost in proportion to its size. For instance, a building
20' by 20' has 400 square feet of floor space and requires 80 feet
of outside wall, while one 10' by 40' will, with the same floor
space, require 100 feet of wall. Accordingly more material
and work will be required for the longer one.
In many cases, of course, there are objections to having a
building square. The longer building, for instance, gives
more wall space and more light, and these may be desired
items. The roof and floor items are about the same in either
case.
Preparation of Wood for Building Purposes
In winter the forest trees are cut and in the spring the logs are floated
down the rivers to sawmills, where they are sawed into boards of different
thicknesses. To square the log, four slabs are first sawed off. After these
slabs are off, the remainder is sawed into boards.
As soon as the boards or planks are sawed from the logs, they are piled
on prepared foundations in the open air to season. Each layer is sepa-
rated from the one above by a crosspiece, called a strip, in order to allow
free circulation of air about each board to dry it quickly and evenly. If
lumber were piled up without the strips, one board upon another, the
ends of the pile would dry and the center would rot. This seasoning or
drying out of the sap usually requires several months.
Wood that is to be subject to a warm atmosphere has to be artificially
dried. This artificially dried or kiln-dried lumber has to be dried to a
point in excess of that of the atmosphere in which it is to be placed after
being removed from the kiln. This process of drying must be done grad-
ually and evenly or the boards may warp and then be unmarketable.
CONSTRUCTION OF A HOUSE 131
Definitions
Board Measure. — A board one inch or less in thickness is said
to have as many board feet as there are square feet in its surface.
If it is more than one inch thick, the number of board feet is
found by multiplying the number of square feet in its surface
by its thickness measured in inches and fractions of an inch.
The number of board feet — length (in feet] x width (in feet} x thick-
ness (in inches').
Board measure is used for plank measure. A plank 2" thick, 10" wide,
and 15' long, contains twice as many square feet (board measure) as a
board 1" thick of the same width and length.
Boards are sold at a certain price per hundred (C) or per thousand (M)
board feet.
The term lumber is applied to pieces not more than four inches thick ;
timber to pieces more than four inches thick ; but a large amount taken
together often goes by the general name of lumber. A piece of lumber
less than an inch and a half thick is called a board and a piece from one
inch and a half to four inches thick is called a plank.
Rough Stock is lumber the surface of which has not been dressed or
planed.
The standard lengths of pieces of lumber are 10, 12, 14, 16, 18 feet, etc.
EXAMPLES
1. How many board feet in a board 1 in. thick, 15 in. wide,
and 15 ft. long ?
2. How many board feet of 2-inch planking will it take to
make a walk 3 feet wide and 4 feet long ?
3. A plank 19' long, 3" thick, 10" wide at one end and 12"
wide at the other, contains how many board feet ?
4. Find the cost of 7 2-inch planks 12 ft. long, 16 in. wide
at one end, and 12 in. at the other, at $ 0.08 a board foot.
5. At $ 12 per M, what will be the cost of 2-inch plank for
a 3 ft. 6 in. sidewalk on the street sides of a rectangular corner
lot 56 ft. by 106 ft. 6 in. ?
132 VOCATIONAL MATHEMATICS FOR GIRLS
Frame and Roof
After the excavation is finished and the foundation laid, the construc-
tion of the building itself is begun. On the top of the foundation a large
timber called a sill is placed. The timbers running at right angles to the
front sill are called side sills. The sills are joined at the corners by a
half -lap joint and held together by spikes.
a. Outside studding
6. Rafters
c. Plates
d. Ceiling joists
de. Second-floor joists i.
def. First-floor joists j.
g. Girder or cross sill k.
h. Sills /.
Sheathing
Partition studs
Partition heads
Piers
The walls of the building are framed by placing corner posts 4" by 6"
on the four corners. Between these corner posts there are placed smaller
timbers called studding, 2" by 4", 16" apart. Later the laths, 4' long, are
nailed to this studding. The upright timbers are often mortised into the
sills at the bottom. When these uprights are all in position, a timber, called
a plate, is placed on the top of them and they are spiked together.
On the top of the plate is placed the roof. The principal timbers of
the roof are the rafters. Different roofs have a different pitch or slope —
that is, form different angles with the plate. To obtain the desired pitch
the carpenter uses the steel square.
CONSTRUCTION OF A HOUSE 133
A roof with one half pitch means that the height of the ridge of the
roof above the level of the plate is equal to one half the width of the
building.
This illustrates a roof with one-half pitch.
EXAMPLES
Give the height of the ridge of the roof above the level of
the plate of the following building :
PITCH WIDTH OF BUILDING
1. One-half 32'
2. One-fourth 40'
3. One-third 36'
4. One-sixth 48'
Building Materials
Besides wood many materials enter into the construction of
buildings ; among these are mortar, cement, stone, bricks,
marble, slate, etc.
Mortar is a paste formed by mixing lime with water and sand in the
correct proportions. (Common mortar is generally made of 1 part of
lime to 5 parts of sand.) It is used to hold bricks, etc., together, and
when stones or bricks are covered with this paste and placed together,
the moisture in the mortar evaporates and the mixture " sets " by the
absorption of the carbon dioxide from the air. Mortar is strengthened
by adding cow's hair when it is used to plaster a house ; in such mortar
there is sometimes half as much lime as sand.
Plaster is a mixture of a cheap grade of gypsum (calcium sulphate),
sand, and hair. When the plaster is mixed with water, the water com-
bines with the gypsum and the minute crystals in forming interlace and
cause the plaster to " set."
When masons plaster a house, they estimate the amount of
work to be done by the square yard. Nearly all masons use
the following rule : Calculate the total area of walls and ceil-
134 VOCATIONAL MATHEMATICS FOR GIRLS
ings and deduct from this total area one-half the area of open-
ings such as doors and windows. A bushel of mortar will
cover about 3 sq. yd. with two coats.
EXAMPLE. — How many square yards of plastering are nec-
essary to plaster walls and ceiling of a room 28' by 32' and 12'
high?
Areas of the front and back walls are 28 x 12 x 2 = 672 sq. ft.
Areas of the side walls are 32 x 12 x 2 = 768 sq. ft.
Area of the ceiling is 28 x 32 = 896 sq. ft.
"2336 sq. ft.
2336 sq. ft. = **-£& sq. yd. = 259f sq. yd.
260 sq. yd. Ans.
EXAMPLES
1. What will it cost to plaster a wall 10 ft. by 13 ft. at
$ 0.30 per square yard ?
2. What will it cost to plaster a room 28' 6" by 32' 4" and
9' 6" high, at 29 cents a square yard, if one-half its area is
allowed for openings and there are two doors 8' by 3^' and
three windows 6' by 3' 3" ?
3. What will it cost to plaster an attic 22' 4" by 16' 8" and
9' 4" high, at a cost of 32 cents a square yard ?
Bricks used in Building
Brickwork is estimated by the thousand, and for various
thicknesses of wall the number required is as follows :
8^-inch wall, or 1 brick in thickness, 14 bricks per superficial foot.
12f-inch wall, or \\ bricks in thickness, 21 bricks per superficial foot.
17-inch wall, or 2 bricks in thickness, 28 bricks per superficial foot.
21^-inch wall, or 2| bricks in thickness, 35 bricks per superficial foot.
• EXAMPLES
From the above table solve the following examples :
1. How much brickwork is in a 17" wall (that is, 2 bricks
in thickness) 180' long by 6' high ?
CONSTRUCTION OF A HOUSE 135
2. How many bricks in an 8J" wall, 164' 6" long by 6' 4" ?
3. How many bricks in a 17" wall, 48' 3" long by 4' 8" ?
4. How many bricks in a 211" wall, 36' 4" long by 3' 6" ?
5. How many bricks in a 12f" wall, 38' 3" long by 4' 2"?
6. At $ 19 per thousand find the cost of bricks for a build-
ing 48' long, 31' wide, 23' high, with walls 12f" thick. There
are 5 windows (V x 3') and 4 doors (4' x 81').
To estimate the number of bricks in a wall it is customary
to find the number of cubic feet and then multiply by 22,
which is the number of bricks in a cubic foot with mortar.
7. How many bricks are necessary to build a partition wall
36' long, 22' wide, and 18" thick ?
8. How many bricks will be required for a wall 28' 6"
long, 16' 8" wide, and 6' 5" high?
9. How many cubic yards of masonry will be necessary to
build a wall 18' 4" long and 12' 2" wide and 4" thick?
10. At $ 19 per thousand, how much will the bricks cost to
build an 8^", or one-brick wall, 28' 4" long and 8' 3" high ?
11. At $ 20.50 per thousand, how much will the bricks cost
to build a 12f " wall, 52' 6" long and 14' 8" high ?
12. A house is 45' x 34' x 18', the walls are 1 foot thick,
the windows and doors occupy 368 cu. ft. ; how many bricks
will be required to build the house ?
13. What will it cost to lay 250,000 bricks, if the cost per
thousand is $ 8.90 for the bricks ; $ 3 for mortar ; laying, $ 8 ;
and staging, $ 1.25 ?
Stonework
Stonework, like brickwork, is measured by the cubic foot
or by the perch (161' x !£' X 1') or cord. Practical men usu-
ally consider 24 cubic feet to the perch and 120 cubic feet to
the cord. The cord and perch are not much used,
136 VOCATIONAL MATHEMATICS FOR GIRLS
The usual way is to measure the distance around the cellar on the out-
side for the length. This includes the corners twice, but owing to the
extra work in making corners this is considered proper. No allowance is
made for openings unless they are very large, when one-half is deducted.
The four walls may be considered as one wall with, the same
height.
EXAMPLE. — If the outside dimensions of a wall are 44' by
31', 10' 6" high and 8" thick, find the number of cubic feet.
44 2r
?! % 2
'5 m x — x 4- = 1050 cu. ft. Ans.
150 ft. length. £
Cement
Some buildings have their columns and beams made of
concrete. Wooden forms are first set up and the concrete is
poured into them. The concrete consists of Portland cement,
sand, and broken stone, usually in the proportion of 1 part
cement to 2 parts sand and 4 parts stone. The average weight
of this mixture is 150 pounds per cubic foot. After the con-
crete has " set," the wooden boxes or forms are removed.
Within a few years twisted steel rods have been placed in the forms
and the concrete poured around them. This is called reenforced con-
crete and makes a stronger and safer combination than the whole concrete.
It is used in walls, sewers, and arches. It takes a long time for the con-
crete to reach its highest compressive and tensile strength.
Cement is also used for walls and floors where a waterproof surface is
desired. When the cement "sets," it forms a layer like stone, through
which water cannot pass. If the cement is inferior or not properly made,
it will not be waterproof and water will pass through it and in time
destroy it.
EXAMPLES
1. If one bag (cubic foot) of cement and one bag of sand
will cover 2-| sq. yd. one inch thick, how many bags of cement
and of sand will be required to cover 30 sq. yd. 2£" thick ?
CONSTRUCTION OF A HOUSE 137
2. How many bags of cement and of sand will be required
to lay a foundation V thick on a sidewalk 20' by 8' ?
3. How many bags of cement and of sand will it take to
cover a walk, 34' by 8' 6", I" thick ?
4. If one bag of cement and two of sand will cover 5^ sq. yd.
f" thick, how much of each will it take to cover 128 sq. ft. ?
5. How much of a mixture of one part cement, two parts
sand, and four parts cracked stone will be needed to cover a
floor 28' by 32' and 8" deep ? How much of each will be used ?
Shingles
Shingles for roofs are figured as being 16" by 4" and are
sold by the thousand. The widths vary from 2" upward.
They are put in bundles of 250 each. When shingles are laid
on the roof of a building, they overlap so that only part of
each is exposed to the weather.
EXAMPLES
1. How much will it cost for shingles to shingle a roof
50 ft. by 40 ft., if 1000 shingles are allowed for 125 sq. ft.
and the shingles cost $ 1.18 per bundle ?
2. Find the cost of shingling a roof 38 ft. by 74 ft., 4" to
the weather, if the shingles cost $ 1.47 a bundle, and a pound
and a half of cut nails at 6 cents a pound are used with each
bundle.
3. How many shingles would be needed for a roof having
four sides, two in the shape of a trapezoid with bases 30 ft. and
10 ft., and altitude 15 ft., and two (front and back) in the
shape of a triangle with base 20 ft. and altitude 15 ft.?
(1000 shingles will cover 120 sq. ft.)
Slate Roofing
In order to make the exterior of a house fireproof the roof
should be tile or slate. Slates make a good-looking and durable
138 VOCATIONAL MATHEMATICS FOR GIRLS
roof. They are put on, like shingles, with nails. Estimates
for slate rooting are made on 100 sq. ft. of the roof.1
The following are typical data for building a slate roof :
A square of No. 10 x 20 Monson slate costs about $ 8.35.
Two pounds of galvanized nails cost $0.16 per pound.
Labor, $ 3 per square.
Tar paper, at 2f cents per pound, 1| Ib. per square yard.
EXAMPLES
Using the above data, give the cost of making slate roofs
for the following :
1. What is the cost of laying a square of slate ?
2. What is the cost of laying slate on a roof 112' by 44' ?
3. What is the cost of laying slate on a roof 156' by 64'?
4. What is the cost of laying slate on a roof 118' by 52' ?
5. What is the cost of laying slate on a roof 284' by 78' ?
Clapboards
Clapboards are used to cover the outside walls of frame
buildings. Most clapboards are 4' long and 6" wide. They
are sold in bundles of twenty-five. Three bundles will cover
100 square feet if they are laid 4" to the weather.
To find the number of clapboards required to cover a given
area, find the area in square feet and divide by 1-J. Allowance
may be made for openings by deducting from area.
EXAMPLES
1. How many clapboards will be required to cover an area
of 40 ft. by 30 ft.?
2. How many clapboards will be necessary to cover an area
of 38' by 42' if 56 sq. ft. are allowed for doors and windows ?
3. How many clapboards will a barn 60 ft. by 50 ft. require
if 10 % is allowed for openings and the distance from founda-
tion to the plate is 17 ft. and the gable 10 ft. high ?
1 Called a square.
CONSTRUCTION OF A HOUSE 139
Flooring
Most floors in houses are made of oak, maple, birch, or pine.
This flooring is grooved so that the boards fit closely together
without cracks between them.
The accompanying figure shows the ends of i=] c; c L^
pieces of matched flooring. Matched boards are
also used for ceilings and walls. In estimating for matched flooring
enough stock must be added to make up for what is cut away from the
width in matching. This amount varies from \" to |" on each board ac-
cording to its size. Some is also wasted in squaring ends, cutting up, and
fitting to exact lengths. A common floor is made of unmatched boards
and is usually used as an under floor. Not more than \" per board is
allowed for waste.
EXAMPLE. — A room 12 ft. square is to have a floor laid of
unmatched boards I!" wide ; one-third is to be added for waste.
What is the number of square feet in the floor ? What is the
number of board feet required for laying the floor ?
12 x 12 = 144 sq. ft. = area. 144 x \ = 48
144. Ans. 144
192 board measure for
unmatched floor.
192. Ans.
EXAMPLES
1. How much -J in. matched flooring 3" wide will be re-
quired to lay a floor 16 ft. by 18 ft. ? One-fourth more is al-
lowed for matching and 3 °/0 for squaring ends.
2. How much hard pine matched flooring -|" thick and 1^"
wide will be required for a floor 13' 6" x 14' 10" ? Allow \ for
matching and add 4 % for waste.
3. An office floor is 10' 6" wide at one end and 9' 6" wide at
the other (trapezoid) and 11' 7" long. What will the material
cost for an unmatched maple floor -J-" thick and 1?" wide at
$ 60 per M, if 4 sq. ft. are allowed for waste ?
140 VOCATIONAL MATHEMATICS FOR GIRLS
4. How many square feet of sheathing are required for the
outside, including the top, of a freight car 34' long, 8' wide,
and 7-J-' high, if 37-^-% covers all allowances ?
5. In a room 50' long and 20' wide flooring is to be laid ;
how many feet (board measure) will be required if the stock
is y X 3" and \ allowance for waste is made ?
Stairs
The perpendicular distance between two floors of a building
is called the rise of a flight of stairs. The width of all the
steps is called the run.
The perpendicular dis-
tance between steps is
called the width of risers.
Nosing is the slight pro-
jection on the front of
each step. The board on
each step is the tread.
To find the number of
stairs necessary to reach
from one floor to another :
Measure the rise first.
STAIRS Divide this by 8 inches,
which is the most comfortable riser for stairs. The run should
be 81- inches or more to allow for a tread of 9| inches with
a nosing of 1 \ inches.
EXAMPLE. — How many steps will be required, and what
will be the riser, if the distance between floors is 118 inches ?
118 -f- 8 = 14| or 15 steps.
118 -r- 15 = 7-J-f inches each riser. Ans.
EXAMPLES
1. How many steps will be required, and what will be the
riser, (a) if the distance between floors is 8' ? (6) If the dis-
tance is 9 feet ?
CONSTRUCTION OF A HOUSE 141
2. How many steps will be required, and what will be the
riser, (a) if the distance between floors is 12' ? (b) If the dis-
tance is 8' 8"?
Lathing
Laths are thin pieces of wood, 4 ft. long and 11 in. wide,
upon which the plastering of a house is laid. They are usu-
ally put up in bundles of one hundred. They are nailed J in.
apart and fifty will cover about 30 sq. ft.
EXAMPLES
1. At 30 cents per square yard what will it cost to lath and
plaster a wall 12 ft. by 15 ft. ?
2. At 45 cents per square yard what will it cost to lath and
plaster a wall 18 ft. by 16 ft. ?
3. What will it cost to lath and plaster a room (including
walls and ceiling) 16 ft. square by 12 ft. high, allowing 34 sq. ft.
for windows and doors, at 40 cents per square yard ?
4. What will it cost to lath and plaster the following rooms
at 411 cents per square yard ?
a. 16' x 14' xir high with a door 8' x2£' and 2 windows 2£' x 5'.
6. 18' x 15' xir high with a door 10' X 3' and 4 windows 2|' X 5'.
c. 20' x 18' x 12' high with a door 11' x 3' and 4 windows 2f x 4'.
d. 28' x 32' x 16' high with a door 10' x 3' and 4 windows 3' x5'.
e. 28' x 30' x 15' high with a door 10' x 3' and 3 windows 3' x5'.
Painting
Paint, which is composed of dry coloring matter or pigment mixed
with oil, drier, etc., is applied to the surface of wood by means of a
brush to preserve the wood. The paint must be composed of materials
which will render it impervious to water, or rain would wash it from the
exterior of houses. It should thoroughly conceal the surface to which
it is applied. The unit of painting is one square yard. In painting
wooden houses two coats are usually applied.
142 VOCATIONAL MATHEMATICS FOR GIRLS
It is often estimated that one pound of paint will cover 4 sq. yd. for
the first coat and 6 sq. yd. for the second coat. Some allowance is made
for openings ; usually about one-half the area of openings is deducted,
for considerable paint is used in painting around them.
TABLE
1 gallon of paint will cover on concrete . . . 300 to 375 superficial feet
1 gallon oi' paint will cover on stone or brick
work 190 to 225 superficial feet
1 gallon of paint will cover on wood .... 375 to 525 superficial feet
1 gallon of paint will cover on well-painted sur-
face or iron 600 superficial feet
1 gallon of tar will cover on first coat ... 90 superficial feet
1 gallon of tar will cover on second coat . . 160 superficial feet
EXAMPLES
1. How many gallons .of paint will it take to paint a fence
6' high, and 50' long, if one gallon of paint is required for
every 350 sq. ft.?
2. What will be the cost of varnishing a floor 22' long and
16' wide, if it tak^s a pint of varnish for every four square
yards of flooring and the varnish costs $2.65 per gallon ?
3. What will it cost to paint a ceiling 36' by 29' at 21 cents
per square yard ?
4. What will be the cost of painting a house which is 52'
long, 31' wide, 21' high, if it takes one gallon of paint to cover
300 sq. ft. and the paint costs $ 1.65 per gallon ? (House has
a flat roof.)
Papering
Wall paper is 18" wide and may be bought in single rolls
8 yards long or double rolls 16 yards long. When you get a
price on paper, be sure that you know whether it is by the
single or double roll. It is usually more economical to buy a
double roll. There is considerable waste in cutting and match-
ing paper, hence it is difficult to estimate the exact amount.
CONSTRUCTION OF A HOUSE 143
A fraction of a roll is not sold, — there are various rules pro-
vided. The border, called frieze, is usually sold by the yard.
Find the perimeter of the room in feet, and divide this by
the width of the paper (which is 18" or li'). The quotient
obtained equals the number of strips of paper required. Then
divide the length of the roll by the height of the room in order
to obtain the number of strips in the roll. The number of
rolls required is found by dividing the strips in the room by the
strips in the roll.
Another rule is : Find the perimeter of the room in yards,
multiply that by 2, and you have the number of strips. Find
the length of each strip. How many whole strips can you cut
from a double roll ? How many rolls will it take ? To allow
for doors and windows deduct 1 yard from the perimeter for
each window and each door.
EXAMPLES
1. A paper hanger is asked to paper a square room 18' by
18' with a door and three windows. The door is 3' by 7' and
the windows 2' by 4'. How many double rolls of paper will
he use ? (Consider all rooms 9' high.)
2. How much paper will be required to paper a room 18'
by 14' ?
3. How much paper will be required to paper a room
18' 6" by 16' 4" with 2 doors and 2 windows ?
4. How much will it cost to paper a room 19' 6" by
16' 4" with 2 doors and 2 windows. The paper costs 49^
a roll to place it on the wall.
Taxes
Find out where the money comes from to support the
schools, police, library, etc. in your city or town. How is it
obtained ? What is real estate ? What is personal property ?
What is a poll tax ? A tax is the sum of money assessed on
persons and property to defray the expenses of the community.
144 VOCATIONAL MATHEMATICS FOR GIRLS
The tax rate is usually expressed as so many dollars per
thousand of valuation, generally between $ 10 and $ 20. In
some places it is expressed as a certain number of mills on $ 1
or cents on $ 100.
The tax rate, or the amount on each thousand dollars of
property, is determined by dividing the whole tax by the num-
ber of thousand dollars of taxable property in the community.
To illustrate :
In a certain community the whole tax is $1,942,409.73.
The taxable property is $ 97,945,162.00.
$1,942,409.73 983
97,945
EXAMPLES
1. If the tax rate is $ 21.85, what are the taxes paid by a
family of women owning property worth $ 16,000 ?
2. What is the tax on $ 34,697 in your town or city ?
3. A man owns real estate worth $ 84,313, and has personal
property worth $ 16,584. What is his tax bill, if the tax rate
is $ 1.75 per hundred and a poll tax is $ 2 ?
4. A dwelling house is valued at $ 8500 and the tax rate
is $ 17.52 per thousand. What is the tax ?
5. Wrhat is the tax on a house valued at $ 3500, if the tax
rate is $ 23.45 ?
6. The taxable property of a city is $ 97,945,162.00 ; and the
expenses (taxes) necessary to run the city are $ 1,900,136.14.
Obtain the tax rate.
United States Revenue
The town or city derives revenue from taxes levied on real
and personal property. The county and state derive part of
their revenue from a tax imposed upon the towns and cities.
The United States government derives a great part of its rev-
CONSTRUCTION OF A HOUSE 145
enue from a tax placed on tobacco and liquor sold within its
boundaries and from a tax, called customs duties, imposed upon
articles imported from other countries.
Some articles are admitted into the country free; these are said to
be on the free list. The others are subject to one or both of the follow-
ing duties : a duty placed on the weight or quantity of an article without
regard to value (called specific duty), or a duty based upon the value of
the article (expressed in per cent and called ad valorem duty).
When goods are received into this country, they are examined by an
officer (called a customs officer). The goods are accompanied by a
written statement of the quantity and value (called manifest or invoice).
Sometimes the goods are liquid, and in this case the weight of the bar-
rel (called tare) must be subtracted from the total weight to obtain the
net weight on which duty is imposed.
In case bottles are broken and liquids have escaped, due allowance
must be made before imposing duty. This is called leakage or breakage.
EXAMPLES
1. What is the duty on bronze worth $ 8760 at 45 % ?
2. What is the duty on goods valued at $ 3115 at 35 % ?
3. What is the duty on 3843 sq. ft. of plate glass, duty
$ 0.09 per square foot ?
4. What is the duty on jewelry valued at $ 8376 at 40 % ?
5. What is the duty on cotton handkerchiefs valued at
$ 834 at 45 % ?
6. What is the duty on woolen knit goods valued at $ 1643,
41 cts. per pound plus 50 °/0 ?
7. What is the duty on rugs (Brussels), 120 yards, 27" wide,
invoiced at $ 1.80 a yard, at 29 cts. per square yard and 45 %
ad valorem ?
CHAPTER VII
COST OF FURNISHING A HOUSE
WHEN about to furnish a house, one of the first things to
consider is the amount of money to be devoted to the purpose.
This amount should depend on the income. A person with
a salary of $ 1000 a year should have saved at least $ 250
toward the equipment of his home before starting house-
keeping. This is sufficient to purchase the essentials of a
simply furnished apartment or small house.
After one has lived in the house for a short time, it will be easy
to study the possibilities and necessities of each room, and as time,
opportunity, and money permit, one can add such other things as are
needed. In this way the purchase of undesirable and inharmonious
articles may be avoided.
There are many different styles and grades of furniture. The cost
depends upon the kind of wood used, and the care with which it is put
together and finished. The most inexpensive furniture is not the
cheapest in the end. It is made of inferior wood and with so little care
that it is neither durable nor attractive. The medium grades are gen-
erally made of birch, oak, or willow, are durable, and may be found
in styles that are permanently satisfactory. The best grades are made of
mahogany and other expensive woods, and those whose income consists
only of wages or a salary cannot usually afford to buy more than a few
pieces of this kind.
Furniture that is well made, of good material, and free from striking
peculiarities of design and of decoration is chosen by all people of good
taste and good judgment.
Furnishing the Hall
The only furniture necessary for the vestibule is a rack for umbrellas.
The walls should be painted with oil paint in some warm color, and the
floor should be tiled or covered with inlaid linoleum in tile or mosaic
146
COST OF FURNISHING A HOUSE
147
design. If the vestibule serves also as the only hall, it should contain a
rug, a small table or chair, and a mirror. A panel of filet lace is suitable
to use across the glass in the front door.
Through the front door one gets one's first impression of the occupants
of the house. The furnishings of the hall should therefore be carefully
chosen. It is a passageway rather than a room, and requires very little
furniture. The walls may be done in a landscape paper, if one wishes to
make the room appear larger, or in plain colonial yellow, if a bright effect
is desired. If the size of the hall will permit, it is best to furnish it as a
reception room; it may be made an attractive meeting place for the
family and friends ; but if it is one of the narrow passages so often found
in city houses, one must be content with the regulation hall stand, or a
mirror and a narrow table, and possibly one chair.
PRICE LIST OF HALL FURNITURE
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Umbrella rack .
§ 1.25
$6.00
§7.25
68.50
$10.00
§5.00
§7.60
Table ....
3.75
6.75
8.25
9.75
20.00
10.00
37.50
Mirror . .
3.00
3.00
3.40
3.75
30.00
7.50
Straight chair .
2.75
4.50
5.50
6.60
25.00
6.50
8.00
Chest ....
13.50 i 13.50
16.50
19.50
50.00
40.00
Sofa ....
50.00
35.00
16.00
Tall clock . .
60.00
60.00
150.00
75.00
Settle ....
18.00
18.00
22.50
27.00
32.00
32.00
21.00
Telephone stand
6.75
6.75
8.25
9.75
10.60
5.50
15.00
Clothes rack
3.50
3.50
4.15
4.90
5.00
7.00
8.26
EXAMPLES
1. What is the complete cost of furnishing a hall with
willow furniture ?
148 VOCATIONAL MATHEMATICS FOR GIRLS
2. Compare the cost of furnishing a hall with mahogany
or birch.
3. If a family receives an income of $ 1400 a year and
lives in a single cottage house, what kind of furniture should
be selected? What should the cost not exceed for the hall
furniture ?
4. A hall was furnished with the following articles. What
did it cost ? What kind of furniture was probably purchased ?
Seat, $11.85 Rug, $9.85
Mirror, $ 2.15 China umbrella stand, $ 2.10
Table, $2.20 Table cover (one yard of felt), -$1.15
Two chairs, $ 7.40 Pole, $ 2.20
With hardwood or stained floors the furnishing and care of a house are
much simplified. If one must have carpets, the colors should be neutral.
The best quality of Canton or Japanese matting is satisfactory ; it is
a yard wide and costs fifty cents a yard. Next to matting, the most sani-
tary and economical carpet is good body Brussels. It wears well, and the
dust does not get under it. A cheap, loosely woven matting or woolen
carpet is always unsatisfactory.
FLOOR COVERINGS
In selecting floor coverings there are several important considerations.
The design and quality should be governed by the treatment the rug will
necessarily have.
Hall
A hall rug or carpet will receive hard wear ; therefore, the quality
should be good. A small all-over symmetrical design in two tones of one
color or in several harmonizing colors will show dust and wear less than
a plain surface would do.
Rag rug, machine made, 3 by 6 feet $2.18
Hand-woven rag rug, 3 by 6 feet 7.50
Scotch wool rug, 3 by 6 feet 4.00
Hand-woven wool rug, 3 by 6 feet 6.00
East India drugget, 3 by 6 feet 8.00
Saxony, 3 by 6 feet 9.00
Brussels rug, 3 by 6 feet 9.00
Oriental rug, 3 by 6 feet 35.00
COST OF FURNISHING A HOUSE 149
Living Room
In a living room the floor covering will be worn all over equally.
Since there is always a variety of colors and forms in a living room, it is
well to keep the floor covering as plain as possible. A rug with a plain
center and a darker border of the some color is excellent in this room,
particularly if the walls or hangings are figured. If they are plain,
the rug or carpet may have a small, indefinite figure. If several domestic
rugs are used in the same room, they should be exactly alike in design and
color. If small Oriental rugs are used, they will, of course, differ in
design, but they should be as nearly as possible in the same tone.
Good Living-room Rugs
Crex or grass rug, 9 by 12 feet $ 8. 50
Rag rugs, 9 by 12 feet 310.00 to 45.00
Scotch wool rug, 9 by 12 feet $14.50 to 25.00
Brussels, 9 by 12 feet 32.75
Hand- woven wool rug, 9 by 12 feet 36.00
East India drugget, 9 by 12 feet 43.00
Saxony, 9 by 12 feet 50.00
Oriental, 9 by 12 feet 200.00 up
Dining Room
A dining-room rug gets very hard wear in spots. It should, therefore,
be selected in as good quality as one can afford. It is not well to have a
perfectly plain rug in a dining room, as a plain surface shows crumbs and
spots too readily. There is no objection to having a dining-room floor
quite bare, if the floor is well finished. Inlaid linoleum also makes an
excellent floor covering for a dining room that receives very hard usage.
The best coverings for this room are :
Crex ingrain rug, 9 by 12 feet ....... $8.50
Rag rug, 9 by 12 feet $ 10.00 to 45.00
Brussels, 9 by 12 feet 32.75
East India drugget, 9 by 12 feet 36.00
Saxony, 9 by 12 feet 50.00
Oriental, 9 by 12 feet 200.00 up
Bedroom and Sewing Room
On account of the lint which accumulates in bedrooms, it is a good plan
to keep the space under the beds bare, so that it may be dusted every
day. Small rugs laid where most needed are more hygienic in sleeping
150 VOCATIONAL MATHEMATICS FOR GIRLS
rooms than are large rugs and carpets. Plain Chinese matting makes a
clean floor covering when the boards are not in good condition. Although
it is in good taste to use a carpet or one large rug in a bedroom, the
preference lies among the following :
Small rag rugs, 3 by 6 feet $1.75
Oval braided rag rugs, 3 by 6 feet 2.50
East India drugget, 3 by 6 feet 8.00
Saxony, 3 by 6 feet 8.00
Oriental, 3 by 6 feet 35.00
EXAMPLES
1. A family has an income of $ 1400. They buy a Brussels
rug 3' x 6' for $ 9. Are they extravagant?
2. How much cheaper is a crex rug, 9 by 12 feet, than a
Brussels the same size ? What per cent cheaper ?
3. A dining-room rug is purchased for $ 49.75. What kind
of a rug is it ? Is it suitable for a family with an income of
$ 2500 ?
4. An oval braided rag rug 3' x 6' costs $ 2.50 and will last
twice as long as a small rag rug that costs $ 1.75 for the bed-
room. Which is more economical to purchase? How much
more economical is it?
The Living Room
In houses or apartments of but five or six rooms there is
usually but one living room. This room should represent the
tastes which the members of the family have in common. The
first requisite of such a room is that it should be restful. It
is, therefore, advisable to use a wall covering that is plain in
effect. Tan is good in a room that is inclined to be dark ;
gray-green or gray itself in a very bright living room. One
large rug in two tones of one color, preferably the same color
as the walls, is better than a figured rug for this room.
Chairs are an important part of the furnishing of a living
room. It is well to have comfortable armchairs, upholstered
COST OF FURNISHING A HOUSE 151
in plain material, or willow chairs with cushions of chintz, if
this material is used as curtains. A roomy table with a good
reading lamp is essential, while open bookshelves, a writing
desk or table, a sofa, a sewing table, and a piano are all appro-
priate furnishings for this room.
A HARMONIOUSLY FURNISHED LIVING ROOM
The curtains may be of figured materials, such as chintz or
cretonne. Plain scrim or net curtains may be used over cur-
tains of plain-colored material or of chintz simply to give the
necessary warmth and color to the sides of the room. Valances
are used to reduce the apparent height of a window and to give
a low cozy look to the room. Plants are always appropriate to
use in sunny windows, and pictures of common interest, framed
in polished wood or dull gilt frames, help to make the living
room attractive. Use very little bric-a-brac. Nothing which
does not actually contribute to the beauty of the room should
be allowed to find a place there.
152 VOCATIONAL MATHEMATICS FOR GIRLS
PRICE LIST OF LIVING-KOOM FURNITURE
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$4.50
$15.00
$17.00
$50.00
$35.00
$59.00
$12.00
Chair ....
Sofa . .
17.00
22.50
25.00
45.00
55.00
30.50
68.00
50.00
100.00
12.75
23.50
Armchair . . .
20.00
38.00
32.00
65.00
9.75
Desk chair . .
2.75
6.75
7.75
15.00
4.75
15.00
8.25
Desk ....
9.75
19.50
21.75
90.00
28.00
90.00
37.50
Bookcase .
9.00
9.00
11.25
100.00
25.00
100.00
13.50
Sewing table . .
5.00
5.00
6.00
17.00
18.50
13.50
Tea table . . .
1.50
1.50
2.00
35.00
12.00
7.25
Footstool . . .
2.25
3.75
3.00
6.00
4.50
6.00
5.25
Wood box or rack
5.00
5.00
5.00
5.00
3.50
Magazine stand .
6.00
6.75
8.25
10.00
8.50
8.50
12.75
Piano ....
200.00
250.00
450.00
450.00
Music cabinet
6.75
6.75
8.25
28.00
10.00
c. /Gas, $5.00
Stoves{ Coal, 17.00
Wood . . $15.00
Wood or coal 25.50
Franklin grate or andirons,
wood or coal $35.00
EXAMPLES
1. How much more will it cost to furnish a living room with
library furniture than with willow furniture ?
2. How much more will it cost to furnish a living room with
hand-made oak furniture than with colonial designs in oak ?
3. A living room was furnished with the following furni-
ture. Ascertain from the price list what kind of furniture it is.
Large round table and small Curtains and shades for three
table, $7.95
Six chairs and couch, $ 51.15
Bookcase or shelves, $ 9.85
What is the cost ?
windows, $6.15
Rug and draperies, $34.15
Incidentals, $ 24.65
COST OF FURNISHING A HOUSE 153
The Bedroom
When one stops to think that about one-third of one's life is
spent in sleep, it is easy to understand that the first requisite
in the furnishing of the bedroom is that it be fresh and clean.
A COMFORTABLE BEDROOM
Unless the room must be used as a study or sitting room in
the daytime, the amount of furniture should be reduced as
much as possible. The necessary pieces are a bed, a dressing
case which should be generous in drawers and mirror, a wash-
stand, a toilet set, towel-rack, one easychair and one plain one,
a small table, a rug, and window shades. If space and money
permit, a couch is desirable. Naturally, a writing desk, book-
shelves, and pictures all add to the attractiveness of such a
room. If one cannot have bare floors, the next best thing is
good matting. A woolen carpet is not desirable for a sleeping
room. All draperies should be of materials that will hold
neither dust nor odor.
154 VOCATIONAL MATHEMATICS FOR GIRLS
The bed is the most important article in the room. The
springs and mattress should be firm enough to support all parts
of the body when it is in a horizontal position.
The walls should be light in. color and the woodwork white
if possible. The furniture also may be white, although dull-
finished mahogany in colonial designs, with small rugs on
the floor, makes a charming bedroom. One set of draw cur-
tains, of figured chintz if the walls are plain, and of plain-colored
material if the walls have a small figure, is enough for each
window.
The furnishings of a young girl's bedroom should be carried
out in her favorite color, and to the usual bedroom furniture
should be added a desk, lamp, worktable, and bookshelves.
The bedroom for a growing boy should be his own sitting
room and study as well ; a place where he can entertain his
friends, do his studying, and develop his hobbies. The walls,
hangings, couch cover, etc., should be very plain, as a boy
usually has a collection of trophies which need' the plainest
sort of a background in order to prevent the room from looking
cluttered. Instead of the usual bed he should have an iron-
framed couch, which in the daytime may be made up with a
plain dark cover with cushions, to be used as a couch. A chif-
fonier, an armchair, bookshelves, writing table, and one or two
small rugs will complete the furnishings of the boy's bedroom.
EXAMPLES
1. Sheets should be of ample length and breadth. The
finished sheets should be nearly three yards long. How many
inches long ?
2. The supply of bedroom linen, blankets, and counterpanes
for a small house is as follows :
12 sheets @ §0.85 4 pairs blankets @ $8.00
12 pillow cases @ $ .40 2 counterpanes @ $ 2.50
24 towels @$ 0.50
What is the total cost ?
COST OF FURNISHING A HOUSE
PRICE LIST OF BEDROOM FURNITURE
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$18.75
$21.00
$55.00
$30.00
$56.00
Mattress . .
3.35
16.00
16.00
16.00
36.00
36.00
36.00
to
to
to
to
16.00
25.00
25. CO
25.00
Box spring . .
20.00
20.00
20.00
20.00
20.00
20.00
20.00
Crib (iron) . .
12.75
12.75
12.75
12.75
12.75
12.75
Crib mattress .
3.75
9.00
9.00
9.00
9.00
9.00
Pillows (pair) .
1.25
2.10
2.10
2.10
6.00
5.25
5.25
Bureau . . .
9.75
22.50
25.00
27.50
75.00
50.00
67.50
Washstand
1.50
2.00
2.75
3.50
6.00
10.00
(enamel
iron)
Dressing table
9.00
12.57
14.25
15.75
55.00
26.00
48.00
Chiffonier (no
9.00
12.00
14.25
16.'50
100.00
39.00
60.00
mirror) . .
(high-
boy)
Chair . . .
2.75
4.50
5.25
6.00
10.00
6.50
8.00
Rocking chair .
2.75
6.75
7.75
8.75
9.00
6.50
8.25
Waist box . .
Home-
made
2.50
3.50
4.50
20.00
16.00
4.50
Desk ....
4.50
9.75
10.75
11.75
60.00
20.00
28.50
Armchair .
6.75
7.75
8.75
24.00
8.00
7.50
Couch . . .
5.00
13.25
60.00
50.00
25.00
(iron
frame)
(box)
Bookshelves
Home-
made
9.00
10.50
12.00
(built in)
21.50
13.50
Cheval glass .
11.25
15.50
16.50
18.00
50.00
25.00
ctovpsfGas, $5.00 Wood. . .$15.50 Franklin grate or andirons,
a \Coal, 17.00 Wood or coal 25.50 wood or coal . $35.00
156 VOCATIONAL MATHEMATICS FOR GIRLS
3. If a person spends one-third of a life in a bedroom, how
many hours a day are spent in the bedroom ?
4. A bedroom is furnished with the following furniture :
Enameled bedstead withsprings, Dimity for draping bed, washstand
$7.50 and two windows, twenty-one
A dressing case, $ 15.00 yards, $ 3.15
A plain wooden table to be Enameled cloth for washstand, $.55
used as washstand, $ 1.00 Two pillows, $4.00
A small table $ 2.00 Toilet set, $ 3.00
• Chair, $ 2.00 Shades for two windows, 1 1.00
Mattress, $ 5.00 Towel rack, $ .75
Rug, $3.00
What is the total cost ?
5. What will it cost to furnish a bedroom with simple cot-
tage furniture as provided above ?
6. What will it cost to furnish a bedroom with the good
grade of oak furniture in gloss enamel ? What is the least
income a family should have in order to buy this furniture ?
7. What will it cost to furnish a bedroom with real mahog-
any furniture ? What is the least income one should have in
order to buy this furniture ?
The Dining Room
The dining room does not require a great deal of furniture,
but what there is should be of the most substantial kind.
Mahogany and oak are the woods to be preferred. The table
should be broad, stand well, with the legs so placed that they
will not interfere with the comfort of any one seated at the
table. The chairs should be well made, with broad, deep
seats and high, straight backs. Unless one can afford the
right kind of a sideboard it is better to purchase a sideboard
table in simple design. A piece of Japanese matting in the
center of the dining room floor is quite satisfactory when the
floor is stained.
COST OF FURNISHING A HOUSE
157
The room in which the family assembles several times each
day to enjoy its meals together should be the most cheerful
room in the house.
AN ATTRACTIVE DINING ROOM
Because there is so much attractive blue-and-white china in use,
many persons want dining rooms with blue walls. This is usually a mis-
take, as blue used in large quantities absorbs the light and makes a room
gloomy, particularly on dark days and at night. By using colonial yel-
low on the walls, with hangings, rug, and decorative china in blue and
white, one has an almost ideal arrangement. There are many charming
landscape and foliage papers on the market which, used without pictures
against them, but with bulbs or plants blooming on the windowsills and
with hangings of plain, semitransparent, colored material make most
delightful rooms.
Plate rails or racks reduce the apparent height of an over-high ceiling.
It is better to use a simple flat molding than to crowd a plate rail full of
inharmonious objects.
Ugly glass domes on lamps are being replaced by silk ones with deep
silk fringe or, better still, the center light is abandoned in favor of side
wall fixtures in all of the rooms. Candles, prettily shaded, are used on
the table at night, with a jar of flowers or fruit as a centerpiece.
158 VOCATIONAL MATHEMATICS FOR GIRLS
EXAMPLES
1. What will the following cottage dining-room furniture
cost ? (Include the items given in the price list below.)
2. What will the following oak dining-room furniture cost ?
3. What will the following real mahogany dining-room fur-
niture cost ?
PRICE LIST OF DINING-ROOM FURNITURE
w a as
co a
HI
»*~
~
;
w
^ •<
2; < j
M
tJ C3 -J
2o
2c w
O P4
M
|
««u
™ u
w fc
Q
ft s i
ft " *
c 2
fc
£
as
u * £
P=< ° •«
o
j:§^
, §0
a
M
^o
»55
o 3
ill
O ^ ~;
o £9
|p
||i
s ^
is
|
o5£
OPH 0
OM |
C>£H 1
owS
K H
^
Table ....
$0.00
$30.00
$10.50
$12.00
$85.00
$21.00
$16.00
Chair ....
2.75
4.50
5.50
6.50
10.00
6.50
8.25
Armchair .
2.75
6.75
7.75
8.75
15.00
10.00
Serving table . .
8.25
9.00
10.50
12.75
35.00
18.00
28.00
Buffet ....
18.00
27.50
21.00
24.00
125.00
34.00
82.50
China closet .
15.00
30.00
34.50
39.00
60.00
45.00
Serving table on
wheels . . .
16.75
16.75
30.50
34.00
27.00
27.00
24.00
Screen ....
3.75
5.00
4.50
5.25
25.00
20.00
High chair
2.50
2.50
4.15
5.50
10.00
9.00
8.00
Stoves / Gas' ^5-°° Wood • • • $15.50 Franklin grate or andirons,
\Coal, 17.00 Wood or coal 25.00 wood or coal . . $35.00
4. What will it cost to furnish a home on a moderate scale
with china of the following amounts and kinds :
I dozen soup plates (to be used for cereals also) . . $2.35
\ dozen dinner plates 2.25
1 dozen lunch plates (used also for breakfast and for
salads) 3.85
\ dozen dessert plates 1.60
COST OF FURNISHING A HOUSE 159
£ dozen bread-and-butter plates § 0. 70
\ dozen coffee cups and saucers 3.30
^ dozen tea cups and saucers 2.80
\ dozen after-dinner coffee cups and saucers . . . 2.35
1 teapot 1.90
1 coffee pot 2.00
1 covered hot-milk jug or chocolate pot 2.60
1 large cream pitcher .70
1 small platter or chop platter 2.50
3 odd plates for cheese, butter, etc .95
Covered dish 2.80
\ dozen egg cups 1.50
5. What will it cost to furnish, a home on a moderate scale
with glass, colonial period, of the following amounts and
kinds :
\ dozen tumblers -$0.50
\ dozen sherbet glasses .35
\ dozen dessert plates 1.25
\ dozen ringer bowls .75
Sugar bowl and cream pitcher .50
Dish for lemons 50
Dish for nuts .25
Pitcher 50
Candlesticks .65
Vinegar and oil cruets .50
Berry dish 25
I dozen iced-tea glasses .75
\ dozen individual salt cellars .60
6. What will it cost to furnish a home on a moderate scale
with silver, pilgrim pattern, of the following amounts and
kinds :
1 dozen teaspoons $14.00
\ dozen dessert spoons (used for soup also) . . . 9.50
4 tablespoons 9.50
1 dozen dessert forks (used also for breakfast, lunch,
salad, pie, fruit, etc.) 19.00
\ dozen dessert knives 11.00
160 VOCATIONAL MATHEMATICS FOR GIRLS
\ dozen table knives with steel blades and ivoroid
handles $2.00
Carving set to match steel knives 4.00
\ dozen table forks 12.00
2 fancy spoons for jellies, bonbons, etc. ($ 1.50 each) 3.00
2 fancy forks for olives, lemons, etc. ($1.50 each) . 3.00
\ dozen after-dinner coffee spoons 5.00
\ dozen bouillon spoons 8.00
\ dozen butter spreaders 1.50
1 gravy ladle 4.75
Saltspoon - .... .20
Sugar tongs 2.25
7. What will it cost to furnish, a home on a moderate scale
with silver-plated ware of the following amounts and kinds :
Covered vegetable dish (cover may be used as a
dish by removing handle) $10.00
Platter 11.50
Pitcher 12.00
Coffee pot 12.50
Toast rack 4.50
Small tray 6.50
Sandwich plate 6.00
Silver bowl 9.00
Egg steamer 8.00
Bread or fruit tray 5.50
Tea strainer 1.00
Candlesticks, each 3.75
Household Linen
The quality of linen in every household should be the best that one
can possibly afford. The breakfast runners and napkins are to be made
by hand, of unbleached linen such as one buys for dish towels. With
insets of imitation filet lace these are very attractive, durable, and easy
to launder.
1. What is the cost of supplying the following amount of
table and bed linen for a couple with an average income of
% 1400, who are about to begin housekeeping ?
COST OF FURNISHING A HOUSE 161
Table Linen
2 dozen 22-inch napkins, at $3.00 a dozen.
2 dozen 12-inch luncheon napkins, at $4.50 a dozen.
(Luncheon napkins at $1.00 a dozen if made by hand of coarse linen.)
2 two-yard square tablecloths, at $1.25 a yard.
Two-yard square asbestos or cotton flannel pad for table, at $ 1.00.
\ dozen square tea cloths, $12.00.
^ dozen table runners for breakfast, at $2.40.
1 dozen white fringed napkins, at $1.20.
4 tray covers, at 65 cts.
1 dozen finger-bowl doilies, at $3.00.
1 dozen plate doilies, at $ 3.00.
Bed Linen
4 sheets (extra long) for each bed, at $ 1.10.
4 pillow cases for each pillow, at 20 cts.
1 mattress protector for each bed, with one extra one in the house,
at $1.50.
2 spreads for each bed, at $ 2.50.
1 down or lamb's-wool comforter for each bed, at $ 6.
1 pair of blankets for each bed, with 2 extra pairs in the house, at $ 8.
£ dozen plain huckaback towels for each person, at 25 cts.
3 bath towels for each person, at 30 cts.
£ dozen washcloths for each person, at 11 cts.
1 bath mat in the bathroom, 2 in reserve, at $1.50.
•
The Sewing Room
Even in a small house there is sometimes an extra room which may be
fitted up as a sewing room in such a way as to be very convenient and
practical, and at the same time so attractive as to serve occasionally as an
extra bedroom. This room should be kept as light as possible and should
be so furnished that it may easily be kept clean.
EXAMPLE
1. What will it cost to furnish a sewing room with the fol-
lowing articles ?
Sewing machine with flat top to be Used as a dressing table . . $ 20.00
Chair 1.25
Box couch .... 13.25
Chiffonier . 9.00
162 VOCATIONAL MATHEMATICS FOR GIRLS
Mirror against a door $11.25
Low rocking-chair without arms 1.50
Cutting table, box underneath ; tilt top to be used 6.75
Clothes tree 3.38
The Kitchen
The room in which the average housekeeper spends the
greater part of her time is usually the least attractive room
in the house, whereas it should be made — and we learn by
visiting foreign kitchens that it may be made — a picturesque
setting for one of the finest arts — the art of cookery.
A CONVENIENT KITCHEN
The woodwork should be light in color, the walls should be painted
with oil paint, or covered with washable material, this also in a light
color. A limited number of well-made, carefully selected utensils will
be found more useful than a large supply purchased without due con-
sideration as to their real value and the need of them. Of course, the
style of living and the size of the family must to some extent control the
number, size, and kind of utensils that are required in each kitchen. As
in all the other furnishings, the beginner will do well to purchase only
the essential articles until time demonstrates the need of others.
COST OF FURNISHING A HOUSE
163
EXAMPLES
•> i
1. What will it cost to furnish your kitchen V
Stoves — Gas 12.50, $ 10.00, $30.00
Blue-flame kerosene 10.25
Coal, wood, gas 86.00
Coal and wood 49.75
Small electric 33.00
Table . . . $2.10; .-$9.00 (drop leaf) ;$ 11.25 (white enamel on steel)
Chair $1.87, $6.75
Ice chest $7.00, $ 15.00, $40.00 (white enamel)
Kitchen cabinet $28.00, $29.00 (white enamel on steel)
Linoleum . . . 60c. square yard, printed; $ 1.60 square yard, inlaid
2. What will the following small kitchen furnishings cost ?
Small-sized ironing board . $0.35
Small glass washboard . . .35
Clothesline and pins ... .59
2 irons, holder and stand . .70
2-gallon kerosene can . . .45
Small bread board ... .15
Hack for dish towels ... .10
6 large canisters 60
Wooden salt box 10
1 iron skillet 30
1 double boiler 1.00
Dish drainer 25
2 dish mops 10
Wire bottle washer . . . .10
• Small rolling pin 10
Chopping machine ... 1.10
Large saucepan 30
3 graduated copper, enam-
eled or nickel handled
dishes 50
2 covered earthenware or
enameled casseroles . . 1.50
2 pie plates enameled . . .20
Alarm clock 1.00
Small covered garbage pail . .35
Scrubbing brush 20
Broom and brushes . . . .60
1 quart ice-cream freezer . 1.75
Roller for towel 10
Bread box 50
4 small canisters 40
2 sheet-iron pans to use as
roasting pans 20
Dishpan (fiber) , . . . .50
Plate scraper 15
Soap shaker 10
Vegetable brush 05
Muffin tins 25
Granite soup kettle . . . .45
3 graduated small saucepans .30
Glass butter jar 35
6 popover or custard cups . .30
Soap dish 25
Knives, forks, egg beater,
lemon squeezer, etc. . . 5.50
Sink strainer, brush, and
shovel 50
Galvanized-iron scrub pail . .30
1 Consider income of family aud size of kitchen.
164 VOCATIONAL MATHEMATICS FOR GIRLS
EXAMPLES IN LAYING OUT FURNITURE
Considerable practice should be giveu in laying out furniture according
to scale.
1. A bedroom 12' x 10' 6" faces the south, and has 2 win-
dows, 3' 6" wide, 1 window, 3' 6", two feet from corner of west
sidej and a door 3' wide two feet from east wall. This room
is to contain the following furniture :
1 bed, 6' 6" x 4'
1 dresser, 3' x 1'6"
E
&
SOLUTION
1 dining table, 5' in diameter
1 buffet, 4' x 2'
1 table, 2' 6" x 3'
2 chairs, 1' 6" x 2'
Draw a plan showing the
most artistic arrangement of
furniture. Scale -J-" = V.
2. A dining room 15' x 18'
faces the east, and has two
windows 3' 6" wide on the
east side, 2 windows 3' 6" on
the north side, folding doors 6
wide in the center, on the
south side. Draw a plan and
place the following furniture in
it in the most artistic manner :
6 chairs, 2' x 1' 6"
Scale, \" = I'
3. A living room 15' x 18' faces the north and has 2 win-
dows 3' 6" wide on the north side, 2 windows 3' 6" on the
west side, and folding doors on the south side. Draw a plan
and place the following furniture in the most artistic manner :
1 settee
1 table
1 desk and chair
2 easy-chairs
2 rockers
Scale i" = I'
4. A kitchen 12' x 10' 6" faces the south and has 2 windows
3' 6" wide on the south side, 1 on the west side, two feet
COST OF FURNISHING A HOUSE 165
from the north comer, a door 3' wide, two feet from the north-
east corner that leads into the dining room. Draw a plan and
place the furniture in proper places :
1 kitchen range 1 table
. 1 sink 2 chairs
• 2 set tubs Scale \" = V
REVIEW EXAMPLES
1. A living room was fitted out with the furniture in the list
below. What kind of furniture is it ? What is the cost ?
Large round table and small Curtains and shades for three
table, $8.00 windows, § 6.30
Six chairs and couch, $ 50.00 Rug and draperies, $ 34.00
Bookcase or shelves, $ 10.00 Incidentals, $25.00
2. A hall was furnished with the following articles. What
was the total cost ? What kind of furniture was used ?
Seat, $12.00 Rug, $10.00
Mirror, $2.00 Umbrella stand, $2.00
Table, $ 2.00 Table cover, $ 1.00
Two chairs, $7.50 Pole, $3.00
3. A family of seven — three grown people and four chil-
dren — lived in a southern city on $ 600 a year. The monthly
expense was as follows :
House rent, $ 12.00 Bread, $ 3.50
Groceries, $ 12.00 Beef, $3.50
Washing, $ 5.00 Vegetables, $3.00
What is the balance from the monthly income of $ 50 for
clothing and fuel ?
4. What is the cost of the following kitchen furniture ?
1 kitchen chair, $ 1.25 1 broom, 50 cents
1 table, $ 1.50 Kitchen utensils, $ 8.50
166 VOCATIONAL MATHEMATICS FOR GIRLS
5. What is the cost of the following living-room furniture ?
How much income should a family receive to buy this furniture ?
Overstuffed chair, $12.50
2 willow chairs, $ 6 each
1 willow stool, $4.25
1 rag rug, $ 9.50
1 newspaper basket, $2.25
12 yards of cretonne, 35 cents a
yard
1 green pottery lamp bowl, $ 3.00
1 wire shade frame, 50 cents
7 yards of linen, at 50 cents a yd.
10 yards of cotton fringe, at 5
cents a yd.
6 yards of net, at 25 cents a yd.
Table, 48 by 30 inches, $ 7.00
6. What is the cost of the following bedroom furniture?
How much income should a family have to buy this furniture ?
1 bed pillow, $1.00
10 yards of white Swiss, at 25 cents
a yd.
8 yards of pink linen, at 50 cents
a yd.
1 comfortable, $4.25
Sheets and blankets for one bed,
$6.00
3 yards of cretonne, at 35 cents a yd.
7. What is the cost of the following bedroom furniture ?
How much income should a family have to warrant buying
this furniture ?
1 bed spring, $3.50
1 single cotton mattress, $4.25
1 chiffonier, $6.50
1 dressing table, $2.25
1 mirror, $2.75
1 armchair, $4.00
1 rag rug, $3.25
2 pillows, 75 cents each
2 white iron beds, at $4.25 each
2 single springs, at $2.50 each
2 cotton mattresses, at $4.25
each.
2 bed pillows, at $ 1.00 each
1 dressing table, $ 5.50
1 white desk, $6.75
1 chiffonier, $6.50
1 dressing-table mirror, $3.25
1 chiffonier mirror, $1.50
1 rag rug, $3.25
I wastepaper basket, .50
II yards of cretonne, at 35 cents
a yd.
5 yards of yellow sateen, at 25
cents a yd.
2 comfortables, at $ 4.25 each
10 yards of cream sateen, at 25
cents a yd.
15 yards of cotton fringe, at 5 cents
a yd.
1 willow chair, $ 6.00
1 cushion, 75 cents
4 yards of net, at 25 cents a yd.
Sheets and blankets for two beds,
$ 12.00
1 dressing table chair, $4.50
HEAT AND LIGHT 167
8. What is the cost of the following dining-room furniture ?
What income should one receive to buy this furniture ?
6 dining-room chairs, $4.50 lOyardsof cretonne, at 35 cents a yd.
1 dining table, $6.75 One wire shade frame, 50 cents
1 serving table, $ 6.25 Table linen, $ 8.00
1 rag rug, $9.50 Silverware, $7.50
1 set of dishes, $ 9.75 1 willow tray, $3.25
HEAT AND LIGHT
Value of Coal to Produce Heat
Several different kinds of coal are used for fuel. Some grades of the
same coal give off more heat in burning than others. The heating value
of a coal may be determined in three ways : (1) by chemical analysis to
determine the amount of carbon ; (2) by burning a definite amount in a
calorimeter (a vessel immersed in water) and noting the rise in tempera-
ture of the water ; (3) by actual trial in a stove or under a steam boiler.
The first two methods give a theoretical value ; the third gives the real
result under the actual conditions of draft, heating surface, combustion,
etc.
The coal generally used for household purposes in the Eastern states
comes from the anthracite fields of Pennsylvania. This coal, as shipped
from the mines, is divided into several different grades according to size.
The standard screening sizes of one of the leading coal-mining districts are
as follows :
Broken, through 4|" round Pea, through f" square
Egg, through 2f" square Buckwheat, through ±" square
Stove, through 2" square Rice, through f " round
Nut, through 1|" square Barley, through \" round
The last three sizes given above are too small for household use and
are usually purchased for generating steam in large power-plant boilers.
Coke is used to some extent in localities where it can be obtained at a
reasonable price in sizes suitable for domestic purposes. The grades of
coke generally used for this purpose are known as nut and pea. The use
of coke in the household has one principal objection. It burns up quickly
and the fires, therefore, require more attention. This is due to the fact
that a given volume of coke weighs less and therefore contains less heat
than other fuel occupying the same space in the stove or furnace.
168 VOCATIONAL MATHEMATICS FOR GIRLS
The chief qualities which determine the value of domestic coal are its
percentage of ash and its behavior when burned. Coal may contain an
excessive amount of impurities such as stone and slate, which may be easily
observed by inspection of the supply. The quality of domestic coke
depends entirely upon the grade of coal from which it has been made,
and may vary as much as 100 °fo in the amount of impurities contained.
Aside from the chemical characteristics of domestic coal, the most im-
portant factor to consider in selecting fuel for a given purpose is the size
which will best suit the range or heater. This depends on the amount
of grate surface, the size of the fire-box, and the amount of draft.
EXAMPLES
1. Hard coal of good quality has at least 90 % of carbon.
How much carbon in 9 tons of hard coal ?
2. A common coal hod holds 30 pounds of coal. How many
hods in a ton ?
3. If coal sells for $ 8.25 in June and for $ 9.00 in January,
what per cent is gained by buying it in June rather than in
January ? When is the most economical time to buy coal ?
4. The housewife buys kerosene by the gallon. If the price
per gallon is 13 cts. and live gallons cost 55 cts., what is the per
cent gained by buying in 5-gallon can lots ?
5. If kerosene sells for $4.60 a barrel, what is the price per
gallon by the barrel ? What per cent is gained over single
gallons at 13 cts. retail ? What is the most economical way
to buy kerosene ? (A barrel contains 42 gallons.)
How to Read a Gas Meter 1
1. Each division on the right-hand circle denotes 100 feet ;
on the center circle 1000 feet ; and on the left-hand circle
10,000 feet. Read from left-
hand dial to right, always tak-
ing the figures which the hands
have passed, viz. : The above
dials register 3, 4, 6, adding
1 Gas is measured in cubic feet.
HEAT AND LIGHT
169
two ciphers for the hundreds, making 34,600 feet registered.
To ascertain the amount of gas used in a given time, deduct
the previous register from the present, viz. :
Register by above dials 34,600
Register by previous statement 18,200
Given number of feet registered 16,400
16,400 feet @ 90 cts. per 1000 costs what amount ?
2. If a gas meter at the pre-
vious reading registered 82,700 ^ ^ ^^ ^
feet, and to-day the dials read
as follows,
what is the cost of the gas at 95 cts. per 1000?
3. What is the cost of the gas used during the month from
the reading on this meter, if
the previous reading was 6100
feet ? The rate is $ 1.00 per
1000 cu. ft. less ten per cent,
if paid before the 12th of the
month. Give two answers.
4. What is the cost of gas
registered by this meter at
85 cts. per 1000 cu. ft.?
How to Read an Electric Meter
(See the subject of the electricity in the Appendix)
There are three terms used in connection with electricity
which it is important to understand ; namely, the volt, the
ampere, and the watt or kilowatt.
(1) The volt is the unit of Electromotive Force or electrical
pressure. It is the pressure necessary to force a current of
one ampere through a resistance of one ohm.
(2) The unit of electric current strength is the ampere. It
170 VOCATIONAL MATHEMATICS FOR GIRLS
is the amount of current flowing through a resistance of one
ohm under a pressure of one volt.
(3) The watt is the unit of electrical power ; it is the prod-
uct of volts (of electromotive force) and current (amperes) in
the circuit, when their values are respectively one volt and
one ampere. That is to say, if we have an electrical device
operated at 3 amperes, on a line voltage of 115 volts, the
amount of current consumed is equal to 115 x 3 = 345 watts,
which, if operated continuously for one hour, will register on
the electric meter as 345 watt hours, or .345 kilowatt hours
(a kilowatt hour being equal to 1000 watt hours).
All electrically operated devices are stamped with the ampere and
voltage rating. This stamping may be found on the name-plate or bottom
of the device. By multiplying the voltage of the circuit upon which the
device is to be operated by the amperes as found stamped on the device,
we can quickly determine the wattage consumption of the latter, as ex-
plained under the definition of the watt, and as shown above. The line
voltage which is most extensively supplied by Electric Lighting com-
panies in this country is 115 volts, and where this voltage is in operation,
the devices are stamped for voltage thus: V. 110-125. This means
that the device may be used on a circuit where the voltage does not drop
below 110 volts or rise above 125 volts. By operating a device with the
above stamping on a circuit of 106 volts the life of the device would be
very much longer, but the results desired from it would be secured much
more slowly. Again, if the same device were used on a circuit oper-
ating at 130 volts, the life of the device would be very short, although the
results desired from it would be brought about much more quickly. Be-
fore attempting to operate an electrically heated or lighted device, if in
doubt about the voltage of the circuit, it is best to call upon the Electric
Company with which you are doing business and ask the voltage of their
lines.
Incandescent electric lamps, while known to the average user as
lamps of a certain "candle-power," are all labeled with their proper
wattage consumption. Mazda lamps, suitable for household use and
obtainable at all lighting companies, are made in 15, 25, 40, 60, and 100
watt sizes. For commercial use, lamps of 1000 watts and known as the
nitrogen-filled lamps are on the market. Nitrogen lamps are made in
sizes of 200 watts and upwards.
HEAT AND LIGHT 171
The rate by which current consumed for lighting and small heating is
figured in some cities is known as the "sliding scale rate," and current
is charged for each month, as follows :
The first 200 kw. hrs. used @ 10^ per kilowatt hour.
The next 300 kw. hrs. used @ 8^ per kilowatt hour.
The next 500 kw. hrs. used @ 7 0 per kilowatt hour.
The next 1000 kw. hrs. used @ 6 ^ per kilowatt hour.
The next 3000 kw. hrs. used @ 5 ^ per kilowatt hour.
All over 5000 kw. hrs. used @ 4 ^ per kilowatt hour.
Less 5% discount, if bill is paid within 15 days from date of issue.
Under the sliding-scale rate the more electricity that is consumed, the
cheaper it becomes. But it is also readily seen that the customer who
uses a large amount of electricity pays in exactly the same way as the
small consumer pays for his consumption.
If a person uses less than 200 kw. hrs. per month, he pays for his con-
sumption at the rate of 10 ^ per kilowatt hour ; if he uses 201 kw. hrs. of
electricity per month, he pays for his first 200 kw. hrs. at the first step,
namely 10 ^, and for the remaining 1 kw. hr. he pays 8 fi per kilowatt
hour.
If a meter reads " 1000 kw. hrs.," the bill is not figured at 6^ direct,
but must be figured step by step as shown in the examples below.
For convenience in figuring, the amount of power used by various
electrically operated devices is given in the following table. By figuring
the cost of each per hour, it will be seen that these electric servants work
very cheaply.
APPARATUS WATTS USED
WHAT is COST
PER HOUR 1
(a) Disk stove 200 • ?
(&) 6 Ib. iron 440 ?
(c) Air heater, small 1000 ?
(d) Toaster-stove 500 ?
(e) Heating pad 55 ?
(/) Sewing-machine motor 50 (average) ?
(#) 25 watt (16 c p.) lamp 25 ?
(ft) Chafing dish 500 ?
(0 Washing-machine motor 200 (average) ?
EXAMPLE. — Suppose a customer in one month used 6120
kilowatt hours of electricity, what is the amount of his bill
1 Based on 10 cents per kilowatt hour.
172 VOCATIONAL MATHEMATICS FOR GIRLS
with 5 % deducted if the bill is paid within the discount
period of 15 days from date of issue ?
SOLUTION. — 6120 kw. hrs. = total amount used.
First 200 kw. hrs. @ 10^ = $ 20.00
5920
Next _300kw. hrs. @ 8?= 24.00
5620
Next 500 kw. hrs. @ 7 / = 35.00
5120
Next 1000 kw. hrs. @ §t= 60.00
4120
Next 3000 kw. hrs. @ 5/= 150.00
We have now figured for 5000 kw. hrs., and as our rate states that all
over 5000 kw. hrs., is figured at 4 ^ per kilowatt hours, we have
1120 kw. hrs. @ 4? = $ 44.80
$333.80 = gross bill
Assuming that the bill is paid within the given discount period, we
deduct 5 % from the
gross bill, which equals I 16.69
$317. 11 = net bill
EXAMPLES
1. A customer uses in one month 300 kw. hr. of electricity.
What is the amount of his bill if 5 % is deducted for payment
within 15 days ?
2. What is the amount of bill, with 5 % deducted, for 15
kw. hr. of electricity ?
An electric meter is read in the same way that a gas meter is read.
In deciding the reading of a pointer, the pointer before it (to the right)
must be consulted. Unless the pointer to the right has reached or passed
zero, or, in other words, completed a revolution, the other has not com-
pleted the division upon which it may appear to rest. Figure 1 reads
11 kw. hrs., as the pointer to the extreme right has made one complete
revolution, thus advancing the second pointer to the first digit and has
itself passed the first digit on its dial.
HEAT AND LIGHT
173
FIG. 1. — READING 11 KW. MRS.
Fia. 2. — WHAT is THE READING?
Q/r^>\
f ¥ " \
ILOWATT- HOUC.S
FIG. 3. — READING 424 KW. HRS.
FIG. 4. — WHAT is THE READING?
FIG. 5. — WHAT is THE READING ?
174 VOCATIONAL MATHEMATICS FOR GIRLS
1. What is the cost of electricity in Fig. 1, using the rates
on page 171 ?
2. What is the cost of electricity in Fig. 2, using the rates
on page 171, with the discount ?
3. What is the cost of electricity in Fig. 3, using the rates
on page 171, with the discount ?
EXAMPLES
1. What is the cost of maintaining ten 25-watt Mazda lamps,
burning 30 hours at 10 cents per kw. hr. ?
2. What will it cost to run a sewing machine by a motor
(50 watts) for 15 hours at 9 cents per kw. hr. ?
3. A 6-lb. electric flatiron is marked 110 V. and 4 amperes.
What will it cost to use the iron for 20 hours at 8 cents per
kw. hr. ?
4. An electric washing machine is marked 110 V. and
2 amperes. What will it cost to run it 15 hours at 81 cents
per kw. hr. ?
5. An electric toaster stove is marked 115 volts and 3J am-
peres. What will it cost to run it for a month (thirty break-
fasts) 15 hours at 8J cents per kw. hr. ? If a discount of
5 % is allowed for prompt payment, what is the net amount
of the bill ?
Methods of Heating
Houses are heated by hot air, hot water, or steam. In the
hot-water system of heating, hot water passes through coils
of pipes from the heater in the basement to radiators in the
rooms. The water is heated in the boiler, and the portion of
the fluid heated expands and is pushed upward by the adjacent
colder water. A vertical circulation of the water is set up
and the hot water passes from the boiler to the radiators and
gives off its heat to the radiators, which in turn give it off to
the surrounding air in the room. The convection currents
HEAT AND LIGHT
175
carry heat through the room and at the same time provide for
ventilation.
In the hot-air method the heat passes from the furnace
through openings in the floor called registers. This method
frequently fails to heat
a house uniformly be-
cause there is no way for
the air in certain rooms
to escape so as to per-
mit fresh and heated
air to enter.
Steam heating consists
in allowing steam from
a boiler in the basement
to circulate through coils
or radiators. The steam
gives off its heat to the
HOT AIR HEATING SYSTEM
radiators, which in turn
give it off to the surrounding air.
Room-heating Calculations
In order to insure comfort
and health, every housewife
should be able to select an
efficient room-heating appli-
ance, or be able to tell whether
the existing heating appara-
tus is performing the required
service in the most econom-
ical manner. In order to do
this, it is necessary to know
how to determine the re-
quirements for individual
room heating.
HOT WATER HEATING SYSTEM
176 VOCATIONAL MATHEMATICS FOR GIRLS
For Steam Heating
Allow 1 sq. ft. of radiator surface for each
80 cu. ft. of volume of room.
13 sq. ft. of exposed wall surface.
3 sq. ft. of exposed glass surface (single window).
6 sq. ft. of exposed glass surface (double window).
For Hot-water Heating
Add 50 per cent to the amount of radiator surface obtained by the
above calculation.
For Gas Heaters having no Flue Connection
Allow 1 cu. ft. of gas per hour for each
215 cu. ft. of volume of room.
35 sq. ft. of exposed wall surface.
9 sq. ft. of exposed glass surface (single window).
18 sq. ft. of exposed glass surface (double window) .
The results obtained must be further increased by one or more of the
following factors if the corresponding conditions are present.
Northern exposure 1.3
Eastern or western exposure 1.2
Poor frame construction 2.5
Fair frame 2.0
Good frame or 12-inch brick 1.2
Room heated in day time only 1.1
Room heated only occasionally 1.3-1.4
Cold cellar below or attic above . . . . . 1.1
EXAMPLE. — How much radiating surface, for steam heating,
is necessary to heat a bathroom containing 485 cu. ft. ? The
bathroom is on the north side of the house.
-4F8ff5- = 6^ sq. ft. of radiating surface
6TV X 1.3 = ft x ft = 7tfi sq. ft.
6% + 7|ft = 6TVo + 7 J|i = 13}f£ sq. ft. or approx. 14 sq. ft. Ans.
EXAMPLES
1. How much radiating surface, for steam heating, is re-
quired for a bathroom 12' X 6' x 10' on an eastern exposure ?
2. How much radiating surface, for hot-water heating, is
required for the bathroom in example 1 ?
COST OF FURNISHING A HOUSE 177
3. How large a gas heater should be used for heating the
bathroom in example 1 ?
4. (a) How much radiating surface is required for steam
heating, in a living room 18' x 16^' x 10', with three single
windows 2' x 5^ ' ? The room is exposed to the north.
(6) How much radiating surface for hot- water heating ?
(c) How much gas should be provided to heat the room in
example (a) ?
5. (a) How much radiating surface is required for steam
heating a bedroom 19' x 17' x 11' with two single windows
2' x 5y ? The house is of poor frame construction.
(b) How much radiating surface for hot- water heating ?
(c) How much gas should be provided to heat room in
example (a) ?
CHAPTER VIII
THRIFT AND INVESTMENT
IT is not only necessary to increase your earning capacity, but
also to develop systematically and regularly the saving habit.
A dollar saved is much more than two dollars earned. For a
dollar put at interest is a faithful friend, earning twenty-four
hours a day, while a spent dollar is like a lost friend — gone
forever. Histories of successful men show that fortune's
ladder rests on a foundation of small savings ; it rises higher
and higher by the added power of interest. The secret of
success lies in regularly setting aside a fixed portion of one's
earnings, for instance 10 % ; better still, 10 % for a definite
object, such as a home or a competency.
In every community one will find various agencies by which
savings can be systematically encouraged and most success-
fully promoted. These institutions promote habits of thrift,
encourage people to become prudent and wise in the use of
money and time. They help people to buy or build homes for
themselves or to accumulate a fund for use in an emergency or
for maintenance in old age.
Banks
Working people should save part of their earnings in order to have
something for old age, or for a time of sickness, when they are unable to
work. This money is deposited in banks — savings, National, cooper-
ative, and trust companies.
National Banks
National banks pay no interest on small deposits, but give the depositor
a check book, which is a great convenience in business. National banks
require that a fixed sum should be left on deposit, $ 100 or more, and
some of them charge a certain amount each month for taking care of the
money.
178
THRIFT AND INVESTMENT 179
Trust Companies
Trust companies receive money on deposit and allow a customer to
draw it out by means of a check. They usually pay a small interest on
deposits that maintain a balance over $ 500.
Cooperative Banks
When a person takes out shares in a cooperative bank, he pledges him-
self to deposit a fixed amount each month. If he deposits $5, he is said
to have five shares. No person is permitted to have more than twenty-
five shares. The rate of interest is much higher than in other banks, and
when the shares mature, which is usually at the end of about eleven
years, all the money must be taken out. Many people build their home
through the cooperative bank, for, like every other bank, it lends money.
When a person borrows money from a cooperative bank, he has to give a
mortgage on real estate as security, and must pay back a certain amount
each month.
Savings Banks
The most common form of banking is that carried on by the Savings
Sank. People place their money in a savings bank for safe keeping and
for interest. The bank makes its money by lending at a higher interest
than it pays its depositors. There is a fixed date in each bank when
money deposited begins to draw interest. Some banks pay quarterly and
some semi-annually. At different times banks pay different rates of
interest; and often in the same community there are different rates of in-
terest paid by different banks.
Every bank is obliged to open its books for inspection by special
officers who are appointed for that work. If these men did their work
carefully and often enough, there would be almost no chance of loss in
putting money in a bank. Banks fail when they lend money to too many
people who are unable to pay it back.
EXAMPLES
(Review interest on page 50)
1. I place $ 400 in a savings bank that pays 4 % on Jan. 1,
1916. Money goes on interest April 1 and at each successive
quarter. How much money have I to my credit at the begin-
ning of the third quarter ?
2. A man with a small business places his savings, $ 1683,
180 VOCATIONAL MATHEMATICS FOR GIRLS
in a trust company so he can pay his bills by check. The
bank pays 2 % for all deposits over $ 500. He draws checks
for $ 430 and $ 215 within a few days. At the end of a
month he will receive how much interest ?
3. Practically 10 % of the entire population of the United
States, including children, have savings-bank accounts. If the
population is 92,818,726, how many people have savings bank
deposits ?
4. On April 1, 1910, a woman deposited $ 513 in a savings
bank which pays 4 % interest. Interest begins April 1 and at
each succeeding quarter. Dividends are declared Jan. 1 and
July 1. What is the total amount of her deposit at the present
date?
The savings bank is not adapted to the needs of those with large sums
to place at interest. It is a place where small sums may be deposited
with absolute safety, earn a modest amount, and be used by the depositor
at short notice. The savings bank lends money on mortgages and re-
ceives about 5 °fo. It pays its depositor either 3£ % or 4 %. The differ-
ence goes to pay expenses and to provide a surplus fund to protect
depositors.
The question may be asked, " Why cannot the ordinary depositor lend
his money on mortgages and receive 5 °fo ? " He can, if he is willing to
assume the risk. When you receive 4 % interest, you are paying 1% to
1| °fo in return for absolute safety and freedom from the necessity of
selecting securities.
Mortgages
A mortgage is the pledging of property as a security for a debt. Mr.
Allen owns a farm and wants some money to buy cattle for it. He goes
to Mr. Jones and borrows $ 1000 from him, and Mr. Jones requires him
to give as surety a mortgage on his farm. That is, Mr. Allen agrees that
if he does not pay back the $ 1000, the farm, or such part as is necessary
to cover the debt, shall belong to Mr. Jones.
Under present law, if a man wishes to foreclose a mortgage, — that is,
compel its payment when due, — he cannot take the property, but it must
be sold at public auction. From the money received at the sale the man
who holds the mortgage receives his full amount, and anything that is
left belongs to the man who owned the property.
THRIFT AND INVESTMENT 181
Notes
A promissory note is a paper signed by the borrower promising to
repay borrowed money. Notes should state value received, date, the
amount borrowed (called the face), the rate, to whom payable, and
the time and place of payment. Notes are due at the expiration of
the specified time.
The rate of interest varies in different parts of the country. The
United States has to pay about 2 % . Savings banks pay 3 % or 4 °fo .
Individuals borrowing on good security pay from 4 % to 6 °fo.
In order to make the one who loans the money secure, the borrower,
called the maker of the note, often has to get a friend to indorse or sign
this note. The indorser must own some sort of property and if, at the
end of six months or the time specified, the maker cannot pay the note,
he is notified by written order, called a protest, and may, later, be called
upon to repay the note.
A man is asking a great deal when he asks another man to sign a note
for him. Unless you have more money than you need, it is better busi-
ness policy to refuse the favor.
Always be sure that you know exactly what you are signing and that
you know the responsibility attached. If you are a stenographer or a
clerk in an office, you will often be called upon to witness a signature and
then to sign your own name to prove that you have witnessed it. Always
insist upon reading enough of the document to be sure that you know
just what your signature means.
EXAMPLES
1. My house is worth $4000 and the bank holds a mortgage
on it for one-half its value. They charge 5 % interest, which
must be paid semi-amiually. How much do I pay each time ?
2. A bank holds a mortgage of $ 2500 on a house. The in-
terest is 5 % payable semi-annually. How much is paid for
interest at the end of three years ?
3. A man buys property worth $ 3000. He gives a $ 2000
mortgage and pays 5-J- % interest. What will be the interest
on the mortgage at the end of the year ? Suppose he does not
pay the interest, how long can he hold the property ?
182 VOCATIONAL MATHEMATICS FOR GIRLS
DIFFERENT KINDS OF PROMISSORY NOTES
$_ Montgomery, Ala —
after date for value received ^promise
to pay to the order of
.Dollars
at fHed)amc0 National
No
A COMMON NOTE
St. Paul, Minn 19
.after date for value received we jointly and
severally promise to pay to the order o
.Dollars
at $fledjamc0 National Bank.
No. _ _ Due
JOINT NOTE
$ FALL RIVER, MASS. 191
after date for value received
promise to pay to the order of THE MECHANICS NATIONAL BANK of Fall River,
Mass. _ DOLLARS,
at said Bank, and interest for such further time as said principal sum or any
part thereof shall remain unpaid at the rate of per cent per annum,
having deposited with the said Mechanics National Bank, as GENERAL COL-
LATERAL SECURITY, for the payment of any of liabilities to said
Bank due, or to become due, direct or indirect, joint or several, individual or
firm, now or hereafter contracted or incurred, at the option of said Bank,
the following property, viz. :
and hereby authorize said Bank or its assigns to sell and transfer said
property or any part thereof without notice, at public or private sale, at the
THRIFT AND INVESTMENT 183
option of said Bank or its assigns, on the non-payment of any of the liabili-
ties aforesaid, and to apply the proceeds of said sale or sales, after deducting
all the expenses thereof, interest, all costs and charges of enforcing this
pledge and all damages, to the payment of any of the liabilities aforesaid,
giving credit for any balanqe that may remain. Said Bank or its
assigns shall at all times have the right to require the undersigned to deposit
as general collateral security for the liabilities aforesaid, approved additional
securities to an amount satisfactory to said Bank or its assigns, and
hereby agree to deposit on demand (which may be made by notice in writing
deposited in the post office and addressed to at last known
residence or place of business) such additional collateral. Upon fail-
ing to deposit such additional security, the liabilities aforesaid shall be deemed
to be due and payable forthwith, anything hereinbefore or elsewhere ex-
pressed to the contrary notwithstanding, and the holder or holders may
immediately reimburse themselves by public or private sale of the security
aforesaid; and it is hereby agreed that said Bank or any of its officers,
agents, or assigns may purchase said collateral or any part thereof at such
sale. In case of any exchange of or addition to the above described collateral,
the provisions hereof shall apply to said new or additional collateral.
COLLATERAL NOTE
4. On Jan. 2, 1915, Mr. Lewis gave his note for $2400,
payable on Feb. 27, with interest at 6 %. On Feb. 2, he paid
$ 600. How much was due Mar. 2, 1915 ?
SOLUTION. — In the case of notes running for less than a year, exact
days are counted ; from Jan. 2 to Feb. 2 is 31 days.
Interest Jan. 2 to Feb. 2, 31 days,
$ 12.00 for 30 days
.40 for 1 day
.$ 12.40 31 days
Amount due Feb. 2, $ 2400 + 12.40 = $ 2412.40.
$ 2412.40 - 600 = $ 1812.40.
Interest Feb. 2 to March 2, 28 days,
§6.0413 20 days
1.8124 6 days
.6041 2 days
$8.4578 or $8.46
1812.40
Amount due March 2, $ 1820.86 Ans.
184 VOCATIONAL MATHEMATICS FOR GIRLS
Money lenders may discount their notes at banks and thus obtain their
money before the note comes due. But the banks, in return for this serv-
ice, deduct from the full amount of the note interest at a legal rate on
the full amount for such time as remains between the day of discount
and the day when the note comes due.
To illustrate : A man has a note for $ 600 due in three months at 6 %
interest. At the end of a month he presents the note at a bank and
returns the difference between the amount at maturity, $ 609, and the
interest on $609 for two months, the remaining time, at legal rate 6%,
$6.09 or $609 - 6.09 = $602.91.
5. On June 1, 1914, Mr. Smith gives his note for $ 1200,
payable on demand with interest at 6%. The following pay-
•ments are made on the note : Aug. 1, 1914, $ 140 ; Oct. 1, 1914,
$100; Dec. 1, 1914, $100; and Feb. 1, 1914, $160. How
much was due May 1, 1915 ?
6. A merchant buys paper amounting to $ 945. He gives
his note for this amount, payable in three months at 6 % .
The paper dealer desires to turn the note into cash immedi-
ately. He therefore discounts it at the bank for 6 % . How
much does he receive ?
Stocks
It often happens that one man or a group of men desire to engage in a
business that requires more money than they alone are able or willing to
invest in it. They obtain more money by organizing a stock company,
in which they themselves buy as many shares as they choose, and then
they induce others to pay for enough more shares to make up the capital
that is needed or authorized for the business.
A stock company consists of a number of persons, organized under a
general law or by special charter, and empowered to transact business as
a single individual. The capital stock of a company is the amount named
in its charter. A share is one of the equal parts into which the capital
stock of a company is divided (generally $ 100).
The par value of a share of stock is its original or face value ; the
market value of a share of stock is the price for which the share will sell
in the market. The market values of leading stocks vary from day to
day, and are quoted in the daily papers; e.g. "N. Y. C., 131" means
that the stock of the New York Central R. R. Co. is selling to-day at
$ 131 a share.
THRIFT AND INVESTMENT
185
Dividends are the net profits of a stock company divided among the
stockholders according to the amount of stock they own.
Stock companies often issue two kinds of stock, namely : preferred
stock, which consists of a certain number of shares on which dividends
are paid at a fixed rate, and common stock, which consists of the re-
maining shares, among which are apportioned whatever profits there are
remaining after payment of the required dividends on the preferred stock.
CAZENDV1A NATIONAL BANK
CERTIFICATE OF STOCK
Stocks are generally bought and sold by brokers, who act as agents
for the owners of the stock. Brokers receive as their compensation a
certain per cent of the par value of the stock bought or sold. This is
called brokerage. The usual brokerage is | % of the par value ; e.g. if a
broker sells 10 shares of stock for me, his brokerage is | °/o of $ 1000, or
$1.25.
EXAMPLE. — What is the cost of 20 shares of No. Butte 301 ?
$ 30^ + $ \ i = $ 30|, cost of 1 share.
X 20 = $600 + $ 12£ = §612.50, total cost.
i of 1 % of $ 100 = 1 of $1, broker's charge per share.
186 VOCATIONAL MATHEMATICS FOR GIRLS
EXAMPLES
1. The par value of a certain stock is $ 100. It is
quoted on the market at $ 87|-.. What is the difference in
price per share between the market value and the par
value ?
2. What is the cost of 40 shares of Copper Range at 53 ?
3. What is the cost of 53 shares of Calumet and Hecla
at 680 ?
4. I have 50 shares of Anaconda. How much shall I re-
ceive if I sell at 661 ?
5. I buy 60 shares of Anaconda at 66J. It pays a quarterly
dividend of $ 1.50. What interest am I receiving on my
money ?
Bonds
Corporations and national, state, county, and town governments often
need to borrow money in order to meet extraordinary expenditures.
When a corporation wishes to borrow a large sum of money for several
years, it usually mortgages its property to a person or bank called a trustee.
The amount of the mortgage is divided into parts called bonds, and these
are sold to investors. The interest on the bonds is at a fixed rate and is
generally payable semi-annually. Shares of stock represent the property
of a corporation, while bonds represent debts of the corporation ; stock-
holders are owners of the property of the corporation, while bondholders
are its creditors.
Bonds of large corporations whose earnings are fairly stable and regu-
lar, like steam railroads, street railways, and electric power and gas
plants, whose property must be employed for public necessities regardless
of the ability of the managers, are usually good investments. Well-secured
bonds are safer than stocks, as the interest on the bonds must be paid re-
gardless of the condition of the business.
For the widow who is obliged to live on the income from a moderate
amount of capital, it is better to invest in bonds and farm mortgages than
in stock.
THRIFT AND INVESTMENT
187
A SAMPLE BOND
188 VOCATIONAL MATHEMATICS FOR GIRLS
EXAMPLES
1. A man put $ 200 in the Postal Savings Bank and received
2 % interest. What would have been the difference in his
income for a year if he had taken it to a savings bank that
paid 3| % ?
2. A widow had a principal of $ 18,000. She placed it in a
group of savings banks that paid 3f %,. The next year she
purchased farm mortgages and secured 51 %. What was the
difference in her income for the two years ?
3. Two sons were left $ 15,000 each. One placed it in first-
class bonds paying 5^ %. The other placed it in savings banks
and averaged 41 % . WThat was the difference in income per
year?
Fire Insurance
Household furniture, books, apparel., etc., can be insured at a low rate.
While it will not make a man less careful in protecting his home from fire,
it will make him more comfortable in the thought that if fire should come,
the family will not be left without the means of clothing themselves and
refurnishing the house. One of the first duties then, after the home is
established, is to secure insurance.
Insurance companies issue a policy for 1, 3, or 5 years. There is an
advantage in buying a policy for more than one year, for on the 3- or 5-
year policy there is a saving of about 20 % in premiums. Rules of per-
centage apply to problems in insurance.
EXAMPLE. — A house worth $ 8400 is insured for its full
value at 28 cents per $ 100. WThat is the cost of premium ?
SOLUTION.
$ 8400 is the value of the policy or base.
28 cents is the rate of premium or rate.
The premium or interest is the amount to be found.
84 x $0.28 = $23.52, premium.
EXAMPLES
1. Find the insurance upon a dwelling house valued at
$ 3800 at $ 2.80 per $ 1000 if the policy is on 80 % of the value
of the house.
THRIFT AND INVESTMENT 189
2. Mr. Jones takes out $ 800 insurance on his automobile at
2 °fc . What is the cost of the premium ?
3. The furniture in one tenement of a three-family house is
valued at $ 1000. What premium is paid, if it is insured at
the rate of 1 % for 5 years ?
4. If the premium on the same furniture in a two-family
house in a different city is $ 7.50, what is the rate, expressed
in per cent ?
Life Insurance
Every industrious and thrifty person lays aside a certain amount
regularly for old age or future necessities, or in case of death to provide
sufficient amount for the support of the family. This is usually done by
taking out life insurance from a corporation called an insurance company.
This corporation is obliged to obtain a charter from the state, and is
regularly inspected by a proper state officer.
The policy or contract which is made by the company with the member,
fixing the amount to be paid in the event of his death, is called a life
insurance policy, and the person to whom the amount is payable is
termed the beneficiary. The contribution to be made by the member to
the common fund, as stipulated in the policy, is termed the premium, and
is usually payable in yearly, half-yearly, or quarterly installments.
There are different kinds of insurance policies : the simplest is the
ordinary life policy. Before entering into a contract of this kind, it
is necessary to fix the amount of the premium, which must be large
enough to enable the company to meet the necessary expense of conduct-
ing the business and to accumulate a fund sufficient to pay the amount of
the policy when the latter matures by the death of the insured.
Making the Premium. — If it were known to a certainty just how long
the policy holder would live, anyone could compute the amount of the
necessary premium. Let us suppose, for illustration, that the face of the
policy is $ 1000, and that the policyholder will live just twenty years. Let
us assume that the business is conducted without expense, and that the
premiums are all to be invested at interest from date of payment. We
do not know to a certainty what rate of interest can be earned during the
whole period, and we shall therefore assume one that we can safely
depend upon, say three per cent. A yearly payment of .$36.13 invested
at three per cent compound interest will amount to $ 1000 in twenty
years.
190 VOCATIONAL MATHEMATICS FOR GIRLS
No. >213649 $5000
gto* fjmrtfe # tar ptutttal gif* Iwstxmtxtje
Cfompaug
In Consideration of the application for this Policy, a copy of which
is attached hereto and made a part hereof, and in further consideration of
the payment of
ffiunflrrti ffln'rtg^cight^ 2^_ Dollars,
100
the receipt whereof is hereby acknowledged, and of the &ttttuat payment
of a like sum to the said Company, on or before the __ day of
Januarg jn every year during the continuance of this Policy, promises
to pay at its office in Milwaukee, Wisconsin, unto _
_, Beneficiar 2.
of gofrn Boe— _the insured, Of
®cs jffloines jn t]ie state of.
subject to the rigfrt of the gnstircfr, frcrcfag rcscrbcfr, to flange the 33cncficiarg
or iScncficiarics the sum of_ ^tbj ^Ti)ousanU— ^Dollars,
upon receipt and approval of proof of the death of said Insured while this
Policy is in full force, the balance of the year's premium, if any, and any
other indebtedness on account of this Policy being first deducted there-
from ; provided, however, that if no Beneficiary shall survive the said
Insured, then such payment shall be made to the executors, administra-
tors or assigns of the said Insured.
In Witness Whereof, THE NORTH STAR MUTUAL LIFE INSURANCE
COMPANY, at its office in Milwaukee, Wisconsin, has by its President and
Secretary, executed this contract, this FirKt day of January Qne
thousand nine hundred and sixteen. — _
S. A. Hawkins, Secretary. L. H. Perkins, President.
ORDINARY LIFE INSURANCE POLICY
THRIFT AND INVESTMENT 191
If it were certain that the policyholder would live just twenty years,
and that his premiums would earn just three per cent interest, and that
the business could be conducted without expense, the necessary premium
would be $36.13. But there are certain other contingencies that should
be provided for ; such as, for example, a loss of invested funds, or a
failure to earn the full amount of three per cent interest.
To meet these expenses and contingencies something should be added
to the premium. Let us estimate as sufficient for this purpose the sum of
$7. This will make the gross yearly premium $43.13, the original pay-
ment ($36.13) being the net premium, while the amount added thereto
for expenses, etc. ($7.00), is termed the loading.
The net premium is the amount which is mathematically necessary
for the creation of a fund sufficient to enable the company to pay the
policy in full at maturity. The loading is the amount added to the net
premium to provide for expenses and contingencies. The net premium
and loading combined make up the gross premium, or the total amount to
be paid each year by the insured.
Mortality Tables. — Although it is impossible, as in the illustration
given above, to predict in advance the length of any individual life, there
is a law governing the mortality of the race by which we may determine
the average lifetime of a large number of persons of a given age. We
cannot predict in what year the particular individual will die, but we may
determine with approximate accuracy how many out of a given number
will die at any specified age. By means of this law it becomes possible to
compute the premium that should be charged at any given age with
almost as much exactness as in the example given, in which the length of
life remaining to the individual was assumed to be just twenty years.
Let us suppose, for example, that observations cover a period of time
sufficient to include the history of 100,000 lives. Of these, you will find a
certain number dying at the age of thirty, a larger number at the age of
forty, and so on at the various ages, the extreme limit of life reached
being in the neighborhood of one hundred years. The mortuary records
of other groups of 100,000. living where conditions are practically the
same, would give approximately the same results — the same number of
deaths at each age in 100,000 born. The variation would not be great,
and the larger the number of lives under observation, the nearer the
number of deaths at the several ages by the several records would ap-
proach to uniformity.
In this manner mortality tables have been constructed which show how
many in any large number of persons born, or starting at a certain age,
will live to age thirty, how many to age forty, how many to any other
192 VOCATIONAL MATHEMATICS FOR GIRLS
age, and likewise the number that will die at each age, with the average
lifetime remaining to those still alive. The insurance companies from
these tables construct tables of premiums, varying according to the amount
and kind of insurance and the age at which the policy is taken out.
Kinds of Policies. — An endowment policy is essentially for persons who
must force themselves to save. It is an expensive form of insurance, but
one that affords the young man or woman an incentive for saving, and
that matures at a time when the individual has, as a result of long
experience, better opportunities to make profitable investments. This
policy also has a larger loan value than any other, and this sometimes be-
comes an advantage to the young person. However, the chief advantage
of the endowment policy is its incentive to save.
A limited payment policy, such as the twenty-payment life, appeals
most directly to those who desire to pay for life insurance only within the
productive period of their life. This policy should attract the young man
who is uncertain of an income after a given period, or who does not wish
insurance premiums to be a burden upon him after middle life. Out of
the relatively large and certain income of his early productive years he
pays for his insurance. This policy also appeals to the man of middle
age who has neglected to purchase life insurance but who wishes to buy
it and pay for it before be becomes actually old.
The Annuity
An annuity is a specific sum of money to be paid yearly to some
designated person. The one to whom the money is to be paid is termed
the annuitant. If the payment is to be made every year until the annui-
tant dies, it is termed a life annuity. For example, a life insurance
company or other financial institution, in consideration of the payment
to it of a specified amount, say § 1000, will enter into a contract to pay
a designated annuitant a stated sum, say $ 70, on a specified day in every
year so long as the annuitant continues to live. The latter may live to
draw his annuity for many years, until he has received in aggregate
several times the original amount paid by him, or he may die after having
collected but a single payment. In either case, the contract expires and
the annuity terminates with the death of the annuitant.
The amount of the yearly income or annuity which can be purchased
with $ 1000 will depend, of course, upon the age of the annuitant. That
sum will buy a larger income for the man of seventy than for one of
fifty-six, for the reason that the former has, on the average, a much
shorter time yet to live. The net cost of an annuity, that is, the net
THRIFT AND INVESTMENT 193
amount to be paid in one sum, and which is termed the value of the
annuity, is not a matter of estimate, but, like the life insurance premium,
is determined by mathematical computation, based upon the mortality
table. The process is quite as simple as the computation of the single
premium.
Many men who insure their lives choose a form of policy under which
the beneficiary, instead of receiving the full amount of the insurance at
the death of the insured, is paid an annuity for a period of years or
throughout life. The amount of annuity paid in such cases is exactly
equal to the amount that could be bought for a sum equal to the value of
the policy when it falls due.
EXAMPLES
1. A young man at 26 years of age takes out a straight life
policy of $ 1000, for which he pays $ 17.03 a year as long as
he lives, and his estate receives $ 1000 at his death. If he
dies at 46 years of age, how much has he paid in ? How much
more than he has paid does his estate receive then ?
2. Another young man at the same age takes out a twenty-
payment life policy and pays $ 24.85 for twenty years. At
the end of the twenty years, how much has he paid in ? Does
he receive anything in return at the end of the twenty years ?
3. Another form of insurance, called an endowment, is taken
out by another young man at twenty-six years of age. He
pays $ 41.94 a year. At the end of twenty years he receives
$ 1000 from the insurance company. How much has he paid
in ? Where is the difference between these two amounts ?
Exchange
Exchange is the process of making payment at a distant place without
the risk and expense of sending money itself. Funds may be remitted
from one place to another in the same country in six different ways :
Postal money order, express money order, telegraphic money order, bank
draft, check, and sight draft.
The largest amount for which one can obtain a postal money order is
$ 100. It is drawn up by the postmaster after an application has been
duly made out.
An express money order is similar to a postal money order, but may be
194 VOCATIONAL MATHEMATICS FOR GIRLS
drawn for any number of dollars at the same rate as the post office order.
This is issued at express offices.
A telegraphic money order is an order drawn by a telegraph agent at
any office, instructing the agent at some other office to pay the person
named in the message the sum specified. The rates are high, and in
addition one must pay the actual cost of sending the telegram according
to distance and number of words.
A bank draft is an order written by one bank directing another bank
to pay a specified sum of money to a third party. This order looks much
like a check.
A check is an order on a bank to pay the sum named and deduct the
amount from the deposit of the person who signs the check.
A sight draft is an order on a debtor to pay to a bank the sum named
by the creditor who signs the draft.
Foreign exchange is a system for transmitting money to another country.
By this means the people of different countries may pay their debts.
The most common methods of foreign exchange for an ordinary
traveler are letters of credit or travelers' cheques.
A letter of credit is a circular letter issued by a banking house to a
person who desires to travel abroad. The letter directs certain banks in
foreign countries to furnish the traveler such sums as he may require up
to the amount named in the letter.
Fees For Money Orders
Domestic Bates
When payable in Bahamas, Bermuda, British Guiana, British Hon-
duras, Canada, Canal Zone, Cuba, Martinique, Mexico, Newfoundland,
The Philippine Islands, The United States Postal Agency at Shanghai
(China), and certain islands in the West Indies, listed in the register of
money order offices.
For Orders from
$00.01 to $2.50 . Scents
From $ 2.51 to $ 5 5 cents
From $ 5.01 to $ 10 8 cents
From $ 10.01 to $ 20 10 cents
From $ 20.01 to $ 30 12 cents
From $ 30.01 to $ 40 15 cents
From $ 40.01 to $ 50 18 cents
From $ 50.01 to $ 60 20 cents
From $ 60.01 to $75 , 25 cents
From $ 75.01 to $ 100 , . . 30 cents
THRIFT AND INVESTMENT 195
International Rates
When payable in Asia, Austria, Belgium, Bolivia, Chile, Costa Rica,
Denmark, Egypt, France, Germany, Great Britain and Ireland, Greece,
Honduras, Hongkong, Hungary, Italy, Japan, Liberia, Luxemburg,
Netherlands, New South Wales, New Zealand, Norway, Peru, Portugal,
Queensland, Russia, Salvador, South Australia, Sweden, Switzerland,
Tasmania, Union of South Africa, Uruguay, and Victoria.
For Orders from
$00.01 to $10 10 cents
From $ 10.01 to $ 20 20 cents
From $ 20.01 to $ 30 30 cents
From $ 30.01 to $ 40 40 cents
From $ 40.01 to $ 50 50 cents
From $50.01 to $60 60 cents
From $ 60.01 to $ 70 70 cents
From $ 70.01 to $80 . 80 cents
From $80.01 to $90 90 cents
From $ 90.01 to $ 100 1 dollar
Rates for Money Transferred by Telegraph
The Western Union charges for the transfer of money by telegraph to
its offices in the United Stales the following :
First : For $ 25.00 or less 25 cents
$ 25.01 to $ 50.00 35 cents
$50.01 to $ 75.00 60 cents
$75.01 to $100.00 85 cents
For amounts above $ 100.00 add (to the $ 100.00 rate) 25 cents per hundred
(or any part of $ 100.00) up to $ 3000.00.
For amounts above $ 3000.00 add (to the $ 3000.00 rate) 20 cents per hundred
(or any part of $ 100.00).
Second : To the above charges are to be added the tolls for a fifteen word
message from the office of deposit to the office of payment.
Express rates are the same as postal rates.
EXAMPLES
1. A young woman in. California desires to send $ 20 to her
mother in Maine. What is' the most economical way to send
it, and what will it cost ?
2. A young lady, traveling in this country, finds that she
196 VOCATIONAL MATHEMATICS FOR GIRLS
needs money immediately. What is the quickest and most
economical way for her to obtain $ 275 from her brother who
lives 1000. miles distant ?
3. A merchant in Boston buys a bank draft of $ 3480 for
Chicago. The bank charges J of 1 % for exchange. How
much must he pay the bank ?
4. A domestic in this country sends to her mother in Ireland
5 pounds for a Christmas present. What will it cost her, if
$ 4.865 = £ 1 ? A commission of i of 1 °/0 is charged.
Claims
If a person traveling by boat, electric or steam railway is injured by an
accident which is the fault of the company, it is bound to repair the finan-
cial loss. The company is not responsible for the carelessness of passen-
gers or for the action of the elements. When an accident occurs, the
injured persons are interviewed by a claim agent, whom all large com-
panies employ, and he offers to settle with you for a certain amount. If
you are not satisfied with this amount, you may put in your claim and
the case goes to court, where you may lose or win according to the decision
of the jury. When a wreck occurs on a railroad, a claim agent and a
doctor are brought to the scene as soon as possible. They take the name
and address of each person in the accident and try to settle the case at
once, because it is expensive to go to court and the newspaper notoriety
injures the reputation of the company. If you are not seriously hurt, the
claim agent tries to persuade you to sign a paper which relieves the Com-
pany from any responsibility forever after. For instance, in a collision
you seem to be only shaken up, not injured. The claim agent perhaps
offers to pay you $ 25. You think that is an easy way to get $ 25, so you
take it, but in turn you must sign a paper which states that the company
has settled in full with you for any claim that you may have against it for
that accident. Now it may prove later that you have an internal injury
which you did not realize at the time, and that an operation costing $ 500
is necessary. Can you compel the company to pay the bill ? People
who are not hurt at all in an accident and to whom the claim agent offers
nothing are also asked to sign a paper relieving the company from all
responsibility. Do not sign such a paper. The company cannot compel
you to, you gain nothing by it, and may lose much if it proves later that
you are internally injured.
THRIFT AND INVESTMENT 197
EXAMPLES
1. A woman was riding in an electric car that collided with
another. She was cut with flying glass and was obliged to hire
a servant for four weeks at $8. Doctor's bills amounted to
$24.50, medicine, etc., $8.75. She settled at the time of the
accident for $50. Did she lose or gain ?
2. A man working in a mill was injured in an elevator acci-
dent. The insurance company paid his wages and medical
bills for 8 weeks at $13.50 per week. A year later he was out
of work for three weeks for the same injury and did not receive
any compensation. Would it have been better for him to have
settled for $100 at the beginning?
3. A saleslady tripped on a staircase and sprained her ankle.
She was out of work for two weeks and two days at $8.75 per
week. Her medical supplies cost $9.75. She settled for $45.
How much did she gain ?
PART III — DRESSMAKING AND MILLINERY
CHAPTER IX
PROBLEMS IN DRESSMAKING
THE yardstick is much used for measuring cloth, carpets,
and fabrics. The yardstick is divided into halves, quarters,
and eighths. Dressmakers should know the fractional equiva-
lents of yards in inches and the fractional equivalents of
dollars in cents.
It is wise to buy to the nearest eighth of a yard unless the
cost per yard is so small that an eighth would cost as much as
a quarter.
EXAMPLES
1. Give the equivalent in inches of the following :
(a) 1 yd. (/) 4f yd. (fc) 1 yd.
(b) 21 yd. (g) 61 yd. (/) 1 yd.
(c) 11 yd. (ft) li yd. (m) TV yd.
(d) 21 yd. (0 1} yd. (n) A yd.
00 3f yd. CO * yd. 00 A yd.
2. A piece of cloth is 12 yd. long. How many pieces are
needed for 16 aprons requiring 11 yd. each ?
3. A piece of lawn cloth is 28 yd. long. How many pieces
are needed for 20 aprons requiring 1| yd. each ?
4. Give the value in cents of the following fractions of a
dollar :
(«) if 00 it -CO A « A
P) i (n i oo t « A
W i to) I (*) A (<0 i
W ii W T76 (0 A 00 t
198
ARITHMETIC FOR DRESSMAKERS 199
5. If 16" is cut from 1| yd. of cloth, how much remains ?
6. If J of a yard of lawn is cut from a piece 40 in. long,
what part of a yard is left ?
7. I bought 9f yd. of silk for a dress. If If yd. remained,
how much was used ?
8. A towel is 33 inches long and and a dishcloth 13 inches.
(a) Find the length of both. (Allow |" for each hem.)
(b) Find the number of yards used for both.
(c) Find the number of inches used by a class of 24.
(d) Find the number of yards used by a class of 24.
(e) Find the cost per pupil at 6 cts. per yard.
(/) Find the cost for a class of 24 at 6 cts. per yard.
9. If it took 72 yards of material for a dishcloth and towel
for two classes of 24 (48 in all), find the amount used by each
pupil.
10. If 45f yards of material were used for a class of 42, find
the amount used by each pupil.
11. (a) Reduce 75 inches to yards, (b) Find the number of
inches in 3^- yards, (c) From 2J yards cut 40 inches.
Tucks
A tuck is a fold in the cloth
for the purpose of shortening
garments or for trimming or dec~
oration. A tuck takes up twice
its own depth ; that is, a V tuck
takes up 2" of cloth.
EXAMPLES
1. Before tucking, a piece of
goods was I- yd. long : after tuck-
MEASURING FOR TUCKS FROM
ing, it was | yd. long. How FoLD T0 FOLD
many y tucks were made ?
200 VOCATIONAL MATHEMATICS FOR GIRLS
2. How much lawn is taken up in 3 groups of tucks, the
first group containing 6 one-inch tucks, the second group 6 one-
half-inch tucks, and the third group 12 one-eighth-inch tucks ?
3. A piece of muslin 29 inches wide was tucked and when
returned to the teacher was only 14 inches wide. How many
yr tucks were made in it ?
4. Before tucking, a piece of goods was f yd. long ; after
tucking, it was % yd. long. How many y tucks were made ?
Hem
HEM TURNED
A hem on a piece of cloth is
an edge turned over to form a
border or finish. In making a
hem an edge must always be
turned to prevent fraying ; ex-
cept for very heavy or very
loosely woven cloth this is usu-
ally y. For an inch hem you
would have to allow 1-".
EXAMPLES
1. I wish to put three V' tucks in a skirt- which is to be 40"
long. How long must the skirt be cut to allow for the tucks
and 31" hem ?
2. My cloth for a ruffle is 10" deep. It is to have a 1^"
hem, and five 1" tucks. How long will it be when finished ?
3. If. a girl can hem 21 inches in five minutes, how long
will she take to hem 2 yards ?
4. At the rate of f of an inch per minute, how long will it
take a girl to hem 2 yards ? 10 yards ?
5. At the rate of 51 inches per ten minutes, how long will
it take to hem 3^ yards ?
ARITHMETIC FOR DRESSMAKERS 201
6. A girl can hem 3 inches in five minutes. How much in
an hour ?
7. How long will she take to hem 90 inches ?
8. At 6 cents per hour, how much can she earn by hemming
190 inches ?
9. How long will it take a girl to hem 2J yards if she can
hem 5| inches in ten minutes ?
Ruffle
A ruffle is a strip of cloth gathered in narrow folds on one
edge and used for the trimming or decoration. Different pro-
portions of material are allowed
according to the use to which it
is to be put. For the ordinary
ruffle at the bottom of a skirt,
drawers, apron, etc., allow once and
a half. Once and a quarter is RUFFLE
enough to allow for trimming for
a corset cover or for other places where only a scant ruffle is
desirable. A plaiting requires three times the amount.
EXAMPLES
1. How much hamburg would you buy to make a ruffle for
a petticoat which measures 3 yd. around, if once and a half
the width is necessary for fullness ?
2. How much lace 2^ inches wide would you buy to have
plaited for sleeve finish, if the sleeve measures 8 inches around
the wrist — allowing three times the amount for plaiting ?
3. A skirt measuring 3J yd. around is to have two 5-inch
ruffles of organdie flouncing. Allowing twice the width of
skirt for lower ruffle, and once and three quarters for the
upper one, how much flouncing would you buy, and what
would be the cost at $ .87-J per yard for organdie ?
202 VOCATIONAL MATHEMATICS FOR GIRLS
4. How deep must a ruffle be cut to be 6" deep when
finished, if there is to be a 1-J" hem on the bottom and three
i" tucks above the hem ?
5. How deep a ruffle can be made from a strip of lawn 16"
deep, if a 2" hem is on the bottom and above it three \" tucks ?
6. How many yards of cloth 36" wide
are needed for 3i yd. of ruffling which is
to be cut 6" deep?
7. How many widths for ruffling can be
cut from 4 yd. of lawn 36" wide, if the
ruffle is 6" finished, and has a J" hem
and five J" tucks ?
NOTE. — Allowance must be made for joining a ruffle to a skirt, usu-
ally y>.
8. How deep must a ruffle be cut to be 5" deep when
finished, if there is to be a 11" hem on the bottom, and
five i" tucks above the hem ?
9. How many yards of ruffling are needed for a petticoat
21 yd. around the bottom ?
EXAMPLES IN FINDING COST OF MATERIALS
1. What is the cost of hamburg and insertion for one pair
of drawers ?
32 in. around each leg.
Hamburg at 16 cents a yard.
Insertion at 15 cents a yard.
2. What is the cost of hamburg and insertion for one pair
of drawers ?
36 in. around each leg.
Hamburg at 18^ cents a yard.
Insertion at 16| cents a yard.
3. What is the cost of hamburg and insertion for a petticoat ?
5 yd. around.
Hamburg at 25 cents a yard.
Insertion at 15 cents a yard.
ARITHMETIC FOR DRESSMAKERS 203
4. What is the cost of hamburg and insertion for a petti-
coat?
5 1 yd. around.
Hamburg at 27| cents a yard.
Insertion at 16| cents a yard.
5. What is the cost of trimming for a corset cover ?
38 in. around top.
13 in. around armhole.
Lace at 10 cents a yard.
6. What is the cost of trimming for a corset cover ?
41 in. around top.
13^ in. around armhole.
Lace at 12£ cents a yard.
7. What is the cost of lace for neck and sleeves at 121 cents
a yard ?
Neck, 13 in., sleeves, 8 in.
8. What is the cost of lace for neck and sleeves at 15 cents
a yard ?
Neck, 14 in., sleeves, 8J in.
9. What is the cost of a petticoat requiring 21 yd. long-
cloth at 121 cents a yard, and 2J yd. hamburg at 151 cents
a yard?
10. What is the cost of a petticoat requiring 2f yd. long-
cloth at 13 i cents a yard, and 21 yd. hamburg at 151 cents a
yard?
11. What is the cost of a nightdress requiring 31 yd. of
cambric at 25 cents a yard and 3 skeins of D. M. C. em-
broidery cotton which sells at 5 cents for 2 skeins, and 11 yd.
i-inch ribbon at 9 cents a yard ?
12. What is the cost of the following material for a corset
cover ?
1£ yd. longcloth at 15 cents a yard.
2£ yd. hamburg at 8 cents a yard.
6 buttons at 12 cents a dozen.
204 VOCATIONAL MATHEMATICS FOR GIRLS
13. What is the cost of the following material for a skirt ?
7 yd. silk at 79 cents a yard.
1-| yd. lining at 35 cents a yard.
14. What is the cost of the following material for a corset
cover ?
1$ yd. longcloth at 15 cents a yard.
2£ yd. hamburg at 8^ cents a yard.
4 buttons at 12| cents a dozen.
15. What is the cost of the following material for a corset
cover ?
14 yd. longcloth at 16| cents a yard.
2f yd. hamburg at 25| cents a yard.
2f yd. insertion at 191 cents a yard.
4 buttons at 15 cents a dozen.
16. What is the cost of the following material for a corset
cover ?
1| yd. longcloth at 14$ cents a yard.
If yd. hamburg at 17$ cents a yard.
17. What is the cost of the following material for a skirt ?
7$ yd. silk at 83$ cents a yard.
1$ yd. lining at 37$ cents a yard.
18. Find the cost of a corset cover that requires
1 yd. cambric at 12$ cents a yard,
f yd. bias binding at 2 cents a yard,
i doz. buttons at 12 cents a dozen.
If yd. lace at 10 cents a yard.
\ spool thread at 5 cents a spool.
19. Find the cost of an apron that requires
1 yd. lawn at 12$ cents a yard.
2^ yd. lace at 10 cents a yard.
\ spool thread at 5 cents a spool.
ARITHMETIC FOR DRESSMAKERS 205
20. Find the cost of a nightgown containing
3| yd. cambric at 12| cents a yard.
2 yd. lace at 5 cents a yard.
3 yd. ribbon at 3 cents a yard.
\ spool thread at 5 cents a spool.
21. Find the cost of drawers containing
2 yd. cambric at 12| cents a yard.
1^ yd. finishing braid at 5 cents a yard.
\ spool thread at 5 cents a spool.
2 buttons at 10 cents a dozen.
22. What is the cost of a waist made of the following ?
2f yd. shirting, 32 inches wide, at 23 cents a yard.
Sewing cotton, buttons, and pattern, 25 cents.
23. What is the cost of 1\ yd. chiffon faille, 36 inches wide,
at $ 1.49 a yard ?
24. How many yards of ruffling are needed for 1 dozen aprons
if each apron is one yard wide and half the width of the apron
is added for fullness ?
25. How many pieces of lawn-36 inches wide are needed for
the ruffle for one apron ? For eight aprons ?
26. A skirt measures 2| yards around the bottom. How
much material is needed for ruffling if the material is one yard
wide and ruffle is to be cut 7 inches wide ?
27. How deep would you cut a cambric ruffle that when
finished will measure 121", including the hamburg edge which
measures 4", two clusters of 5 tucks -J-" deep, and allowing V
for making ?
28. Find the cost of a poplin suit made of the following :
Silk poplin, 40 inches wide : 5| yards, at $ 1.79 a yard.
Satin facing for collar, revers, and cuffs, 21 inches wide : 1 yard, at
$1.25 a yard.
Coat lining, 36 inches wide : 2f yards, at $ 1.50 a yard.
Buttons, braid, sewing silk, two patterns, $ .64.
206 VOCATIONAL MATHEMATICS FOR GIRLS
Cloths of Different Widths
There are in common use cloths of several different widths
and at various prices. It is often important to know which is
the most economical cloth to buy. This may be calculated by
finding the cost per square yard, 36" by 36". To illustrate :
which is less expensive, broadcloth 56" wide, at $2.25 per
yard, or 50" wide, at $1.75 per yard ?
36 x ^ x 2.25 = $ 1.44& per square yard.
56 x 3^
36 x W x 1.75 = $ 1.26 per square yard.
EXAMPLES
Find the cost per square yard and the relative economy in
purchasing :
(a) Prunella, 46" wide, at $ 1.50 a yard.
Prunella, 44" wide, at $ 1.35 a yard.
(6) Serge, 54" wide, at $ 1.25 a yard.
Poplin, 42" wide, at $ 1.00 a yard.
(c) Serge, 42" wide, at 49 cents a yard.
Serge, 37" wide, at 39 cents a yard.
(<Z) Shepherd check, 54" wide, at $ 1.75 a yard.
Shepherd check, 52" wide, at $ 1.50 a yard.
Shepherd check, 42" wide, at $ 1.00 a yard.
(e) Taffeta, 19" wide, at 89 cents a yard.
Taffeta, 36" wide, at $ 1.25 a yard.
(/) Cashmere, 42" wide, at $ 1.00 a yard.
Nuns veiling 44" wide, at 75 cents a yard.
($r) Cheviot, 57" wide, at $1.50 a yard.
Diagonal, 54" wide, at $2.00 a yard.
(A) Messaline, 26 " wide, at 59 cents a yard.
Messaline, 36 " wide, at $ 1.25 a yard.
ARITHMETIC FOR DRESSMAKERS 207
PROBLEMS IN TRADE DISCOUNT
ILLUSTRATIVE EXAMPLE. — A dressmaker bought $ 125 worth
of material, receiving 6 % discount for cash. She sold the
material for 20 % more than the original price. What was the
gain?
SOLUTION. — $ 125.00 original price $125.00
.06 7.50
$ 7.50 discount $ 117.50 price paid for material.
5[$125 original cost $ 150.00 selling price
$25 20 % gain 117.50 price paid
$ 150 selling price $32. 50 gain. Am.
EXAMPLES
1. A dressmaker bought 25 yd. of hamburg at 50 cents per
yard, receiving 6 % discount for cash. She then sold the ham-
burg to her customers at 60 cents per yard. What was the
price paid for hamburg, and what per cent did she make ?
2. A dressmaker bought $ 325 worth of goods, receiving 6 °/0
discount for cash. She sold the goods for 25 % more than the
original price. What was the gain ?
3. A milliner bought $200 worth of ribbons, velvets, and
flowers, receiving 5 % discount for cash. She then sold the
materials for 30 % more than the original price. What was
the gain ?
REVIEW EXAMPLES
1. A dressmaker bought 30 yd. of silk at $ 1.25 per yard.
She received a discount of 10 %. She sold the silk for $ 1.39
per yard. How much did she gain on the 30 yards ?
2. A merchant bought 50 yd. of lawn at 12^ cents a yard,
and received a discount of 6 % for cash. How much did the
lawn cost ?
3. A piece of crinoline containing 45 yd. was bought for
$ 18. It was made into dress models of 5 yd. each. What was
the cost of the crinoline in each model ?
208 VOCATIONAL MATHEMATICS FOR GIRLS
4. A dressmaker bought $ 175 worth of silk, receiving 6 %
discount for cash. She sold the silk for 25 % more than the
original price. What was the gain per cent ?
5. A dressmaker bought 24J yd. of silk, at $ 1.10 per yard.
From it she made three dresses, and had 13f yd. left. How
much did the silk for one dress cost ?
6. Find the cost of ,36 yd. of Valenciennes lace at 7^ cents
a yard, 12 yd. of insertion at 6|- cents a yard, and 12 yd. of
beading at 7 cents a yard. What is the net cost, when 2 %
discount is given ?
7. How many lingerie shirtwaists, each containing 2| yd.,
can be made from 49 yd, of batiste ? What is the cost of
material for one waist, if the whole piece cost $ 9.80, less 5 %
discount ?
8. A dressmaker bought 2i yd. of crepe at 29 cents a yard,
for a shirtwaist, 3 yd. of beading at 121 cents a yard, 6 crochet
buttons at 35 cents a dozen. What did the material for the
waist cost ?
9. A woman bought 91 yd. of foulard silk, at $ 1.10 a yard,
for a dress, If yd, of net at $ 1.50 a yard, and J yd. of plain
silk at $ 1.25 a yard. What was the cost of material ?
10. A dressmaker bought 50 yd, of taffeta silk for $ 45.00.
She sold 81 yd. to one customer for $ 1.25 a yard, 151 yd. at
$ 1.00 a yard to another customer, and the remainder at cost.
What did she gain on the entire piece ? What was the gain
per cent ?
11. Two and one-half yards of cloth cost $ 2.75. What was
the price per yard ?
12. A dressmaker bought 50 yd. of handmade lace abroad
and paid $ 75 for it. She paid 60 % duty on the lace and sold
it at a gain of 33^ % • What was the selling price per yard ?
ARITHMETIC FOR DRESSMAKERS 209
13. A dressmaker bought 20 yd. of foulard silk at 90 cents
a yard. She received 6 % discount. She sold it for 10£ %
more than the original price. How much did she gain on the
sale ? What per cent did she gain ?
14. A dressmaker bought the following materials for a
customer : 4^ yd. of broadcloth at $ 2.75 a yard, 6-J- yd. of silk
at $3.75 a yard, 2-^ yd. of trimming at $2.50 a yard. She
received a dressmaker's discount of 6 % , and 5 % discount for
cash payment. What did she pay for the materials? She
charged the retail price for them. How much did she gain ?
What per cent ?
15. A dressmaker bought a 7^-yd. remnant of broadcloth
for $ 22.50. She sold 6 yd. to a customer at $ 3.50 a yard, but
the remainder could not be sold. Did she gain or lose ? What
per cent ?
16. A dressmaker bought in France three 15-yd. pieces of
dress silk at 25^ cents a yard. After paying 60 % duty
on them, she sold two pieces to one customer at 48 % gain, and
the third piece to another customer at 35 % gain. What was
the gain on the three pieces ?
• 17. A dressmaker furnished the materials for a lingerie
dress and charged $25 for it. For the materials she paid
the following : 10 yd. of dimity at 45 cents a yard, 12i yd.
Cluny insertion at 25 cents a yard, findings, $2. If she
charged $12 for making, how much did she gain on the
material ? Make a bill for the same and receipt it.
18. The materials for a dress cost a dressmaker $ 14.50.
She sold them for 10% more than cost and charged $15 for
making. She paid her helper 20 % of the amount received.
What was the gain f>er cent ?
19. If it takes 6^ yards of cloth 52 inches wide to make
a dress, how many yards of cloth 22 inches wide will be
needed to make the same dress ?
210 VOCATIONAL MATHEMATICS FOR GIRLS
20. A dressmaker agreed to make a dress for a customer for
$ 25. She paid 2 assistants $ 1.25 a day each for 31 days
of work. The dress was returned for alterations, and the
assistants were paid for one more day's work. How much did
the dressmaker receive for her own work ?
21. A dressmaker bought $1.50 worth of silk, receiving
6 % discount for cash. She sold the silk for 40 % more than
the original price. What was the gain per cent ?
22. A dressmaker has an order for three summer dresses,
for which 31 J yd. of batiste are needed. She can buy three
remnants of 101 yd. each for 25 cents a yard, or she can buy a
piece of 35 yd. for 25 cents a yard and receive 4 % discount
for cash. Which is the better plan ?
23. (a) How many inches in f yd. ? (b) How many inches
in I yd. ? (c) How many inches in f yd. ? (d) How many
inches in J yd. ? (e) How many inches in -J yd. ? (/) How
many inches in J yd. ? (g) How many inches in |- yd. ?
24. Find the cost of each of the above lengths in lace at
$ .121 a yard.
25. Find the cost of 4^ yd. of lace at $1.95 per piece (one
piece = 12 yd.).
26. A dressmaker bought 2 pieces of white lining taffeta,
one piece 42 yd. and another 48 J yd., at $ .421 a yard. What
was the total cost ?
27. A piece of crinoline containing 421 yd. that cost $ 1.70
a yard was made into dress models of 81 yd. each. What
was the cost of the crinoline in each model ?
28. What is the cost of a child's petticoat containing :
2J yd. longcloth at 15 cents a yard,
If yd. hamburg at 19 cents a yard,
1£ yd. insertion at 15 cents a yard ?
ARITHMETIC FOR DRESSMAKERS 211
29. What is the cost of two petticoats requiring for one :
2£ yd. longcloth at 19 cents a yard,
3 yd. hamburg at 25 cents a yard,
2£ yd. insertion at 19 cents a yard ?
30. What is the cost of a petticoat requiring :
3 yd. longcloth at 12£ cents a yard,
3£ yd. hamburg at 17 cents a yard ?
31. What is the total cost of the following ?
Wedding gloves, $ 2.75.
Slippers and stockings, $5.00.
Six undervests, at 19 cents each.
Six pairs of stockings, at 33£ cents a pair.
Two pairs of shoes, at $ 5.00 a pair.
One pair of rubbers, 75 cents.
One pair long silk gloves, §2.00.
One pair of long lisle gloves, $ 1.00.
Two pairs of short silk gloves, $ 1.00.
Veils and handkerchiefs, $5.00.
Two hats, $ 10.00.
Corsets, $3.00.
Wedding veil of 3 yards of tulle, 2 yards wide, at 89 cents a yard.
32. What is the cost of the following material for a top
coat?
Cotton corduroy, 32 inches wide : 4| yards at 75 cents a yard.
Lining, 36 inches wide : 4] yards at $ 1.50 a yard.
Buttons, sewing silk, pattern, 27 cents.
Velvet for collar facing, J yard, at $1.50 a yard.
33. What is the cost of the following dressmaking supplies ?
| yard of China silk, 27 inches wide, at 49 cents a yard (for the lining).
If yard of mousseline de soie interlining 40 inches wide, at 80 cents
a yard.
f yard of all-over lace 36 inches wide, at $1.48 for front and lower
back.
\ yard of organdie at $1.00, 32 or more inches wide, for collar and vest.
Sewing silk, hooks and eyes, pattern, at 32 cents.
212 VOCATIONAL MATHEMATICS FOR GIRLS
34. What is the cost of the following ?
Cotton gabardine, 36 inches wide : 5f yards at 39 cents a yard.
Sewing cotton, braid, buttons, pattern, at 35 cents.
35. Which of the following fabrics is the most economical
to buy ?
Crepe meteor, 44" wide, at $3.25 a yard.
Faille Franchise, 42" wide, at $3.00.
Charmeuse, 40" wide, at $2.25.
Louisine, 38" wide, at $2.00.
Armure, 20" wide, at $1.50.
Satin duchesse, 21" wide, at $1.25.
MILLINERY PROBLEMS
1. What would a hat cost with the following trimmings ?
1| yd. velvet, at $2.50 a yard.
£ yd. satin for facing, at $ 1.98 a yard.
2 feathers, at $ 5.50 each.
Frame and work, at $2.50.
Make out a bill. (See lesson on Invoice, Chapter XI, page
243.)
2. A leghorn hat cost $6.98. Four bunches of fadeless
roses at $2.98, 2 bunches of foliage at $.98, and 11 yd. of
velvet ribbon at $ 1.49 were used for trimming. The milliner
charged 75 cents for her work. How much did the hat cost ?•
3. A milliner used the following trimmings on a child's
bonnet :
1 piece straw braid, at $1.49.
2 yd. maline, at 25 cents a yard.
4 bunches flowers, at 69 cents each.
4 bunches foliage, at 49 cents each.
Work, at $2.00.
What was the total cost of the hat ? Make out a bill and
receipt it.
ARITHMETIC FOR MILLINERS 213
4. An old lady's bonnet was trimmed with tlie following :
3 yd. silk, at $ 1.50 a yard.
1 piece of jet, 83.00.
2 small aigrettes, at $ 1.50 each.
Ties, 75 cents.
Work, $ 1.50.
How much did the finished bonnet cost ?
5. What was the total cost of a hat with the following trim-
mings ?
2 pieces straw braid, at §2.50 each.
2 yd. velvet ribbon, at 98 cents a yard.
5 flowers, at 59 cents.
4 foliage, at 49 cents.
Frame and work, at $2.50.
6. A milliner charged $ 2.00 for renovating an old hat.
She used 2 yd. satin at $ 1.50 a yard and charged $ 2.25 for
an ornament. . How much did the hat cost ?
7. The following trimmings were used on a child's hat :
3 yd. velvet, at 8 1.50 a yard.
8 yd. lace, at 15 cents a yard.
2 bunches buds, at 49 cents a bunch.
Work, §2.00.
How much did the hat cost ?
8. A milliner charged $ 6.00 for renovating three feathers,
$ 2.50 for a fancy band, $ 4.75 for a hat, and 75 cents for
work. How much did the customer pay for her hat ?
9. A lady bought a hat with the following trimmings :
2 yd. satin, at $ 1.75 a yard.
2 bunches grapes, at $ 1.59 a bunch.
2| yd. ribbon, at 69 cents a yard.
Work, 75 cents.
How much did the hat cost ?
214 VOCATIONAL MATHEMATICS FOR GIRLS
10. What would a hat cost with the following trimmings ?
2 pieces straw braid, at $ 1.98 each.
3 yd. ribbon, at 89 cents a yard.
Fancy feather, $6.98.
Frame and work, $ 2. 50.
11. Estimate the cost of a hat using the following materials :
2£ yd. plush, at $2.25 a yard.
2 yd. ribbon, at 25 cents a yard,
f yd. buckram, at 25 cents a yard.
^ yd. tarlatan, at 10 cents a yard.
1 band fur, 75 cents.
Foliage, 10 cents.
Labor, $2.00.
12. If the true bias from selvedge to selvedge is about ^
longer than the width of the goods, how many bias strips must
be cut from velvet 18" wide in order to have a three-yard bias
strip ?
13. The edge of a hat measures
45 inches in circumference ; the
velvet is 16 inches wide. How
many bias strips of velvet would
it take to fit the brim ?
WIRE HAT FRAME ^ what amount of velyet wou]d
be needed to cover brim if each strip cut measured f of a yard
along the selvedge ?
15. Give the number of 13^-in. strips that can be cut from
3|- yards of material ; also the number of inches of waste.
16. How many 22^-in. strips can be cut from 2J yd. of
material ?
17. What length bias strip can be made from 11 yd. of silk,
each strip 1 yd. 10 in. long and 1^ in. wide?
18. How many six-petal roses can be made from 1 yard of
velvet 18 inches wide, each petal cut 3 inches square ?
ARITHMETIC FOR MILLINERS 215
19. Estimate the total cost of roses, if velvet is $ 1.50 a
yard, centers 18 cents a dozen, sprays 12 cents a dozen, stem-
ming 6 cents a yard, using 1 of a yard for each flower.
20. Find the cost of one flower ; the cost of ^ of a dozen
flowers, using the figures given above.
21. What amount of velvet will be needed to fit a plain-top
facing and crown of hat, width of brim 5 inches, diameter of
headsize 7£ inches, diameter of crown 151 inches, allowing 81
inches on brim for turning over edges ?
22. If the circumference of the brim measures 56 inches,
what amount of silk will it take for a shirred facing made of
silk 22 inches wide, allowing twice around the hat for fullness,
and also allowing 1 inch oil depth of silk for casings ?
23. At the wholesale rate of eight frames for one dollar,
what is the cost of five dozen frames ? of twelve dozen ?
24. A milliner had 2J dozen buckram frames at $ 3.60 a
dozen. She sold |- of them at 75 cents each, but the others
were not sold. Did she gain or lose and what per cent ?
25. Flowers that were bought at $ 5.50 a dozen bunches
were sold at 75 cents a bunch. What was the gain on 11
dozen bunches ?
26. A milliner bought ten rolls of ribbon, ten yards to the
roll, for $ 8.50. Ten per cent of the ribbon was not salable.
The remainder was sold at 19 cents a yard. How much was
the gain ? what per cent ?
27. A piece of velvet containing twelve yards was bought
for $28.20 and sold for $2.75 per yard. How much was
gained on the piece ?
28. A thirty-six yard piece of maline cost $ 7.02 and was
sold at 29 cents a yard. One yard was lost in cutting. How
much was gained on the piece ?
216 VOCATIONAL MATHEMATICS FOR GIRLS
29. Find cost of a velvet hat requiring
1| yd. of velvet, at $ 1.50 a yard.
| yd. of fur band, at 34.00 a yard.
1 feather ornament, at $ 3.00.
Hat frame, 50 cents.
Edge wire, 10 cents.
Taffeta lining, 25 cents.
Making, $ 2.50.
30. A milliner charged $ 8.37 for a hat. She paid 37 cents
for the frame, $ 2.80 for the trimming, and $ 1.50 for labor.
What was the per cent profit ?
31. A child's hat of organdie has two ruffles edged with
Valenciennes lace. The lower ruffle is 3" wide ; the upper ruffle,
2i-". 2J yd. lace edging cost 12^ cents a yard,
2 yd. of 3" ribbon cost 25 cents a yard, 11 yd.
of organdie cost 25 cents a yard, the hat frame
cost 35 cents, and the lining cost 10 cents.
Find the total cost.
32. How much velvet at $ 2.00 a yard would you buy to
put a snap binding on a hat that measures 43" around the
edge ? Should the velvet be bias or straight ?
CHAPTER X
CLOTHING
SINCE about one aighth of the income in the average working-
man's family is spent for clothing, this is a very important
subject. The housewife purchases the linen for the house and
her own wearing apparel. It is not uncommon for her to have
considerable to say about the clothing of the men, particularly
about the underclothing. Therefore she should know some-
thing about what constitutes a good piece of cloth, and be able
to make an intelligent selection of the best and most economical
fabric for a particular purpose. The cheapest is not always
the best, although it is in some cases.
All kinds of cloth are made by the interlacing (weaving) of
the sets of thread (called yarn). The thread running length-
wise is the strongest and is called the warp. The other thread
is called the filling. Such fabrics as knitted materials and lace
are made by the interlacing of a single thread. Threads
(yarn) are made by lengthening and twisting (called spinning)
short fibers. Since the fibers vary in such qualities as firmness,
length, curl, and softness, the resulting cloth varies in the
same way. This is the reason why we have high-grade, medium-
grade, and low-grade fabrics.
The principal fabrics are wool, silk, mohair, cotton, and flax
(linen).
The consumer is often tempted to buy the cheaper fabrics
and wonders why there is such a difference in price. This
difference is due in part to the cost of raw material and in part
to the care in manufacturing. For example, raw silk costs
from $ 1.35 to $ 5.00 a pound, according to the nature and
217
218 VOCATIONAL MATHEMATICS FOR GIRLS
quality of the silk. The cost of preparing the raw silk aver-
ages about 55 cents a pound, according to the nature of the
twist, which is regulated by the kind of cloth into which it is
to enter. The cost of dyeing varies from 55 cents to $ 1.50 a
pound. Weavers are paid from 2 to 60 cents a yard for weav-
ing, the price varying according to the desirability of the cloth.
When we compare the relative values of similar goods
produced by different manufacturers, there are a few general
principles by which good construction can easily be determined.
The density of a fabric is determined by the number of warp
yarn and filling yarn to the inch. This is usually determined
by means of a magnifying glass with a \" opening. To illus-
trate : If there are 36 threads in the filling and 42 threads in
the warp to J", what is the density of the cloth to the inch ?
SOLUTION. —
36 x 4 = 144 threads in the filling.
42 x 4 = 168 threads in the warp.
EXAMPLES
1. A 25-cent summer undervest (knitted fabric) will outwear
two of the flimsy 15-cent variety in addition to retaining better
shape. What is the gain, in wear, over the 15-cent variety ?
2. A 50-cent undervest will outwear three of the 25-cent
variety. What is gained by purchasing the 50-cent style ?
3. A cotton dress for young girls, costing 75 cents ready
made, will last one season. A similar dress of better material
costs 94 cents, but will last two seasons. Why is the latter
the better dress to buy ? What is gained ?
4. A linen tablecloth (not full bleached) costing $1.04 a
yard, will last twice as long as a bleached linen at $ 1.25 a
yard. Which is the better investment ?
5. A sheer stocking at 50 cents will wear just half as long
as a thicker stocking at 35 cents. What is gained in wear ?
What kind of stockings should be selected for wear ?
CLOTHING 219
SHOES
Our grandfathers and grandmothers wore handmade shoes,
and wore them until they had passed their period of usefulness.
At that time the consumption of leather did not equal its pro-
duction. But, since the appearance of machine-made shoes,
different styles are placed on the market at different seasons
to correspond to the change in the style of clothing, and are
often discarded before they are worn out. Thus far we have
not been able to utilize cast-off leather as the shoddy mill uses
cast-off wool and silk. The result is that the demand for
leather is above the production ; therefore, as in the case of
textiles, substitutes must be used. In shoe materials there is
at present an astonishing diversity and variety of leather and
its substitutes. Every known leather from kid to cowhide is
used, and such textile fabrics as satins, velvets, and serges
have rapidly grown in favor, especially in the making of
women's and children's shoes. Of course, we must bear in
mind that for wearing qualities there is nothing equal to
leather. In buying a pair of shoes we should try to combine
both wearing qualities and simple style as far as possible.
EXAMPLE
1. A pair of shoes at $ 1.75 was purchased for a boy. The
shoes required 80 cents worth of mending in two months. If
a $3.00 pair were purchased, they would last three times as
long with 95 cents worth of mending. How much is gained
by purchasing a $ 3.00 pair of shoes ?
YARNS
Worsted Yarns. — All kinds of yarns used in the manufacture
of cloth are divided into sizes which are based on the relation
between weight and length. To illustrate : Worsted yarns are
made from combed wools, and the size, technically called the
220 VOCATIONAL MATHEMATICS FOR GIRLS
counts, is based upon the number of lengths (called hanks) of
560 yards required to weigh one pound.
ROVING OR YARN SCALES
These scales will weigh one pound by tenths of grains or one seventy-thou-
sandth part of one pound avoirdupois, rendering them well adapted for use
in connection with yarn reels, for the numbering of yarn from the weight
of hank, giving the weight in tenths of grains to compare with tables.
Thus, if one hank weighs one pound, the yarn will be number one
counts, while if 20 hanks are required for one pound, the yarn is the 20's,
etc. The greater the number of hanks necessary to weigh one pound, the
higher the counts and the finer the yarn. The hank, or 560 yards, is the
unit of measurement for all worsted yarns.
LENGTH FOR WORSTED YARNS
No.
YARDS
PER LH.
No.
YARDS
PER Lu.
No.
YARDS
PER LB.
No.
YARDS
PER LB.
1
560
5
2800
9
5040
13
7280
2
1120
6
3360
10
5600
14
7840
3
1680
7
3920
11
6160
15
8400
4
2240
8
4480
12
6720
16
8960
Woolen Yarns. — In worsted yarns the fibers lie parallel to
each other, while in woolen yarns the fibers are entangled.
CLOTHING
221
This difference is due entirely to the different methods used
in working up the raw stock.
In woolen yarns there is a great diversity of systems of grading, vary-
ing according to the districts in which the grading is done. Among the
many systems are the English skein, which differs in various parts of Eng-
land ; the Scotch spyndle ; the American run ; the Philadelphia cut ; and
others. In these lessons the run system will be used unless otherwise
stated. This is the system used in New England. The run is based upon
100 yards per ounce, or 1600 yards to the pound. Thus, if 100 yards of
woolen yarn weigh one ounce, or if 1600 yards weigh one pound, it is
technically termed a No. 1 run ; and if 300 yards weigh one ounce, or 4800
yards weigh one pound, the size will be No. 3 run. The finer the yarn,
or the greater the number of yards necessary to weigh one pound, the
higher the run.
YARN REEL
For reeling and measuring lengths of cotton, woolen, and worsted yarns.
LENGTH FOR WOOLEN YARNS (RUN SYSTEM)
No.
YARDS
PER 1,1$.
No.
YARDS
PER LB.
No.
YARDS
PER LB.
No.
YARDS
PER LB.
i
200
1
1600
2
3200
3
4800
*
400
u
2000
2*
3600
8*
5200
\
800
11
2400
2^
4000
3|
5600
1
1200
if
2800
2f
4400
222 VOCATIONAL MATHEMATICS FOR GIRLS
Raw Silk Yarns. — For raw silk yarns the table of weights
is:
16 drams = 1 ounce
16 ounces = 1 pound
256 drams = 1 pound
The unit of measure for raw silk is 256,000 yards per pound. Thus, if
1000 yards — one skein — of raw silk weigh one dram, or if 256,000 yards
weigh one pound, it is known as 1-dram silk, and if 1000 yards weigh
two drams, the yarn is called 2-dram silk ; hence the following table is
made:
1-dram silk = 1000 yards per dram, or 256,000 yards per Ib.
2-dram silk = 1000 yards per 2 drams, or 128,000 yards per Ib.
4-dram silk = 1000 yards per 4 drams, or 64,000 yards per Ib.
DRAMS PER 1000 YARDS
YARDS PER POUND
YARDS PER OUNCE
1
256,000
16,000
1*
204,800
12,800
1*
p
?
If
146,286
9143
2
128,000
8000
2i
113,777
7111
*i
102,400
6400
2f
93,091
5818
3
?
?
8*
78,769
4923
8*
73,143
4571
Linen Yarns. — The sizes of linen yarns are based on the lea
or cuts per pound and the pounds per spindle. A cut is 300
yards and a spindle 14,000 yards. A continuous thread of
several cuts is a hank, as a 10-cut hank, which is 10 X 300 =
3000 yards per hank. The number of cuts per pound, or the
leas, is the number of the yarn, as 30's, indicating 30 x 300 =
9000 yards per pound. Eight-pound yarn means that a spindle
weighs 8 pounds or that the yarn is 6-lea (14,400 -5- 8) -s- 300 = 6.
CLOTHING 223
Cotton Yarns. — The sizes of cotton yarns are based upon the
system of 840 yards to 1 hank. That is, 840 yards of cotton
yarn weighing 1 pound is called No. 1 counts.
Spun Silk. — Spun silk yarns are graded on the same basis
as that used for cotton (840 yards per pound), and the same
rules and calculations that apply to cotton apply also to spun
silk yarns.
Two or More Ply Yarns. — Yarns are frequently produced in
two or more ply ; that is, two or more individual threads are
twisted together, making a double twist yarn. In this case
the size is given as follows :
2/30's means 2 threads of 30's counts twisted together, and 3/30's
would mean 3 threads, each a 30's counts, twisted together.
(The figure before the line denotes the number of threads twisted to-
gether, and the figure following the line the size of each thread.)
Thus when two threads are twisted together, the resultant
yarn is heavier, and a smaller number of yards are required to
weigh one pound.
For example : 30's 'worsted yarn equals 16,800 yd. per lb., but a two-
ply thread of 30's, expressed 2/30?s, requires only 8400 yards to the pound,
or is equal to a 15's ; and a three-ply thread of 30's would be equal to a
10's.
When a yarn is a two-ply, or more than a two-ply, and made
up of several threads of equal counts, divide the number of the
single yarn in the required counts by the number of the ply,
and the result will be the equivalent in a single thread.
To Find the Weight in Grains of a Given Number of Yards
of Worsted Yarn of a Known Count
EXAMPLE. — Find the weight in grains of 125 yards of 20's
worsted yarns.
No. 1's worsted yam = 560 yards to a lb.
No. 20' s worsted yarn = 11,200 yards to a lb.
1 lb. worsted yarn = 7000 grains.
224 VOCATIONAL MATHEMATICS FOR GIRLS
If 11,200 yards of 20's worsted yarn weigh 7000 grains, then — -
1 1 j^UU
of 7000 = — 5— x 7000 = — = 78.125 grains.
11,200 8
NOTE. — Another method: Multiply the given number of yards by
7000, and divide the result by the number of yards per pound of the
given count.
125 x 7000 = 875,000.
1 pound 20 's= 11,200.
875,000 -T- 11,200 = 78. 125 grains. Ans.
To Find the Weight in Grains of a Given Number of Yards
of Cotton Yarn of a Known Count
EXAMPLE. — Find the weight in grains of 80 yards of 20's
cotton yarn.
No. 1's cotton = 840 yards to a Ib.
No. 20's cotton = 16,800 yards to a Ib.
1 Ib. = 7000 grains.
lyd. 20's cotton =J grains.
80 yd. 20's cotton = x 80 = — = 33.33 grains. Ans.
16,800 21
It is customary to solve examples that occur in daily practice
by rule.
The rule for the preceding example is as follows :
Multiply the given number of yards by 7000 and divide the
result by the number of yards per pound of the given count.
80 x 7000 = 560,000.
560,000 -*- (20 x 840) = 33.33 grains. Ans.
NOTE. — 7000 is always a multiplier and 840 a divisor.
To find the weight in ounces of a given number of yards of
cotton yarn of a known count, multiply the given number of
yards by 16, and divide the result by the yards per pound of
the known count.
To find the weight in pounds of a given number of yards
of cotton yarn of a known count, divide the given number of
yards by the yards per pound of the known count.
CLOTHING 225
To find the weight in ounces of a given number of yards of
woolen yarn (run system), divide the given number of yards
by the number of runs, and multiply the quotient by 100.
NOTE. — Calculations on the run basis are much simplified, owing to
the fact that the standard number (1600) is exactly 100 times the number
of ounces contained in 1 pound.
EXAMPLE. — Find the weight in ounces of 6400 yards of
5-run woolen yarn.
6400- (5 x 100)= 12.8 oz. Ans.
To find the weight in pounds of a given number of yards of
woolen yarn (run system) the above calculation may be used,
and the result divided by 16 will give the weight in pounds.
To find the weight in grains of a given number of yards of
woolen yarn (run system), multiply the given number of yards
by 7000 (the number of grains in a pound) and divide the
result by the number of yards per pound in the given run,
and the quotient will be the weight in grains.
EXAMPLES
1. How many ounces are there (a) in 6324 grains ? (6) in
341 pounds ?
2. How many pounds are there in 9332 grains ?
3. How many grains are there (a) in 168J pounds ? (6) in
2112 ounces ?
4. Give the lengths per pound of the following worsted
yarns : (a) 41's ; (6) 55's ; (c) 105's ; (d) 115's ; (e) 93's.
5. Give the lengths per pound of the following woolen
yarns (run system): (a) 9J's ; (6) 6's ; (c) 19's ; (d) 17's ;
(e) li's.
6. Give the lengths per pound of the following raw silk
yarns : (a) li's ; (6) 3's ; (c) 3J's ; (d) 20's ; (e) 28's.
7. Give the lengths per ounce of the following raw silk
yarns : (a) 4J's ; (b) 6|'s ; (c) 8's ; (d) 9's ; (e) 14's.
226 VOCATIONAL MATHEMATICS FOR GIRLS
8. What are the lengths of linen yarns per pound : (a) 8's ;
(b) 25's ; (c) 32's ; (cf) 28's ; (e) 45's ?
9. What are the lengths per pound of the following cotton
yarns : (a) 10's ; (b) 32's ; (c) 54's; (d) 80's; (e) 160;s?
10. What are the lengths per pound of the following spun
silk yarns : (a) 30's ; (b) 45's ; (c) 38's ; (d) 29's ; (e) 42's ?
11. Make a table of lengths per ounce of spun silk yarns
from 1's to 20's.
12. Find the weight in grains of 144 inches of 2/20's worsted
yarn.
13. Find the weight in grains of 25 yards of 3/30's worsted
yarn.
14. Find the weight in ounces of 24,000 yards of 2/40's
cotton yarn.
15. Find the weight in pounds of 2,840,000 yards of 2/60's
cotton yarn.
16. Find the weight in ounces of 650 yards of li-run woolen
yarn.
17. Find the weight in grains of 80 yards of ^-run woolen
yarn.
18. Find the weight in pounds of 64,000 yards of 5-run
woolen yarn.
Solve the following examples, first by analysis and then by
rule :
19. Find the weight in grains of 165 yards of 35's worsted.
20. Find the weight in grains of 212 yards of 40's worsted.
21. Find the weight in grains of 118 yards of 25's cotton.
22. Find the weight in grains of 920 yards of 18's cotton.
23. Find the weight in pounds of 616 yards of 16^'s woolen.
24. Find the weight in grains of 318 yards of 184's cotton.
25. Find the weight in grains of 25 yards of 30's linen.
CLOTHING 227
26. Find the weight in pounds of 601 yards of 60's spun
silk.
27. Find the weight in grains of 119 yards of 118's cotton.
28. Find the weight in grains of 38 yards of 64's cotton.
29. Find the weight in grains of 69 yards of 39's worsted.
30. Find the weight in grains of 74 yards of 40's worsted.
31. Find the weight in grains of 113 yards of 1^'s woolen.
32. Find the weight in grains of 147 yards of l|^s woolen.
33. Find the weight in grains of 293 yards of 8's woolen.
34. Find the weight in grains of 184 yards of 16^ 's worsted.
35. Find the weight in grains of 91 yards of 44's worsted.
36. Find the weight in grains of 194 yards of 68's cotton.
37. Find the weight in pounds of 394 yards of 180's cotton.
38. Find the weight in pounds of 612 yards of 60's cotton.
39. Find the weight in grains of 118 yards of 44's linen.
40. Find the weight in pounds of 315 yards of 32's linen.
41. Find the weight in grains of 84 yards of 25's worsted.
42. Find the weight in grains of 112 yards of 20's woolen.
43. Find the weight in grains of 197 yards of 16's woolen.
44. Find the weight in grains of 183 yards of 18's cotton.
45. Find the weight in grains of 134 yards of 28's worsted.
46. Find the weight in grains of 225 yards of 34's linen.
47. Find the weight in pounds of 369 yards of 16's spun silk.
48. Find the weight in pounds of 484 yards of 18's spun silk
To Find the Size or the Counts of Cotton Yam of Known
Weight and Length
EXAMPLE. — Find the size or counts of 84 yards of cotton
yarn weighing 40 grains.
228 VOCATIONAL MATHEMATICS FOR GIRLS
Since the counts are the number of hanks to the pound,
™0°. x 84 = 14,700 yd. in 1 Ib.
40
14,700 -4- 840 = 17.5 counts. Ans.
RULE. — Divide 840 by the given number of yards ; divide
7000 by the quotient obtained ; then divide this result by the
weight in grains of the given number of yards, and the
quotient will be the counts.
840 -=- 84 = 10.
7000 -f- 10 = 700.
700 -f- 40 = 17.5 counts. Ans.
To Find the Run of a Woolen Thread of Known Length
and Weight
EXAMPLE. — If 50 yards of woolen yarn weigh 77.77 grains,
what is the run ?
1600 + 50 = 32.
7000-32 = 218.75.
218.75 -f- 77.77 = 2.812-run yarn. Ans.
RULE. — Divide 1600 (the number of yards per pound of 1-
run woolen yarn) by the given number of yards ; then divide
7000 (the grains per pound) by the quotient ; divide this
quotient by the given weight in grains and the result will be
the run.
To Find the Weight in Ounces for a Given Number of Yards of
Worsted Yarn of a Known Count
EXAMPLE. — What is the weight in ounces of 12,650 yards of
30's worsted yarn ?
12,650 x 16 = 202,400.
202,400 - 16,800 = 12.047 oz. Ans.
RULE. — Multiply the given number of yards by 16, and
divide the result by the yards per pound of the given count,
and the quotient will be the weight in ounces.
CLOTHING 229
To Find the Weight in Pounds for a Given Number of Yards
of Worsted Yarn of a Known Count
EXAMPLE. — Find the weight in pounds of 1,500,800 yards
of 40's worsted yarn.
1,500,800 -4- 22,400 = 67 Ib. Am.
RULE. — Divide the given number of yards by the number
of yards per pound of the known count, and the quotient will
be the desired weight.
EXAMPLES
1. If 108 inches of cotton yarn weigh 1.5 grains, find the
counts.
2. Find the size of a woolen thread 72 inches long which
weighs 2.5 grains.
3. Find the weight in ounces of 12,650 yards of 2/30's
worsted yarn.
4. Find the weight in ounces of 12,650 yards of 40's worsted
yarn.
5. Find the weight in pounds of 1,500,800 yards of 40's
worsted yarn.
6. Find the weight in pounds of 789,600 yards of 2/30's
worsted yarn.
7. What is the weight in pounds of 851,200 yards of 3/60's
worsted yarn ?
8. If 33,600 yards of cotton yarn weigh 5 pounds, find the
counts of cotton.
Buying Yarn, Cotton, Wool, and Rags
Every fabric is made of yarn of definite quality and quan-
tity. Therefore, it is necessary for every mill man to buy
yarn or fiber of different kinds and grades. Many small mills
buy cotton, wool, yarn, and rags from brokers who deal in
these commodities. The prices rise and fall from day to day
230 VOCATIONAL MATHEMATICS FOR GIRLS
according to the law of demand and supply. Price lists
giving the quotations are sent out weekly and sometimes
daily by agents as the prices of materials rise or fall. The
following are quotations of different grades of cotton, wool,
and shoddy, quoted from a market list :
QUANTITY PRICE PER LB.
8103 lb. white yarn shoddy (best all wool) $0.485
3164 Ib. white knit stock (best all wool) 365
2896 Ib. pure indigo blue 315
1110 Ib. fine dark merino wool shoddy 225
410 Ib. fine light merino woolen rags 115
718 Ib. cloakings (cotton warp) , . . . . .02
872 Ib. wool bat rags 035
96 Ib. 2/20's worsted (Bradford) yarn 725
408 Ib. 2/40's Australian yam 1.35
593 Ib. 1/50's delaine yarn 1.20
987 Ib. 16-cut merino yarn (50 % wool and 50 % shoddy) . . .285
697 Ib. carpet yarn, 60 yd. double reel, wool filling 235
Find the total cost of the above quantities and grades of
textiles.
EXAMPLES
1. The weight of the fleece on the average sheep is 8 Ib.
Wyoming has at least 4,600,000 sheep ; what is the weight of
wool raised in a year in this state ?
2. A colored man picks 155 Ib. of cotton a day ; how much
cotton will he pick in a week (6 days) ?
3. The average yield is 558 Ib. per acre ; how much cotton
will be raised on a farm of 165 acres ?
4. The standard size of a cotton bale in the United States
is 54 x 27 x 27 inches ; what is the cubical contents of a bale ?
5. In purchasing cotton an allowance of 4 % is made for
tare. How much cotton would be paid for in 96 bales, 500 Ib.
to each bale ?
CLOTHING 231
6. Broadcloth was first woven in 1641. How many years
has it been in use ?
7. The length of "Upland" cotton varies from three-
fourths to one and one-sixteenth inches. What is the differ-
ence in length from smallest to largest ?
8. If a sample of 110 Ib. of cotton entered a mill and 68 Ib.
were made into fine yarn, what is the per cent of waste ?
9. If a yard of buckram weighs 1.8 ounces, how many
yards to the pound ?
10. If a calico printing machine turns out 95 fifty-yard
pieces a day, how many are printed per hour in a ten-hour day ?
11. If a sample of linen weighing one pound and a half
absorbs 12 % moisture, what is the weight after absorption ?
12. A piece of silk weighing 3 Ib. 4 oz. is " weighted " 175% ;
what is the total weight ?
13. If the textile industry in a certain year pays out
$ 500,000,000 to 994,875 people, what is the wage per capita ?
14. How much dyestuff, etc., will be required to dye 5 Ib. of
cotton by the following receipt ?
6 °/o brown color, afterwards treated with
1.5 % sulphate of copper,
1.5 % bichromate of potash,
3 °/o acetic acid.
15. How many square yards of cloth weighing 8 oz. per sq.
yd. may be woven from 1050 Ib. of yarn, the loss in waste be-
ing 5 per cent ?
16. A piece of union cloth has a warp of 12's cotton and is
wefted with 30's linen yarn, there being the same number of
threads per inch in both warp and weft ; what percentage of
cotton and what of linen is there in the cloth ?
17. A sample of calico 3 in. by 4 in. weighs 30 grains.
What is the weight in pounds of a 70-yard piece, 36 in. wide ?
232 VOCATIONAL MATHEMATICS FOR GIRLS
18. 4 yd. of a certain cloth contains 2 Ib. of worsted at 67
cents a pound and 1£ Ib. of cotton at 18 cents a pound. Each
is what per cent of the total cost of material ?
19. A bale of worsted weighing 75 Ib. loses 8 oz. in reeling
off ; what is the per cent of loss ?
20. If Ex. 19 worsted gains 0.45 Ib. to the 75 Ib. bale in dye-
ing, what is the per cent of gain ?
21. This 75 Ib. cost $ 50.25 and it lost 4 oz. in the fulling
mill, what was the value of the part lost ?
22. The total loss is what per cent of the original weight ?
What is its value at 67 cents a pound ?
PART IV — THE OFFICE AND THE STORE
CHAPTER XI
ARITHMETIC FOR OFFICE ASSISTANTS
EVERY office assistant should be quick at figures — that 'is,
should be able to add, subtract, multiply, and divide accurately
and quickly. In order to do this one should practice addition,
subtraction, multiplication, and division until all combinations
are thoroughly mastered.
An office assistant should make figures neatly so that there
need be no hesitation or uncertainty in reading them.
Rapid Calculations
Add the following columns and check the results. Compare
the time required for the different examples.
1. 27 2. 37 3. 471 4. 568 5. 1,039
12 20 295 284 579
8 11 194 187 381
18 20 327 341 668
12 16 287 272 559
8 12 191 184 375
8 16 237 193 430
8, 9 194 156 350
7 12 169 166 335
11 15 247 232 479
12 13 194 180 374
2 3 27 25 52
12 17 253 240 493
11 14 241 212 453
12 20 355 367 722
12 14 244 222 466
8 11 93 79 172
10 15 208 213 421
233
234 VOCATIONAL MATHEMATICS FOR GIRLS
. 7
7. 7
8. 159
9. 152
10. 311
2
5
60
78
138
4
7
111
88
199
6
10
173
121
294
4
6
112
84
196
4
4
88
76
164
4
5
104
83
187
4
6
96
104
200
4
7
120
97
217
8
9
144
123
267
4
5
60
101
161
4
5
73
92
165
8
10
186
176
362
4
4
64
75
139
4
6
114
113
227
4
4
89
88
177
6
7
91
80
171
8
9
204
170
374
4
13
175
166
341
4
4
73
77
150
4
7
119
127
246
4
5
84
103
187
8
11
177
165
342
6
8
156
136
292
3
4
94
61
155
12
18
310
293
603
8
12
191
189
380
8
13
268
198
466
2
2
17
17
34
4
8
122
137
259
8
0
193
185
378
1
1
4
6
10
1
1
9
15
24
1
2
16
16
32
1
1
11
15
26
1
2
34
44
78
1
2
27
34
61
1
2
26
53
79
1
2
36
41
76
1
2
17
10
27
1
2
38
22
60
ARITHMETIC FOR OFFICE ASSISTANTS
235
11. $162.24
12. $37,000.00
13. §31.25
14. $8,527.08
15. $630.33
266.45
300,000.00
73.70
2,907.31
408.32
277.56
410,000.00
2.00
3,262.68
399.99
12,171.44
82,000.00
425
8,096.90
28.00
17.72
.89
9,359.21
644.15
6.00
51,000.00
31.15
2,177.30
18,000.00
33.15
40,000.00
3.20
8,385.50
32.85
23.65
16.75
7,229.20
154.65
3.18
34,500.00
4.51
8,452.38
82.35
3,066.34
1,758.13
517.50
17,000.00
2,665.76
5,236.32
25.00
1128.13
6,147.42
639.24
36.00
15,500.00
3.20
4,443.88
2.60
30.00
3,386.72
79.90
4.00
5,500.00
3,927.78
1,143.00
289.22
1,000.00
29.12
4,797.46
265.50
2,612.00
727.00
17.82
70,500.00
1.00
2,476.31
141.33
199.87
33.27
3,705.00
2314.76
10,000.00
19.09
6,417.42
3,091.72
2.40
12,500.00
720.00
1,574.50
1,049.95
9.25
1,500.00
28.80
3,121.97
166.64
55.80
300.00
96.00
120.00
494.03
3.41
26,146.93
1,483.84
18.00
800.00
5.00
51.397.19
657.62
1.55
50.00
7.37
99.55
1,416.68
3.15
100.00
3.60
3,605.93
135.50
2.55
200.00
22,830.14
208.33
4,010.92
250.00
9.08
85,706.13
42.84
126.45
300.00
36,361.19
362.25
2.25
4.50
39,056.23
234.47
152.70
2,000.00
30,000.00
31.50
10.25
35.84
179,346.77
49.76
3.62
1,000.00
3,375.31
150.22
4.00
2.00
12,638.85
2.64
111.10
1,200.00
3.50
30,992.76
2.40
324.83
11.06
179,346.77
22.50
302.10
114,350.00
.74
3,375.31
8.92
345.04
40,000.00
7.25
12,638.85
176.91
301.10
120,000.00
6.00
30,992.76
11.30
1.20
9,476.00
3.00
16,503.48
17.00
236 VOCATIONAL MATHEMATICS FOR GIRLS
16.
$437.58
2.75
1.40
70.06
3.54
396.89
33.00
18.24
6.75
68.70
1.53
9.20
.90
98.95
117.13
192.71
58.43
2.11
2.92
43.34
5.80
108.81
1.75
10.10
3.25
881.69
82.80
.75
3.00
26.50
19.04
2.24
19.50
2,676.35
25.25
.70
36.53
3.60
3.00
168.66
67.60
17.
$81.33
18. $144.40
19. $61.45
31.66
15.00
14.50
9.91
1,124.04
1.80
20.00
110.59
2.00
23.25
44.83
24.17
129.99
318.40
272.90
9.01
22.35
5.13
208.01
757.00
482.09
150.98
674.37
.50
14.50
220.50
10.60
27.30
36.60
280.00
6.50
3.60
83.78
.32
2.50
36.90
216.60
31.00
245.00
40.00
91.87
481.30
542.25
18.97
57.96
25.49
59.35
53.07
3.75
2.54
8.14
1,863.74
36.08
155.70
21.25
22.38
1,076.82
6.47
8,699.46
449.85
132.28
4,437.97
4.00
391.00
394.48
3.00
72.00
24.00
35.00
85.12
10.00
310.49
47.90
10.40
1,078.50
31.68
.85
49.50
37.70
77.91
39.76
64.43
17.21
2.20
158.26
185.99
1.50
2.40
6.00
53.49
8.62
2.50
7.50
3.85
1.70
5.05
23.65
2.00
7.60
259.00
.70
2.00
701.47
92.00
11,50
3,148.00
ARITHMETIC FOR OFFICE ASSISTANTS 237
Horizontal Addition
Reports, invoices, sales sheets, etc., are often written in such
a way as to make it necessary to add figures horizontally. In
adding figures horizontally, it is customary to add from left to
right and check the answer by adding from right to left.
EXAMPLES
Add the following horizontally :
1. 38 + 76 + 49 =
2. 11 + 43 + 29 = -
3. 27 + 57 + 15 =
4. 34 + 16 + 23 =
5. 47 + 89 + 37 =
6. 53 + 74 + 42 =
7. 94 + 17 + 67 =
8. 79 + 37 + 69 =
9. 83 + 49 + 74 =
10. 19 -f 38 + 49 =
Add the following and check by adding the horizontal and
vertical totals :
11. 36 + 74 -|- 19 + 47 =
29 + 63 -f 49 + 36 =
+ + 4- =
12. 74 + 34 + 87 + 27 =
37 + 19 + 73 + 34 =
+ + 4- =
13. 178+ 74 + 109+ 83 =
39 + 111 + 381 + 127 =
+ + + =
14. 217+589 + 784 =
309 + 611 + 983 =
+ + =
238 VOCATIONAL MATHEMATICS FOR GIRLS
15.
1118 + 3719 + 8910 =
3001 + 5316 + 6715 =
+ + =
Add the following and check by adding horizontal and verti-
cal totals. Compare the time required for the different examples.
16. $702,000 $14,040 $370,000
$6,475.00
$1,072,000
$20,515.00
525,000
10,500
20,000
350.00
565,000
11,300.00
1,267,500
25,350
447,250
7,826.88
1,724,750
33,401.88
333,000
6,660
340,000
5,950.00
833,000
16,022.50
380,000
7,600
351,000
6,142.50
790,000
15,070.00
1,077,000
21,540
50,000
875.00
1,127,000
22,415.00
702,000
14,040
370,000
6,475.00
1,072,000
20,515.00
525,000
10,500
20,000
350.00
565,000
11,300.00
1,264,500
25,290
447,250
7,826.87
1,721,750
33,341.87
333,000
6,660
200,000
3,500.00
693,009
13,572.50
355,000
7,100
348,000
6,090.00
758,000
14,427.50
1,072,000
21,440
50,000
875.00
1,122,000
22,315.00
17. 318,143
28,760
9.04
491.86
189.54
77,751,393
295,187
18,363
6.22
498.23
188.74
78,426,000
300,789
23,398
7.95
479.80
187.88
75,180,746
279,735
22,290
7.97
511.43
187.24
79,864,039
302,737
28,699
9.48
523.55
187.80
82,001,180
302,338
22,149
7.33
578.00
188.83
91,025,879
341,085
27,765
8.14
554.30
192.87
89,161,101
335,775
24,080
7.17
534.23
192.13
85,603,137
311,739
20,356
6.53
521.79
192.17
83,627,195
335,350
21,299
6.35
524.17
192.76
84,266,576
281,481
18,032
6.41
500.09
194.89
81,283,747
305,370
20,865
6.83
496.12
196.06
81,122,570
18. 380,782,151
451,880
,223 520
,781,017 389,692,401 1
,743,135,792
452,491,808
480,722
,907 537
,837,574 481
,528,491 1
,952,580,780
71,709,667
28,842,684 17
,056,557 91
,836,090
209,444,988
1,585
600
317
1,907
1,102
283,448,988
282,640,795 326
,233,015 291
,835,151 1
,184,157,949
. 6,264
5
,879
6,066
6,061
6,068
97,333,163
169,239
,428 194
,548,002 97
,857,250
558,977,843
ARITHMETIC FOR OFFICE ASSISTANTS
239
19. 3,200,000
17,000,000
28,000,000
7,000,000
55,700,000
27,200,000
25,000,000
31,400,000
23,000,000
106,600,000
6,100,000
6,100,000
850,000
65,100,000
64,200,000
12,300,000
142,450,000
3,500,000
12,000,000
15,500,000
625,000
5,200,000
2,900,000
8,725,000
1,416,353
7,263,712
2,000,000
11,866,463
22,546,528
665,907
542,539
443,392
415,531
1,967,359
3,500,000
11,200,000
13,200,000
7,400,000
35,300,000
12,500,000
2,500,000
3,500,000
2,600,000
21,100,000
20. 29,000,000
22,500,000
14,200,000
16,600,000
82,300,000
13,500,000
10,200,000
9,600,000
8,600,000
41,900,000
327,998
330,915
508,266
358,262
1,525,441
1,122,905
1,222,262
1,296,344
1,317,004
4,958,515
2,400,000
1,100,000
1,650,000
1,800,000
6,950,000
1,500,000
850,000
900,000
900,000
4,150,000
250,000
305,000
350,000
300,000
1,205,000
Add the following decimals and check
the answer
:
21. 21.51
35.21
36.17
20.32
28.30
18.91
12.42
5.95
20.95
14.56
15.85
6.00
3.17
19.07
11.02
22. 44.33
73.15
71.59
14.36
8.15
43.20
47.14
126.04
85.05
70.42
93.35
80.13
31.15
62.51
49.17
49.17
- 29.37
47.25
31.10
206.38
37.59
47.25
35.59
50.47
73.26
23. On the following page is an itemized list of invest-
ments.
What is the total amount of investments ?
What is the average rate of interest ?
Review Interest, page 50.
240 VOCATIONAL MATHEMATICS FOR GIRLS
List of Investments Held by the Sinking Funds of Fall Ewer, Mass.
January 1, 1913
NAME
KATE
MATURITY
AMOUNT
City of Boston Bonds
34
July 1, 1939
$15,000
City of Cambridge Bonds
»i
Nov. 1, 1941
25,000
City of Chicago Bonds
4
Jan. 1, 1921
27,500
City of Chicago Bonds
4
Jan. 1, 1922
100,000
City of Los Angeles Bonds
4*
June 1, 1930
50,000
City of So. Norwalk Bonds
4
July 1, 1930
23,000
City of So. Norwalk Bonds
4
Sept. 1, 1930
22,000
City of Taunton Bonds
4
June 1, 1919
39,000
Town of Revere Note
4.35 disc.
Aug. 13, 1913
10,000
Boston & Albany R. R. Bonds
4
May 1, 1933
57,000
Boston & Albany R. R. Bonds
4
May 1, 1934
57,000
Boston Elevated R. R. Bonds
4
May 1, 1935
50,000
Boston Elevated R. R. Bonds
44
Oct. 1, 1937
68,000
Boston Elevated R. R. Bonds
44
Nov. 1, 1941
50,000
Boston & Lowell R. R. Bonds
4
April 1, 1932
16,000
Boston & Maine R. R. Bonds
*4
Jan. 1, 1944
150,000
Boston & Maine R. R. Bonds
4
June 10, 1913
20,000
C. B. & Q. R. R. Bonds (111. Div.)
4
July 1, 1949
50,000
C. B. & Q. R. R. Bonds (111. Div.)
3|
July 1, 1949
55,000
Chi. & N. W. R. R. Bonds
7
Feb. 1, 1915
92,000
Chi. & St. P., M. & O.. R. R. Bonds
6
June 1, 1930
20,000
Cleveland & Pittsburg R. R. Bonds
44
Jan. 1, 1942
35,000
Cleveland & Pittsburg R. R. Bonds
4£
Oct. 1, 1942
10,000
Fitchburg R. R. Bonds
34
Oct. 1, 1920
50,000
Fitchburg R. R. Bonds
34
Oct. 1, 1921
20,000
Fitchburg R. R. Bonds
44
May 1, 1928
50,000
Fre. Elk. & Mo. Val. R. R. Bonds
6
Oct. 1, 1933
85,000
Great Northern R. R. Bonds
4J
July 1, 1961
25,000
Housatonic R. R. Bonds
5
Nov. 1, 1937
46,000
Louis. & Nash. R. R. Bonds
(N. O. & M.)
6
Jan. 1, 1930
20,000
Louis. & Nash. R. R. Bonds
(St. L. Div.)
6
March 1, 1921
5,000
Louis. & Nash. R. R. Bonds
(N. & M.)
44
Sept. 1, 1945
10,000
Louis. & Nash. R. R. Bonds
5
Nov. 1, 1931
35,000
Mich. Cent. R. R. Bonds
5
March 1, 1931
37,000
Mich. Cent. R. R. Bonds
(Kal. & S. H.)
2
Nov. 1, 1939
50,000
ARITHMETIC FOR OFFICE ASSISTANTS
241
24. What is total amount of the following water bonds?
What is the average rate of interest ?
Water Bonds of Fall River, Mass.
DATE OF ISSUE
KATE
TERM
MATURITY
AMOUNT
June 1, 1893
4
30 years
June 1, 1923
$ 75,000
May 1, 1894
4
30 years
May 1, 1924
25,000
Nov. 1, 1894
4
29 years
Nov. 1, 1923
25,000
Nov. 1, 1894
4
30 years
Nov. 1, 1924
25,000
May 1, 1895
4
30 years
May 1, 1925
25,000
June 1, 1895
4
30 years
June 1, 1925
50,000
Nov. 1, 1895
4
30 years
Nov. 1, 1925
25,000
May 1, 1896
4
30 years
May 1, 1926
25,000
Nov. 1, 1896
4
30 years
Nov. 1, 1926
25,000
April 1, 1897
4
30 years
April 1, 1927
25,000
Nov. 1, 1897
4
30 years
Nov. 1, 1927
25,000
April 1, 1898
4
30 years
April , 1928
25,000
Nov. 1, 1898
4
30 years
Nov. , 1828
25,000
May 1, 1899
4
30 years
May , 1929
50,000
Aug. 1, 1899
4
30 years
Aug. , 1929
150,000
Nov. 1, 1899
3*
30 years
Nov. , 1929
175,000
Feb. 1, 1900
3*
30 years
Feb. , 1930
100,000
May 1, 1900
3*
30 years
May , 1930
20,000
April 1, 1901
3|
30 years
April ,1931
20,000
April 1, 1902
3*
30 years
April , 1932
20,000
April 1, 1902
3*
30 years
April , 1932
50,000
Dec. 1, 1902
3|
30 years
Dec. 1, 1932
50,000
April 1, 1903
3*
30 years
April 1, 1933
20,000
Feb. 1, 1904
3£
30 years
Feb. 1, 1934
175,000
May 2, 1904
4
30 years
May 2, 1934
20,000
1. 33 2. 35
3. 37 4. 3f
7 9
8 1
8. 42
9. 49 10. 46
17
18 19
SUBTRACTION
DRILL EXERCISE
5. 36
9 7
6. 32
4
7. 26
9
11. 43 12. 41
16 15
242 VOCATIONAL MATHEMATICS FOR GIRLS
13. 45 14. 44 15.
17 17
364 16. 468
126 329
17. 566
328
18. 661 19. 363
324 127
20. 465
228
24. 200,000
121,314
21. 362
129
22. 865,900 23. 891,000
697,148 597,119
25. 30,071
28,002
26. 581,300 27. 481,111
391,111 310,010
28. 681,900
537,349
29. 868,434
399,638
30. 753,829 31. 394,287
537,297 277,469
32. 567,397
297,719
33. 487,196
311,076
34.
38.
291,903 35.
187,147
$ 835.00
119.00
36. $1100.44
835.11
37. $2881.44
1901.33
$ 3884.59 39.
1500.45
$ 4110.59
1744.43
40. $2883.40
1918.17
41. $3717.17
1999.18
42. $1911.84
1294.95
43. $
2837.73 44.
1949.94
$ 5887.93
4999.99
MULTIPLICATION
DRILL EXERCISE
By inspection, multiply the following numbers :
1. 1600x900.
2. 800 x 740.
3. 360 x 400.
4. 590 x 800.
5. 1700 x 1100.
6. 1900x700.
7. 788,000 x 600.
8. 49,009x400.
9. 318,000x4000.
10. 988,000 x 50,000.
11. 80 x 11.
12. 79 x 11.
13. 187 x 11.
14. 2100 x 11.
15. 2855 x 11.
16. 84x25.
17. 116 x 50.
18. 288 x 25,
19. 198x25.
20. 3884 x 25.
"Review rules on multiplication, pages 8-9.
ARITHMETIC FOR OFFICE ASSISTANTS 243
BILLS (Invoices)
When a merchant sells goods (called merchandise), he sends
a bill (called an invoice) to the customer unless payment is
made at the time of the sale. This invoice contains an itemized
list of the merchandise sold and also the following :
The place and date of the sale.
The terms of the sale (usually in small type) — cash or a
number of days' credit. Sometimes a small discount is given
if the bill is paid within a definite period.
The quantity, name, and price of each item is placed on the
same line. The entire amount of each item, called the exten-
sion, is placed in a column at the right of the item.
Discounts are deducted from the bill, if promised.
Extra charges, such as cartage or freight, are added after
taking off the discount.
Make all Checks payable to We handle only highest grades
Union Coal Company of Anthracite and Bitu-
of Boston minous Coals
UNION COAL COMPANY
40 CENTER STREET
BRANCH EXCHANGE TELEPHONE
CONNECTING ALL WHARVES AND OFFICES
SOLD TO L. T. Jones,
5 Whitney St. ,
Mattapan, Mass.
BOSTON, Sept. 3, 1914.
6000 Ib. Stove Coal 7.00 $21.00
4000 " Nut 7.25 14.50
35.50
REC'D PAYMENT
SEPT. 28, 1914
UNION COAL CO.
244 VOCATIONAL MATHEMATICS FOR GIRLS
When the amount of the bill or invoice is paid, the invoice
is marked.
Received payment,
Name of firm.
Per name of authorized person.
This is called receipting a bill.
Ledger
Whenever an invoice is sent to a customer, a record of the
transaction is made in a book called a ledger. The pages of
this book are divided into two parts by means of red or double
lines. The left side is called the debit and the right side the
credit side. At the top of each ledger page the name of a
person or firm that purchases merchandise is recorded. The
record on this page is called the account of the person or firm.
When the person or firm purchases merchandise, it is recorded
on the debit side. When merchandise or cash is received, it is
recorded on the credit side. The date, the amount, and the
word Mdse. or cash is usually written.
We debit an account when it receives value, and credit an
account when it delivers value.
E. D. REDINGTON
1917
1917
Jan. 2
Cash
109
1000
Jan. 1
Acc't to Perkins
114
810
Note, 60 ds.
114
1500
2
Mdse.
100
3057
9
Page's Order
115
575
10
"
100
575
25
Cash
109
500
22
Order to Jenness
115
375
27
Mdse.
93
157
~>0
688.05
1+81S
31
Browne's Ace.
115
397
53
1130
OS
SPECIMEN LEDGER PAGE
ARITHMETIC FOR OFFICE ASSISTANTS
245
A summary of the debits and credits of an account is called
a statement. The difference between the debits and credits
represents the standing of the account. If the debits are
greater than the credits, the customer named on the account
owes the merchant. If the credits are greater than the debits,
then the merchant owes the customer.
EXAMPLES
Balance the following accounts :
BLANEY, BROWN & CO.
1917
1917
Jan. 14
Cons't #7
177
669
98
Jan. 6
Mdse.
171
1303
" Co. #/
179
386
25
30
Dft. favor Button
180
900
28
" #1 53.23
179
1200
7,~>
LUDWIG & LONG
1917
1917
Jan. 6
Cons' t #2
177
1939
60
Jan. 6
Cash
172
1000
20
" #2 327.50
177
1327
50
15
"
172
939
28
«
172
1000
CHARLES N. BUTTON
1917
1917
Jan. 7
Mdse.
168
651
88
Jan. 9
Ship't Co. #2
177
856
67
12
Cash
173
1000
24
Cons't #2 208.51
176
4699
0!t
20
"
173
2000
29
Ship't Co. #1
179
795
37
30
Dft. on Blaney, B.
180
900
246 VOCATIONAL MATHEMATICS FOR GIRLS
D. K. REED & SON
1917
1917
Jan. 8
Cons 't #1
177
525
42
Jan. 8
Note at 60 ds.
180
525
17
Mdse.
170
202
50
17
Cash
172
202
26
Cons' t £1
177
243
7~>
" CO. #7
179
206
PROFIT AND LOSS
(Review Percentage on pages 50-56)
A merchant must sell merchandise at a higher price than he
paid for it in order to have sufficient funds at the end of the
transaction to pay for clerk hire, rent, etc. Any amount above
the purchasing price and its attendant expenses is called
profit ; any amount below purchasing price is called loss.
A merchant must be careful in figuring his profit. He
must have a set of books so arranged as to show what caused
either an increase or reduction in the profits.
There are certain special terms used in considering profit
and loss. The first cost of goods is called the net or prime
cost. After the goods have been received and unpacked, and
the freight, cartage, storage, commission, etc. paid, the cost
has been increased to what is called gross or full cost. The
total amount received from the sale of goods is called gross
selling price. The sum of expenses connected with the sale of
goods subtracted from the gross selling price is called the net
selling price. A merchant gains or loses according as the net
selling price is above or below the gross cost.
There are two methods of computing gain or loss, each
based on the rules of percentage. In the first method, the
gross cost is the base, the per cent of gain or loss the rate, the
gain or loss the percentage. The second method considers
the selling price the base and will be explained in detail later.
ARITHMETIC FOR OFFICE ASSISTANTS
247
EXAMPLES
1. Make extensions after deducting discounts and give total :
Credit not allowed on goods returned without our permission
PETTINGELL-ANDREWS COMPANY
ELECTRICAL MERCHANDISE
General Offices and Warerooms
156 to 16O PEARL STREET and 491 to 511 ATLANTIC AVENUE
Terms : 30 Days Net
NEW YORK, Nov 17 1911
SOLD TO City of Lowell School Dept, Lowell, Mass.
SHIPPED TO Same Lowell Industrial School, Lowell, Mass.
SHIPPED BY B &. L 11/14/11 OUR REG. NO. 3786
ORDER REC'D 1 1/13/1 I REELS COILS BUNDLES CASES BBLS.
tl
H
00
^ "D
o>$>
ORDER No. 78158 REG. No. 52108
PRICE
1
\
#4 Comealong# 11293 Ea
4 00
15%
1
\
#14492 16" Extension Bit Ea
2 00
50%
36
36
2 oz cans Nokorode Soldering Paste Doz
2 00
50%
15
15
#8020 Cutouts Ea
36
40%
2
2
#322 H & H Snap Sws Ea
76
30%
125
125
#9395 Pore Sockets Ea
30
»
45%
125
125
# 1999 Fuseless Rosettes Ea
08
45%
100
100
0 C Ball Adjusters for Lp Cord M
7 00
50
50
I" Skt Bushings C
50
200
200
Pr #43031 Std #328 #1 Single Wire Cleats M Pr
2668
40 o£)
200
200
Pr #43033 Single Wire Cleats M Pr
i U /O
40 00
40%
2
2
Lb White Exemplar Tape Lb
45
248 VOCATIONAL MATHEMATICS FOR GIRLS
2. Make extensions on the following items and give total :
Goods are Charged for the Convenience of Customers and Accounts are Rendered Monthly
R A. McWniRR Co.
DEPARTMENT STORE
FALL RIVER, MASS.
A. A. MILLS, Pres't & Treas.
J. H. MAHONEY, Supt.
R. S. THOMPSON, Sec'y.
Purchases for . Fall River Technical High School
September, 1913 City
No. Order Number 719
DATB
ARTICLES
AMOUNTS
DAILY TOTAL
CREDITS
Sept 4
2 Doz C Hangers
2 " Skirt "
90
45
5
6
120 Long Cloth
34$ Cambric
522 B Cambric
15
18
100 B Nainsook
16
24 Doz Kerr L Twist
120
8 Doz Tape Measures
84 " W Thread
25
51
9
1 10/12 Doz Tape
1 Gro Tambo Cotton
25
520
£ Doz Bone Stillettos
46
I " Steel
46
40 Paper Needles
20 "
8
2 Doz M Plyers
2 Boxes Edge Wire
12 " Even Tie Wire
600
125
180
24 " Brace
225
2 " Lace
160
2 Pk Ribbon
125
2 Rolls Buckram
90
13
48 Yd Cape Net
100 Crinoline
15
5
125
5
ARITHMETIC FOR OFFICE ASSISTANTS 249
3. Make extensions on the following items and give total :
Goods are Charged for the Convenience of Customers and Accounts are Rendered Monthly
R A. McWniRR Co.
DEPARTMENT STORE
FALL RIVER, MASS.
A. A. MILLS, Pres't & Treas.
J. H. MAHONEY, Vice-Pres't.
R. S. THOMPSON, Sec'y.
Purchases for Fall River Public Buildings
September, 1913 City
No. For Technical High School
DATE
ARTICLES
AMOUNTS
DAILY TOTAL
CKEDITS
Sept 4
1 Dinner Set
1700
100 Knives
9
100 Forks
9
100 D Spoons
10
100 Tea Spoons
09
1 Doz Glasses
90
8J Doz Tumblers
45
8£ " Bowls
96
54 Crash
111
50 "
31
^ Doz Napkins
270
i „
415
2 Table Cloths
360
12
120 Crash
"i
15
2 Stock Pots
325
1 Lemon Squeezer
14
1 Doz Teaspoons
500
1 Butter Spreader
75
£ Doz Forks
625
250 VOCATIONAL MATHEMATICS FOR GIRLS
EXAMPLE. — A real estate dealer buys a house for $ 4990
and sells it to gain $ 50. What is the per cent of gain over
cost?
SOLUTION. ^L x 100 = — = UiT %. Ana.
4990 499
DRILL EXERCISE
Find per cent of gain or loss :
1.
Cost
$1660
Gain
$175
6.
Cost
$6110
Loss
$112
2.
$1845
$135
7.
$5880
$ 65
3.
$ 1997.
75
$ 412.50
8.
$ 3181.
10
$108.
75
4.
$ 2222.
50
$ 319.75
9.
$ 7181.
49
$213.
(>0
5.
$ 3880.
11
$ 610.03
10.
$ 3333.
19
$ 28.
<)<)
EXAMPLES
1. A dealer buys wheat at 91 cents a bushel and sells to
gain 26 cents. What is the per cent of gain?
2. A farmer sold a bushel of potatoes for 86 cents, and gained
20 cents over the cost. What was the per cent of gain ?
3. Real estate was sold for $ 19,880 at a profit of $ 3650.
What was the per cent of gain ?
4. A provision dealer buys smoked hams at 19 cents a pound
and sells them at 31 cents a pound. What is the per cent of
gain?
5. A grocer buys eggs at 28 cents a dozen and sells them
at 35 cents a dozen. What is the per cent gain ?
6. A dealer buys sewing machines at $22 each and sells
them at $ 40. What is the per cent gain ?
7. A dealer buys an automobile for $ 972 and sells it for
$ 1472 and pays $ 73.50 freight. What is the per cent gain ?
ARITHMETIC FOR OFFICE ASSISTANTS 251
DRILL EXERCISE
Find the per cent gain or loss on both cost and selling price :
1.
Cost
$1200
Selling Price
$1500
6.
Co»t
$2475
Selling Price
$2360
2.
$1670
$1975
7.
$1650
$1490
3.
$2325
$2980
8.
$ 4111.50
$ 2880.80
4.
$ 4250.50
$ 5875.75
9.
$4335.50
$4660.60
5.
$ 3888.80
$ 4371.71
10.
$ 2880.17
$ 2551.60
REVIEW EXAMPLES
1. A dealer buys 46 gross of spools of. cotton at $11.12.
He sells them at 5 cents each. What is his profit ? What is
the per cent of gain on cost ? on selling price ?
2. Hardware supplies were bought at $ 119.75 and sold for
$ 208.16. What is the per cent of gain on cost and on selling
price ?
3. A grocer pays $ 840 f.o.b. Detroit for an automobile
truck. The freight costs him $ 61.75. What is 'the total cost
of automobile truck ? What per cent of the total cost is
freight ?
4. A dry goods firm buys 900 yards of calico at 5 cents a
yard, and sells it at 9 cents. What is the profit ? What per
cent of cost and selling price ?
5. A grocer buys a can (81 qt.) of milk for 55 cents and sells
it for 9 cents a quart. What is the per cent of gain ?
EXAMPLES
1. A dealer sold a piano at a profit of $ 115, thereby gaining
18 % on cost. What was the selling price ?
SOLUTION. — If $ 115 = 18 % of cost, which is 100 %,
1 % = JjJff5_ = 6.3889
100% =$ 638. 89 cost
Adding 115.00 profit
$ 753.89 selling price.
252 VOCATIONAL MATHEMATICS FOR GIRLS
2. A dealer sold furniture at a profit of $ 98. What was
the cost of the furniture, if he sold to gain 35 % ?
3. A coal dealer buys coal at the wharf and sells it to gain $ 2
per ton. What is the cost per ton if he gains 31 % ?
4. A shoe jobber buys a lot of shoes for $ 1265 and sells to
gain 26 % . What is the selling price ?
5. An electrician buys a motor for $ 48 and sells it to gain
18 %. What is the selling price?
6. A pair of shoes was sold to gain 26 %, giving the shoe
dealer a profit of 97 cents. What was the cost price ? What
was the selling price ?
FORMULAS
Gain or loss = Cost x rate of gain or loss
Gain or loss
Cost =
Rate of gain or rate of loss
Selling Price = Cost (100 % + rate of gain) or (100 % — rate of loss)
Cost = Selling Price Selling Price
~ 100 % + rate of gain ( r 100 °/o - rate of loss
DRILL EXERCISE
Find the selling price in each of the following problems :
Sold to Lose
Coat
Sold to Gain
Cost
1. 16|%
$96
6. 37%
$250
2. 20%
$115
7. 33%,
$ 644.50
3. 30%
$48
8. 41%
$ 841.75
4. 19%
$ 112.50
9. 29%
$ 108.19
5. 201%
$ 187.75
10. 221%
$237.75
COMPUTING PROFIT AND LOSS
Second MetTiod. — Many merchants find that it is better busi-
ness practice to figure per cost profit on the selling price rather
than on the cost price. Many failures in business can be
ARITHMETIC FOR OFFICE ASSISTANTS 253
traced to the practice of basing profits on cost. We must bear
in mind that no comparison can be made between per cents of
profit or cost until they have been reduced to terms of the
same unit value or to per cents of the same base.
To illustrate : It costs $ 100 to manufacture a certain article. The
expenses of selling are 22 %. For what must it sell to make a net
profit of 10%? Most students would calculate $132, taking the first
cost as the basis of estimating cost of sales and net profit. The average
business man would say that the expenses of selling and cost should be
quoted on the basis of the selling price.
SOLUTION. — Expenses of selling = 22 %
Profit = 10%
32 % on selling price.
. •. Cost on $ 100 = 68 % selling price.
100 % = .$ 147 selling price.
EXAMPLE 1. — An article costs $ 5 and sells for $ 6. What
is the percentage of profit? Ans. 16| %.
Process. — Six dollars minus $5 leaves $1, the profit. One dollar
divided by $6, decimally, gives the correct answer, 16|%.
EXAMPLE 2. — An article costs $ 3.75. What must it sell
for to show a profit of 25 % ? Ans. $ 5.
Process. — Deduct 25 from 100. This will give you a remainder of
75, the percentage of the cost. If $3.75 is 75%, 1% would be $3.75
divided by 75 or 5 cents, and 100% would be $5. Now, if you marked
your goods, as too many do, by adding 25 % to the cost, you would ob-
tain a selling price of about $4.69, or 31 cents less than by the former
method.
EXAMPLES
1. What is the percentage of profit, if an article costs $ 8.50
and sells for $ 10 ?
2. What is the percentage of profit on an automobile that
cost $ 810 and sold for $ 1215 ?
3. An article costs $ 840. What must I sell it for to gain
30 % ?
254 VOCATIONAL MATHEMATICS FOR GIRLS
4. A case of shoes is bought for $ 30. For what must I sell
them to gain 25 % ?
TABLE FOR FINDING THE SELLING PRICE OF ANY ARTICLE
COST
TO DO
NET PER CENT PROFIT DESIRED
BUSINESS
1
2
8
9
10
11
12
13
14
15
20
25
30
35
40
50
15%
84
88
82
81
80
79
78
77
76
7.-)
74
78
7-2
71
7o
65
60
55
50
4;,
35
16%
88
82
81
so
79
7^
77
76
7r»
74
78
72
71
7(i
til)
64
59
64
49
44
84
17 %
82
81
80
79
7s:
77
76
75
74
78
7'2
71
70
69
(is
68
58
58
4s
48
88
18%
si
SI)
79
7s
77
76
7.')
74
78
72
71
7o
(','.»
68
(17
62
57
52
47
42
'52
19%
80
79
7s
77
76
75
74
7:',
72
71
7o
69
68
67
66
61
56
51
4f,
41
81
20%
79
78
77
76
75
74
7:i
1-1
71
70
69
68
67
66
65
60
55
50
4,r>
40
80
21%
78
77
76
78
74
73
72
71
70
69
68
67
66
66
64
59
54
49
44
89
•2!)
22%
77
76
7:>
74
78
72
71
70
69
68
67
66
65
64
68
58
58
4s
4:',
88
•2s
23%
76
75
74
78
7'2
71
7o
69
68
67
66
65
(14
€3
62
57
52
47
4-2
87
27
24%
75
74
7:',
7'.'
71
7<i
69
68
67
66
65
64
68
62
61
56
51
46
41
86
2t>
25%
74
73
72
71
7(i
f-9
68
67
66
65
64
68
62
(11
tin
55
50
45
40
85
25
The percentage of cost of doing business and the profit are
figured on the selling price.
Rule
Divide the cost (invoice price with freight added) by the
figure in the column of " net rate per cent profit desired " on
the line with per cent it cost you to do business.
EXAMPLE. — If a wagon cost
Freight . . .
$60.00
1.20
$ 61.20
You desire to make a net profit of 5 per cent
It costs you to do business 19 per cent
Take the figure in column 5 on line 19, which is 76.
76 j $6 1.2000 [ $80. 52, the selling price.
608
400
380
200
152
ARITHMETIC FOR OFFICE ASSISTANTS 255
Solve the following examples by table :
1. I bought a wagon for $84.00 f.o.b. New York City.
Freight cost $ 1.05. I desire to sell to gain 8 %. If the cost
to do business is 18 %, what should be the selling price?
2. I buy goods at $ 97 and desire a net profit of 7 % . It
costs 16 % to do business. What should be my selling price ?
3. Hardware supplies are purchased for $489.75. If it
costs 23 % to do the business, and I desire to make a net profit
of 11 %, for what must I sell the goods?
EXAMPLES
1. I bought 15 cuts of cloth containing 40^ yd. each, at
7 cents a yd., and sold it for 9 cents a yd. What was the
gain?
2. A furniture dealer sold a table for $ 14.50, a couch for
$ 45.80, a desk for $ 11.75, and some chairs for $ 27.30. Find
the amount of his sales.
3. Goods were sold for $367.75 at a loss of $125. Find
the cost of the goods.
4. Goods costing $ 145.25 were sold at a profit of $ 28.50.
For how much were they sold ?
5. A woman bought 4± yards of silk at $ 1.80 per yard, and
gave in payment a $ 10 bill. What change did she receive.?
6. I bought 25 yards of carpet at $ 2.75 per yard, and 6
chairs at $ 4.50 each, and gave in payment a $ 100 bill.
What change should I receive ?
TIME SHEETS AND PAY ROLLS
Office assistants must tabulate the time of the different em-
ployees and compute the individual amount due each week.
In addition, they must know the number of coins and bills of
different denominations required so as to be able to place the
exact amount in each envelope. This may be done by making
out the following pay roll form.
256 VOCATIONAL MATHEMATICS FOR GIRLS
FORM USED TO DETERMINE THE NUMBER OF DIFFERENT DENOMINATIONS
No, Persons
Amt, Rec'd
$10
$5
$2
$1
g
50?
g
25?
10?
5?
1?
2
13.50
g
g
3
<?.8<5
S
6
3
3
3
9-
r.y-s
¥•
V-
¥•
S
/g
2
<?./8
2
V-
2
g
6
Total Number Coins
2
<?
/6
2
5
7
/3
2
IS
TIME CARD
Week Ending-
No.
NAME
.191
1
Mou
Tue
Wed
MORNING
AFTERNOON
LOST OR
OVERTIME
_i
<t
IN
OUT
IN
OUT
IN
OUT
Thu
Fri
Sat
Sun
Total Time-- Mrs.
Rate
Total Wages for Week $
FORM USED TO SEND TO THE BANI
FOR THE MONEY FOR PAY ROLL
MEMORANDUM OF
CASH FOR PAY ROLL
WANTED BY
-19-
Twenties ....
Tens
Fives
Twos
Ones
Halves
Quarters ....
Dimes
Nickels ....
Pennies ....
Total
ARITHMETIC FOR OFFICE ASSISTANTS 257
TABLE OF WAGES1
To find the amount due at any rate from 30 cents to. 55
cents per hour, look at the column containing the number of
hours and the amount will be shown. Time and a half is
counted for overtime on regular working days, and double
time for Sundays and holidays.
M
M
P9
n
H
H
w
H
K
W
H
i
s
g
H
-
I
3
H
s
fa
X
H
p
S
S
1
1
I
g
M
M
§
i
S
M
i
e
«
H
a
K
M
M
0
Q
K
6
Q
tt
M
0
Q
%...
$0 30
$0 15
$0 22*
$0 30
$0 32*
$0 16}
$024|
$0 32*
$0 45
$0 22*
$0 33f
$0 45
1...
30
30
45
60
32*
32*
48}
65
45
45
67*
90
2...
30
60
90
1 20
32*
65
97*
1 80
45
90
1 35
1 80
8...
30
90
1 35
1 80
32*
97*
1 46£
1 95
45
1 35
2 02*
2 70
4...
30
1 20
1 80
2 40
32*
1 30
1 95
2 60
45
1 80
2 70
3 60
5...
30
1 50
2 25
3 00
32*
1 62*
2 43|
3 25
45
2 25
3 37*
4 50
6...
30
1 SO
2 70
8 60
82*
1 95
2 92*
3 90
45
2 70
4 05
5 40
7...
30
2 10
3 15
4 20
82*
2 27*
3 41J
4 55
45
3 15
4 72*
6 80
8...
30
2 40
3 60
4 80
32*
2 60
3 90
5 20
45
3 60
5 40
7 20
9...
30
2 70
4 05
5 40
32*
2 92*
4 38|
5 85
45
4 05
6 07*
8 10
10...
30
3 00
4 50
6 00
32*
3 25
4 87*.
6 50
45
4 50
6 75
9 00
%...
$047*
$023f
$0 35f
$0 47*
$0 50
$0 25
$0 37*
$0 50
$0 55
$0 27*
$0 41J
$0 55
1...
47*
47*
71}
95
50
50
75
1 00
55
55
82*
1 10
2...
47*
95
1 42*
1 90
50
1 00
1 50
2 00
55
1 10
1 65
2 20
8...
47*
1 42*
2 13J
2 85
50
1 50
2 25
3 00
55
1 65
2 47*
3 30
4...
47*
1 90
2 85
3 80
50
2 00
3 00
4 00
. 55
2 20
8 30
4 40
5...
47*
2 37*
3 56J
4 75
50
2 50
3 75
5 00
55
2 75
4 12*
550
6...
47*
2 85
4 27*
5 70
50
3 00
4 50
6 00
55
3 30
4 95
6 60
7...
47*
3 32*
4 98}
6 65
50
3 50
5 25
7 00
55
3 85
5 77*
7 70
8...
47*
3 80
5 70
7 60
50
4 00
6 00
8 00
55
4 40
6 60
8 80
9...
47*
4 27*
6 41i
8 55
50
4 50
6 75
9 00
55
4 96
7 42*
9 90
10...
47*
4 75
7 12*
9 50
50
5 00
7 50
10 00
55
5 50
8 25
11 00
EXAMPLES
1. Find the amount due a carpenter who has worked 8
hours regular time and 2 hours overtime at 55 cents per hour.
1 Similar tables may be constructed for other rates.
258 VOCATIONAL MATHEMATICS FOR GIRLS
2. A plasterer worked on Sunday from 8 to 11 o'clock. If
his regular wages are 45 cents per hour, how much will he
receive ?
3. A machinist's regular wage is 55 cents an hour. How
much money is due him for working July 4th from 8-12 A.M.
and 1^.30 P.M. ?
4. A plumber works six days in the week ; every morning
from 7.30 to 12 M. ; three afternoons from 1 to 4.30 P.M. ; two
afternoons from 1 to 5.30 ; and one from 1 until 6 P.M. What
will he receive for his week's wages at 50 cents per hour ?
WAGES OF EMPLOYEES
Superintendent $1,200.00 per annum
Matron 700.00 per annum
Nurses, 2 at 45.00 per month
Nurses, 1 at 40.00 per month
Nurses, 3 at 35.00 per month
Attendant 6.00 per week
Cook 12.00 per week
Assistant cook 1.00 per day
Kitchen maid 6.00 per week
Ward maids, 4 at 6.00 per week
Waitresses, 2 at 6.00 per week
Laundress 8.00 per week
Washwomen, 2 at 6.00 per week
Janitors, 1 day and 1 night 16.00 per week
Barber . . 6.00 per week
5. Find the total of coins and bills of all different denomi-
nations necessary to make up the weekly pay roll (52 weeks
= a year) of the above. Assume full time for a week. Make
out the currency memorandum for bank.
6. Find the total of coins and bills of the different denomi-
nations necessary to make up the following pay roll :
47£ hours, at 30 cents.
48 hours, at 45 cents.
48 hours, at 47| cents.
46 hours, at 32^ cents.
ARITHMETIC FOR OFFICE ASSISTANTS
259
7. Make a pay roll memorandum for the following pay
roll:
48 at 42|, 39 at 45, 46| at 48£.
TEMPORARY LOANS
The following is a statement of the temporary loans of a
New England city negotiated during the year, — amount, time,
rates.
DATE
AMOUNT OF
LOAN
TIME
RATE OF
INTEREST
AMOUNT OF
INTEREST
Months
Ehys
Feb. 28
$50,000
243
2.76
Feb. 28
25,000
243
2.76
Feb. 28
25,000
243
2.76
June 6
100,000
5
3.25
June 19
25,000
126
3.52
June 19
25,000
126
3.52
June 19
25,000
126
3.52
June 19
25,000
126
3.52
July 3
25,000
124
3.55
July 3
25,000
124
3.55
July 3
25,000
124
3.55
July 3
25,000
124
3.55
July 3
25,000
124
3.55
July 3
25,000
124
3.55
Aug. 14
25,000
2
4.
Aug. 20
25,000
80
4.07
Aug. 20
25,000
80
4.07
Aug. 20
25,000
80
4.07
Aug. 20
25,000
80
4.07
Sept. 4
25,000
40
4.
Sept. 4
25,000
40
4.
Sept. 4
16,000
40
4.
Write in a column after each loan, as suggested above, the
amount of interest on each loan for the time and at the rate.
CHAPTER XII
ARITHMETIC FOR SALESGIRLS AND CASHIERS
THE majority of employees in a department store are sales-
girls. It may be well to describe briefly the method of
operation of such a store and to indicate what part a salesgirl
has in it.
A department store is a combination of a number of distinct
stores or departments under one roof and general manage-
ment. It is organized in this way for the purpose of economy.
Each department is conducted as a separate store, and is
in charge of a buyer, who both buys and plans the sales
for his department. His department is charged for rent,
according to its location, and must also pay for overhead
charges.
The buyer in charge of each department has under him
salesgirls or saleswomen, who sell the goods. Each salesgirl
has a book containing sales slips in duplicate and a card to
show the amount of sales.
The sales slip shows the name and address of the purchaser
if the merchandise is to be sent to the customer's home. In
the case of a charge account a special form of sales slip is
used. The name and quality of the article purchased are
written in large space and the amount extended to the right.
The amount of money received from the purchaser is placed
at the top of the sales slip.
A carbon copy of each sales slip is made. The carbon copy
is given to the customer and the original is sent with the money
to the cashier. It is then used to tabulate data in regard to
sales, etc.
260
ARITHMETIC FOR SALESGIRLS AND CASHIERS 261
J. *
860(
Name
Addrei
sou
BY
[. EMKRSON CO.
; Dallas, Tex., 19 --
In Case of Error Please Return Goods and Bill
J- »
8606
Name
Addre&
SOLI
BY
[. 1C MIC It SON CO.
Dallas, Tex., 19 --
S- -
D
E
) P AM'T
T REC'D
s
D
E
> P AM'T
__T REC'D
Pur. by \
$1
Pur. by
~ 1u
.fl'O.O
So32
E
oo"S
3a8-
^
C0? fl
SS1_
Is4*00
ce£2
Am't Rec'd Sold by Am't of Sale
Customers tcill please report any failure
to deliver bill with goods
8606 1
This Slip must go in Customer's Parrel. Violation
of this Rale is cause for Instant Dismissal
1
SAI^KSMATS'S VOUCHER.
DEPARTMENT
SALESMAN DATE--
Cash Sales
Charge Sales
Cash Sales
Charge Sales
1
Forward
2
10
3
11
4
12
5
13
6
14
7
15
8
16
9
17
262 VOCATIONAL MATHEMATICS FOR GIRLS
Salesgirls should be able to do a great many calculations at
sight. This ability comes only through practice.
EXAMPLES
Find the amount of the following :
1. 10 yd. percale at 121 cents.
2. 12 yd. voile at 16 1 cents.
3. 27 yd. silesia at 331 cents.
4. 50 yd. serge at $ 1.50.
5. 28 yd. mohair at $ 1.25.
6. 48 yd. organdie at 37^- cents.
7. 911 yd. gingham at 10 cents.
8. 112 yd. calico at 41 cents.
9. 36 yd. galatea at 15 cents.
10. 11 yd. lawn at 19 cents.
11. 64 yd. dotted muslin at 621 cents.
12. 24 yd. gabardine at $1.75.
13. 18 yd. poplin at 29 cents.
14. 16 yd. hamburg at 15 cents.
15. 12 yd. lace at 871 cents.
16. 19 yd. val lace at 9 cents.
17. 26 yd. braid at 25 cents.
18. 48 dz. hooks and eyes at 12 cents0
19. 19 yd. cambric at 15 cents.
20. 18 pc. binding at 16 cents.
21. 6 yd. canvas at 24 cents.
22. 56 yd. linen at 621 cents.
23. 18 yd. albatross at $ 1.50.
24. 22 yd. silk at $ 2.25.
ARITHMETIC FOR SALESGIRLS AND CASHIERS 263
PROBLEMS
1. I bought cotton cloth valued at $ 6.25, silk at $ 13.75,
handkerchiefs for $ 2.50, and hose for $ 2.75. What was the
whole cost ?
2. Ruth saved $ 15.20 one month, $ 20.75 a second month,
and the third month $ 4.05 more than the first and second
months together. How much did she* save in the three
months ?
3. Goods were sold for $ 367.75, at a loss of $ 125. Find
the cost of the stock.
4. Goods costing $ 145.25 were sold at a profit of $ 28.50.
For how much were they sold ?
5. A butcher sold 8J pounds of meat to one customer,
9J- pounds to a second, to the third as much as the first plus
1£ pounds, to a fourth as much as to the second. How many
pounds did he sell ?
6. Edith paid $42.75 for a dress, one-half as much for a
cloak, and $ 7.25 for a hat. How much did she pay for all ?
7. A merchant sold four pieces of cloth ; the first piece
contained 24 yards, the second 32 yards, the third 16 yards,
and the fourth five-eighths as many yards as the sum of the
other three. How many yards were' sold?
8. From a piece of cloth containing 65f yards, there were
sold 23J yards. How many yards remained ?
9. A merchant sold goods for $ 528.40 and gained $ 29.50.
Find the cost.
10. From 11 yards of cloth, 3| were cut for a coat, and
6J yards for a suit. How many yards remained ?
11. I bought 15 cuts of cloth containing 40^ yards each at
7 cents a yard and sold it for 9 cents a yard. What was the
gain?
264 VOCATIONAL MATHEMATICS FOR GIRLS
12. What is the cost of 13-J yards of silk at $ 3.75 per yard ?
13. What is the cost of 16^ yards of broadcloth at $ 2.25
per yard ?
14. What is the cost of 3 pieces of cloth containing 12|,
14^, and 15^ yards at 121 cents per yard ?
15. What will 5| yards of velvet cost at $ 2.75 per yard ?
16. What is the cost of three-fourths of a yard of crgpe de
chine at $ 1.75 per yard ?
17. A saleslady is paid $ 1.00 per day for services and a
bonus of 2 % on all sales over $ 50 per week. If the sales
amount to $ 175 per week, what will be her salary ?
18. At $ 1.33^ a yard, how much will 15 yards of lace cost ?
19. At $ 1.16 J a yard, how much will 9 yards of silk cost ?
20. At $ 1.12^ per yard, how much will 6 yards of velvet
cost?
21. At 33^ cents each, find the cost of 101 handkerchiefs.
22. A salesgirl sold 141 yards of gingham at 25 cents, 9
yards of cotton at V2I\ cents, 101 yards of Madras at 35 cents.
Amount received, $ 10. How much change will be given to the
customer ?
23. Sold 6£ yards of cheviot at $ 1.10, 5f yards of silk at
$ 1.25, 91 yards of velveteen at 98 cents. Amount received,
$ 25.00. How much change will be given to the customer ?
24. Sold 111 yards of Persian lawn at $ 1.95, 6| yards of
dimity at 25 cents, 12J' yards of linen suiting at 75 cents.
Amount received, $ 40. How much change will be given to
the customer ?
25. Sold 9| yards of Persian lawn at $ 1.371, 5J- yards of
cheviot at $ 1.25, 15 yards of cotton at 121 cents. Amount
received, $ 30. How much change will be given to the cus-
tomer ?
ARITHMETIC FOR SALESGIRLS AND CASHIERS 265
26. Sold 7 yards of muslin at 25 cents, 12^ yards of lining
at 11 cents, 6| yards of lawn at $ 1.50, 7 yards of suiting at
75 cents. Amount received, $ 20. How much change will be
given to the customer ?
27. Sold 16 yards of velvet at $ 2.25, 14J yards of suiting
at 48 cents, 23 yards of cotton at 15 cents, 6| yards of dimity
at 24 cents, 7| yards of ribbon at 25 cents. Amount received,
$ 50. How much change will be given to the customer ?
28. At 121 cents a yard, what will 8J yards of ribbon cost ?
29. At $ 2.50 a yard, what will 2.8 yards of velvet cost ?
30. If it takes 5^ yards of cloth for a coat, 3i yards for a
jacket, and 1 a yard for a vest, how many yards will it take
for all ?
31. I gave $ 16.50 for 33 yards of cloth. How much did
one yard cost ?
32. Mary went shopping. She had a $ 20 bill. She bought
a dress for $ 9.50, a pair of gloves for $ .75, a fan for $ .87,
two handkerchiefs for $ .37 each, and a hat for $ 4.50. How
much money had she left ?
33. Emma's dress cost $ 11.25, and Mary's cost f as much.
How much did Mary's cost ? How much did both cost ?
34. What is the cost of 16f yards of silk at $ 2.75 a yard ?
35. What is the cost of 14| yards of cambric at 42 cents a
yard?
36. If 5J yards of calico cost 33 cents, how much must be
paid for 14f yards ?
37. One yard of sheeting cost 22| cents. How many yards
can be bought for $ 15.15 ?
38. From a piece of calico containing 33| yards there have
been sold at different times 11J, 7|, and 1£ yards. How many
yards remain ?
266 VOCATIONAL MATHEMATICS FOR GIRLS
39. I bought 16 \ yards of cloth for $ 3J per yard, and sold
it for $ 4J per yard. What was the gain ?
40. A merchant has three pieces of cloth containing, respec-
tively, 28|, 35 L, and 41 f yards. After selling several yards
from each piece, he finds that he lias left in the three pieces
67 yards. How many yards has he sold ?
ARITHMETIC FOR CASHIER
How to Make Change. — Every efficient cashier or saleslady
makes change by adding to the amount of the sale or purchase
enough change to make the sum equal to amount presented.
The change should be returned in the largest denominations
possible.
To illustrate : A young lady buys dry goods to the amount
of $ 1.52. She gives the saleslady a $ 5 bill. What change
should she receive ?
The saleslady will say: $1.52, $1.55, -$1.65, $1.75, $2.00, $4.00,
$5.00. That is, $ 1.52 + $ .03 = $ 1.65 ; $ 1.55 + $.10 = $ 1.65 ; $1.65
+ $.10= $1.75; $1.75 + $.25 = $2.00; $2.00 + $2.00 = $4.00 ; $4.00
+ $1.00 = $5.00.
EXAMPLES
1. What change should be given for a dollar bill, if the
following purchases were made ?
a. $.87 c. $.43 e. $.20
b. $.39 d. $.51 /. $.23
2. What change should be given for a two-dollar bill, if the
following purchases were made ?
a. $1.19 d. $1.57 g. $.63
6. $.89 e. $1.42 h. $.78
c. $1.73 /. $1.12 i. $.27
ARITHMETIC FOR SALESGIRLS AND CASHIERS 267
3. What change should be given for a five-dollar bill, if the
following purchases were made ?
a. $3.87 d. $2.81 g. $1.93
6. $2.53 e. $3.74 h. $.17
c. $4.19 /. $4.29 i. $.47
4. What change should be given for a ten-dollar bill, if the
following purchases were made ?
o. $8.66 d. $6.23 g. $3.16
6. $9.31 e. $5.29 ft. $2.29
c. $7.42 /. $4.18 t. $1.74
5. What change should be given for a twenty-dollar bill, if
the following purchases were made ?
a. $18.46 c. $17.09 e. $8.01
b. $ 19.23 d. $ 12.03 /. $ 6.27
CHAPTER XIII
CIVIL SERVICE
ALMOST every government position open to women has to be
obtained through an examination. In most cases Arithmetic
is one of the subjects tested. It is wise to know not only the
subject, but also the standards of marking, and for this reason
some general rules on this subject follow.
Marking Arithmetic — Civil Service Papers
1. On questions of addition, where sums are added across and the
totals added, for each error deduct 16| %.
2. .For each error in questions containing simple multiplication or
division, as a single process, deduct 50 % ; as a double process, deduct
25%.
3. In questions involving fractions and problems other than simple
computation, mark as follows :
(a) Wrong process leading to incorrect result, credit 0.
(6) For inconvenient or complex statement, process, or method, giving
right result, deduct from 5 to 25 °fc .
(c) If the answer is correct but no work is shown, credit 0.
(d) If the answer is correct and the process is clearly indicated, but
not written in full, deduct 25 °fo .
(e) If no attempt is made to answer, credit 0.
(/) If the operation is incomplete, credit in proportion to the work
done.
(gr) For the omission of the dollar sign ($) in final result or answer,
deduct 5.
(ft) In long division examples, to be solved by decimals, if the answer
is given as a mixed number, deduct 25.
4. For questions on bookkeeping and accounts, mark as follows :
(a) For omission of total heading, deduct 25 ; for partial omission, a
commensurate deduction.
(6) For every misplacement of credits or debits, deduct 25.
CIVIL SERVICE 269
(c) For every omission of date or item, deduct 10.
(d) For omissions or misplacement of balance, deduct 25.
NOTE. — Hard and fast rules are not always applicable because the impor-
tance of certain mistakes differs with the type of example. Before a set of
examples is marked, the deductions to be made for various sorts of errors
are decided upon by the examiners. In general, examples in arithmetic for
high-grade positions are marked on practically the same basis as clerical
arithmetic. Arithmetic in lower-grade examinations, such as police and fire
service and the like, is marked about 60% easier than clerical.
CIVIL SERVICE EXAMPLES
(Give the work in full in each example.)
1. Multiply 83,849,619 by 11,079.
2. Subtract 16,389,110 from 48,901,001.
3. Divide 18,617.03 by .717.
4. At $ 0.37 per dozen, how many dozen eggs can be
bought for $ 33.67 ?
5. What would 372 pounds of corn meal cost if 4 Ib. cost
12 cents ?
6. If a man bought 394 cows for $ 23,640 and sold 210
for $ 14,700, what was the profit on each cow ?
7. What is the net amount of a bill for $ 93.70, subject to
a discount of 37^- % ?
8. How many pints in a measure containing 14,784 cubic
inches ?
9. What number exceeds the sum of its fourth, fifth, and
sixth by 23 ?
10. If a man's yearly income is $ 1600, and he spends $ 25
a week, how much can he save in a year ?
11. What will 16|- pounds of butter cost at 34 cents a pound ?
12. How many hogs can be bought for $ 1340 if each hog
averages 160 pounds and costs 9 cents a pound ?
13. How many tons of coal can be bought for $446.25, if
each ton costs $ 8.75 ?
270 VOCATIONAL MATHEMATICS FOR GIRLS
14. A young lady can separate 38 letters per minute. If a
letter averages 6^ ounces, how many pounds of mail does she
handle in an hour ?
15. Multiply 53 J by 9f and divide the product by 2^.
(Solve decimally.) *
16. Roll matting costs 73 £ cents per sq. yd. What will be
the cost of 47 rolls, each roll 60 yd. long and 36 in. wide ?
17. A man paid $ 5123.25 for 27 mules and sold them for
$ 6500. How much did he gain by the transaction ?
18. A wheel measures 3' 1" in diameter. What is the dis-
tance around the tire ?
19. A bricklayer earns 70 cents an hour. If he works 129
days, 8 hours a day, and spends $ 50 a month, how much does
he save a year ?
20. A rectangular courtyard is 48' 5" long and 23' 1" wide.
How many square yards is it in area ?
21. How many days will it take a ship to cross the Atlantic
Ocean, 2970 miles, if the vessel sails at the rate of 21 miles an
hour ?
22. Eleven men bought 7 tracts of land with 22 acres in
each tract. How many acres will each man have ?
23. A merchant sends his agent $ 10,228 to buy goods.
What is the value of the goods, after paying $ 28 for freight
and giving the agent 2 % for his commission ?
24. If milk costs 6 cents a quart, and you sold it for 9 cents
a quart, and your profit for the milk was $48, how many
quarts of milk did you sell ?
25. A traveler travels llf miles a day for 8 days. How
many more miles has he yet to travel if the journey is 134
miles ?
26. What is the net amount of a bill for $29.85, subject to
a discount of 16| % ?
CIVIL SERVICE 271
27. Add across, placing the totals in the spaces indicated ;
then add the totals and check :
TOTALS
8,431 17,694 18,630 91 707
5,912 305 3,777 871 8,901
6,801 29,006 5,891 30 16,717
5,008 10,008 7,771 144 9,001
13,709 10,999 39 1,113 3,444
28. Divide 37,818.009 by .0391.
29. A pile of wood is 136 ft. long, 8 ft. wide, and 6 ft. high,
and is sold for $ 4.85 per cord, which is 20 % more than the
cost. What is the cost of the pile ?
30. Add the following column and from the sum subtract
81,376,019 :
80,614,304
68,815,519
32,910,833
54,489,605
96,315,809
75,029,034
21,201,511
31. A man bought 128 gal. cider at 23 cents a gallon ; he
sold it for 28 cents a gallon. How much did he make ?
32. A laborer has $48 in the bank. He is taken sick and
his expenses are $ 7.75 a day. How many days will his fund
last?
33. In paving a street If mi. long and 54 ft. wide, how
many bricks 9 in. long and 4 in. wide will be required ?
34. Find the simple interest on $ 841.37 for 2 yr. 3 mo. 17 da.
at 5%.
35. Find the simple interest on $ 367.49 for 1 yr. 7 mo. 19 da.
at 4%.
272 VOCATIONAL MATHEMATICS FOR GIRLS
SPECIMEN ARITHMETIC PAPERS
CLERKS, MESSENGERS, ETC.
Rapid Computation
1. Add these across, placing the totals in the spaces in-
dicated ; then add the totals :
TOTALS
15,863
3,175
368
51,461
35,196
27,368
7,242
82,463
24,175
52,837
3,724
51,493
68,317
58,417
41,582
4,738
16,837
5,281
52,683
26,364
73,642
25,164
42,525
70,463
1,476
18,572
7,368
15,726
71,394
62,958
2. Multiply 82,473,659 by 9874. Give the work in full.
3. From 68,515,100 subtract 24,884,574. Give the work in
full. 4. Divide 29,379.7 by .47. Give the work in full.
5. What is the net amount of a bill for $19.20, subject to a
discount of 16f % ? Give the work in full.
Arithmetic
1. How many times must 720 be added to 522 to make
987,642 ? Give the work in full. 2. If the shadow of an up-
right pole 9 ft. high is 8^ ft. long, what is the height of a church
spire which casts a shadow 221 ft. long ? Give the work in full.
3. How many sods, each 8 in. square, will be required to sod a
yard 24 feet long and 10 feet 8 inches wide ? Give the work
in full. 4. A retired merchant has an income of $ 25 a day,
his property being invested at 6 % . What is he worth ? Give
the work in full. 5. Find the principal that will yield $ 38.40
in 1 yr. 6 mo. at 4 % simple interest. Give the work in full.
6. If the time past noon increased by 90 minutes equals -f^
of the time from noon to midnight, what time is it ? Give the
work in full. 7. A merchant deducts 20 % from the marked
price of his goods and still makes a profit of 16 %. At what
CIVIL SERVICE 273
advance on the cost are the goods marked ? Give the work
in full. 8. If a grocer sells a tub of butter at 22 cents a pound,
he will gain 168 cents, but if he sells it at 17 cents a pound, he
will lose 112 cents. Find (a) the weight of the butter and (b)
the cost per pound. Give the work in full. 9. The product of
four factors is 432. Two of the factors are 3 and 4. The other
two factors are equal. What are the equal factors ? Give the
work in full.
STENOGRAPHER-TYPEWRITER
1. From what number can 857 be subtracted 307 times and
leave a remainder of 49 ? Give the work in full.
2. What number exceeds the sum of its fourth, fifth, sixth,
and seventh parts by 101 ? Give the work in full.
3. A sells to B at 10'% profit; B sells to C at 5 % profit;
if C paid $ 5336.10, what did the goods cost A ? Give the
work in full.
4. Find the simple interest of $ 297.60 for 3 yr. 1 mo. 15 da.
at 6 %. Give the work in full.
5. A man sold \ of his farm to B, f of the remainder to C,
and the remaining 60 acres to D. How many acres were in
the farm at first ? Give the work in full.
SEALERS OF WEIGHTS AND MEASURES
(Review Weights and Measures, pages 43, 276)
1. A measure under test is found to have a capacity of
332.0625 cu. in. What is its capacity in gallons, quarts, etc. ?
Give the work in full.
2. How many quarts, dry measure, would the above meas-
ure hold ? Give the work in full, carrying the answer to four
decimal places.
3. What is the equivalent of 175 Ib. troy in pounds avoir-
dupois ? Give the work in full. 1 av. Ib. = 7000 grains.-
274 VOCATIONAL MATHEMATICS FOR GIRLS
4. How many grains in 12 Ib. 15 oz. avoirdupois ? Give
the work in full.
5. Reduce 15 Ib. 10 oz. 20 grains avoirdupois to grains
troy weight. Give the work in full.
6. What part of a bushel is 2 pecks and 3 pints ? Give
the work in full and "the answer both as a decimal and as a
common fraction.
7. What will 10 bushels 3 pecks and 4 quarts of seed cost
at $ 2.10 per bushel ? Give the work in full.
8. What part of a troy pound is 50 grains, expressed both
decimally and in the form of a common fraction ?
9. A strawberry basket was found to be 65.2 cubic inches
in capacity. (a) How many cubic inches short was it ?
(&) WThat percentage of a full quart did it contain ? Give the
work in full.
10. In testing a spring scale it was found that in weighing
22 Ib. of correct test weights on same, the scale indicated
22 Ib. 101 oz. What was the percentage of error in this scale
at this weight ? Give the work in full.
VISITOR
1. A certain "home" had at the beginning of the year
$ 693.07, and received during the year donations amounting
to $ 1322.48. The expenses for the year were : salaries,
$387.25 ; printing, etc., $175 ; supplies, $651.15 ; rent, $104.25
heat, etc., $ 75 ; interest, $ 100 ; miscellaneous, $ 72.83. Find
the cash on hand at the end of the year. Give the work in
full.
2. Of the 72,700 persons relieved in a certain state at public
expense in the year ending March 31, 1912, 76 % were aided
locally, and the remainder by the state. Find the number
relieved by the state. Give the work in full.
CIVIL SERVICE 275
3. There was spent in state, city, and town public poor relief
in Massachusetts in one year the sum of $3,539,036. The
number of beneficiaries was 72,700. What was the average
sum spent per person ? Give the work in full.
4. Of the 72,900 persons aided by public charity in this
state in a certain year y9-^ were classed as sane. Of the re-
mainder, ± were classed as insane, ^ as idiotic, and the rest as
epileptic. How many epileptics received public aid? Give,
the work in full.
PART V — ARITHMETIC FOR NURSES
CHAPTER XIV
A NURSE should be familiar with the weights and measures
used in dispensing medicines. There are two systems used —
the English, based on the grain, and the Metric system, based
on the meter.
APOTHECARIES' WEIGHT
(Troy Weight)
20 grains (gr.) = 1 scruple (3)
33 =1 dram ( 3 ) = 60 gr.
83 =1 ounce ( 3 ) = 24 3 =480 gr.
123 =1 pound (ft) =96 3 =288 3 = 5760 gr.
The number of units is often expressed by Roman numerals
written after the symbols. (See Roman Numerals, p. 2.)
EXAMPLES
1. How many grains in iv scruples ?
2. How many grains in iii drams ?
3. How many grains in iv ounces ?
4. How many scruples in lb i ?
5. How many grains in lb iii ?
6. How many drams in lb iv ?
7. How many grains in 3 ii ?
8. How many scruples in 3 v ?
9. How many drams in 5 vii ?
10. How many ounces in lb viii ?
276
ARITHMETIC FOR NURSES 277
11. Salt 5 i will make how many quarts of saline solution,
gr. xc to qt. 1 ?
12. How many drains of sodium carbonate in 10 powders of
Seidlitz Powder ? Each powder contains gr. xl.
APOTHECARIES' FLUID MEASURE
60 minims (m) = 1 fluid dram = (f 3 ).
8 f 3 =1 fluid ounce (f 3).
16 f 3 =1 pint (0) = 128 f 3 = 7680 m.
8 O =1 gallon (C) = 128 f 3 =1024 f 3 .
EXAMPLES
1. How many minims in f 3 iv ?
2. How many minims in f 3 iii ?
3. How many fluid drams in 1 0 ?
4. How many minims in 5 pints ?
5. How many pints in 8 gallons ?
6. How many fluid drams in 0 ii ?
7. How many minims in f 5 viii ?
8. How many fluid drams in C vii ?
9. How many pints in C v ?
10. How many minims in f 5 ix ?
11. If the dose of a solution is m xxx and each dose contains
-^g- gr. strychnine, how much of the drug is contained in f 5 i
of the solution ?
12. 3 ii of a solution contains gr. i of cocaine. How much
cocaine is given when a doctor orders m x of the solution ?
Approximate Measures of Fluids
(With Household Measures)
An ordinary teaspoonful is supposed to hold 60 minims of
pure water and is approximately equal to a fluid dram. A
278 VOCATIONAL MATHEMATICS FOR GIRLS
A GRADUATE.
drop is ordinarily considered equivalent
to a minim, but this is only approxi-
mately true in the case of water. The
specific gravity, shape, and surface from
which the drop is poured influence the
size. In preparing medicines to be
taken internally, minims should never
be measured out as drops. There are
minim droppers and measures for this
purpose.
A level teaspoonful of either a fluid
or solid preparation is equal to a dram.
Level spoonfuls are always considered
in measurements.
1 teaspoonful = 1 fluid dram.
1 dessertspoonful = 2 fluid drains.
1 tablespoonful = 4 fluid drams or J fluid ounce.
1 wineglassf ul = 2 fluid ounces.
1 teacupful = 6 fluid ounces.
1 tumblerful = 8 fluid ounces.
EXAMPLES
1. How many dessertspoonfuls in 8 fluid ounces ?
2. How many wineglassfuls in 2 tumblerfuls ?
3. How many tablespoonfuls in 3 fluid drams ?
4. How many teaspoonful s in 5 fluid ounces ?
5. How many teacupfuls in 4 fluid drams ?
6. How many dessertspoonfuls in 6 fluid drams ?
7. How many teaspoonfuls in 1 gallon ?
8. How many drops of water in 1 quart ?
9. How many teaspoonfuls in 3 ounces ?
10. How many minims in 3 pints ?
ARITHMETIC FOR NURSES 279
11. What household measure would you use to make a solu-
tion, 3 i to a pint ?
12. Read the following apothecaries' measurements and give
their equivalents :
a. 3 iv. /. 3 ss.1
b. gr. v. g. 0 iv.
c. 0 ii. h. 3 ii.
d. 5 ii- *"• 3 iv.
e. § ij. j. I ss.
Metric System of Weights and Measures
(Review Metric System in Appendix.)
The metric system of weights and measures is used to a
great extent in medicine. The advantage of this system over
the English is that, in preparing solutions, it is easy to change
weights to volumes and volumes to weights without the use of
common fractions.
In medicine the gramme (so written in prescriptions to
avoid confusion with the dram) and the milligramme are the
chief weights used.
1 gramme = wt. of 1 cubic centimeter (cc. ) of water at 4° c.
1000 grammes = 1 kilogram or " kilo."
1 kilogram of water = 1000 cc. = 1 liter.
CONVERSION FACTORS
1 gramme = 15.4 or approx. 15 grains.
1 grain = 0.064 gramme.
1 cubic centimeter = 15 minims.
1 minim = 0.06 cc.
1 liter = 1 quart (approx.).
The liter and cubic centimeter are the principal units of
volume used in medicine.
1 ss means one-half.
280 VOCATIONAL MATHEMATICS FOR GIRLS
A micro-millimeter is used in measuring microscopical dis-
tances. It is r^ mm. and is indicated by the Greek letter /A.
To convert cc. into minims multiply by 15.
To convert grammes into drams divide by 4.
To convert cc. into ounces divide by 30.
To convert minims into cc. divide by 15.
To convert grains into grammes divide by 15.
To convert fluid drams into cc. multiply by 4.
To convert drams into grammes multiply by 4.
1 grain = 0.064 gramme.
2 grains = 0.1 gramme.
5 grains = 0.3 gramme.
8 grains = 0.5 gramme.
10 grains = 0.6 gramme.
15 grains = 1 gramme.
1 milligramme = 0.0154 grain.
Review Troy (apothecary) and avoirdupois weights, pages
43 and 276.
EXAMPLES
1. A red corpuscle is 8 /x, in diameter. Give the diameter in
a fraction of an inch.
2. A microbe is 25000" inch in diameter. What part of a
millimeter is it ?
3. Another form of microbe is -Q^-^Q of an inch in diameter.
What part of a millimeter is it ?
4. A bottle holds 48 cc. What is the weight of water in the
bottle when it is filled ?
5. How many liters of water in a vessel containing 4831
grams of water ?
6. Give the approximate equivalent in English of the
following :
a. 48 grammes d. 8 kilos
b. 3.6 kilograms e. 3:9 grammes
c. 3.5 liters /. 53 milligrammes
ARITHMETIC FOR NURSES 281
7. Give the approximate equivalents in the metric system
of the following :
a. 39 grains e. 13 quarts
b. 4 drams /. 2 gallons
c. 1 fluid drams g. 39 minims
d. 47 flb h. 8321 grains
Approximate Equivalents between Metric and Household
Measures
1 teaspoon ful = 4 cc. or 4 grams of water.
1 dessertspoonful = 8 cc. or 8 grams of water.
1 tablespoonful = 16+ cc. or 15+ grams of water.
1 wineglassful = 60 cc. or 60 grams of water.
1 teacupful = 180 cc. or 180 grams of water.
1 glassful = 240 cc. or 240 grams of water.
EXAMPLES
(Give approximate answers.)
1. What is the weight of two glassfuls of water in the
metric system ?
2. What is the weight of a gallon of water in the metric
system ?
3. What is the weight of three liters of water in the
English system ?
4. What is the volume of four ounces of water in the
metric system ?
5. What is the volume of twelve cubic centimeters of water
in the English system ?
6. What is the volume of f 3 iii in the metric system ?
7. What is the volume of eighty grammes of water ?
8. What is the weight of 360.1 cc. of water ?
9. What is the volume of 4 kilos of water ?
10. What is the weight of 6.1 liters of water ?
11. With ordinary household measures how would you
obtain the following : 5 gm., m xv, 1.5 L., 25 cc., 5 ii, f 5 ss ?
282 VOCATIONAL MATHEMATICS FOR GIRLS
METRIC SYSTEM
EXAMPLES
1. Change the following to milligrammes :
8 gin., 17 dg., 13 gm.
2. Change the following to grammes :
13 mg., 29 dg., 7 dg., 21 mg.
3. Add the following :
11 mg., 18 dg., 21 gm., 4.2 gm.
Express answer in grammes.
4. Add the following :
25 mg., 1.7 gm., 9.8 dg., 21 mg.
Express answer in milligrammes.
5. The dose of atropine is 0.4 mg. What fraction of a
gramme is necessary to make 25 cc. of a solution in which 1 cc.
contains the dose ?
6. Give the equivalent in the metric system of the following
doses :
a. Extract of gentian, gr. ^.
b. Tincture of quassia, 3 i.
c. Tincture of capsicum, m iii.
d. Spirits of peppermint, 3 i.
e. Cinnamon spirit, m x.
/. Oil of cajuput, m xv.
g. Extract of cascara sagrada, gr. v.
h. Fluid extract of senna, 3 ii.
i. Agar agar, 5 ss.
7. Give the equivalent in the English system of the follow-
ing doses :
a. Ether, 1 cc.
6. Syrup of ipecac, 4 cc.
c. Compound syrup of hypophosphites, 4 cc.
ARITHMETIC FOR NURSES 283
d. Pancreatin, 0.3 gm.
e. Zinc sulphate, 2 gm.
/. Copper sulphate, 0.2 gm.
g. Castor oil, 30 cc.
h. Extract of rhubarb, 0.6 gm.
i. Purified aloes, 0.5 gm.
DOSES
Since all drugs are harmful or poisonous in sufficiently large
quantities, it is necessary to know the least amount needed to
produce the desired change in the body — the minimum dose.
This has been ascertained by careful and prolonged experiments.
Similar experiments have told us the largest amount of drug
that one can take without producing dangerous effect — the
maximum dose.
On the average, children under 12 years of age require smaller
doses than adults. To determine the fraction of an adult dose
of a drug to give to a child, let the child's age be the numer-
ator, and the sum of the child's age plus twelve be the denomina-
tor of the fraction. For infants under one year, multiply the
adult dose by the fraction ^ in months _
lou
To illustrate : How much of a dose should be given to a
child of four ?
Age of child = 4.
Age of child + 12 = 16.
Fraction of dose T\ = \, Ans. £ of a dose.
EXAMPLES
1. What is the fraction of a dose to give to a child of 8 ?
2. What is the fraction of a dose to give to a child of 6 ?
3. What is the fraction of a dose to give to a child of 3 ?
4. What is the fraction of a dose to give to a child of 10 ?
284 VOCATIONAL MATHEMATICS FOR GIRLS
5. If the normal adult dose of aromatic spirits of ammonia
is 1 dram, what is the dose for a child of 7 ?
6. If the normal adult dose of castor oil is one-half ounce,
what is the dose for a child of 6 ?
7. If the normal adult dose of epsom salts is 4 drams,
what is the dose for a child of 4 ?
8. If the normal adult dose of strychnine sulphate is ^ gr.,
what is the dose for a child of 8 ?
9. If the normal adult dose of ipecac is 15 grains, what is
the dose for a child of 11 ?
10. If the normal adult dose of aromatic spirits of ammonia
is 4 grammes, what is the dose for a child of 5 m'onths ?
11. If the normal adult dose of ipecac is 1 gramme, what is
the dose for a child 10 months old ?
12. The normal adult dose of strychnine sulphate is 3.2 mg.
How much should be given to a child 2 years old ?
STRENGTH OF SOLUTIONS
A nurse should know about the strength of substances used
in treating the sick. Most of these substances are drugs which
are prepared according to formulas given in a book called a
Pharmacopoeia. Preparations made according to this standard
are called official preparations, and often have the letters
U. S. P. written after them to distinguish them from patented
preparations prepared from unknown formulas.
Drugs are applied in the following forms : solutions, lini-
ments, oleates, cerates, powders, lozenges, plasters, ointments,
etc.
An infusion is a liquid preparation of the drug made by
extracting the drug with boiling water. The strength of an
infusion is 5% of the drug, unless otherwise ordered by the
physician.
ARITHMETIC FOR NURSES 285
The strength of a solution may be written as per cent or in
the form of a ratio. A 10 % solution means that in every
100 parts by weight of water or the solvent there are 10 parts
by weight of the substance. This may be written in form
of a fraction — y1^- or y1^-. In other words, for every ten parts
of solvent there is one part of substance. Since a fraction may
be written as a ratio, it may be called a solution of one to ten,
written thus, 1 : 10.
EXAMPLES
1. Express the following per cents as ratios: 5%, 20%,
2%, 0.1%, 0.01%.
Since per cent represents so many parts per hundred, a
ratio may be changed to per cent by putting it in the form
of a fraction and multiplying by 100. The quotient is the per
cent.
2. Express the following in per cents : 1 : 4, 1 : 3, 1 : 6,
1 : 15, 1 : 25, 1 : 40.
3. Arrange the following solutions in the order of their
strength : 3 %, 8 %, 24 %, 6 %, 1 : 10, 1 : 14, 1 : 50, 40 %, 1 : 45,
50%.
4. Express the strength of the following solutions as per
cents, and in ratios.
a. 80 ounces of dilute alcohol contains 40 ounces of absolute
alcohol.
6. 6 pints of dilute alcohol contains two pints of absolute
alcohol.
5. Change the following ratios into per cents : 1 : 18, 1 : 20,
1:5, 1 : 35, 1 : 100. Arrange in order, beginning with the
highest.
6. Change the following per cents to ratios : 33 % , 12 % ,
15%, .5%,!%.
7. Is it possible to make an 8 % solution from 4 % ? Ex-
plain.
286 VOCATIONAL MATHEMATICS FOR GIRLS
8. Express the following strengths in terms of ratio :
a. 25 cc. of alcohol in 100 cc. solution.
b. 5 pints of alcohol in 3 qts.
c. f § i contains f 3 iii.
9. Express the following strengths in terms of per cent :
a. 50 cc. of ^solution containing 5 cc. of peroxide of hydrogen.
b. f 5 iii of dilute alcohol containing ^ ii of pure alcohol.
How to Make Solutions of Different Strengths from Crude
Drugs or Tablets of Known Strengths
Exact Method
ILLUSTRATIVE EXAMPLE. — How much water will be neces-
sary to dissolve 5 gr. of powdered bichloride of mercury to
make a solution of 1 part to 2000 ?
Since the whole powder is dissolved,
1 part is 5 gr.
2000 parts = 10,000 grains.
480 gr. = 1 oz.
32 oz. = 1 qt.
10000 _ 20f . Approx. 21 oz. or 1£ pints of water should be used to
dissolve it.
The above example may be solved by proportion, when x = no. oz. of
water necessary to dissolve powder ; then wt. of powder : drug : : x : water.
f „ : 1 : : X : 2000.
> = 20|oZ. Approx. 21o,
EXAMPLES
Solve the following examples by analysis and proportion :
1. How much water will be required to dissolve 5 gr. of
powdered corrosive sublimate to make a solution of 1 part to
1000?
ARITHMETIC FOR NURSES 287
2. How much water will be required to dissolve a 7^-grain
tablet of corrosive sublimate to make a solution 1 part to 2000 ?
ILLUSTRATIVE EXAMPLE. — How much of a 40 % solution
of formaldehyde should be used to make a pint of 1 : 500
solution ?
480 minims = 1 oz.
7680 minims = 1 pint.
= 15295 minims = amt. of pure formaldehyde necessary to make a
pint of 1 : 500.
Since the strength of the solution is 40%, 15^ minims represents but
A°o or f °f the actual amount necessary. Therefore, the full amount of
40 °/o solution is obtained by dividing by f .
192
?j* x I = — = 38§ minims to a pint.
To Determine the Amount of Crude Drug Necessary to Make a
Certain Quantity of a Solution of a Given Strength
To illustrate : To make a gallon of 1 : 20 carbolic acid solu-
tion, how much crude carbolic acid is necessary ?
1 : 20 : : x : 1 gal.
1 : 20 : : x : 8 pints or 128 ounces.
20 x = 128 ounces.
x = 6f ounces crude carbolic acid.
EXAMPLES
1. How much crude boric acid is necessary to make 6 pints
of 5 % boric acid ?
5 : 100 : : x : 6 pts.
6 : 100 : : x : 576 drams.
100 x = 2880.
x - 28.8 drams.
2. How much crude boric acid is necessary to make 2 quarts
of 1 : 18 boric acid ?
288 VOCATIONAL MATHEMATICS FOR GIRLS
3. How much crude drug is necessary to make f 3iii of 2 %
cocaine ?
4. How many 7J~grain tablets are necessary to make 2 gal-
lons of 1 : 1000 bichloride of mercury ? 1
5. How much crude drug is necessary to make 0 vi of 1 : 20
phenol solution ?
6. How much crude drug is necessary to make 0 vii of 1 : 500
bichloride of mercury ?
7. How much crude drug is necessary to make 0 iii of 1 : 10
chlorinated lime ?
Hypodermic Doses
Standard strong solutions and pills are kept on hand in a
hospital and from these weaker solutions are made as required
by the nurse for hypodermic use. This is done by finding out
what part the required dose is of the tablet or solution on
hand. The hypodermic dose is not administered in more than
25 or less than 10 minims. The standard pill or solution is
dissolved or diluted in about 20 minims and the fractional
part, corresponding to the dose, is used for injection.
To illustrate : A nurse is asked to give a patient -^ir gr.
strychnine. She finds that the only tablet on hand is -£$ gr.
How will she give the required dose ?
•dro -s- sV = imr x 30 = -^
The required dose is ^ of the stock pill. Therefore she dissolves the
pill in 80 minims of water and administers 12 minims. The reason for
dissolving in 80 rather than in 20 minims is to have the hypodermic
dose not less than 10 minims.
EXAMPLES
1. Express the dose, in the illustrative example, in the
metric system.
1 Hospitals usually use 1 tablet for a pint of water to make 1 : 1000 solution.
ARITHMETIC FOR NURSES 289
2. How would you give a dose g-1^ gr. strychnine sulphate
from stock tablet -^ gr. ?
3. How would you give gr. -J^, if only -^ grain were on
hand?
4. How would you give gr. y1^, if only ^--grain tablets were
on hand ?
5. How would you give gr. -£$, if only ^y-grain tablets were
on hand ?
6. How would you give gr. g-1^, if only T^7-grain tablets were
on hand ?
7. How would you give gr. y^ of atropine sulphate, if only
in tablets were on hand ?
8. How would you give gr. -^ of apomorphine hydrochloride
if only y^grain tablets were on hand ?
To Estimate a Dose of a Different Fractional Part of a Grain
from the Prepared Solution
Nurses are often required to give a dose of medicine of a
different fractional part of a grain from the drug they have.
To illustrate : Give a dose of -^ gr. of strychnine when the
only solution on hand is one containing -fa gr. in every 10
minims.
Since ^ grain is contained in 10 minims,
1 grain or 30 x -^ grain is contained in 300 minims.
Then, ^ of a grain is ^ of 300 = ^ x 300 = 12 m.
EXAMPLES
1. What dose of a solution of 60 minims containing -^ gr.
will be given to get T^ gr. ?
2. Reckon quickly and accurately how much of a tablet
gr. i should be given to have the patient obtain a dose gr. y1^.
290 VOCATIONAL MATHEMATICS FOR GIRLS
3. What dose of a solution of in x containing gr. i morphine
sulphate will be given to give gr. 1 ?
4. What dose of a solution of m xx containing gr. -^ strych-
nine sulphate will be given to give gr. -fa ?
5. What dose of a solution of 1 cc. containing 0.1 cc. of the
fluid extract of nux vomica will be given to give 0.06 cc. ?
To Obtain a Definite Dose from a Stock Solution
of Definite Strength
To illustrate : To give a patient a yL-grain dose when the
stock solution has a strength of 1%.
1 °IG solution means that each drop of the solution contains T^7 part or
— ££-' of strychnine.
100
2*5 gr. is contained in as many drops as T^ is contained in it.
A * T*<T = h x 100 = 4.
Therefore 4 drops of the 1 °fo solution contains "^ gr.
EXAMPLES
1. To give fa gr. strychnine from 2 % solution.
2. To give 2T gr. strychnine from solution containing in
ten minims ^ gr.
3. To give 3 gr. of caffeinic sodium benzoate from a 25 %
solution.
4. To give 2-J-Q- gr. of atropine from 1 % solution.
5. To give Y^-Q gr. of strychnine from l °/0 solution.
6. To give ^ gr. atropine from solution containing in ten
minims -^ gr.
ARITHMETIC FOR NURSES 291
Temperature
The temperature of the body is due to the combined activity
of all its various systems but is regulated chiefly by the skin and
circulatory system. It remains very nearly constant in the nor-
mal person, in spite of the variations of the outdoor temperature.
A variation of more than one degree from the normal tempera-
ture, that is, above 991° F. or below 971° F., may be regarded as
a sign of a disease. The temperature is obtained
by means of a small thermometer — called a clinical
thermometer. See Appendix, page 337, for descrip-
tion of the different thermometers.
Temperature readings are usually expressed in
the Fahrenheit scale, but scientific data gathered
in laboratories are expressed according to the Centi-
grade scale. Therefore, we should be able to change
readings from one scale to another.
Fahrenheit readings may be obtained by adding
32° to f of the Centigrade reading. This rule may '
be abbreviated into a formula as follows :
^=1(7+32°,
where F = Fahrenheit reading,
C — Centigrade reading.
Centigrade readings may be obtained by sub
tracting 32° from the Fahrenheit and taking -| of
the remainder. This may be abbreviated into a
formula as follows : CLINICAL
THBRMOM-
320.
EXAMPLES
1. Albumin is coagulated by heat at 155° F. What is the
degree Centigrade ?
292 VOCATIONAL MATHEMATICS FOR GIRLS
2. When milk is heated above 170° F., the albumin coagulates
and forms a scum on the milk. To what degree on Centigrade
scale does this correspond ?
3. Egg albumin (white of egg) coagulates at 138° F. At
what degree on the Centigrade scale ?
4. Milk is pasteurized by bringing milk in the bottle to a
temperature of 165° F. To what degree on the Centigrade
scale ?
5. " Gentle heat " is a term used to denote the temperature
between 32° to 38° C. What are the corresponding degrees on
the Fahrenheit scale ?
Baths
(Change the following temperatures to Centigrade scale.)
A bath with a temperature between 33° and 65° F. is known
as a cold bath.
A bath with a temperature between 65° and 75° F. is known
as a cool bath.
A bath with a temperature between 75° and 85° F. is known
as a temperate bath.
A bath with a temperature between 85° and 92° F. is known
as a tepid bath.
A bath with a temperature between 92° and 98° F. is known
as a warm bath.
A bath with a temperature between 98° and 112° F. is known
as a hot bath.
i
Medical Chart (Graph)
(See Graphs in the Appendix.)
In order to follow the condition of a patient from day to
day, the temperature, the pulse beats, and respirations are
recorded morning and night on a special ruled chart. The
name of the patient is placed on each chart.
ARITHMETIC FOR NURSES
293
NAME.....
Day of
Month
HoMf
101'
i=t
I I
! I
TT
294 VOCATIONAL MATHEMATICS FOR GIRLS
EXAMPLES
Chart the following case of pneumonia :
2 day
3 day
4 day
5 day
6 day
7 day
8 day
9 day
10 day
11 day
12 day
13 day
14 day
Morning
102°
102.6°
102.4°
102.4°
102.4°
102.4°
101.8°
102.9°
102°
98.4°
97.4°
97.4°
98.2°
Evening
104°
105°
104.2°
103.6°
104°
104.4°
103°
104°
102.8°
98.5°
98.2°
98.2°
98.4°
PROBLEMS IN HOUSEHOLD CHEMISTRY
Bacteria are low forms of vegetable and animal life, and some
are capable of producing disease.
Chemicals that are employed to destroy bacteria are known
as germicides. Those which limit the growth or destructive
power of bacteria are called antiseptics. Deodorants remove or
neutralize unpleasant odors.
1. Bacteria multiply in all temperatures between 2° and
70° C. What are the temperatures in the Fahrenheit scale
within which bacteria will grow ?
2. Creolin is used as a germicide and deodorant for offen-
sive wounds in solutions of from 2 to 5 °/0. The creolin must
never be added to water over 98° C., as its strength is impaired.
What is the corresponding temperature on the Fahrenheit
scale ?
3. The most important medium or preparation for growing
bacteria is nutrient bouillon. It is made of the following :
ARITHMETIC FOR NURSES 295
Meat extract 5 grams
Peptone 10 grams
Salt 5 grams
Water 1 liter
What per cent of each ?
4. A sugar bouillon culture is used for artificially cultivat-
ing bacteria. It is made by adding 1 % of glucose to nutrient
bouillon. How many grams of glucose to a liter of solution?
5. Carbolic acid is bought by hospitals in a 95 % solution
and diluted as required. A solution of carbolic acid 1 : 20 is
used to destroy germs. How much 95 % solution will be
required to make 5 gallons 1 : 20 ?
6. 1 : 1000 solution means how many grams to the gallon ?
7. A normal salt solution is made by dissolving 9 grams
of salt to the quart. How many teaspoonfuls to the quart?
How many grains to the quart ?
8. What is the ratio of a pure drug ? What is the per-
centage of purity of a pure drug ?
9. If I desire to make a lotion of 1 : 1000 corrosive subli-
mate, how much of the substance would be added and how
much water used ?
10. How much water and corrosive sublimate are required
for a gallon of the following strengths?
a. 1:2000. d. 1:20,000.
b. 1:4000. e. 1:100,000.
c. 1:10,000. /. 1:150,000.
11. A saturated solution of boric acid may be made by dis-
solving 3 v to pint (0 i) of water. What is the per cent of the
saturated solution?
12. A saturated solution of KMn04 may be made by dis-
solving § i to 0 i ? What is the per cent ?
13. How much of the saturated solution should be added to
water 0 i to make 1 °/0 solution ?
296 VOCATIONAL MATHEMATICS FOR GIRLS
Water Analysis
Every nurse should be able to interpret a biological and
chemical analysis of water.
Terms used in Chemical and Bacteriological Reports
The following brief explanation of the terms used in chemi-
cal and bacteriological examinations of water is given in order
that the reports of analyses of samples may be clearly under-
stood. As the quantities to be obtained by analyses are usu-
ally very small, they are ordinarily expressed in parts per
million (p. p. m.), and always by weight.
Turbidity of water is caused by fine particles such as clay,
silt, and microscopic organisms.
Sediment is self-explanatory. The amount and nature of
the sediment are usually noted.
Color is measured by comparing the sample with artificial
standards made by dissolving certain salts in distilled water,
or sometimes with colored glass disks. The color of large
lakes is usually below 0.10.
Odor. This requires no explanation.
Residue on Evaporation, or Total Solids, indicates the total
solid matter, both organic and inorganic, in 1,000,000 parts
of water. The determination is made by placing about 100
grams of water in a platinum dish and weighing the whole
accurately. The water is then evaporated to dryness by mod-
erate heat and the dish again weighed ; the difference between
this and the weight of the empty dish gives the total solids in
the water. The dish is then heated red hot, to burn out the
organic matter, when the weight of the remaining ash gives
the inorganic or fixed solids. The loss on ignition, sometimes
reported, is a measure of the organic solids.
Ammonia. Ammonia in water indicates the presence of
organic matter in an advanced stage of decay, and although
ARITHMETIC FOR NURSES 297
the amount is small, it affords a valuable indication of what
is going on in the water. It is determined in two forms,
called " free " and " albuminoid."
Free Ammonia is that which has actually been set free in
the water in the process of decay of organic matter, while
Albuminoid Ammonia is that which has not yet been set free,
but which is liable to be freed under the action of the oxygen
in the water. The sum of the two gives an indication of the
total amount of organic matter in the water.
Water which has 0.05 p. p. m. of free ammonia is probably
pure, while if it has more than 0.1 p. p. m., it is perhaps dan-
gerous. A low figure for albuminoid ammonia is 0.05 p. p. m.,
and a high one is 0.50.
Chlorine in water usually represents sodium chloride, or
common salt. It may be due to sewage pollution or to near-
ness to the ocean. It is always found in natural waters, the
normal amount decreasing from the seacoast inland. If the
amount exceeds 20 p. p. m., it may cause corrosion in boilers
and plumbing fixtures. Properly interpreted, the chlorine
content is one of the most useful indexes of the extent of
sewage pollution.
Nitrogen is usually determined in the form of nitrates and
nitrites, the former being the final result of decomposition,
while the latter is the incomplete result of the same action.
If an analysis shows the ammonia to be low and the nitrates
high, it indicates that the water has become completely puri-
fied, while the reverse indicates that the decaying process
is going on and the water is dangerous. In good drinking
water the nitrates may be as high as 1 or 2 p. p. m., while
the nitrites, if present, are practically always a sign of pol-
lution.
Oxygen Consumed. This is the amount of oxygen absorbed
by the water from potassium permanganate. As the oxygen
is absorbed by the organic matter present in the water, the
amount consumed gives a measure of the amount of impurities
298 VOCATIONAL MATHEMATICS FOR GIRLS
contained in it. Less than 1 p. p. m. indicates probable pur-
ity, while as high as 4 or 5 p. p. m. indicates danger in drink-
ing water.
Hardness. A water is said to be " hard " when it contains
in solution the carbonates and sulphates of calcium or mag-
nesium. When a hard water is used for washing, these salts
have to be decomposed by soap before a lather can be formed.
In boilers, a hard water forms scale. Hardness is expressed
by the number of parts of calcium carbonate in 1,000,000
parts of water. Rain water has a hardness of about 5, and
river waters of from 50 to 100.
Iron may be troublesome in a water used for domestic pur-
poses if it is present in quantities greater than 0.3 to 0.5
p. p. m.
Alkalinity or Temporary Hardness is that part of the total
hardness which is due to carbonates removable by boiling, thus
causing the formation of scale. For purposes of softening
water for boiler use, it is necessary to know both the total
hardness and the alkalinity.
Bacteria. While it is obvious that the quality of a water of
turbid appearance and unpleasant odor is suspicious, it does
not follow that it is dangerous, nor is a water which is entirely
free from color and odor necessarily a safe drinking water, for
epidemics of typhoid have been caused by such. The bacteri-
ological examination of water, by which the number of bacteria
present in one cubic centimeter (1 cc.) is determined, is there-
fore an important part of an analysis.
As a general statement, it may be said that fresh water con-
taining less than 100 bacteria per cc. is pure, that water contain-
ing 500 bacteria per cc. should be viewed with suspicion, and
that water containing 1000 bacteria per cc. is undoubtedly con-
taminated. In considering these figures with relation to a
water supply, it must be remembered that all natural surface
waters contain some bacteria and that, except where there
is pollution, the greater part of them are absolutely harmless.
ARITHMETIC FOR NURSES 299
The bacteria are so small that they may be seen only with
the aid of a high-powered microscope. In order to count them
a culture jelly of gelatine, albumin, and extract of beef is
prepared and 1 cc. of the water is thoroughly mixed with
10 cc. of the culture jelly, a small measured portion of this
mixture then being poured in a thin layer on a sterilized plate
to harden. Each bacterium eats and multiplies to such an
extent that in about forty-eight hours a visible colony is
produced. From a count of these colonies within a measured
area of the plate the number of bacteria in the original 1 cc.
of water is determined.
Different species of bacteria may be detected by the use of
different media for development, or they may be found by
further examination with the microscope. The well-known
colon bacillus (B. coli), which, although harmless itself, is an
indication of sewage pollution, is detected by the gas which it
produces in a closed tube. As B. coli are found in practically
all warm-blooded animals and sometimes in fish and elsewhere,
the finding of a few in large samples of water, or their occa-
sional discovery in small samples, is of no special significance ;
but if they are found in a larger proportion in small samples
and in considerable numbers in larger ones, sewage pollution is
indicated.
EXAMPLES
1. If a bacterium multiplies tenfold every half hour in a
person's mouth, how many will be produced in twenty-four
hours ?
2. A sample of water contains 0.24 parts per million of free
ammonia. How many parts per 100,000 ?
3. A sample of water contains 1.1 parts per million of iron.
How many parts per 100,000 ?
4. A sample of water contains 10 parts per million of lime.
How many parts per 10,000 ?
300 VOCATIONAL MATHEMATICS FOR GIRLS
EXAMPLES ON ANALYSES OF WATER
(Parts in 100,000)
KESIDUE ON
EVAPORATION
AMMONIA
NITROGEN
AS
Albuminoid
Loss
H
Z:
. 3
1
Total
on
Igni-
tion
Fixed
Free
Total
In
Solu-
tion
In
Sus-
pen-
sion
CHLORI
Nitrate
Nitrites
ii
II
HARDN
1
a.
4.00
1.65
2.35
.0026
.0190
.0156
.0034
.72
.0030
.0001
.31
1.3
.0160
b.
4.65
2.00
2.65
.0028
.0172
.0148
.0024
.68
.0030
.0000
.28
1.3
.0080
c.
3.85
1.15
2.70
.0014
.0148
.0130
.0018
.68
.0000
.0000
.31
1.1
.0080
d.
4.20
1.50
2.70
.0052
.0140
.0128
.0012
.71
.0000
.0000
.32
1.1
.0050
c,.
4.15
1.35
2.80
.0018
.0170
.0152
.0018
.71
.0000
.0000
.26
1.0
.0080
f.
5.00
1.75
3.25
.0014
.0162
.0142
.0020
.73
.0000
.0000
.36
1.0
.0120
'/•
4.35
1.60
2.75
.0020
.0178
.0150
.0028
.70
.0010
.0001
.24
1.3
.0080
h.
4.10
1.15
2.95
.0018
.0162
.0136
.0026
.71
.0010
.0001
.24
1.3
.0100
1. Give the number of parts of free ammonia in 10,000 in a.
2. Give the number of parts of nitrates in 10,000 in b.
3. Give the number of parts of nitrites in 10,000 in d.
REVIEW EXAMPLES
1. Give the number of cubic centimeters of water you
would measure out to get the following :
a. 70 gm. b. 11 kg. c. 0.4 gm. d. 61 mg.
2. How much would the following amounts of water weigh ?
a. 9 1. b. 4.7 cc. c. 1 1. d. 48 cc.
3. If the dose of aromatic spirits of ammonia is 30 minims,
what is the dose for a child 6 years old ?
4. Give the approximate equivalents in household measures
of the following :
a. 7 drams c. 4 ounces e. 12 fluid ounces
b. 36 grams d. 90 minims /. 3 fluid drams
ARITHMETIC FOR NURSES 301
5. Give the approximate equivalents in household measures
of the following :
a. 1500 cc. c. 3 liters e. 1 gramme
6. 11 cc. d. 0.003 grain /. 0.008 gramme
6. How many grammes in 3 ounces of 1 % solution ?
7. How many drams in 1 gallon of 1 : 50 solution ?
8. How many grammes in a liter of 10 % solution ?
9. How many grammes in 5 liters of 1 : 25 solution ?
10. How many teaspoonfuls of pure carbolic acid in a gallon
of 1 % solution ?
11. How many drops (minims) of carbolic acid in a quart
of 1 : 1000 solution ?
12. A basin of rain water has a temperature of 94° F. Give
the equivalent on the Centigrade scale.
13. A cool bath registers a temperature of 26° C. Give the
equivalent on the Fahrenheit scale.
14. A dose of ipecac is 20 to 30 grammes. What is the dose
for a child of seven years ?
15. A dose of 1 : 500 solution means how many grammes to
a quart?
16. Given a 5% solution of silver nitrate, how would you
make a gallon of 1 : 5000 solution ?
17. How would you make a gallon of 3 % solution of acetic
acid from the pure acid ?
18. How would you make two quarts of 5 % solution of car-
bolic acid from pure acid ? (Consider pure acid 95 %.)
19. A 1 : 50 solution is used for disinfecting wounds. How
would you make a gallon of this fluid from standard solution ?
(Consider standard strength about 40 %.)
302 VOCATIONAL MATHEMATICS FOR GIRLS
20. A 2 % solution of boric acid is used for eye and ear
irrigations. How much boric acid will be necessary to make a
quart of the solution ?
21. Give the approximate equivalents of metric and
apothecaries' measures of the following :
a. 31 cc. d. 50 minims
b. yi¥ gram e. 5 pints
c. 1.6 gram /. 101 cc.
22. If the pharmacy nurse buys 3 oz. of trional, how many
powders of 10 grains each can she make ?
23. The dose of the tincture of opium is 0.5 cc. ; 10 cc. of
the tincture contains 1 gm. of opium ; 12 % of opium is
morphine. How many milligrams of morphine in one dose of
the tincture ?
24. a. Convert the following to milligrams : 5 dg. and 0.27 gm.
6. Convert the following to grams : 483 dg. and 7 mg.
25. How much alcohol (15 °/G strength) will be necessary to
make a quart of alcohol containing 80 % volume of absolute
alcohol? 66%? 37%? 75%?
26. If lactic acid is composed of 75 % of absolute acid, how
much absolute acid in a pound of the official preparation ?
27. Diluted alcohol contains 41.5 % absolute alcohol. How
much absolute alcohol in a gallon of dilute alcohol ?
28. The dose of morphine sulphate is 0.008 gm. What is the
dose for a baby 7 months old ?
29. The dose of camphorated tincture of opium (paregoric)
is f 3 i. What is the dose for a baby 6 months old ?
30. Hands and arms are often disinfected by washing in a
solution of permanganate of potash (two ounces to four quarts
of water) followed by immersion in a solution of oxalic acid
(eight ounces to four quarts of water). What is the percentage
of each ?
ARITHMETIC FOR NURSES 303
31. Adhesive iodoform gauze is made by saturating sterilized
gauze in the following solution :
Iodoform 22 grams
Resin 10 grams
Glycerine 5 cc.
Alcohol 26 cc.
(Consider specific gravity of alcohol and glycerine as 1.)
What per cent of each ? Give quantity in English system.
32. How much of each ingredient should be used in prepar-
ing a pound of the following mass ?
Zinc oxide 5 parts
Gelatine 5 parts
Glycerine 12 parts
Water 10 parts
33. What per cent of the following solution is atropine
sulphate ?
Atropine sulphate 1^ gr.
Water \ fluid ounce
34. What amount of carbolic acid crystals is used to make
4 oz. of 3 % carbolized petrolatum ?
35. What per cent of the following solution is boric acid ?
Boric acid 18 gr.
Water 1 oz.
36. How much bichloride of mercury is required to make
1 qt. of a 1 : 25,000 solution ?
37. How much potassium permanganate will be necessary
to make a pint of a 1 : 1000 solution ?
PART VI — PROBLEMS ON THE FARM
CHAPTER XV
EVERY young person who lives on the farm has more or less
to do with the bookkeeping and the arithmetic connected with
the selling of the eggs, milk, and other products. Very few
of the men on the farm have the time or the inclination to do
this work, and it is usually performed by the wife or daughter.
EXAMPLES
1. I sold 16 dozen eggs at 30 cents a dozen and took my
pay in butter at 40 cents a pound. How many pounds did I
receive ?
2. A dealer bought 16 cords of wood at $ 4 a cord and sold
it for $ 96. Find the gain.
3. Three men bought a farm. Henry paid $ 1135.75,
Philip $2400.25, and Carl as much as Henry and Philip.
Find the value of the farm.
4. A farmer divided his farm as follows : to his elder son
he gave 257-f acres, to his younger son 200fV acres, and to his
wife as many acres as to his two sons. How many acres in
the farm ?
5. One farm contains 287f acres and another 244J acres.
Find the difference between them.
6. One bin contains 165^ bushels of grain and the other
bin 184^ bushels. How many bushels more does the larger
bin contain than the smaller ?
304
PROBLEMS ON THE FARM 305
7. From a farm of 375J acres, 84^ acres were sold. How
many acres remained ?
8. A farmer owning 57f acres of land sold 281 acres and
afterwards bought 141 acres. How many acres did he then
own?
9. A farm contained 132 acres, one-eighth of which is
woodland, one-sixth is pasture, and the remainder is culti-
vated. What part of the farm is cultivated ? How many
acres are cultivated ?
10. From four trees, 14J barrels of apples were gathered.
One man bought 5^ bbl., another 31 bbl. How many barrels
remained ?
11. I owned two-fifths of a farm and sold three-fourths of
my share for $ 1350. Find value of the whole farm.
12. I bought 5 loads of potatoes containing 331 bushels,
27f bushels, 40J bushels, 35^ bushels, and 29^ bushels. I
sold 12J bushels to each of three men, and 25^- bushels to each
of four men. How many bushels were left ?
13. If two-thirds of a farm costs $ 2480, what is the cost
of the farm ?
14. Mr. Thomas bought 168 sheep at $5.50 a head. He
sold three-sevenths of them at $ 6 a head, and the remainder
at $ 7 a head. Find the gain.
15. A farm is divided into four lots. The first contains
SOy7-^ acres, the second 4211 acres, the third 35^ acres, the
fourth 28|- acres. How many acres in the farm ?
16. A farmer sold sheep for $ 62.50, cattle for $ 102.60, a
horse for $ 125.75, and a plow for $ 18.25. How much did he
receive ?
17. Farmer Blake raised 114 bushels of apples and 73f
bushels of pears. How many more bushels of apples than
pears did he raise ?
306 VOCATIONAL MATHEMATICS FOR GIRLS
18. A farmer paid $ 78 for a cow, $ 165 for a horse. How
much more did the horse cost than the cow?
19. Mr. Borden has 450T7¥ acres of woodland and sells 304f
acres. How much has he left ?
20. Mr. Sherman bought ten acres of land at $ 65 an acre
and sold it for $ 24.60 an acre. How much did he lose ?
21. A's farm contains 265f- acres, B's 43^ acres. What is
the difference in the size of their farms ?
22. Mr. Grover had 110 acres of land, and sold 7^%- acres.
' ID
How many acres had he left ?
23. Mr. Dean sold one-third of his farm to one man, one-
fourth to another, and one-eighth to another. What part had
he left ?
24. I paid $ 365.75 for a horse, and sold him for four-fifths
of what he cost. What was the loss ?
25. How many bushels of grain can be put into 16 bags, if
they hold 2| bushels each ?
26. A farmer carries 35 bushels of apples to market. What
is half this load worth at 75 cents a bushel ?
27. I paid $76.50 for 18 sheep. What was the average
price ?
28. Mr. Platt gave 435 acres of land to his sons, giving each
721 acres. How many sons had he ?
29. If 41 bushels of potatoes were bought for $ 3.60, how
many bushels can be bought for $ 10.80 at the same price per
bushel ?
30. Mr. White paid $ 16.25 for 2^ cords of wood. How
many cords could he buy for $ 74.75 at the same price per
cord?
31. A father divided 183 acres of land equally among his
sons, giving to each 45 J acres. How many sons had he ?
PROBLEMS ON THE FARM 307
FARM MEASURES
(Review Mensuration and Table of Measures.)
1. If a bushel of shelled com contains 1J cubic feet, how
many bushels in a bin 8' x 4' x 2' 6"?
2. A bushel of ear com contains 21 cubic feet. How many
bushels in a crib 10' x 4' 3" x 2' 4" ?
3. A ton of tame hay contains 512 cubic feet. How many
tons in a space 14' x 12' x 13' ?
4. A ton of wild hay contains 343 cubic feet. How many
tons in a space 28' 6" x 18' 9" x 13' 5" ?
5. A bushel of potatoes contains 11 cubic feet. How
many bushels in a bin 8' 6" X 7' 5" x 9' 3" filled with
potatoes ?
6. How many bushels of com on the ear in a pointed heap
12' x 8' and 6' high?
7. How many bushels of corn in a circular crib with a
diameter 12' 6" and a height 8'?
8. How many gallons of water in a rectangular trough
6' 3" x 2f 6" x 3' 4" ? (Consider a gallon | cubic foot.)
9. How many acres in 694 sq. rods ?
10. A 60-acre piece of land, half a mile across, is 6' 8"
higher on one side than the other. How much of a fall (grade)
to the rod ?
11. How many bushels of corn in a rectangular crib with
sloping sides 16' long, 1' high and 4' 6" wide at the bottom and
6' 8" wide at the top ?
ENSILAGE PROBLEMS
1. A farmer with the purpose of filling his silo with corn
began the preparation of one acre of land for planting : 8 loads
of stable fertilizer were used in dressing the land. What is
the average number of square rods a load will fertilize ?
308 VOCATIONAL MATHEMATICS FOR GIRLS
2. The field was plowed in a day. Mr. A receives, when
working for others, 20 cents per hour for his horses and 15
cents per hour for his own work. How much is his time
worth for the day of 10 hours ?
3. Mr. A paid $ 12 for his plow and two extra points. The
regular price without extras was $ 10.50. Mr. A broke a plow
point on a rock. How much was the loss ?
4. It took three-fifths as long to harrow the field (see ex-
ample 2) as to plow it. If the work was begun at 7 o'clock in
the morning, at what time would the corn piece be harrowed ?
(Noon hour from 12 M. to 1 P.M.)
5. Mr. A bought seed com at $ 1.25 per bushel. What
did the seed cost, 12 quarts being the amount used ?
6. He bought 4 one-hundred-pound bags of fertilizer at
$ 1.40 per hundred. How much did the fertilizer cost ?
7. Mr. A is agent for Bradley fertilizers and receives a
commission of 10 % on what he sells for the company. How
much must he sell to receive a commission equal to the cost of
fertilizer used on his own corn piece, and also the expense of
hauling from the railroad station, which amounted to $ 2.50 ?
8. Mr. A hires a man to plant his corn with a horse planter.
He pays $ 2 for the planting, which is at the rate of 30 ^ per
hour. How long did it take ?
9. Mr. A cultivated his corn three times, each time requir-
ing about 8 hours. Besides this he and his hired man spent 3
days hoeing the corn once. Which was more expensive, the
hoeing or the cultivating ? How much ?
10. The corn was planted June 1st. It was ready for cut-
ting September 1st. Some of the stalks had grown to a height
of 6 ft. What was the average weekly growth ?
11. Mr. A's silo is rectangular, 10 ft. long, 10 ft. wide, and
20 ft. deep. The floor is cemented. How many sq. yd. of
cement in the floor ?
PROBLEMS ON THE FARM 309
12. If the lumber is 1 inch thick, how many board feet in
one thickness of the walls ?
13. How many cubic feet of ensilage will the silo hold ?
How many cubic feet below the level of the bam floor, which
is 5 ft. higher than the cemented floor of the silo ?
14. How many bushels of the cut and compressed corn
stalks must have been produced on the acre of land to fill the
space ?
15. On September 1st a gang of men helped Mr. A fill the
silo. Two men worked in the field cutting down the stalks at
$ 1.50 per day each. Two men hauled to the barn with teams at
$ 3 per day each. Two men, a cutting machine, and horses for
power cost $ 7. One man leveled corn in the silo at $ 1.50
per day. What did Mr. A pay these men for the work of the
day ? The next day the men with the cutting machine, one
man with a team, and the man for the silo worked two hours
to finish the work. Add this expense to that of the previous
day.
16. A week later the ensilage had settled 8 feet and Mr. A
filled the space with surplus corn. He and a helper hoisted it
with a pulley in a two-bushel basket. How many times must
he fill the basket ?
17. The mass was left to the fermenting process for two
months. When Mr. A opens the silo, he begins feeding regu-
larly to his 10 cows, giving each one-half bushel twice a day.
At this rate when will the silo be emptied? When should
the ensilage be even with the barn floor ?
18. It is estimated that 1 ton of ensilage is equal in value
to one-third of a ton of hay. If ensilage weighs 50 pounds
per bushel, how many pounds of hay equal a feed of ensilage ?
19. How many tons of hay is the ensilage worth ? What
is the value at $ 15 per ton ?
310 VOCATIONAL MATHEMATICS FOR GIRLS
20. Does it pay the farmer to raise ensilage ?
NOTE. — Ensilage could not be used as a substitute for hay, but is ex-
cellent as a milk producer when fed in moderate quantities. Cows like
it better than hay.
DAIRY PRODUCE
Milk is graded according to the amount of cream (fat) in it.
In addition to cream, it contains casein (cheese), milk sugar,
and about 84 % water. Milk is usually sold by the farmer by
weight and the per cent cream.
To illustrate : A sample of milk from a large can weighing
50 Ib. contains 4 % cream. The large can contains 2 Ib. of
butter fat.
EXAMPLES
1. A sample of milk from a cooler containing 48-J- Ib. tested
3J % butter fat. How much butter fat in the cooler ?
2. A cow gives 3J gallons of milk per day. If a gallon
weighs 8f Ib., what is the weight of milk per day ? per week ?
per month ?
3. If the milk in example 2 contains 4.7 per cent of cream,
how much butter fat does it yield per week ? per month ?
4. If butter fat is worth 26 cents a pound, how much is ob-
tained per week from the butter fat in example 3 ? How much
per month ?
5. Another cow gives 3-| gallons of milk a day that tests
4.6 % butter fat. Is it more profitable to keep this cow or the
one in examples 2 and 3, and by how much ?
6. Skim milk from the butter fat is usually sold to feed the
pigs at 5J cents a gallon. Is it cheaper to sell milk at 5^ cents
a quart or to make butter and sell it at 26 cents a pound and
give the skim milk to the pigs ?
PROBLEMS ON THE FARM 311
PROBLEMS ON EQUIPPING A COOPERATIVE CHEESE FACTORY
Seventy-three farmers came together, and after the election
of officers it was decided that a stock company of seventy-three
shares should be formed, and each member bought a share at
the rate of $ 75. Part of the money was used to erect a cheese
factory, and the rest was deposited in a bank and drawn out
as it was needed to run the business until the sale of the
products should be sufficient to supply money for carrying on
the business and paying a small per cent on the money invested
by each man. The following are the items of expense :
Half acre of land at $0.03 a square foot.
The building cost $ 2000 for material and work.
Three large vats, $ 50 each ; 4 % discount.
A Babcock tester $30 ; 2| % discount.
A small engine, belts, etc., $53.85.
Whey trough and leads, $ 54 ; 4 °Jo discount.
Cheese press, $ 28 ; 3 % discount.
Rennet, salt, coloring, wood, cheesecloth, boiler, and piping, $15.
Boxes, acid for test, etc., $ 57 ; 3 % commission.
Scales, weights, and weighing can, $27.85.
A year's salary to the cheese maker, $ 520.
The money left after these expenses was put at 3 % interest.
PROBLEMS
1. How much money was put into the business ?
2. How much did the cheese maker average a week ?
3. What was the cost of the land ?
4. The man who sold the boiler and piping and also the
engine and belt to the company received 6 % commission.
What did he receive for his sales ?
5. The man who bought the whey trough, the leads, and
the large vats received the discount as his commission. How
much did he receive ?
6. How much did the company that sold the whey trough,
leads, and vats receive ?
312 VOCATIONAL MATHEMATICS FOR GIRLS
7. How much did the buying company pay out for the whey
trough, leads, and vats ?
8. An agent sold the tester ; his commission was 5 %. How
much did he receive ?
9. How much did the tester cost the company ?
10. The man who bought the press and material received
3 °/o commission. How much did the press and materials cost
the company?
11. How much did it cost to buy the land, build the fac-
tory, and equip the plant ?
12. How much was left at interest ?
13. How much would the interest be for 3 years 6 months
and 16 days ?
14. What is the interest for one year ?
15. What per cent of the whole investment is this interest ?
16. What per cent of the whole was left at interest ?
PROBLEMS ON POULTRY
One hen has to have five square feet of room in the house.
It costs about ten cents a month to feed one hen.
One dozen eggs sell on the average for 30 cents.
One hen lays about 100 eggs per year.
Broilers are sold at 25 cents a pound.
Hens are sold for 15 cents a pound.
An incubator costing $ 20 holds 150 eggs.
Setting eggs cost $ 1.00 a dozen.
Brooders cost $7.50.
A small chicken coop costs $ 8.00.
One hen costs 60 cents.
Little chickens 1 week old cost 10 cents.
Chicken wire, 6 ft. wide, costs 4J- cents per foot.
PROBLEMS ON THE FARM 313
PROBLEMS
1. How large would the floor of my poultry house have to
be for 30 chickens ? for 50 ? 80 ? 200 ?
2. If I have a poultry house the floor of which is 30 ft. by
50 ft., how many chickens can I put in it?
3. How much will it cost to keep the chickens one month ?
one year ?
4. If I have one hen, how much does it cost me to feed her
one year? Suppose she lays 90 eggs, how much will I receive
for them ? Does it pay me to keep the hen ?
5. Suppose I sold 25 broilers, 12 weighing 3 lb., 5 weigh-
ing 5 lb., and the rest an average of 4 lb. How much would I
receive for them ? If I had kept them 14 weeks, how much
would they have cost me ? Would I gain or lose in keeping
them ? How much ?
6. If I bought an incubator for $ 20, a brooder for $ 7.50,
a chicken coop for $ 8, and 150 eggs to put into the incubator,
how much did I pay in all ?
7. If from the 150 eggs only 139 were hatched and lived,
how much would I receive for the little chickens when I sold
them?
8. Suppose I had a chicken yard 100 by 250 ft. How many
feet of wire would I need to fence it in ? How much would
it cost me to put wire around it ?
9. How many eggs would I receive from 60 hens in one
year ? If I sold all from 40 hens, how much would I receive
for them ?
10. If I bought 50 little chickens, kept them 16 weeks, and
then sold them, each weighing on the average 3 lb., how much
profit did I make on them ?
314 VOCATIONAL MATHEMATICS FOR GIRLS
POULTRY RAISING
1. A man wishes to build and stock a henhouse for $ 125.
If he has $ 75, how much will he have to borrow ? How much
interest will -he have to pay for 1 year at 5 % ?
2. If he pays $ 20 for labor, three times as much for mate-
rial, one-fourth as much for apparatus as for material, how
much will he have left ? How many hens could he buy with
the remainder if each hen cost 50 cents ?
3. If it cost $ 1 per year to keep one hen, how much would
it cost to keep all of his hens for 1 year ? for 5 years ?
4. If each hen lays 100 eggs a year, how many eggs would
they yield in one year ? how many dozen ?
5. If he sold 400 dozen at 25 cents per dozen, how much
would he receive for them ?
6. If he sold the remaining dozen " for setting " at 50 cents
a dozen, how much would he receive for these ? How much
did he receive for all his eggs ?
7. If it cost him the above amount to keep the hens for a
year, how much did he gain from selling his eggs ?
8. If, from 100 dozen eggs sold for " setting," 9 chickens
were hatched from each dozen, how many chickens were
hatched in all ?
9. If it cost him 27 cents to raise one broiler,' how much
would it cost to raise all the chickens for broilers ?
10. If for each pair of broilers he received $ 1.50, how
much would his entire stock net him ?
11. After considering the cost of raising the broilers and
the price received for them, what was his profit ?
12. After he had paid his interest, what was his net
profit ?
PROBLEMS ON THE FARM 315
REVIEW EXAMPLES
1. A crib of corn is 12' wide, 34' long, and has an average
depth of 11' of corn in it. How many bushels ?
2. How many bushels of oats in a bin 12' wide, 12' long,
18' deep ?
3. A freight car is 8' x 32' x 11'. If it is filled 3f ' deep
with apples for the cider mill, how many bushels in the car ?
4. At 26 cents a barrel, what is the car of apples worth ?
(2-i bu. = 1 bbl.)
5. A field of hay is 88 rods long and 64 rods wide. How
much is it worth at $ 98 an acre ?
6. A cow gives 3f gallons of milk a day. The milk tests
4.2 % butter fat. At 27 cents a pound for butter, and 5 cents
a quart for skim milk, how much is obtained a week from this
cow?
7. A flock of 200 hens averaged 135 eggs a year, and at
the end of four years were sold for 10| cents a pound, the
average weight being 6J- lb. If the cost of feed for a year is
$ 27.05 for the whole flock, what is the average gain per hen ?
8. A man receives S 35 a month. How much per hour, if
the month contains 26 working days of 10 hours a day ?
9. What is the cost of 963J bushels of oats at 47 cents per
bushel?
10. If I buy 125 bushels of corn at 41| cents per bushel
and sell it at 52^ cents a bushel, how much do I gain ?
APPENDIX
METRIC SYSTEM
THE metric system is used in nearly all the countries of
Continental Europe and among scientific men as the standard
system of weights and measures. It is based on the meter as
the unit of length. The meter is supposed to be one ten-
millionth part of the length of the meridian passing from the
equator to the poles. It is equal to about 39.37 inches. The
unit of weight is the gram1 which is equal to about one
thirtieth of an ounce. The unit of volume is the liter, which
is a little larger than a quart.
Measures of Length
10 millimeters (mm.) . . . . = I centimeter cm.
10 centimeters =1 decimeter dm:
10 decimeters =1 meter m.
10 meters =1 dekameter Dm.
10 dekameters =1 hektometer Hm.
10 hektometers =1 kilometer Km.
Measures of Surface (not Land)
100 square millimeters (mm.) . . =1 square centimeter . . sq. cm.
100 square centimeters ..-.=! square decimeter . . sq. dm.
100 square decimeters . . . . = 1 square meter sq. m.
. Measures of Volume
1000 cubic millimeters (mm.) . . = 1 cu. centimeter . . . cu. cm.
1000 cubic centimeters . . . . = 1 cubic decimeter . . . cu. dm.
1000 cubic decimeters . . . . = 1 cubic meter cu. m.
1 The gram is the weight of one cubic centimeter of pure distilled water at
a temperature of 39.2° F. ; the kilogram is the weight of 1 liter of water ; the
metric ton is the weight of 1 cubic meter of water.
317
318 VOCATIONAL MATHEMATICS FOR GIRLS
Measures of Capacity
10 milliliters (ml.) =1 centiliter
10 centiliters . = 1 deciliter
10 deciliters
= 1 liter
10 liters =1 dekaliter
cl.
dl.
1.
Dl.
10 dekaliters
= 1 hektoliter . HI.
10 hektoliters =1 kiloliter
Kl.
Measures of Weight
10 milligrams (mg.)
10 centigrams . .
10 decigrams . .
10 grams . . .
10 dekagrams . .
10 hektograms .
1000 kilograms
centigram eg.
decigram dg.
gram g.
dekagram ...... Dg.
hektogram ...... Hg.
kilogram Kg.
= 1 ton
1 meter
1 centimeter
1 millimeter
1 kilometer
1 foot
1 inch
METRIC EQUIVALENT MEASURES
Measures of Length
= 39.37 in. = 3.28083 ft.= 1.0936 yd.
= .3937 inch
= .03937 inch, or -fa inch nearly
= .62137 mile
= .3048 meter
= 2.54 centimeters = 25.4 millimeters
Measures of Surface
1 square meter = 10.764 sq. ft. = 1.196 sq. yd.
1 square centimeter = .155 sq. in.
.00155 sq. in.
.836 square meter
.0929 square meter
6.452 square centimeters =645. 2 square millimeters
1 square millimeter =
1 square yard
1 square foot =
1 square inch =
Measures of Volume and Capacity
1 cubic meter = 35.314 cu. ft. = 1.308 cu. yd. = 264.2 gal.
1 cubic decimeter = 61.023 cu. in. = .0353 cu. ft.
1 cubic centimeter = .061 cu. in.
1 The liter is equal to the volume occupied by 1 cubic decimeter.
METRIC SYSTEM 319
1 liter = 1 cubic decimeter = 61.023 cu. in. = .0353 cu. ft. =
1.0567 quarts (U. S.)=.2642 gallon (U. S.) =
2.202 Ib. of water at 62° F.
cubic yard = .7645 cubic meter
cubic foot = .02832 cubic meter = 28.317 cubic decimeters =
28.317 liters
cubic inch = 16.387 cubic centimeters
gallon (British) = 4.543 liters
gallon (U. S.) = 3.785 liters
Measures of Weight
1 gram = 15.432 grains
1 kilogram = 2.2045 pounds
1 metric ton = .9842 ton of 2240 Ib. = 19.68 cwt. = 2204.6 Ib.
1 grain = .0648 gram
1 ounce avoirdupois = 28.35 grams
1 pound = .4536 kilogram
1 ton of 2240 Ib. = 1.016 metric tons = 1016 kilograms
Miscellaneous
1 kilogram per meter = .6720 pound per foot
1 gram per square millimeter = 1.422 pounds per square inch
1 kilogram per square meter = .2084 pound per square foot
1 kilogram per cubic meter = .0624 pound per cubic foot
1 degree centigrade — 1.8 degrees Fahrenheit
1 pound per foot = 1.488 kilograms per meter
1 pound per square foot = 4.882 kilograms per square meter
1 pound per cubic foot = 16.02 kilograms per cubic meter
1 degree Fahrenheit = .5556 degree centigrade
1 Calorie (French Thermal Unit) = 3.968 B. T. U. (British Thermal Unit)
1 horse power = 33,000 foot pounds per minute = 746 watts
1 watt (Unit of Electrical Power) = .00134 horse power = 44.24 foot
pounds per minute
1 kilowatt = 1000 watts = 1.34 horse power =44,240 foot pounds per minute
TABLE OF METRIC CONVERSION
To change meters to feet multiply by 3.28083
feet to meters multiply by .3048
square feet to square meters . . . multiply by .0929
square meters to square feet . . . multiply by 10.764
320 VOCATIONAL MATHEMATICS FOR GIRLS
To change square centimeters to square inches . multiply by .155
square inches to square centimeters . multiply by 6.452
inches to centimeters multiply by 2.54
centimeters to inches multiply by .3937
grams to grains multiply by 15.43
grains to grams multiply by .0648
grams to ounces multiply by .0353
ounces to grams multiply by 28.35
pounds to kilograms multiply by .4536
kilograms to pounds multiply by 2.2045
liters to quarts multiply by 1.0567
liters to gallons multiply by .2642
gallons to liters multiply by 3.78543
liters to cubic inches multiply by 61.023
cubic inches to cubic centimeters . . multiply by 16.387
cubic centimeters to cubic inches . . multiply by .061
cubic feet to cubic decimeters or liters multiply by 28.317
kilowatts to horse power multiply by 1.34
calories to British Thermal Units . . multiply by 3.968
EXAMPLES
1. Change 8 m. to centimeters ; to kilometers.
2. Reduce 4 Km., 6 m., and 2 m. to centimeters.
3. How many square meters of carpet will cover a floor
which is 25.5 feet long and 24 feet wide ?
4. (a) Change 6J5 centimeters into inches.
(6) Change 48.3 square centimeters into square inches.
5. A cellar 18 m. x 37 m. x 2 m. is to be excavated ; what
will it cost at 13 cents per cubic meter to do the work ?
6. How many liters of capacity has a tank containing
5.2 cu. m. ?
7. What is the weight in grams of 31 cc. of water ?
8. Give the approximate value of 36 millimeters in inches.
9. Change 84.9 square meters into square feet.
10. Change 23.6 liters to cubic inches.
METRIC SYSTEM 321
11. Change 7.3 m. to millimeters ; to centimeters ; to
kilometers.
12. Reduce 9.8 m. to kilometers ; to centimeters ; to milli-
meters.
13. What is the difference in millimeters between 2.7 m.
and 48.1 mm. ?
14. What part of a kilometer is 1.8 mm. ?
15. What part of a meter is 1.3 cm. ?
16. How many square centimeters are there in 26 square
kilometers ?
17. How many square meters in 4 rectangular gardens,
3.4 Dm. long and 85.7 dm. wide ?
18. How many cubic meters in a wall 43 m. long, 8.4 dm.
high, and 69 cm. wide ?
19. Reduce 869.7 eg. to milligrams ; to kilograms ; to grams.
20. What is the weight in grams of 48.7 cc. of water ?
What is the weight in kilograms of 43.9 1. of water?
21. Mercury weighs 13.6 times as much as water ; what is
the weight of 87.5 cc. of mercury ? Of 5 1. of mercury ?
22. A tank is 7.9 m. by 4.3 m. by 3.1 m. How many grams
of water will it hold ?
23. What is the weight of 874 cc. of copper, the density of
which is 89 g. per cubic centimeter ?
24. What is the capacity of a bottle that holds 5 kg. of
alcohol, the density of which is 0.8 g. per cubic centimeter ?
25. What is the weight in grams of 56.8 cc. of alcohol?
What is the weight in kilograms of 7 1. of alcohol ?
26. What part of a liter is 1.7 cc. ?
GRAPHS
A SHEET of paper, ruled with horizontal and vertical lines
that are equally distant from each other, is called a sheet of
cross-section, or coordinate, paper. Every tenth line is very
distinct so that it is easy for one to measure off the horizontal
and vertical distances without the aid of a ruler. Ruled or
0
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GRAPH SHOWING THE VARIATION IN PRICE OF COTTON YARN FOR A
SERIES OF YEARS
coordinate paper is used to record the rise and fall of the
price of any commodity, or the rise and fall of the barometer
or thermometer.
Trade papers and reports frequently make use of coordinate
paper to show the results of the changes in the price of com-
modities. In this way one can see at a glance the changes
322
GRAPHS 323
and condition of a certain commodity, and can compare these
with the results of years or months ago. He also can see
from the slope of the curve the rate of rise or fall in price.
If similar commodities are plotted on the same sheet, the
effect of one on the other can be noted. Often experts are
able to prophesy with some certainty the price of a commodity
for a month in advance. The two quantities which must be
employed in this comparison are time and value, or terms
corresponding to them.
The lower left-hand corner of the squared paper is generally
used as an initial point, or origin, and is marked 0, although
any other corner may be used. The horizontal line from this
corner, taken as a line of reference or axis, is called the ab-
scissa. The vertical line from this corner is the other axis,
and is called the ordinate.
Equal distances on the abscissa (horizontal line) represent
definite units of time (hours, days, months, years, etc.), while
equal distances along the ordinate (vertical line) represent
certain units of value (cost, degrees of heat, etc.).
By plotting, or placing points which correspond to a certain
value on each axis and connecting these points, a line is ob-
tained that shows at every point the relationship of the line
to the axis.
EXAMPLES
1. Show the rise and fall of temperature in a day from
8 A.M. to 8 P.M., taking readings every hour.
2. Show the rise and fall of temperature at noon every day
for a week.
3. Obtain stock quotation sheets and plot the rise and fall
of cotton for a week.
4. Show the rise and fall of the price of potatoes for two
months.
5. Show a curve giving the amount of coal used each day
for a week.
FORMULAS
MOST technical books and magazines contain many formulas.
The reason for this is evident when we remember that rules
are often long and their true meaning not comprehended until
they have been reread several times. The attempt to abbre-
viate the length and emphasize the meaning results in the
formula, in which whole clauses of the written rule are ex-
pressed by one letter, that letter being understood to have
throughout the discussion the same meaning with which it
started.
To illustrate : One of the fundamental laws of electricity is that the
quantity of electricity flowing through a circuit (flow of electricity) is
equal to the quotient (expressed in amperes) obtained by dividing the
electric motive force (pressure, or expressed in volts, voltage) of the
current by the resistance (expressed in ohms).
One unfamiliar with electricity is obliged to read this rule over several
times before the relations between the different parts are clear. To show
how the rule may be abbreviated,
Let 7 = quantity of electricity through a wire (amperes)
E — pressure of the current (volts)
E = resistance of the current (ohms)
Then 1= E+ £ = -
It is customary to allow the first letter of the quantity to represent it in
the formula, but in this case I is used because the letter C is used in an-
other formula with which this might be confused.
Translating Rules into Formulas
The area of a trapezoid is equal to the sum of the two parallel
sides multiplied by one half the perpendicular distance between
them.
324
FORMULAS 325
We may abbreviate this rule by letting
A = area of trapezoid
L = length of longest parallel side
M= length of shortest parallel side
JV = length of perpendicular distance between them
Then A = (L + M ) x ^, or
2
The area of a circle is equal to the square of the radius
multiplied by 3.1416. When a number is used in the formula
it is called a constant, and is sometimes represented by a letter.
In this case 3.1416 is represented by the Greek letter TT (pi) .
Let A = area of circle
E = radius of circle
Then A = IT x jR2, or (the multiplication sign is usually left out between
letters)
Thus we see that a formula is a short and simple way of
stating a rule. Any formula may be written or expressed in
words and is then called a rule. The knowledge of formulas
and of their use is necessary for nearly every one engaged in
the higher forms of mechanical or technical work.
* When two or more quantities are to be multiplied or divided or other-
wise operated upon by the same quantity, they are often grouped together
by means of parentheses ( ) or braces { }, or brackets [ ]. Any number
or letter placed before or after one of these parentheses, with no other
sign between, is to multiply all that is grouped within the parentheses.
In the trapezoid case above, — is to multiply the sum of L and Jf, hence
the parentheses. To prevent confusion, different signs of aggregation
may be used for different combinations in the same problem.
For instance,
V= \TrH\*(r* + r'2) + ^2] which equals
o L 2 2 J
V =
326 VOCATIONAL MATHEMATICS FOR GIRLS
EXAMPLES
Abbreviate the following rules into formulas :
1. One electrical horse power is equal to 746 watts.
2. One kilowatt is equal to 1000 watts.
3. The number of watts consumed in. a given electrical,
circuit, such as a lamp, is obtained by multiplying the volts by
the amperes.
4. The number of volts equals the watts divided by the
amperes.
5. Number of amperes equals the watts divided by the
volts.
6. The horse power of an electric machine is found by mul-
tiplying the number of volts by the number of amperes and
dividing the product by 746.
7. The speed at which a body travels is equal to the ratio
between the distance traveled and the time which is required.
8. To find the pressure in pounds per square inch of a
column of water, multiply the height of the column in feet by
0.434.
9. The amount of gain in a business transaction is equal to
the cost multiplied by the rate of gain.
10. The selling price of a commodity is equal to the cost
multiplied by the quantity 100 % plus the rate of gain.
11. The selling price of a commodity is equal to the cost
multiplied by the quantity 100 % minus the rate of loss.
12. The interest on a sum of money is equal to the product
of the principal, time (expressed as years), and the rate (ex-
pressed as hundredths).
FORMULAS 327
13. The amount of a sum of money may be obtained
by adding the principal to the quantity obtained by multi-
plying the principal, the time (as years), and the rate (as
hundredths).
14. To find the length of an arc of a circle : Multiply the
diameter of the circle by the number of degrees in the arc and
this product by .0087266.
15. To. find the area of a sector of a circle : Multiply the
number of degrees in the arc of the sector by the square of the
radius and by .008727; or, multiply the arc of the sector by
half its radius.
Translating Formulas into Rules
In order to understand a formula, it is necessary to be able
to express it in simple language.
1. One of the simplest formulas is that for finding the area
of a circle, A = TT R*
Here A stands for the area of a circle,
E for the radius of the circle.
TT is a constant quantity and is the ratio of the circumference of a
circle to its diameter. The exact value cannot be expressed in figures,
but for ordinary purposes is called 3.1416 or 3^.
Therefore, the formula reads, the area of a circle is equal to
the square of the radius multiplied by 3.1416.
2. The formula for finding the area of a rectangle is
A = Lx W
Here A = area of a rectangle
L = length of rectangle
W = width of rectangle
The area of a rectangle, therefore, is found by multiplying
the length by the width,
328 VOCATIONAL MATHEMATICS FOR GIRLS
EXAMPLES
Express the facts of the following formulas as rules :
1. Electromotive force or voltage of electricity delivered by
a current, when current and resistance are given:
E = RI
2. For the circumference of a circle, when the length of the
radius is given:
O ^— £ 7T MV OF 7T.L/
3. For the area of an equilateral triangle, when the length
of one side is given: a*v'3
T~
4. For the volume of a circular pillar, when the radius and
height are given :
5. For the volume of a square pyramid, when the height
and one side of the base are given :
o^
3
6. For the volume of a sphere, when the diameter is given :
7. For the diagonal of a rectangle, when the length and
breadth are given :
8. For the average diameter of a tree, when the average
girth is known : G
Lf = —
7T
9. For the diameter of a ball, when the volume of it is
known. sf
FORMULAS 329
10. The diameter of a circle may be obtained from the area
by the following formula :
Z> = 1.1283 x VZ
11. The number of miles in a given length, expressed in
feet, may be obtained from the formula
M = .00019 x F
12. The number of cubic feet in a given volume expressed
in gallons may be obtained from the formula
C = .13367 x O-
13. Contractors express excavations in cubic yard s ; the
number of bushels in a given excavation expressed in yards
may be obtained from the formula
C = .0495 x Y
14. The circumference of a circle may be obtained from the
area by the formula
(7= 3.5446 x V2
15. The area of the surface of a cylinder may be expressed
by the formula A = (C X L) + 2a
When C = circumference
L = length
a = area of one end
16. The surface of a sphere may be expressed by the formula
S = D* x 3.1416
17. The solidity of a sphere may be obtained from the
formula
S = D3 X .5236
18. The side of an inscribed cube of a sphere may be ob-
tained from the formula
S = E* 1.1547, where S = length of side,
It = radius of sphere.
330 VOCATIONAL MATHEMATICS FOR GIRLS
19. The solidity or contents of a pyramid may be expressed
by the formula
-pi
S = A x — , where A = area of base,
F = height of pyramid.
20. The length of an arc of a circle may be obtained from
the formula
L = N x .017453 R, where L = length of arc,
N = number of degrees,
R = radius of circle.
21. The loss in a transaction may be expressed by the
formula
L = c x r, where L = loss,
c = cost,
r = rate of loss.
22. The rate of loss in a transaction may be expressed by
the formula
23. The cost of a commodity may be expressed by the
formula
or
c = — — -- , where S = selling price,
lUU -r- T
C = COSt,
r = rate.
24. The volume of a sphere when the circumference of a
great circle is known may be determined by the formula
v-C3 .
~e^
25. The diameter of a circle the circumference of which
is known may be found by the formula
FORMULAS 331
26. The area of a circle the circumference of which is known
may be found by the formula
Coefficients and Similar Terms
When a quantity may be separated into two factors, one of
these is called the coefficient of the other ; but by the coefficient
of a term is generally meant its numerical factor.
Thus, 4 b is a quantity composed of two factors 4 and b ; 4 is a coef-
ficient of b.
Similar terms are those that have as factors the same letters
with the same exponents.
Thus, in the expression, 6 a, 4 b, 2 a, 5 a&, 5 a, 2 b. 6 a, 2 a, 5 a are
similar terms ; 46, 2b are similar terms ; 5 ab and 6 a are not similar
terms because they do not have the same letters as factors. 3 ab, 5 ab,
lab, Sab are similar terms. They may be united or added by simply
adding the letters to the numerical sum, 17 ab.
In the following, 8 6, 5 &, 3 ab, 4 a, ab, and 2 a, 8 b and 5 b are similar
terms ; 3 ab and ab are similar terms ; 4 a and 2 a are similar terms ; 8 b,
3 ab, and 4 a are dissimilar terms.
In addition the numerical coefficients are algebraically added ;
in subtraction the numerical coefficients are algebraically sub-
tracted ; in multiplication the numerical coefficients are alge-
braically multiplied ; in division the nurnerial coefficients are
algebraically divided.
EXAMPLES
State the similar terms in the following expressions :
1. 5 a?, 8 ax, 3x, 2 ax. 6. 15 abc, 2 abc, 4 abc, 2 ab,
2. 8 abc, 7c, 2ab, 3c, Sab, 3a&.
7. 8x, 6x, 13xy, 5x, 1 y.
3. 2pq, 5p, 8 q, 2p, 3 q, 5pq. 8. 7y,2y,2 xy, 3y,2xy.
4. 47/, 5yz, 2y,15z,5z,2yz.
v
_ , Q 9. 2 7T; 5 Trr2, -, Ti-r, 2 TTT.
5. 18 mn, 6 m, 5 rc, 4 mw, 2m. 2
332 VOCATIONAL MATHEMATICS FOR GIRLS
Equations
A statement that two quantities are equal may be expressed
mathematically by placing one quantity on the left and the
other on the right of the equality sign (=). The statement
in this form is called an equation.
The quantity on the left hand of the equation is called the
left-hand member and the quantity on the right hand of the
equation is called the right-hand member.
An equation may be considered as a balance. If a balance
is in equilibrium, we may add or subtract or multiply or divide
the weight on each side of the balance by the same weight and
the equilibrium will still exist. So in an equation we may
perform the following operations on each member without
changing the value of the equation :
We may add an equal quantity or equal quantities to each mem-
ber of the equation.
We may subtract an equal quantity or equal quantities from
each member of the equation.
We may multiply each member of the equation by the same or
equal quantities.
We may divide each member of the equation by the same or
equal quantities.
We may extract the square root of each member of the equation.
We may raise each member of the equation to the same power.
The expression, A = trip is an equation. Why ?
If we desire to obtain the value of R instead of A we may do
so by the process of transformation according to the above
rules. To obtain the value of R means that a series of opera-
tions must be performed on the equation so that R will be left
on one side of the equation.
(1) .A = irIP
(2) — = IP (Dividing equation (1) by the coefficient of .R2.)
(3) -/— = R (Extracting the square root of each side of the equation.)
FORMULAS 333
Methods of Representing Operations
MULTIPLICATION
The multiplication sign ( X ) is used in most cases. It should
not be used in operations where the letter (x) is also to be em-
ployed.
Another method is as follows :
2-3 a-6 2a-3b 4 x • 5 a
This method is very convenient, especially where a number
of small terms are employed. Keep the dot above the line,
otherwise it is a decimal point.
Where parentheses, etc., are used, multiplication signs may
be omitted. For instance, (a + b) x (a — b) and (a + &)(a — b)
are identical ; also, 2 • (x — y) and 2(x —y).
The multiplication sign is very often omitted in order to
simplify work. To illustrate, 2 a means 2 times a ; 5 xyz means
5 • x • y • z ; x(a — b) means x times (a — &), etc.
A number written to the right of, and above, another (x*~) is
a sign indicating the special kind of multiplication known as
involution.
In multiplication we add exponents of similar terms.
Thus, x2 - y? = #2+3 = x5
abc • ab • «26 = a4b3c
The multiplication of dissimilar terms may be indicated.
Thus, a • b • c • x • y • z = abcxyz.
DIVISION
The division sign (-*-) is used in most cases. In many
cases, however, it is best to employ a horizontal line to indicate
division. To illustrate, a means the same as (a + &) -f-
x-y
(x — y) in simpler form. The division sign is never omitted.
334 VOCATIONAL MATHEMATICS FOR GIRLS
A root or radical sign (yH, ^/x2) is a sign indicating the special
form of division known as evolution.
In division, we subtract exponents of similar terms.
Thus, y? + xi = — = y?-2 = x
The division of dissimilar terms may be indicated.
Thus,
xyz
Substituting and Transposing
A formula is usually written in the form of an equation.
The left-hand member contains only one quantity, which is
the quantity that we desire to find. The right-hand member
contains the letters representing the quantity and numbers
whose values we are given either directly or indirectly.
To find the value of the formula we must (1) substitute for
every letter in the right-hand member its exact numerical
value, (2) carry out the various operations indicated, remem-
bering to perform all the operations of multiplication and
division before those of addition and subtraction, (3) if there
are any parentheses, these should be removed, one pair at a
time, inner parentheses first. A minus sign before a parenthesis
means that when the parenthesis is removed, all the signs of
the terms included in the parenthesis must be changed.
Find the value of the expression
3a + 6(2a-& + 18), where a = 5, b = 3.
Substitute the value of each letter. Then perform all addition or
subtraction in the parentheses.
3x5 + 3(10 -3 + 18)
15 + 3(28-3)
15 + 3(25)
15 + 75 = 90
FORMULAS 335
EXAMPLES
Find the value of the following expressions :
1. 2 A x (2 + 3 A) X 8, when A = 10.
2. 8 a X (6 — 2 a) X 7, when a = 7.
4. 8 (a? + y), when x = 9 ; y = 11.
5. 13 (a;— y), when x = 27 ; y = 9.
6. 24 y -f 8 z (2 + y) — 3 y, when ?/ = 8 ; 2 = 11.
7. Q(6 Jf +3^r)-f2 0, when M = 4, ^=5, Q = 6, 0 = 8.
8. Find the value of X in the formula X =
when Jf = 11, N= 9, P = 28.
9. 5
P— Q
10. Find the value of T in the equation
11. 3 a -f- 4 (6 — 2 a + 3 c) — c, when a = 4, 6 = 6, c = 2.
12. 5|> — 8 q (p 4- r — $) — g, when p = 5, g = 7, r = 9, $ = 11.
13. si + p — p^ — S^ + t+p), when p = 5, S = 8, t = 9.
14. a2 — 63 -\- c2, when a = 9, 6 = 6, c = 4.
15. (a + 6) (a + 6 — c), when a = 2, 6 = 3, c = 4.
16. (a2 -62) (a2 + 62), when a = 8, 6 = 4.
17. (c3 + d3) (c3 - d3), c = 9, d = 5.
18. Va2 + 2 a6 + 62, when a = 7, 6 = 8.
19. "v/c3 — 61, when c = 5.
336 VOCATIONAL MATHEMATICS FOR GIRLS
PROBLEMS
Solve the following problems by first writing the formula
from the rule on page 326, and then substituting for the answer.
1. How many electrical horse power in 4389 watts ?
2. How many kilowatts in 2389 watts ?
3. (a) Give the number of watts in a circuit of 110 volts
and 25 amperes.
(b) How many electrical horse power ?
4. What is the voltage of a circuit if the horse power is
2740 watts and the quantity of electricity delivered is 25
amperes ?
5. What is the resistance of a circuit if the voltage is 110
and the quantity of electricity is 25 amperes ?
6. What is the pressure per square inch of water 87 feet
high?
7. What is the capacity of a cylinder with a base of 16
square inches and 6 inches high? (Capacity in gallons is
equal to cubical contents obtained by multiplying base by the
height and dividing by 231 cubic inches.)
8. What is the length of a 30° arc of a circle with 16"
diameter ?
9. What is the area of a sector which contains an arc of
40° in a circle of diameter 18" ?
10. What is the amount of $ 800 at the end of 5 years at 5 % ?
11. What is the amount of gain in a transaction, when a
man buys a house for $ 5000 and gains 10 % ?
12. What is the selling price of an automobile that cost
$ 895, if the salesman gained 33 % ?
13. What is the capacity of a pail 14" (diameter of top),
11" (diameter of bottom), and 16" in height ?
14. What is the area of an ellipse with the greatest length
16" and the greatest breadth 10" ?
FORMULAS
337
Interpretation of Negative Quantities
The quantity or number — 12 has no meaning to us according
to our knowledge of simple arithmetic, but in a great many
problems in practical work the minus sign before a number
assists us in understanding the different solutions.
To illustrate :
FAHRENHEIT THF.BMOMETEB
CENTIGRADE THERMOMETER
Boiling
point of
water
Freezing
point of •
water
212°
Boiling
point of
water
§•§<!
-32°
Freezing
point of •
water
100°
-0°
On the Centigrade scale the freezing point of water is marked
0°. Below the freezing point of water on the Centigrade scale
all readings are expressed as minus readings.
— 30° C means thirty degrees below the freezing point. In
other words, all readings, in the direction below zero are
expressed as — , and all readings above zero are called +.
Terms are quantities connected by a p'lus or minus sign.
Those preceded by a plus sign (when no sign precedes a quan-
tity plus is understood) are called positive quantities, while
those connected by a minus sign are called negative quantities.
338 VOCATIONAL MATHEMATICS FOR GIRLS
Let us try some problems involving negative quantities.
Find the corresponding reading on the Fahrenheit scale cor~
responding to — 18° C.
F = § C + 32°
F = f(- 18°) +.32°
Notice that a minus quantity is placed in parenthesis when it is to be
multiplied by another quantity.
F =- ip> + 32° = - 32f° + 32° ; F =- |°.
The value — f ° is explained by saying it is f of a degree below zero
point on Fahrenheit scale.
Let us consider another problem. Find the reading on the Centi-
grade scale corresponding to — 40° F.
Substituting in the formula, we have
C = I (_ 40° - 32°) = | (- 72°) = - 40°.
Since subtracting a negative number is equivalent to adding
a positive number of the same value, and subtracting a posi-
tive number is equivalent to adding a negative number of the
same value, the rule for subtracting may be expressed as fol-
lows : Change the sign of the subtrahend and proceed as in
addition.
For example, 40 minus — 28 equals 40 plus 28, or 68.
40 minus + 28 equals 40 plus — 28, or 12.
— 40 minus + 32 equals — 40 plus — 32 = - 72.
(Notice that a positive quantity multiplied by a negative quantity or
a negative quantity multiplied by a positive quantity always gives a
negative product. Two positive quantities multiplied together will give
a positive product, and two negative quantities multiplied together will
give a positive product. ) To illustrate :
5 times 5 = 5 x 5 = 25
5 times _ 5 =£ x(- 5) = -26
(-5) times (-5) = +25
In adding positive and negative quantities, first add all the
positive quantities and then add all the negative quantities
FORMULAS 339
together. Subtract the smaller from the larger and prefix the
same sign before the remainder as is before the larger number.
For example, add :
2 a, 5 a, — 6 a, 8 a, - 2 a
2a + 5a-f 8a = 15a ; — Qa-2a = -Ba
15 a — 8 a = 7 a
EXAMPLES
Add the following terms :
1. 3 x, — x, 1 x, 4 Xj — '2 x.
2. 6y, 2y, 9y, -7y.
3. 9 ab, 2 db, 6 ab, - 4 a&, 7 a&, - 5 a&.
Multiplication of Algebraic Expressions
Each term of an algebraic expression is composed of one or
more factors, as, for example, 2 ab contains the factors #, a, and
b. The factors of a term have, either expressed or understood,
a small letter or number in the upper right-hand corner, which
states how many times the quantity is to be used as a factor.
For instance, ab2. The factor a has the exponent 1 understood
and the factor b has the exponent 2 expressed, meaning that a
is to be used once and b twice as a factor. abz means, then,
a X b X b. The rule of algebraic multiplication by terms is as
follows: Add the exponents of all like letters in the terms
multiplied and use the result as exponent of that letter in the
product. Multiplication of unlike letters may be expressed
by placing the letters side by side in the product.
For example : 2 ab x 3 &2 =
4 a x 3 b = 12 ab
Algebraic or literal expressions of more than one term are
multiplied in the following way : begin with the first term to
the left in the multiplier and multiply every term in the multi-
plicand, placing the partial products underneath the line. Then
340 VOCATIONAL MATHEMATICS FOR GIRLS
repeat the same operation, using the second term in the multi-
plier. Place similar products of the same factors and degree
(same exponents) in same column. Add the partial products.
Thus, a + b multiplied by a — b.
a + b
a-b
a?+ ab - 62
-ab
a2 -62
Notice the product of the sum and difference of the quantities is equal
to the difference of their squares.
EXAMPLES
1. Multiply a -f- b by a + b.
State what the square of the sum of the quantities equals.
2. Multiply x — y by x — y.
State what the square of the difference of the quantities equals.
3. Multiply (p + q)(p — q). 7. Multiply (a; — y)(x — y).
4. Multiply (p + g)(jp + g). 8. (x + 2/)2=?
5. Multiply (r -f s)(r - s). 9. (a; - y)2 = ?
6. Multiply (a ± 6)(a ± b). 10. (x + y)(x - y) = ?
USEFUL MECHANICAL INFORMATION
There are certain mechanical terms and laws that every girl
should know and be able to apply to the labor-saving devices
and machines that are used in the home to-day.
Time and Speed
Two important terms are time and speed. Speed is the
name given to the time-rate of change of position. That is,
Sueed — Change of position or distance
Time taken
EXAMPLES
1. A train takes 120 seconds to go one mile ; what is its
speed in miles per hour ?
One hour contains 60 minutes, 1 minute contains 60 seconds, then 1 hour
contains
60 x 60 = 3600 seconds.
If the train goes one mile in 120 seconds, in one second it will go T£7
of a mile and in 3600 seconds it will go
3600 x T£7 = 30 miles per hour. Ans.
2. At the rate of 80 seconds per mile, how fast is a train
moving in miles per hour ?
In a second it will move ^ of a mile ; in 3600 seconds it will move
3600 times as much.
3. At the rate of 55 miles an hour, how many seconds will
it require to travel between mile-posts ?
4. A watch shows 55 seconds between mile-posts ; what is
the speed in miles per hour ?
341
342 VOCATIONAL MATHEMATICS FOR GIRLS
5. What number of seconds between mile-posts will corre-
spond to a speed of 40 miles an hour ?
6. The rim of a fly-wheel is moving at the rate of one mile
a minute. How many feet does it move in a second ?
7. If a train continues to travel at the rate of 44 feet a
second, how many miles will it travel in an hour ?
8. If a train travels at the rate of 3.87 miles in 6 minutes,
how many miles an hour is it traveling ?
Motion and Momentum
Many interesting facts about the motion of bodies can be
understood by the aid of a knowledge of the laws of motion
and momentum.
A body acted upon by some force,1 such as steam or elec-
tricity, starts slowly, increasing its speed under the action of
the force. To illustrate : — when an electric car starts, we
often experience a heavy jarring ; this is due to the fact that
the seat starts before our body, and the seat pushes us along.
There is a tendency of bodies to remain in a state of rest or
motion, which is called inertia, that is, the inability of a body
itself to change its position, to stop itself if moving, or to start
if at rest.
The momentum of a body is defined as the quantity of
motion in a body, and is the product of the mass2 and the
velocity in feet per second (speed).
EXAMPLE. To find the momentum of a body 9 pounds in
weight, when moving with the velocity of 75 feet per second.
If the mass of the body upon which the force acts is given in pounds,
and the velocity in seconds, the force will be given in foot-pounds.
MASS VELOCITY MOMENTUM
9 x 75 675 foot-pounds.
1 Force is that which tends to produce motion.
2 Mass is the quantity of matter in a body.
USEFUL MECHANICAL INFORMATION 343
We may abbreviate this rule by allowing letters to stand for
quantities. Let the mass be represented by M and the veloc-
ity by V.
EXAMPLES
1. What is the momentum of a car weighing 15 tons, mov-
ing 12 miles per hour ?
2. What is the momentum of a motor-car weighing 3 tons,
moving 26 miles per hour ?
3. What is the momentum of a person weighing 135 pounds,
moving 5 miles per hour ?
4. A truck weighing 4 tons has a momentum of 520,000 foot-
pounds. At what speed is it moving ?
Work and Energy
Work is the overcoming of resistance of any kind. Energy
is the ability to do work. Work is measured in a unit called
a foot-pound. It is the work done in raising one pound one
foot in one second. One horse power is 33,000 foot-pounds in
one minute.
EXAMPLES
1. A woman lifts a package weighing 15 Ib. from the floor
to a shelf 5 ft. above the floor in two seconds. How many
foot-pounds of force does she use ?
2. How much work does a woman weighing 130 pounds do
in climbing a 13-story building in 20 minutes ? Each story
is 16' high.
3. If an engine is rated at 5 H. P.,1 how much work will it
do in 8 seconds ? in 3 minutes ?
1 Remember that 1 H. P. means 33,000 ft.-lb. in one minute.
344 VOCATIONAL MATHEMATICS FOR GIRLS
4. Find the horse power developed by a locomotive when it
draws at the rate of 31 miles per hour a train offering a resist-
ance of 130,000 Ib.
Machines
Experience shows that it is often possible to use our strength
to better advantage by means of a contiivance called a
machine. Every home-maker is interested in labor-saving
devices.
The mechanical principles of all simple machines may be
resolved into those of the lever, including the wheel and axle
and pulley, and the inclined plane, to which belong the wedge
and screw.
In all machines there is more or less friction.1 The work
done by the acting force always exceeds the actual work
accomplished by the amount that is transformed into heat.
The ratio of the useful work to the total work done by the
acting force is called the efficiency of the machine.
Em i — ^se^ work accomplished ^
Total work expended
• Levers. — The efficiency of simple levers is very nearly
100 % because the friction is so small as to be disregarded.
Inclined Planes. — In the inclined plane the friction is
greater than in the lever, because there is more surface with
which the two bodies come in contact ; the efficiency is some-
where between 90 % and 100 % .
Pulleys. — The efficiency of the commercial block and tackle
with several movable pulleys varies from 40 % to 60 % .
Jack Screw. — In the use of the jack screw there is neces-
sarily a very large amount of friction so that the efficiency is
often as low as 25 %.
1 Friction is the resistance which every material surface offers to the slid-
ing or moving of any other surface upon it.
USEFUL MECHANICAL INFORMATION 345
EXAMPLES
1. Mention some instances in which friction is of advantage.
2. If 472 foot-pounds of work are expended by a dredge in
raising a load, and only 398 pounds of useful work are accom-
plished, what is the efficiency of the dredge ?
3. If 250 foot-pounds of work are expended at one end of
a lever, and 249 pounds of useful work are accomplished, what
is the efficiency of the lever ?
4. If 589 foot-pounds of work are expended in raising a
body on an inclined plane, and only 584 pounds of useful
work are accomplished, what is the efficiency of the inclined
plane ?
5. If 844 foot-pounds of work are expended in raising a
body by means of pulleys and only 512 pounds of useful work
are accomplished, what is the efficiency of the pulley ?
Water Supply
The question of the water supply of a city or a town is very
important. Water is usually obtained from lakes and rivers
which drain the surrounding country. If a lake is located in
a section of the surrounding country higher than the city
(which is often located in a valley), the water may be obtained
from the lake, and the pressure of the water in the lake may
be sufficient to force it through the pipes into the houses. But
in most cases a reservoir is built at an elevation as high as the
highest portion of the town or city, and the water is pumped
into it. Since the reservoir is as high as the highest point of
the town, the water will flow from it to any part of the town.
If houses are built on the same hill with the reservoir, a stand-
pipe, which is a steel tank, is erected on this hill and the water
is pumped into it.
Water is conveyed from the reservoir to the house by means
346 VOCATIONAL MATHEMATICS FOR GIRLS
of iron pipes of various sizes. It is distributed to the differ-
ent parts of the house by small lead, iron, or brass pipes.
Since water exerts considerable pressure, it is necessary to
know how to calculate the exact pressure in order to have pipes
of proper size and strength.
WATER SUPPLY
The distribution of water in a city during 1912 is as follows
MONTHS
»
w
~ _
O
GALLONS PHR
DAY
ESTIMATED No.
OF CONSUMERS
POPULATION
GALLONS PER
DAY FOR EACH
CONSUMER
GALLONS PER
DAY FOR EACH
INHABITANT
January .
157,866,290
5,092,461
February .
147,692,464
5,092,844
March .
146,933,054
4,739,776
April .
143,066,067
4,768,869
May . .
161,177,486
5,199,274
o
r- 1
o
June . .
176,479,354
5,882,645
O
"^
July . .
189,063,250
6,098,815
1— 1
^
August
179,379,566
5,786,438
September
169,394,758
5,646,492
October .
176,067,571
5,679,599
November
153,484,712
5,116,157
December
151,976,208
4,902,458
What is the number of gallons per day for each consumer ?
What is the number of gallons per day for each inhabitant ?
PLUMBING AND HYDRAULICS 347
EXAMPLES
1. Water is measured by means of a meter. If a water
meter measures for live hours 760 cubic feet, how many gal-
lons does it indicate ?
NOTE. — 231 cubic inches = 1 gallon.
2. If a water meter registered 1845 cubic feet for 3 days,
how many gallons were used ?
3. A tank holds exactly 12,852 gallons ; what is the capacity
of the tank in cubic feet ?
4. A tank holds 3841 gallons and measures 4 feet square on
the bottom ; how high is .the tank ?
Rectangular Tanks. — To find the contents in gallons of a square or
rectangular tank, multiply together the length, breadth, and height in
feet; multiply the result by 7.48.
I = length of tank in feet
b = breadth of tank in feet
h = height of tank in feet
Contents = Ibh cubic feet x 7.48 = 7.48 Ibh
gallons
(NOTE. — 1 cu. ft. = 7.48 gallons.)
If the dimensions of the tank are in inches, multiply the length,
breadth, and height together, and the result by .004329.
5. Find the contents in gallons of a rectangular tank having in-
side dimensions (a) 12' x 8' x 8'; (b) 15" x 11" x 6" ; (c) 3' 4"
X 2' 8" x 8"; (d) 5' 8" x 4' 3" x 3' 5" ; (e) 3' S" x 3' 9" x 2' 5".
Cylindrical Tank. — To find the contents of a cylin-
drical tank, square the diameter in inches, multiply
this by the height in inches, and the result by .0034.
d = diameter of cylinder
h = height of cylinder
Contents = d2h cubic inches x .0034 = d?h .0034 gallons
6. Find the capacity in gallons of a cylindrical tank (a) 14"
in diameter and 8' high; (&) 6" in diameter and 5' high;
348 VOCATIONAL MATHEMATICS FOR GIRLS
(c) 15" in diameter and 4' high; (d) V 8" in diameter and
5' 4" high ; (e) 2' 2" in diameter and 6' 1" high.
Inside Area of Tanks. — To find the area, for lining purposes, of a
square or rectangular tank, add together the widths of the four sides of
the tank, and multiply the result by the height. Then add to the above
the area of the bottom. Since the top is usually open, we do not line
it. In the following problems find the area of the sides and bottom.
7. Find the amount of zinc necessary to line a tank whose
inside dimensions are 2' 4"x 10" x 10".
8. Find the amount of copper necessary to line a tank
whose inside dimensions are 1'9" x 11" X 10", no allowance
made for overlapping.
9. Find the amount of copper necessary to line a tank
whose inside dimensions are 3' 4" x 1' 2" x 11", no allowance
for overlapping.
10. Find the amount of zinc necessary to line a tank
2' 11" x V 4" x 10".
Capacity of Pipes
Law of Squares. — The areas of similar figures vary as the
squares of their corresponding dimensions.
Pipes are cylindrical in shape and are, therefore, similar
figures. The areas of any two pipes are to each other as the
squares of the diameters.
EXAMPLE. — If one pipe is 4" in diameter and another is 2"
in diameter, their ratio is -^, and the area of the larger one is,
therefore, 4 times the smaller one.
EXAMPLES
1. How much larger is a section of 5" pipe than a section
of 2" pipe ?
2. How much larger is a section of 2±ff pipe than a section
of 1" pipe ?
3. How much larger is a section of 5" pipe than a section of
3" pipe ?
PLUMBING AND HYDRAULICS
349
Atmospheric Pressure
The atmosphere has weight and exerts .pressure. The pres-
sure is greatest at sea level, because here the depth of the
atmosphere is greatest. In mathematics the pressure at sea
level is taken as the standard. Men have learned to make
use of the principles of atmospheric pressure in such devices
as the pump, the barometer, the vacuum, etc.
Atmospheric pressure is often expressed as a
certain number of " atmospheres." The pressure
of one " atmosphere " is the weight of a column of
air, one square inch in area.
At sea level the
average pressure of
the atmosphere is
approximately 15
pounds per square
inch.
The pressure of
the air is measured
by an instrument
called a barometer.
The barometer con-
sists of a glass tube,
about 311 inches
long, which has
been entirely filled
with mercury (thus
removing all air from the tube) and inverted in
a vessel of mercury.
The space at the top of the column of mercury
varies as the air pressure on the surface of the
mercury in the vessel increases or decreases. The
BAROMETER pressure is read from a graduated scale which indi-
BAROMETER TUBE
350 VOCATIONAL MATHEMATICS FOR GIRLS
cates the distance from the surface of the mercury in the
vessel to the top of the mercury column in the tube.
QUESTIONS
1. Four atmospheres would mean how many pounds ?
2. Give in pounds the following pressures: 1 atmosphere;
-J atmosphere ; J atmosphere.
3. If the air, on the average, will support a column of
mercury 30 inches high with a base of 1 square inch, what
is the pressure of the air ? (One cubic foot of mercury weighs
849 pounds.)
Water Pressure
When water is stored in a tank, it exerts pressure against
the sides, whether the sides are vertical, oblique, or horizontal.
The force is exerted perpendicularly to the surface on which it
acts. In other words, every pound of water in a tank, at a
height above the point where the water is to be used, possesses
a certain amount of energy due to its position.
It is often necessary to estimate the energy in the tank at
the top of a house or in the reservoir of a town or city, so as
to secure the needed water pressure for use in case of fire. In
such problems one must know the perpendicular height from
the water level in the reservoir to the point of discharge. This
perpendicular height is called the head.
Pressure per Square Inch. — To find the pressure per square
inch exerted by a column of water, multiply the head of water
in feet by 0.434. The result will be the pressure in pounds.
The pressure per square inch is due to the weight of a column of
water 1 square inch in area and the height of the column. Therefore,
the pressure, or weight per square inch, is equal to the weight of a foot of
water with a base of 1 square inch multiplied by the height in feet. Since
the weight of a column of water 1 foot high and having a base of 1 square
inch is 0.434 lb., we obtain the pressure per square inch by multiply-
ing the height in feet by 0.434.
PLUMBING AND HYDRAULICS
351
EXAMPLES
What is the pressure per square inch of a column of water
(a) 8' high? (6) 15' 8" high? (c) 30' 4" high? (d) 18' 9"
high ? (e) 41' 3" high ?
Head. — To find the head of water in feet, if the pressure
(weight) per square inch is known, multiply the pressure by
2.31.
Let p = pressure
h = height in feet
Then p = h x 0.434 Ib. per sq. in.
h — P —
~ 0.434 ~ 0.434
X p=
EXAMPLES
Find the head of water, if the pressure is (a) 49 Ib. per
sq. in. ; (b) 88 Ib. per sq. in. ; (c) 46 Ib. per sq. in. ; (d)
28 Ib. per sq. in. ; (e) 64 Ib. per sq. in.
Lateral Pressure. — To find the lateral
(sideways) pressure of water upon the
sides of a tank, multiply the area of the
submerged side, in inches, by the pressure
due to one half the depth.
EXAMPLE. — A tank 18" long and 12"
deep is full of water. What is the lateral
pressure on one side ?
length depth
18" x 12" = 216 square inches = area of side
depth
1' X 0.434 = .434 Ib. pressure at the bottom of
the tank
0 = pressure at top
2).4341b.
.217 Ib. average pressure due to one half the
depth of the tank
.217 x 216 = 46.872 pounds = pressure on one
side of the tank
— Pressure
is zero
Pressure
is ha If that at
base
LATERAL PRESSURE
352 VOCATIONAL MATHEMATICS FOR GIRLS
Water Traps
The question of disposing of the waste water, called sewage, is of
great importance. Various devices may be used to prevent odors from
the sewage entering the house. In order to prevent the escape of gas
WATER TRAPS
from the outlet of the sewer in the basement of a house or building, a
device called a trap is used. This trap consists of a vessel of water
placed in the waste pipe of the plumbing fixtures. It allows the free pas-
sage of waste material, but prevents sewer gases or foul odors from enter-
ing the living rooms. The vessels holding the water have different forms ;
(see illustration) . These traps may be emptied by back pressure or by
siphon. It is a good plan to have sufficient water in the trap so that it
will never be empty. All these problems belong to the plumber and in-
volve more or less arithmetic.
To determine the pressure which the seal of a trap will resist :
EXAMPLE. — What pressure will a l^-inch trap resist ?
If one arm of the trap has a seal of If inches, both arms will make a
column twice as high, or 3 inches. Since a column of water 28 inches
in height is equivalent to a pressure of 1 pound, or 16 ounces, a column
of water 1 inch in height is equivalent to a pressure of $ of a pound, or
£| = £ ounces, and a column of water 3 inches in height is equivalent to
3 x | = ty = 1.7 ounces.
Therefore, a 1^-inch trap will resist 1.7 ounces of pressure.
PLUMBING AND HYDRAULICS 353
EXAMPLES
1. What back pressure will a f-inch seal trap resist ?
2. What back pressure will a 2-inch seal trap resist ?
3. What back pressure will a 21-inch seal trap resist ?
4. What back pressure will a 4J-inch seal trap resist ?
5. What back pressure will a 5-inch seal trap resist ?
Water Power
When water flows from one level to another, it exerts a
certain amount of energy, which is the capacity for doing
work. Consequently, water may be utilized to create power
by the use of such means as the water wheel, the turbine, and
the hydraulic ram.
Friction, which must be considered when one speaks of
water power, is the resistance which a substance encounters
when moving through or over another substance. The amount
of friction depends upon the pressure between the surfaces in
contact.
When work is done a part of the energy which is put into
it is naturally lost. In the case of water this is due to the
friction. All the power which the water has cannot be used
to advantage, and efficiency is the ratio of the useful work done
by the water to the total work done by it.
Efficiency. — To find the work done upon the water when a
pump lifts or forces it to a height, multiply the weight of the
water by the height through which it is raised.
Since friction must be taken into consideration, the useful
work done upon the water when the same power is exerted
will equal the weight of the water multiplied by the height
through which it is raised, multiplied by the efficiency of the
pump.
EXAMPLE. — Find the power required to raise half a ton
354 VOCATIONAL MATHEMATICS FOR GIRLS
(long ton, or 2240 Ib.) of water to a height of 40 feet, when
the efficiency of the purnp is 75 % .
Total work done = weight x height x efficiency counter
1120 x 40 x V?° = 59,733.3 ft. Ib.
H. P. required = 59^733-3 = 1.8. Ana.
33000
EXAMPLES
1. Find the power required to raise a cubic foot of water
28', if the pump has 80% efficiency.1
2. Find the power required to raise 80 gallons of water 15',
if the pump has 75 % efficiency.
3. Find the power required to raise 253 gallons of water
18', if the pump has 70 % efficiency.
4. Find the power required to raise a gallon of water 16', if
the pump has 85 % efficiency.
5. Find the power required to raise a quart of water 25', if
the pump has 70 % efficiency.
Density of Water
The mass of a unit volume of a substance is called its
density. One cubic foot of pure water at 39.1° F. has a mass
of 62.425 pounds ; therefore, its density at this temperature is
62.425, or approximately 62.5. At this temperature water
has its greatest density. With a change of temperature, the
density is also changed.
With a rise of temperature, the density decreases until at
212° F., the boiling point of water, the weight of a cubic foot
of fresh water is only 59.64 pounds.
When the temperature falls below 39.1° F., the density of
water decreases until we find the weight of a cubic foot of ice
to be but 57.5 pounds.
1 Consider the time 1 minute in all power examples where the time is not given.
PLUMBING AND HYDRAULICS 355
EXAMPLES
1. One cubic foot of fresh water at 62.5° F. weighs 62.355 lb.,
or approximately 62.4 lb. What is the weight of 1 cubic inch ?
What is the weight of 1 gallon (231 cubic inches) ?
2. What is the weight of a gallon of water at 39.1° F. ?
3. What is the weight of a gallon of water at 212° F. ?
4. What is the weight of a volume of ice represented by a
gallon of water ?
5. What is the volume of a pound of water at ordinary
temperature, 62.5° F. ?
Specific Gravity
Some forms of matter are heavier than others, i.e. lead is
heavier than wood. It is often desirable to compare the
weights of different forms of matter and, in order to do this,
a common unit of comparison must be selected. Water is
taken as the standard.
Specific Gravity is the ratio of the mass of any volume of a
substance to the mass of the same volume of pure water at
4° C. or 39.1° F. It is found by dividing the weight of a known
volume of a substance in liquiqL by the weight of an equal
volume of water.
EXAMPLE. — A cubic foot of wrought iron weighs about
480 pounds. Find its specific gravity.
NOTE. — 1 eu. ft. of water weighs 62.425 lb.
Weight of 1 cu. ft. of iron _ 480 _ ,- - ,
Weight of 1 cu. ft. of water ~ 62.425
To find Specific Gravity. — To find the specific gravity of a
solid, weigh it in air and then in water. Find the difference
between its weight in air and its weight in water, which will
be the buoyant force on the body, or the weight of an equal
volume of water. Divide the weight of the solid in air by its
buoyant force, or the weight of an equal volume of water, and
the quotient will be the specific gravity of the solid.
356 VOCATIONAL MATHEMATICS FOR GIRLS
Tables have been compiled giving the specific gravity of different solids,
so it is seldom necessary to compute it.
The specific gravity of liquids is very often used in the
industrial world, as it means the " strength " of a liquid. In
the carbonization of raw wool, the wool must be soaked in
sulphuric acid of a certain strength. This acid cannot be
bought except in its concentrated form (sp. gr. 1.84), and it
must be diluted with water until it is of the required strength.
The simplest way to determine the specific gravity of a liquid is with
a hydrometer. This instrument consists of a closed glass tube, with a
bulb at the lower end filled with mercury. This bulb of mer-
cury keeps the hydrometer upright when it is immersed in a
liquid. The hydrometer has a scale on the tube which can
be read when the instrument is placed in a graduate of the
liquid whose specific gravity is to be determined.
But not all instruments have the specific gravity recorded
on the stem. Those most commonly in use are graduated
with an impartial scale.
In England, Twaddell's scale is commonly employed, and
since most of the textile mill workers are English, we find the
same scale in use in this country. The Twaddell scale bears a
marked relation to. specific gravity and can be easily converted
into it.
Another scale of the hydrometer is the Beaume, but these
readings cannot be converted into specific gravity without
the use of a complicated formula or reference to a table. ^ •• -^
HYDROMETER SCALE FORMULA FOR CONVERTING INTO S. G.
1. Specific gravity hydrometer Gives direct reading
2. Twaddell g G = (.5 x ^) + 100
100
3. Beaume S. G. =
N= the particular degree which is to be converted.
EXAMPLE. — Change 168 degrees (Tw.) into S. G.
=L84>
100
PLUMBING AND HYDRAULICS 357
Another formula for changing degrees Twaddell scale into specific
gravity is : (5 x JV) + 1000 = gpecific gravitVi
1000
In Twaddell's scale, 1° specific gravity = 1.005
2° specific gravity = 1.010
3° specific gravity = 1.015
and so on by a regular increase of .005 for each degree.
To find the degrees Twaddell when the specific gravity is given, multi-
ply the specific gravity by 1000, subtract 1000, and divide by 5. Formula :
(S.G.X 1000) -1000 = degrees T^^H
5
EXAMPLE. — Change 1.84 specific gravity into degrees Twad-
dell,
(1.84x1000) -1000 = 16g degrees Twadden
5
EXAMPLES
1. What is the specific gravity of sulphuric acid of 116° Tw.?
2. What is the specific gravity of acetic acid of 86° Tw. ?
3. What is the specific gravity of a liquid of 164° Be. ?
4. What is the specific gravity of a liquid of 108° Be. ?
5. What is the specific gravity of a liquid of 142° Tw.?
Heat
Heat Units. — The unit of heat used in the industries and
shops of America and England is the British TJiermal Unit
(B. T. U.) and is defined as the quantity of heat required to
raise one pound of water through a temperature of one degree
Fahrenheit. Thus the heat required to raise 5 Ib. of water
through 15 degrees F. equals
5 x 15 = 75 British Thermal Units (B. T. U.)
Similarly, to raise 86 Ib. of water through £° F. requires
86 x i = 43 B. T. U.
The unit used on the Continent and by scientists in America
is the metric system unit, a calorie. This is the amount of
heat necessary to raise 1 gram of water 1 degree Centigrade.
358 VOCATIONAL MATHEMATICS FOR GIRLS
EXAMPLES
1. How many units (B. T. U.) will be required to raise
4823 Ib. of water 62 degrees ?
2. How many B. T, U. of heat are required to change 365
cubic feet of water from 66° F. to 208° F.?
3. How many units (B. T. U.) will be required to raise 785
Ib. of water from 74° F. to 208° F.?
(Consider one cubic foot of water equal to 621 lb.)
4. How many B. T. U. of heat are required to change 1825
cu. ft. of water from 118° to 211° ?
5. How many heat units will it take to raise 484 gallons of
water 12 degrees ?
6. How many heat units will it take to raise 5116 gallons
of water from 66° F. to 198° F.?
Temperature
The ordinary instruments used to measure temperature
are called thermometers. There are two kinds — Fahren-
heit and Centigrade. The Fahrenheit ther-
mometer consists of a cylindrical tube filled
with mercury with the position of the mercury
at the boiling point of water marked 212, and the
position of mercury at the freezing point of
water 32. The intervening space is divided into
180 divisions. The Centigrade thermometer has
the position of the boiling point of water 100
and the freezing point 0. The intervening space
is divided into 100 spaces. It is often necessary
to convert the Centigrade scale into the Fah-
renheit scale, and Fahrenheit into Centigrade.
To convert F. into C., subtract 32 from the F.
degrees and multiply by -|, or divide by 1.8, or
C. = (F. - 32°) f , where C. = Centigrade reading
and F. = Fahrenheit reading.
100
O
17.8
THERMOMETERS
2121
HEAT AND TEMPERATURE 359
To convert C. to F., multiply C. degrees by f or 1.8 and add
32°.
5
EXAMPLE. — Convert 212 degrees F. to C. reading
5(212°- 32 ") 5(180°) 900°
-* =• - • = luu v_y.
9 99
EXAMPLE. — Convert 100 degrees C. to F. reading.
9 x 100° + 32° = — + 32° = 180° + 32° = 212° F.
5 5
If the temperature is below the freezing point, it is usually
written with a minus sign before it : thus, 15 degrees below the
freezing point is written — 15°. In changing — 15° C. into F.
we must bear in mind the minus sign.
Thus, Jp = -+320 ^=~ + 32° =-27° + 32° =5°
5 5
EXAMPLE. — Change - 22° F. to C.
C. = f (F. - 32)
C. = f (- 22° - 32°) = $ ( - 54°) = - 30°
EXAMPLES
1. Change 36° F. to C. 6. Change 225° C. to F.
2. Change 89° F. to C. 7. Change 380° C. to F.
3. Change 289° F. to C. 8. Change 415° C. to F.
4. Change 350° F. to C. 9. Change 580° C. to F.
5. Change 119° C. to F.
Latent Heat
By latent heat of water is meant that heat which water ab-
sorbs in passing from the liquid to the gaseous state, or that
heat which water discharges in passing from the liquid to the
360 VOCATIONAL MATHEMATICS FOR GIRLS
solid state, without affecting its own temperature. Thus, the
temperature of boiling water at atmospheric pressure never
rises above 212 degrees F., because the steam absorbs the
excess of heat which is necessary for its gaseous state. Latent
heat of steam is the quantity of heat necessary to convert a
pound of water into steam of the same temperature as the
steam in question.
COMMERCIAL ELECTRICITY
Amperes. — What electricity is no one knows. Its action,
however, is so like that of flowing water that the comparison is
helpful. A current of water in a pipe is measured by the
amount which flows through the pipe in a second of time, as
one gallon per second. So a current of electricity is measured
WATER ANALOGY OF FALL OF POTENTIAL
by the amount which flows along a wire in a second, as one
coulomb per second, — a coulomb being a unit of measurement
of electricity, just as a gallon is a unit of measurement of
water. -The rate of flow of one coulomb per second is called one
ampere. The rate of flow of five coulombs per second is five
amperes.
Volts. — The quantity of water which flows through a pipe
depends to a large extent upon the pressure under which it
flows. The number of amperes of electricity which flow along
a wire depends in the same way upon the pressure behind it.
COMMERCIAL ELECTRICITY 361
The electrical unit of pressure is the volt. In a stream of
water there is a difference in pressure between a point on the
surface of the stream and a point near the bottom. This is
called the difference or drop in level between the two points.
It is also spoken of as the pressure head, " head " meaning the
difference in intensity of pressure between two points in a body
of water, as well as the intensity of pressure at any point.
Similarly the pressure (or voltage) between two points in an
electric circuit is called the difference or drop in pressure or
the potential. The amperes represent the amount of electricity
flowing through a circuit, and the volts the pressure causing
the flow.
Ohms. — Besides the pressure the resistance of the wire
helps to determine the amount of the current : — the greater
the resistance, the less the current flowing under the same
pressure. The electrical unit of resistance is called an ohm.
A wire has a resistance of one ohm when a pressure of one volt
can force no more than a current of one ampere through it.
Ohm's Law. — The relation between current (amperes),
pressure (volts), and resistance (ohms) is expressed by a law
known as Ohm's Law. This is the fundamental law of the
study of electricity and may be stated as follows :
An electric current flowing along a conductor is equal to
the pressure divided by the resistance.
Current (amperes) ^ (volts
Resistance (ohms)
Letting /= amperes, E = volts, R = ohms,
I=E + Rm / = -
R
E= IR
EXAMPLE. — If a pressure of 110 volts is applied to a re-
sistance of 220 ohms, what current will flow ?
362 VOCATIONAL MATHEMATICS FOR GIRLS
/ = — = — = - = .5 ampere. Ans.
E 220 2
EXAMPLE. — A current of 2 amperes flows in a circuit the resist-
ance of which is 300 ohms. What is the voltage of the circuit ?
IE = E
2 x 300 - 600 volts. Ans.
EXAMPLE. — If a current of 12 amperes flows in a circuit
and the voltage applied to the circuit is 240 volts, find the
resistance of the circuit.
^ = E ?40 _ 2Q ohms_ Ans
I 12
Ammeter and Voltmeter. — Ohm's Law may be applied to a
circuit as a whole or to any part of it. It is often desirable to
know how much current is flowing
in a circuit without calculating it by
Ohm's Law. An instrument called
an ammeter is used to measure the
current. This instrument has a
low resistance so that it will not
cause a drop in pressure. A volt-
meter is used to measure the voltage.
This instrument has high resistance
so that a very small current will
flow through it, and is always placed in shunt, or parallel
(see p. 235) with that part of the circuit the voltage of which
is to be found.
EXAMPLE. — What is the resistance of wires that are carry-
ing 100 amperes from a generator to a motor, if the drop or
loss of potential equals 12 volts ?
Drop in voltage = IE 1= 100 amperes
Drop in volts =12 E = ? ohms
E=— E= — = 0.12 ohm. Ans.
EXAMPLE. — A circuit made up of incandescent lamps and
conducting wires is supplied under a pressure of 115 volts.
COMMERCIAL ELECTRICITY
363
The lamps require a pressure of 110 volts at their terminals
and take a current of 10 amperes. What should be the resist-
ance of the conducting wires in order that the necessary cur-
rent may flow ?
Drop in conducting wires = 115 — 110 = 5 volts
Current through wires = 10 amperes
It = — — — = 0.5 ohm resistance. Ans.
EXAMPLES
1. How much current will flow through an electromagnet
of 140 ohms' resistance when placed across a 100-volt circuit ?
2. How many amperes will flow through a 110-volt lamp
which has a resistance of 120 ohms ?
3. What will be the resistance of an arc lamp burning
upon a 110-volt circuit, if the current is 5 amperes ?
4. If the lamp in Example 3 were to be put upon a 150-volt
circuit, how much additional resistance would have to be put
into it in order that it might not take more than 5 amperes ?
11
Fc-ed Wire ^
§
< }
A^ -f-
Trolley Wire/
Dynamo
ELECTRIC ROAD SYSTEM
5. In a series motor used to drive a street car the resistance
of the field equals 1.06 ohms ; the current going through equals
30 amperes. What would a voltmeter indicate if placed
across the field terminals ?
6. If the load upon the motor in Example 5 were increased
so that 45 amperes were flowing through the field coils, what
would the voltmeter then indicate ?
INDEX
Addition, 3
Compound numbers, 46
Decimals, 33
Fractions, 21
Aliquot parts, 39
Alkalinity of water, 298
Ammeter, 362
Ammonia, 296
Amount, 53
Amperes, 169, 360
Angles, 66
Complementary, 66
Right, 66
Straight, 66
Supplementary, 66
Annuity, 192
Antecedent, 37
Apothecary's weights, 276
Apothem, 72
Approximate equivalents between
metric and household measures,
281
Approximate measures of fluids, 277
Arc, 64
Area of a ring, 65
Area of a triangle, 69
Atmospheric pressure, 349
Avoirdupois weight, 43
Bacteria, 294, 298
Banks, 178
Cooperative, 179
National, 178
Savings, 179
Baths, 292
Bed linen, 161
Beef, 118
Bills, 243
Blue print reading, 80
Board measure, 131
Bonds, 187
Brickwork, 134
Building materials, 133
Buying Christmas gifts, 94
Cotton, 229
Rags, 229
Wool, 229
Yarn, 229
Cancellation, 13
Capacity of pipes, 348
Carbohydrates, 102
Cement, 136
Chlorine, 297
Circle, 64
Circumference, 64
Civil Service, 268
Claims, 196
Clapboards, 138
Clothing, 91
Coefficients, 331
Color of water, 296
Common denominator, 20
Fractions, 17
Multiple, 15
Comparative costs of digestible
nutrients and energy in different
food materials at average prices,
114, 115
Compound numbers, 42, 46
Addition, 46
Division, 47
Multiplication, 47
Subtraction, 46
Computing profit and loss, 252
Cone, 75
Consequent, 57
Construction of a house, 128
Cooperative banks, 179
Cost of food, 105
Cost of furnishing a house, 146
Cost of subsistence, 91
Cotton, 217
Yarns, 223
365
366
INDEX
Counting, 44
Credit account, 244
Cube, 61
Cube Root, 61
Cubic measure, 42
Cuts of Beef, 120
Mutton, 122
Pork, 121
Cylindrical tank, 347
Dairy Products, 310
Debit, 244
Decimal Fractions, 30
Addition, 33
Division, 36
Mixed, 31
Multiplication, 35
Reduction, 32
Subtraction, 34
Denominate fraction, 45
Number, 42, 45
Denomination, 42
Denominator, 17
Density of water, 354
Deodorants, 294
Diameter, 64
Distribution of income, 89
Division, 9
Compound numbers, 47
Fractions, 25
Income, 92
Drawing to scale, 85
Dressmaking, 198
Dry measure, 43
Economical marketing, 125
Uses of Meats, 117
Economy of space, 130
Efficiency of water, 353
Electricity, 360
Ellipse, 72
English system, 276
Ensilage problems, 307
Equations, 332
Substituting, 334
Transposing, 334
Equiangular triangle, 68
Equilateral triangle, 68
Estimating distances, 86
Weights, 87
Evolution, 61
Exchange, 193
Expense account book, 95
Factors, 13
Farm measures, 307
Problems, 305
Filling, 217
Flax, 217
Flooring, 139
Fluid measure, 277
Food, 100
Values, 110
Formulas, 327
For computing profit and loss, 253
Fractions, 17
Addition, 21
Common, 17
Decimal, 30
Division, 25
Improper, 17
Multiplication, 24
Reduction, 17
Subtraction, 22
Frame and roof, 132
Free ammonia, 297
Frustum of a cone, 76
Furnishing a bedroom, 153, 154, 155
Dining room, 156
Hall, 146
Kitchen, 162
Living room, 149, 150, 152
Sewing room, 161
Germicides, 294
Graphs, 322
Greatest common divisor, 15
Hardness of water, 298
Heat, 357
And light, 167
Units, 357
Hem, 200
Hexagon, 72
Horizontal addition, 237
Household linens, 160
Measures, 277
How to make change, 266
Solutions of various strengths from
crude drugs or tablets of known
strength, 286
INDEX
367
How to read an electric meter, 169
Gas meter, 168
Hypodermic doses, 288
Improper fractions, 17
Inclined planes, 344
Income, 89
Inside area of tanks, 348
Insurance, 188
Fire, 188
Life, 189
Integer, 1, 17, 31, 45
Interest, 53
Compound, 56
Simple, 53
Interpretation of negative quantities,
337
Invoice bills, 243
Involution, 61
Iron in water, 298
Isosceles triangle, 68
Jack screw, 344
Kilowatt, 169
Kitchen weights and measures, 103
Latent heat, 359
Lateral pressure, 351
Lathing, 141
Law of squares, 348
Least common multiple, 15
Ledger, 244
Levers, 344
Linear measure, 42
Linen, 217
Yarns, 222
Liquid measure, 43
Lumber, 131
Machines, 344
Measure, of time, 43
Length, 317
Medical chart, 292
Mensuration, 64
Menus, making up, 113
Merchandise, 243
Methods of heating, 174
Methods of solving examples, 87,
88
Metric system, 276, 279, 282, 317-319
Millinery problems, 212
Mixed decimals, 31
Mohair, 217
Momentum, 342
Money orders, 194
Mortar, 133
Mortgages, 180
Motion, 342
Multiplication, 8, 242
Algebraic expressions, 339
Compound numbers, 47
Decimals, 31
Fractions, 24
Mutton, 122, 123
National banks, 178
Nitrogen, 297
Notation, 1
Notes, 181
Numerals, Roman, 2
Numeration, 1
Numerator, 17
Nurses, arithmetic for, 276-303
Nutritive ingredients of food, 101
Octagon, 72
Odor of water, 296
Ohm, 361
Ohm's Law, 361
Oxygen consumed, 297
Painting, 141
Papering, 142
Paper measure, 44
Pay rolls, 255
Pentagon, 72
Percentage, 50
Perimeter, 72
Plank, 131
Plastering, 133
Polygons, 72
Poultry problems, 312
Power, 30
Pressure, lateral, 351
Per square inch, 350
Water, 350
Principal, 53
Profit and loss, 246
Promissory notes, 182
Proper fractions, 17
Proportion, 57, 58, 59
368
INDEX
Protractor, 67
Pulleys, 344
Pyramid, 75
Quadrilaterals, 71
Radius, 64
Rapid calculation, 233
Rate (per cent), 50
Ratio, 57
Raw silk yarns, 222
Reading a blue print, 80
Rectangle, 71
Reduction, 42
Ascending, 42, 44
Descending, 42, 44
Right triangles, 68, 69
Roman numerals, 2
Root, cube, 62
Square, 61
Ruffles, 201
Rule of thumb methods, 88
Savings bank, 179
Interest tables, 56
Scalene triangle, 68
Sector, 64
Sediment in water, 296
Shingles, 137
Shoes, 219
Silk, 217
Similar figures, 77
Terms, 331
Simple interest, 53
Proportion, 59
Slate roofing, 137
Specific gravity, 355
Specimen arithmetic papers, 272
Sealers of Weights and Measures,
273
State visitors, 274
Stenographers, 273
Sphere, 76
Spun silk yarns, 223
Square measure, 42
Square root, 62
Stairs, 140
Steers and beef, 118
Stocks, 184
Stonework, 135
Strength of solutions, 224
Studding, 132
Substituting in equations, 334
Subtraction
Compound numbers, 46
Decimals, 34
Fractions, 22
Supplement, 66
Table linen, 161
Table of metric conversion, 317
Table of wages, 257
Tanks, 347
Taxes, 143
Temperature, 290, 358
Temporary loans, 259
Terms used in chemical and bacterio-
logical reports, 296
Time and speed, 341
Time sheets, 255
Trade discount, 52, 207
Transposing in equations, 334
Trapezium, 71
Trapezoid, 72
Triangles, 68
Equiangular, 68
Equilateral, 68
Isosceles, 68
Right, 68, 69
Scalene, 68
Trust companies, 179
Tucks, 199
Turbidity of water, 296
Two-ply yarns, 223
Unit, 1
United States revenue, 144
Useful mechanical information, 341
Use of tables, 88
Uses of nutrients in the body, 102
Value of coal to produce heat, 167
Volt, 169, 360
Voltmeter, 362
Volume, 74
Warp, 217
Water, alkalinity of, 298
Ammonia in, 296, 297
Analysis of, 296
Bacteria in, 298
Chlorine in, 297
INDEX
369
Water, s — continued.
Color of, 296
Hardness of, 298
Iron in, 298
Nitrogen in, 297
Odor of, 296
Oxygen consumed by, 297
Power, 353
Pressure, 350
Residue on evaporation, 296
Sediment in, 296
Supply, 345
Traps, 352
Turbidity of, 296
Watt, 170
Wool, 211
Woolen yarns, 220
Work, 343
Worsted yarns, 219
Yarns, 217
Cotton, 223
Linen, 222
Raw silk, 222
Spun silk, 223
Two-ply, 223
Woolen, 220
Worsted, 219
ASE TO 5o r« ALTY
DAY AND TO ft, on °CENTS °N THE FOURTH
OVERDUE. $I-°° °N ™E SEVENTH "
2l-50m-l,'?
YB 05200
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