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HARVARD UNIVERSITY
LIBRARY OF THE
Department of Education
COLLECTION OF TEXT-BOOKS
Contributed by the Publishers
TRANSFERRED
TO
LEGE
3 2044 097 004 881
THE
PROGRESSIVE ARITHMETIC
PART III
BY
WILBUR F. NICHOLS, M. A.
Supervising Principal of Eaton District, New Haven, Conn.,
Author of " Graded Lessons in Arithmetic,*' " Arithmetical Problems,"
"Topics in Geography."
THOMPSON, BROWN & CO.,
BOSTON NEW YORK CHICAGO.
. : r
Dept, 0' ; .^
CJOPTRIOHT, 1903,
By WILBUR F. NICHOLS.
H5":VARD COLLEQE UNARY v
-r-~.^^-^r^TT) pi^oim THE
r'^F;^''^Y OF THE
TYPOORAPHY BY C. J. PETERS ft SOW,
nORTON, MASS., U. S. A.
PREFACE.
This is the third in a series of three books.
The plan of development pursued in the previous books of this
series is here continued, giving a review of the subjects previously
presented, and a study of those topics that have not been presented
in the other books. This is followed by a condensed summary,
or concise, topical review of all < subjects. The latter feature
will readily commend itself to those . schools where the whole
subject of arithmetic is carefully reviewed during the last year
of the grammar school.
Throughout the series, principles are taught rather than
rules, and the work is made practical. The shorter and simpler
methods in daily use in the different trades and those used by
merchants and bankers are introduced.
The aim has been to furnish material sufficient for the
demands of any school. This means that the problems are not
so difficult as to discourage the majority of the pupils, or so
easy as to render unnecessary steady, thoughtful work on their
part Any teacher using these arithmetics should omit from
the regular lesson any topic or example which seems unneces-
sary or too difficult for his class.
The lessons upon the different geometrical forms, which have
been an important part of the previous books, are here contin-
ued and extended to include the simpler principles of concrete
and constructive geometry which make the transition to theoret-
ical geometry easier for the pupils.
iii
iv PREFACE.
The first principles of algebra are introduced, and throughout
the book there are many problems that should be solved by
means of algebra. Wherever this method of solution is the
simpler, teachers should encourage the pupils to use it. Put-
ting into algebraic language the conditions of an arithmetical
problem forms an easy transition from the concrete language
of arithmetic to the more abstract language of algebra.
While this book contains much new matter, the author has
drawn largely from his "Graded Lessons in Arithmetic," a
series which has been favorably received, consisting of a sep-
.arate book for each school year.
The author wishes here to express his indebtedness to the
many teachers, supervisors, and superintendents who have
tested the problems, criticized the work, and given many help-
ful suggestions.
W. F. NICHOLS.
New Haven, April 15, 1903.
TABLE OF COI^TENTS.
Addition in Algebra, 241, 242.
Addition of Common Fractions, 266.
Addition of Decimal Fractions, 272.
Algebra, 8, 9, 06, 116, 117, 138, 139,
166, 157, 168, 192, 193, 221, 222,
241, 242, 243, 244, 246, 246, 247,
248.
Algebraic Problems, 8, 9, 96, 116, 192,
222.
Angles, 131, 132, 133.
Application of Rules for Practical
Measurements, 123, 124.
Application of Square Boot, 214, 215,
216.
Area of Circles, 279, 280.
Area of Parallelograms, 277.
Area of Similar Figures, 284.
Area of Triangles, 277.
Bank Discount, 186, 186, 187, 188, 296.
Board Measure, 27, 28.
Business Forms, 297, 298.
Cancellation, 2.
Commercial Discount, 101, 291.
Commission, 94, 181, 182, 288.
Compound Interest, 161, 162.
Cones, 69, 282, 283.
Construction, 108, 131, 132, 133, 168,
169, 194, 196. 209.
Contents of Cones and Pyramids, 69,
283.
Contents of Cylinders, 61, 52, 283.
Concents of Spheres, 165, 281.
Cube Boot, 299, 300, 301.
Cylinders, 61, 62, 282, 283.
Decimals, 23, 37, 48, 271, 272, 273,
274, 276.
Definitions in Common Fractions, 263.
Definitions in Mensuration, 276.
Definitions in Percentage, 286.
Denominate Numbers, 46, 82, 83.
Diagrams, 33, 41.
Difference in Dates, 36.
Division in Algebra, 247, 248.
Division of Decimals, 274.
Division of Denominate Numbers, 82.
Division of Fractions, 3, 4, 268, 269.
Duties or Customs, 134, 136, 136.
Equations, 138, 139, 156, 157, 168, 221.
Evolution, 211, 212, 213.
Exchange, 302, 303.
Fractions, 3, 4, 26, 34, 63, 67, 74, 99,
127, 149, 179, 226, 263-270.
General Summary, 249-298.
Greatest Common Divisor, 81, 262.
Insurance, 93, 289.
Interest, 11, 12, 13, 14, 16, 16, 17, 18,
19, 36, 86, 87, 151, 293, 294.
Involution, 211, 212, 213.
Least Common Multiple, 24, 262.
Longitude and Time, 226, 227, 228,
229.
Measurements, 21, 44, 64, 77, 97, 103,
111, 121, 122, 123, 124, 125, 148,
167, 202, 224, 276-^84.
VI
TABLE OF CONTENTS.
Metric System, 234, 236, 236, 237,
238, 230, 305.
Miscellaneous Facts for Reference,
304.
Miscellaneous Review, 1, 6, 7, 32, 42,
43, 40, 67, 68, 71, 73, 78, 70, 84,
02, 08, 104, 100, 112, 113, 118,
110, 128, 120, 137, 142, 143, 147,
164, 166, 163, 164, 178, 183, 184,
203, 207, 208, 217, 218, 210, 231,
232, 233.
Multiplication in Algebra, 246, 246.
Multiplication of Decimals, 273.
Multiplication of Denominate Num-
bers, 82.
Multiplication of Fractions, 4, 266,
267.
Notation and Numeration, 240, 260,
261, 262.
Oral, 6, 10, 16, 20, 26, 30, 36, 40, 46,
60, 66, 60, 66, 70, 76, 80, 86, 00,
06, 100, 106, 110, 116, 120, 130,
140, 160, 160, 170, 180, 100, 200,
210, 220, 230, 240.
Oral Interest, 16.
Oral Percentage, 26, 06, 130, 100, 220.
Parenthesis, 103.
Partial Payments, 171, 172, 173, 174,
176, 206.
Partnership, 204, 206.
Percentage, 26, 31, 38, 47, 66, 66, 76,
88, 80, 03, 04, 06, 101, 102, 126,
130, 162, 160, 166,' 176, 177, 181,
182, 180, 100, 101, 201, 206, 220,
223, 286-208.
Profit and Loss, 88, 80, 287.
Problems in Algebra, 8, 0, 06, 116,
102, 222.
Problems in Interest, 87.
Properties of Numbers, 261, 262.
Proportion, 50, 60, 61, 62, 63.
Pyramids, 60, 280, 281.
Ratio, 58.
Reduction of Common Fractions, 264.
Reduction of Denominate Numbers,
83.
Review Miscellaneous, 1, 6, 7, 32, 42,
43, 49, 67, 68, 71, 73, 78, 70, 84,
02, 08, 104, 100, 112, 113, 118, 110,
128, 120, 137, 142, 143, 147, 164,
166, 163, 164, 178, 183, 184, 203,
207, 208, 217, 218, 210, 281, 232,
233.
Review of Decimals, 23, 37, 48, 272-276.
Review of Fractions, 26, 34, 63, 67,
74, 00, 127, 140, 170, 226, 263-270.
Review of Interest, 86, 161.
Review of Percentage, 31, 38, 47, 66,
66, 76, 102, 126, 152, 160, 166,
176, 177, 180, 101, 201, 206, 228.
Roman Notation, 262.
Rules for Practical Measurements,
121, 122.
Similar Figures, Area of, 284.
Similar Figures, Volume of, 284.
Sphere, 163, 166, 281.
Square Root, 212, 213, 214, 216, 216.
Standard Time, 220.
Statements, 20, 30, 64, 72, 01, 114,
141.
Stocks and Bonds, 106, 107, 108, 100,
202.
Study of Lines, 106, 107.
Subtraction in Algebra, 243, 244.
Subtraction of Common Fractions,
266.
Subtraction of Decimal Fractions, 272.
Subtraction of Denominate Numbers,
46.
Summary of Addition, 263, 264.
Summary of Common Fractions, 263-
270.
TABLE OF CONTENTS.
VU
Summary of Decimal Fractions, 271-
275.
Summary of Division, 269, 260.
Summary of Mensuration, 276-284.
Summary of Multiplication, 267, 268.
Summary of Percentage, 286-298.
Summary of Subtraction, 265, 266.
Surface of Cone, 282.
Surface of Cylinder, 282.
Surface of Prism, 282.
Surface of Pyramid, 282.
Surface of Sphere, 153, 281.
Tables of the Metric System, 805.
Tables of Weights and Measures, 805.
Taxes, 144, 146, 146, 290.
Trapezoid, 22.
Volume of Cones and Pyramids, 69,
288.
Volume of Cylinders, 51, 62, 283.
Volume of Spheres, 165, 281.
PROGRESSIVE ARITHMETIC.
PART THREE.
MISCELLANEOUS REVIEW.
1. Find 33i% of $28.80. 6k% oi $25.60.
2. $174.04 is 95% of how many dollars?
3. Find the gain per cent when the cost is $600, and the
selling-price $618.
4. Find the cost when the selling-price is $78.84, and the
loss 80%.
5. A man owns a rectangular lot 132 ft. long and 110 ft.
wide. A fence runs from the northeast to the southwest cor-
ner. How many square feet in each part?
6. My garden is 100 ft. long and 25 ft wide. How many
boards, each 12 ft. long, 6 in wide, will it take to build a tight
board fence 6 ft. high across one end and two sides of this
garden ?
7. The selling-price is $175. Find the gain per cent, if
the cost is $150.
8. Make and solve an example illustrating how to find the
number of cords in a pile of wood.
9. Add : Two thousand and four thousandths ; three hun-
dred-thousandths ; six millionths ; two million forty-five ; ten
thousandths.
1
2 C^J^<?Bj:j,ATION.
Cancellation is a method of shortening the work in problems
involving only multiplication and division. Dividing any one
of a series of factors by any number divides their product by
that number. Dividing dividend and divisor by the same num-
ber does not change the quotient.
1. Divide 4x3x6x 12 by 2x8x4x6.
^ X ^ X ^ X 12 This is an example in division in which
2 X ^ X ^ X ^ the dividend and divisor are partially factored.
We shorten the operation by canceling the 3's ; i.e., by dividing
both dividend and divisor by three. We further shorten the
work by canceling the 4's and 6's. The dividend is now 12
and the divisor 2 ; hence the result is 6.
2. Multiply 44 by 8 and the product by 7 ; then divide it by
11 X 16 X 7.
. 8. Divide 40 x 18 x 13 x 8 by 10 x 13 x 16.
4. Divide 15 x 4 x 8 x 9 by 30 x 2 x 6 x 12.
6. Divide 108 x 17 x 9 x 4 by 27 x 3 x 16 x 17.
6. Divide 5 x 25 x 874 by 2 x 437 x 5 x 5 x 5.
7. Divide 376 X 14 X 21 by 7 X 7 X 16 X 3.
8. Divide 10 x 5 x 25 by 2 x 5 x 5 x 5.
9. Divide 120 x 4 x 9 by 3 x 40 x 4 x 3.
10. Divide 60 x 3 x 7 x 21 by 20 x 14 x 3 x 9.
11. Divide 15 x 18 x 21 x 25 by 9 x 3 x 7 x 15.
12. Divide 44 x 8 x 7 by 11 x 16 x 7.
18. Divide 15 x 4 x 8 x 9 by 30 x 2 x 6 x 12.
14. Divide 5 x 25 x 874 by 2 x 437 x 5 x 5 x 5.
15. Divide 11 x 39 x 14 x 96 by 44 x 18 x 26 x 14.
16. Divide 125 x 60 x 24 x 42 by 25 x 120 x 36 x 5.
17. Divide 36 x 21 x 14 by 27 x 7 x 6.
18. Divide 125 x 105 x H by 35 x 33 x 5 x 5.
19. Divide 54 x 3 x 4 x 15 by 18 x 12 x 10.
20. Divide 25 x 160 x 13 x 90 by 51 x 30 x 8 x 15.
21. Divide 400 x 125 x 64 x 72 by 36 x 75 x 32 x 25.
TO FIND A PART OF A FRACTION. 3
1. Divide § by J.
^^^ 3* = J A ^ /, = 8 -^ 9 = f .
For explanation see Part II., page 190.
I divided by 1 is f f divided by J
\P) % ~^ ^ — h must be 4 times J ; hence J divided by }
§ H- :| = 4 X §. must be J of 4 times ), or | x J.
J -1- I = J of 4 X §, or Therefore, dividing f by f is Ihe same
i X 4 = ® ^ multiplying } by }, the divisor in-
^ ^ ^* verted.
NoT£. — In multiplying or dividing fractions, cancel when possible.
Divide ;
2. I by § \l by f A by A l by ^%
•3. A by J A by J J by It A by f
4. i? by 1^ i by A t by f f by f
5. Learn : To divide a fraction by a fraction, divide by the
numerator of the divisor, and then multiply by its denominator,
or invert th**. divisor and proceed as in multiplication of frac-
tions.
6. I -i- ^ may be written i. When written in this form it
is called a complex fraction.
7. A Complex Fraction is a fraction having a mixed num-
ber or a fraction for one or both of its terms.
8. c:^^i,..„. 12^ 2t li 34J 37| 9S
o.
K:Jiiiij^Aj.ijr •
37i If
2i 7J 6f
4^
9.
Simplify :
iride i
171 28A
6H 4A
11§ 125
26H
8A
10.
\ by 2
T^^by i
J byf
A by t.
11.
A by*
Hbyi'^
A by 1
i by §.
12.
Mbyf
A by 1
Abyf
H by f .
13.
5f by 4f
5i by 3i
H by n
2i by U.
14.
34by|
2§ by f
3?byt
6« by #.
MULTIPLICATION AND DIVISION OF FRACTIONS.
1. If f of an acre of land cost $60, what will /s of an acre
Solution. ^^^^
tf 7 If f of an acre cost $60, one acre will cost as
X ^^ 75~* many dollars as f is contained times in $60, or
^ ^f $60 X J. If one acre costs $60 X J, ^^ acres will
r cost $60 X } X A.
2. If 7^ lb. of raisins cost 85 cents, what will 4^ lb.
cost?
8. If 9 oranges cost 22 i cents, what will 16 oranges cost?
5 8
Solution. ^/ x - X ^ = 40 cents.
^ 9 1
4. What will 4J tons of coal cost, if lOj tons cost $75.48 ?
6. What is the cost of a pile of wood 12 feet long, 4 feet
Solution. wide, and 6 feet high, at f 5i
'^ — vgg — ^ ^ ""g ^ vi-^f. The solution explains itself . The
^AA work is all indicated, and cancella-
'^^ tion is then used to lessen the num-
ber of figures required.
6. Find the cost at $3i a cord of wood which fills a shed
24 ft. long, 20 ft. wide, and 16 ft. high.
7. How many gallons of molasses at 37 J/ a gallon are worth
as much as 12| bu. of potatoes at 40/ a bushel?
8. If 6 J cd. of wood cost |41j, what will 2^ cd. cost?
9. The perimeter of a room is 56 ft. 8 in., and its height
8 ft. 9 in. Find the area of the 4 walls.
10. Find the volume of a rectangular solid whose dimen-
sions are 2f ft., 2§ ft., and 4^ ft.
11. What must I pay for 37§ tons of coal, if 12^ tons cost
$74?
12. If 13^ lb. of sugar cost 80 cents, what will 8^ lb. cost?
13. How many times is 4j contained in 7f ?
ORAL. 5
1. In 96 oz. how many haK-pounda ?
2. , When milk is f .05 a quart, what will 5 gallons cost?
3. Find the cost of | of a pound of tea at $1.00 a pound,
and 1 pound of butter at 25/ a pound.
4. If you can buy 6 cards for a cent, how many can you
buy for f 1 ?
6. What will J of a ton of coal cost at $6.30 a ton? At
$5.10?
6. Find the cost of 7 chairs at $3 each, 12 at $2 each, and
a table at $15.
7. A man spent $24 out of $36. What per cent of his
money had he left?
8. If you should put 320 oz. of cloves into quarter pound
packages, how many packages would you have ?
9. How many pint bottles will be required to hold 6 gal. 2
qt. of syrup ?
10. Find the cost of 4 lb. 8 oz. of cheese at 8 cents a half-
pound.
11. How many quarts will it take to fill a bag holding 2 J pk.?
12. If $18 is J of my money, how much money have I?
13. If you gather ^ bu. of walnuts, and sell 1 pk. for 60 cents
and the rest at 10 cents a quart, how much will you receive?
14. Find the cost of 6 J lb. of sugar at 6/ a pound.
16. Find the cost of 6 J lb. of meat at 9 cents a pound.
16. Paid 6 cents a pound for 2 packages of buckwheat, each
package containing 3^ lb. Find the cost.
17. What is the cost of 4 pk. 4 qt. of nuts at 20/ a peck?
18. What 4 equal numbers make 36 ?
19. Nellie had 36 peaches. After eating J of them, and
giving away J of them, how many had she left?
20. What is meant by i of anything ?
«1. What part of 8 is 8 ? What per cent of it?
22. How many days are there in 7 wk. 3 da. ?
6 MISCELLANEOUS REVIEW.
1. Add I, If, A, f , and |^. First reduce eaoH to its lowest
terms.
2. 78« - 49|. 4i + 6§ + 7«.
8. Simplify —
4. What per cent of an acre is a rectangular piece of land
5 rd. long and 4 rd. wide ?
6. Divide 73.8 by .0008. If by h
6. Find the prime factors that are common to 1,156 and
2,206.
7. At $6.40 a ton, how much will 3,675 lb. of coal cost?
8. Add 7.46, 636, 73.76, 7.569, 165, and 46.78.
9. Divide A by Ij. iJ by 5j.
10. 246 X 8 - 169 - 13 - (225 - 15) x 16 + 207 -r- 3.
11. Find the entire surface of a square pyramid whose slant
height is 7 ft., and base a 3 ft. square.
12. What will it cost, at 66/ a square yard, to paint a church
spire whose base is an octagon 7 ft. on each side, and whose
slant height is 95 ft. ?
18. What is the area of a circle, if the distance around it is
314.16 ft.?
Note. — In finding the diameter or circumference, use 3.1416 in place of 3f.
It is more nearly correct.
14. What is the convex surface of a cone, the diameter of
whose base is 6 feet, and whose slant height is 24 ft. ?
16. Of another cone, the circumference of whose base is 16
ft., and whose slant height is 18 ft. ?
16. Find the entire surface of a cylinder whose circumference
is 16 ft., and whose altitude is 25 ft.
17. What will it cost, at $1.25 a square yard, to polish the
convex surface of a cylinder 3 ft. in diameter and 12 ft. long?
18. How many acres are there in a circular field whose diam-<
eter is 22 rods ?
MISCELLANEOUS REVIEW. 7
1. How many square feet in a floor 28 ft long and 19] ft.
wide ?
2. From the middle point of one side of an equilateral tri-
angle to the vertex is 21 feet. The base is 24} ft. Required
the area.
8. If 46 sheep cost $540, what number will $228 purchase?
4. If 12 cd. of wood cost $54, what will 19j cords cost?
6. How much commission shall I pay an auctioneer for sell-
ing my house for $6,600, at 2}% ?
6. From a cask of rice containing 760 lb., 90 lb. were sold
at one time, 60 lb. at another, and 40 lb. at another. What
per cent was sold ?
7. Add 7 rd. 4 yd. 2 ft. 7 in.; 19 rd. 8 yd. 2 ft. 9 in.;
6 rd. 3 yd. 1 ft. 6 in. ; 8 rd. 4 yd. 2 ft. 6 in. ; 1 rd. 8 yd. 1 ft.
11 in.
8. 646J pounds of sugar cost $61.72. The merchant wishes
to gain $3.23 J. How much must he ask a pound to do so?
9. A rectangular block of marble is 7 ft. long, 4 ft. wide,
and 3} ft. high. If it costs 60^ a cubic foot, find its cost.
10. The perimeter of a rectangle is 124 rd., the width is 12
rd. Find the area.
11. A man built a barn 80 ft. long, 45 ft. wide, with 30 ft
posts. The gable is 15 ft. high, and the rafters are 28 ft.
long. Find how many feet of inch boards will be needed for
sides and ends, allowing for 2 double doors 14 ft. by 12 ft., and
2 single doors 9 ft. by 6 ft. Find the feet of lumber needed
for floor, covered with 2-inch planks. Find the feet of roof-
boards needed. Find the feet of lumbey needed for doors, if 1 J
in. boards are used. Find the cost of lumber at $16 per M.
12. A merchant paid $10 for an article, and sold it at an
advance of 20%. Find the selling-price.
18. Find the value of a pile of wood 40 ft. long, 4 ft. wide,
and 4 ft. 6 in. high, at $5.50 a cord.
8 ALGEBRAIC PROBLEMS.
1. A man paid $960 for a carriage and span of horses, pay-
ing three times as much for the horses as for the carriage. How
much did he pay for each ?
In examples like the above use the symbol x. This x stands
for any unknown quantity.
Let X =s the cost of the carriage,
then 3 2; s the cost of the horses.
4 a; = the cost of all, which is $960.
4 a; = $960. x = $240. 3 a; = $720.
2. A and B together had $150, and B had four times as
much money as A. How many dollars had each ?
8. A man bought a harness and a robe for $46. The har-
ness cost 4 times as much as the robe. What was the price of
each?
4. Three men, A, B, and C, form a company with a capital
of $6,000. C put in 2 times as much as A, and B 3 times as
much as A. How many dollars did each put in ?
6. 108 marbles are divided among 3 boys. A takes a certain
sum, B takes 3 times as many as A, and C takes 6 times as
many as A. What is each boy's share ?
6. Divide $81 among A, B, and C, so that B shall have 5
times as much as A, and C 3 times as much as A.
7. A horse and carriage are together worth $460. The
horse is worth twice as much as the carriage. What is each
worth ?
8. A man has 3 houses, which are together worth $6,400.
The second is worth twice as much as the first, and the third is
worth as much as the other two. Find the cost of each.
9. Divide $216 among A, B, and C, so that B may have 3
times as much as A, and C may have 6 times as much as A.
10. Divide the number 136 into three parts such that the sec-
ond shall be 3 times the first and the third as much as the first
and second.
ALGEBRAIC PROBLEMS. 9
. 1. The difference between 5 times a number and 3 times a
number is 40. What is the number ?
Let X = the number The difference between 5 x and 3 x
then 6 X zr 6 times the number is 2 x, hence 2 x must equal 40.
and 3 X = 3 times the number
2x = 40
X = 20.
2. A has three times as much money as B. If the differ-
ence between their shares is |5200, how much money has each?
3. John had a certain number of papers, and George had 12
times as many. If George had 33 papers more than John, how
many papers had each boy ?
4. 6a: — 42; = 10. Make a problem and solve it.
Illustration : If from 6 times a certain number 4 times the
number is taken, the remainder will be 10. Find t£e number.
Make problems for the following and solve them.
6. 9a; — 4^; = 15. 6. 5a: + 3a7 + a: = 81.
7. 6a: + 7a: = 39. 8. 6a: + 4a: - 2a: = 72.
9. 11a: — 6a: = 50. 10. a: + 2a: + 3a: = 48.
11. Margaret is four times as old as Helen, and Ruth is half
as old as Margaret If tlie sum of their ages is 14 years, how
old is each ?
12. Robert has three time as many cents as Amy, and Charles
has four times as many as Amy. If Charles has 12 cents
more than Robert, how many cents has each ?
13. Elizabeth had a certain sum of money when her father
gave her 6 times as much more. She then had 72 cents.
Hov?^ much money had she at first ?
14. Joseph had 7 times as many marbles as Henry, and
Charles had 5 times as many as Henry. Joseph had 18 mar-
bles more tiian Charles. How many marbles had each boy ?
15. A, B, C and D have |140. B has $10 more than A, C
has $10 more than B, and D has $10 more than C. How
many dollars has each ?
10 ORAL.
1. How many feet in two boards 14 ft. long, 12 in. wide?
2. How long is it from 15 minutes of 7 o'clock in the morn-
ing till 16 minutes past 6 o'clock in the evening ?
3. How many cubic feet of earth must be removed to dig a
cellar 30 ft. long, 20 ft. wide, and 4 ft. deep?
4. How many square yards in 27 sq. ft. ? In 64 sq. ft. ?
72 sq. ft. ?
6. How many inches in 4 J ft.?
6. How many inches in 2i yd.?
7. How high is a horse that is 16 J hands high?
Note. — A hand is 4 in., a term used only in speaking of the height of
horses.
8. How many feet deep is water that is 9 J yd. deep ?
9. If by selling a horse for $160 I gain J of the cost, what
did he cost?
10. Read the 9th example, using the equivalent per cent in
place of the fraction.
11. If two-thirds of a dozen oranges cost 16 cents, how much
will 21 dozen cost?
12. I of 72 is I of how many?
13. i of 48 is f of how many ?
14. A farmer sold J of his tobacco crop to one man, and \ to
another, and had 100 lb. remaining. How much had he at first?
16. If 34 lb. of meat cost 50 cents, how much will 2J lb. cost?
16. When f of my monthly salary is f 9, how much do I earn
in § of a month ?
17. What must I pay for 9^ bbl. of apples, if li bbl. cost $6 ?
18. If J of 8 cords of wood costs $12, how much will i of 3
cords cost?
19. What wiU 8 J bbl. of apples cost, if 2i bbl. cost $9?
20. J of 100 is VV of what number?
21. 8 is § of what number? 14 is J of what number?
88. What will 1 gal. of cream cost at 30/ a quart?
INTEREST. . 11
1. If I use another's house, what is the money I pay for the
use of it called ?
2. If I use a man's horse, what is the money I pay for the
use of it called ?
3. If I use another's money, what is the money I pay for
the use of it called ? Interest.
4. Learn : — Interest is money paid for the use of money.
It is usually reckoned at a certain per cent for each year.
NoTB. — Interest differs from house-rent in that it is always a certain per
cent of the money borrowed.
6. I asked a man to lend • me some money. He said he
would at 6%. What did he mean ?
6. When one man borrows money of another, he usually
gives him a paper called a promissory note, to prove that he has
borrowed money.
^100 Boston, Nov. 5, 1902.
On demand, I promise to pay to John Smith, or
order, One Hundred Dollar s^ with interest at G^c
Value received. Greo. Whittaker.
7. In this note who promises to pay? To whom does he
promise to pay ? How much does he promise to pay ?
8. The one who promises to pay is called the promisor or
maker. The one to whom the promise is made is called the
promisee or payee. The sum of money is called the face.
Name the maker, payee, and face in this note.
9. When the words "on demand " are in the note, the payee
can demand the payment of the money at any time. When the
words " or order " are in the note, the payee can order the money
paid to some one besides himself-.
10. Write a demand note, using the following: date, to-day;
maker, yourself ; payee, your teacher ; face, any sum.
12 . INTEREST,
1. Sometimes the words " on demand " are not in the note,
but in their place a certain specified time. The following is a
time note :
^100 Boston, Nov. 5, 1902.
Six months after date I promise to pay John
Smith, or order. One Hundred Dollars, with inter-
est at 6%. Value received. ^ ™., .,, ,
' Geo. Whittaksr.
2. If I must pay 6 cents for the use of $1 for 1 year, what
part of the year can I keep the dollar and pay only 1 cent inter-
est? How many months is J of the year?
8. Then what is the interest of $1 for 2 mo. ?
4. If the interest of $1 for 2 mo. is .01, what will be the
interest of 15 ? $8? $10? $100?
6. If the interest of $1 is .01 for 2 mo., the interest of $100
will be 100 times $.01, which is $1.00. What short method
can you see for multiplying .01 by any number ?
6. Learn : Move the decimal point two j^aces to the left to
find the interest on any sum of money for two months at 6%.
7. Why does pointing off two places give the interest for
2 mo. ?
Find the interest on the following sums of money for 2 mo.
at 6%:
8.
f60
$ 88
1305
$548
$6,678
9.
30
72
475
842
4,978
10.
48
46
267
648
9,876
11.
39
83
894
500
4,748
18.
62
450
412
402
8,649
18.
79
276
516
819
6,891
14.
12
118
618
549
4,206
16.
24
374
■ 728
676
2,060
16.
31
333
176
715
.1,008
INTEREST, IS
Note — . In all examples in Interest on this page 6% is understood.
1. Find the interest of $500 for 6 mo.
||^5^Q() _ Jjii;^ fQj. 2 mo. First find the interest for 2 mo. by pointing
Q off two places; this is $5. If $5 is the interest
for 2 mo., the interest for 6 mo. will be 3 times
$15.00 = int. for 6 mo. as much.
0. How would you find the interest for 4 mo.? 8 mo.? 10
mo.? 1 year (12 mo.)? 1 yr. 4 mo. (16 mo.)?
8. Take the sums of money on page 12, and find the in-
terest on each for 4 mo. 6 mo. 8 mo. 10 mo. 1 year.
4. Find the interest of $600 for 10 mo.
5. Find the interest for 6 mo. on f 1,200. On $2,400.
6. Find the interest fori year on $750. On $300. On $250.
7. Find the interest for 1 yr. 2 mo. on $405. On $145.
On $69.
8. Find the interest for 1 yr. 8 mo. on $1,216. On $2,445.
9. On $1,218 for 1 yr. 10 mo.
10. On $600 for 1 yr. 4 mo.
11. On $436 for 8 mo.
12. Find the interest of $800 for 9 mo.
« Q AA O First find for 8 mo. as usual. Then for the odd month,
f 8.U0 = Z mo. ^hich must be J of the interest for 2 months.
^^1 no I 1 ^^' ^^- ^'^^^ ^^^ interest on $900 for 9 mo.
(fcQA AA ~ Q "'^' ^*- ^^ *^'^^^ *^'" ^ ^^''' (^ ^^- + 1 '^^•)
$36.00 = 9 mo. ^^ ^^ ^2,200 for 1 yr. 3 mo. (14 mo.
+ 1 mo.)
16. On $3,460 for 1 yr. 5 mo. (16 mo. + 1 mo.).
17. On $489 for 1 yr. 2 mo. For 9 mo.
18. On $638 for 1 yr. 7 mo. (18 mo. + 1 mo.).
19. On $1,240 for 11 mo. For 1 yr. 5 mo.
20. On $9,876 for 5 mo. For 1 yr. 3 mo.
21. On $4,675 for 3 mo. For 1 yr. 2 mo.
88. On $4,106 for 7 mo. For 1 yr. 3 mo.
14 INTEREST FOR DAYS,
1. What is the interest on $1 for 2 mo., or 60 days ?
2. 6 days is what part of 60 days ?
8. The interest on $1 for 6 days is what part of the interest
for 60 days?
4. What is iV of .01 ?
6. What is the interest of $1 for 6 days ?
6. If the the interest on $1 for 6 days is 1 mill, what is the
interest on $2 ? $7 ?
7. At the same rate, the interest on $100 is how many times
.001?
8. Multiply .001 by 100 ; by 200 ; by 600 ; by 1000.
9. Tell a short way to multiply .001 by any number.
10. How many places to the left do you move the point?
11. Moving the point three places to the left is the same as
multiplying by what?
12. Learn : Moving the point three places to the left gives
the interest on any sum of money for six days.
13. Find the interest at 6% for 6 days on the sums of money
given on page 12.
14. Tell a short way to find the interest for 12 days.
16. Find the interest on $600 for 12 days.
$.600 interest for 6 days.
2
$1,200 interest for 12 days.
16. How do you find the interest for 18 days? For 24 days?
17. Find the interest on $1,240 for 18 days.
18. Find the interest on $480 for 24 days.
19. Find the interest on $697 for 12 days.
20. Find the interest on $368 for 30 days.
21. Find the interest on $267 for 36 days.
22. Find the interest on $142 for 42 days.
28. Find the interest on $612 for 18 days.
24. Find the interest on $210 for 24 days.
ORAL. 15
The rate of interest in each of the following examples is 6%.
Find the interest of :
1. flOO for 1 yr. $100 for 1 yr. 6 mo.
2. 1100 for 1 yr. 6 mo. 6 da. $200 for 8mo. 12 da.
3. 1200 for 10 mo. 18 da. $200 for 1 yr. 6 mo.
4. $200 for 1 yr. 7 mo. 24 da. $300 for 4 mo.
5. $300 for 9 mo. $300 for 3 mo. 18 da.
6. $400 for 6 mo. 12 da. $400 for 1 yr. 3 mo. 18 da.
7. $400 for 9 mo. 12 da. $400 for 1 yr. 8 mo. 6 da.
8. $500 for 7 mo. $500 for 1 yr. 2 mo.
9. $500 for 1 yr. 4 mo. 12 da. $500 for 60 da.
10. $500 for 36 da. $600 for 1 yr. 1 mo. 6 da.
11. $600 for 10 mo. 24 da. $600 for 48 da.
12. $600 for 1 yr. 2 mo. 12 da. $1,000 for 9 mo.
13. Find the interest of $200 for 2 mo.
14. Find the interest of $300 for 3 mo.
15. Find the interest of $400 for 4 mo.
16. Find the interest of $500 for 5 mo.
17. Find the interest of $600 for 6 mo.
18. Find the interest of $700 for 7 mo.
19. Find the interest of $800 for 8 mo.
20. Find the interest of $900 for 9 mo.
21. Find the interest of $1,000 for 10 mo.
22. Find the interest of $400 for 6 days.
23. Find the interest of $600 for 12 days.
24. Find the interest of $800 for 18 days.
25. Find the interest of $600 for 1 yr. 2 mo. 12 da.
26. Find the interest of ^700 for 24 days.
27. What is the interest of $5 for 1 yr. 10 mo. 12 days?
28. What is the interest of $200 for 1 yr.? For 2 yr.? For
6 mo. ?
29. What is the interest of $500 for 1 yr. ? For 1 yr. 6mo.?
For 2 yr. 6 mo. ?
16 INTEREST,
Find the interest at 6% : —
1. Of 1846 for 1 yr. 2 mo.
2. Of $846 for 12 days.
8. Of $846 for 1 yr. 2 mo. 12 days.
4. Of $1,728 for 1 yr. 4 mo. 6 da. For 1 yr. 8 mo. 18 da.
6. Of $466.56 for 8 mo. 12 da. For 1 yr. 6 mo. 12 da.
6. Of $2,304 for 10 mo. 18 da. For 1 yr. 4 mo. 12 da.
7. Of $450 for 1 yr. 3 mo. 6 da. For 1 yr. 7 mo. 18 da.
8. Of $800 for 1 yr. 5 mo. 12 da. For 1 yr. 9 mo. 12 da.
9. Of $375 for 24 days. For 1 yr. 6 mo. 18 da.
10. Of $323.50 for 1 yr. 10 mo. 18 da. For 11 mo. 12 da.
11. Of $960 for 1 yr. 3 mo. 18 da. For 5 mo. 6 da.
12. Of $842 for 9 mo. 12 da. For 1 yr. 3 mo. 18 da.
13. Of $700 for 6 mo. 18 da. . For 1 yr. 9 mo. 24 da.
14. Of $1,200 for 4 mo. 12 da. For 1 yr. 6 mo. 6 da.
15. Of $900 for 1 yr. 7 mo. 18 da. For 1 yr. 5 mo. 24 da.
16. Of $1,400 for 11 mo. 12 da. For 1 yr. 2 mo. 18 da.
17. Of $976.25 for 4 mo. For 1 yr. 5 mo. 12 da.
18. Of $846.78 for 1 yr. 6 mo. For 1 yr. 9 mo. 12 da.
19. Of $180 for 1 yr. 3 mo. 24 da. For 1 yr. 5 mo. 18 da.
20. Of $680.60 for 10 mo. For 1 yr. 11 mo. 12 da.
21. Of $211.25 for 1 yr. 5 mo. For 1 yr. 6 mo. 18 da.
22. Of $1,234.50 for 1 yr. 3 mo. 18 da. For 1 yr. 2 mo. 24 da.
28. Of $666.60 for 1 yr. 2 mo. 12 da. For 1 yr. 3 mo. 18 da.
24. Of $888.80 for 1 yr. 8 mo. 18 da. For 1 yr. 5 mo. 24 da.
26. Of $555.50 for 1 yr. 4 mo. 6 da. For 1 yr. 7 mo. 30 da.
26. Of $368.60 for 1 yr. 6 mo. 24 da. For 1 yr. 9 mo. 12 da.
27. Of $169.50 for 1 yr. 10 mo. 12 da. For 1 yr. 11 mo. 18 da.
28. Of $444.40 for 1 yr. 2 mo. 6 da. For 1 yr. 8 mo. 24 da.
29. Of $333.30 for 1 yr. 4 mo. 12 da. For 1 yr. 6 mo. 18 da.
80. Of $777.70 for 1 yr. 6 mo. 18 da. For 1 yr. 9 mo. 30 da.
81. Of $999.90 for 1 yr. 8 mo. 24 da. For 1 yr. 9 mo. 24 da.
82. Of $1,224.60 for 1 yr. 10 mo. 12 da. For 1 yr. 11 mo. 18 da.
INTEREST FOR DAYS. 17
1. What is the interest of $600 for 60 days at 6% ? How
did you find it ?
2. What is the interest of $600 for 6 days at 6% ? How
did you find it?
3. Find the interest of $600 for 7 days at 6%.
$.600 = int. for 6 d.
.10 = int. for 1 d.
.70 = int. for 7 d.
4. Find the interest of $400 for 25 d.
$4.00 = int. for 60 d. poi^^ off 2 places to find the interest for 60 d.
1.333 = int. for 20 d. To find the interest for 20 d., take J of the inter-
.333 = int for 5 d. ®®^ ^^^ ^ ^- '^^ ^"^ ^^^ interest for 5 d., take
9IM6 = int.' for 25 d. * ^^ '^^ *'^'"^^''' ^^' ^^ ^•
5. Find the interest on $300 for 19 d.
$3.00 = int. for 60 d. m « ^ .u t * . * .n ^ . , . ^
, To find the interest for 10 dayg take J of
.50 = int. for 10 d. ^^e interest for 60 days. To find the inter-
.30 = int. for 6 d. est for 6 d. take yV of the interest for 60 d.
.15 = int. for 3 d. "^^ ^^^ *'^® interest for 3 d. take J of the inter-
est for 6 d.
.95 = int. for 19 d.
6. To find the interest on any sum of money for a certain number of
days : Move the decimal point two places to the left to find the interest for
60 d., or three places for 6 d., and then take such parts of the interest thus
found, as when united will give the interest for the specified time.
The following table will show the method : —
1 d. = i of 6 days. 14 d. = 12 d. + 2 d.
2 d. = J of 6 days. 15 d. :^ i of 60 d.
3 d. = I of 6 d. 16 d. = 15 d. + 1 d.
4 d. = 3 d. + 1 d. 17 d. = 15 d. + 2 d.
5 d. = tV of 60 d. 18 d. = 15 d. + 3 d. or 3 X 6 d.
7 d. = 6 d. + 1 d. 19 d. = 10 d. -f 6 d. + 3 d.
8 d. = 6 d. + 2 d. 20 d. = i of 60 d.
9 d. = 6 d. + 3 d. 21 d. = 20 d. + 1 d. .
10 d. = ^ of 60 d. 22 d. = 20 d. + 2 d.
11 d. = 10 d. + 1 d. 23 d. = 20 d. + 3 d.
12 d. = ^ of 60 d. 24 d. = 20 d. + ^ of 20 d.
13 d. = 12 d. -f 1 d. 25 d. = 20 d. -f i of 20 d.
18 INTEREST.
Find the interest on the following sums of money at 6%
1. 1475 for 5 days. For 1 mo. 5 da.
8. 368 for 7 days. For 2 mo. 7 da.
8. 694 for 8 days. For 6 mo. 8 da.
4. 176.34 for 9 days. For 3 mo. 9 da.
5. 25.60 for 10 days. For 6 mo. 10 da.
6. 136.50 for 11 days. For 4 mo. 11 da.
7. 97.50 for 13 days. For 5 mo. 13 da.
8. 1,264 for 14 days. For 3 mo. 14 da.
9. 1,478 for 15 days. For 7 mo. 15 da.
10. 672 for 16 days. For 4 mo. 16 da.
11. 47.95 for 17 days. For 9 mo. 17 da.
' 13. 106.25 for 19 days. For 3 mo. 19 da.
13. 16.48 for 20 days. For 10 mo. 20 da.
14. 74.34 for 21 days. For 2 mo. 21 da.
16. 1,497 for 22 days. For 4 mo. 22 da.
16. 748.25 for 23 days. For 8 mo. 23 da.
17. 1,200 for 25 days. For 3 mo. 25 da.
18. 667 for 26 days. For 1 mo. 26 da.
19. 1,463 for 27 days. For 6 mo. 27 da.
80. 1,476.86 for 28 days. For 10 mo. 28 da.
81. 1472.40 for 29 days. For 1 yr. 1 mo. 29 da.
88. 1,491.50 for 13 days. For 1 yr. 3 mo. 17 da.
83. 2,468 for 26 days. For 1 yr. 1 mo. 1 da.
84. 680.50 for 27 days. For 1 yr. 5 mo. 16 da.
86. 746.30 for 11 days. For 1 yr. 2 mo. 13 da.
86. 123.40 for 19 days. For 1 yr. 3 mo. 15 da.
87. 567.80 for 21 days. For 1 yr. 7 mo. 19 da.
88. 912.30 for 17 days. For 1 yr. 2 mo. 21 da.
89. 987.60 for 15 days. For 1 yr. 3 mo. 24 da.
80. 876.50 for 16 days. For 1 yr. 4 mo. 25 da.
81. 765.40 for 19 days. For 1 yr. 5 mo. 26 da.
88. 423.70 for 13 days. For 1 yr. 3 mo. 21 da.
INTEREST AT DIFFERENT RATES. 19
1. Find the interest on f 1,200 for 1 yr. 2 mo. 18 da. at 5%.
$12.00 = 2 mo. First find the interest at 6%, as usual. If
^Q . ^^ z -z $87.60 is the interest at 6%, $14.60 will be the
lll?84.UU = 1 yr. J mo. interest at 1%, and $73.00, the difference be-
3.60 = 18 da. tween 6% and 1%, will be the interest at 5%.
6)187.60 = 6%. ' 8. How would you find the inter-
14.60 = 1 %. est at 7 %? What per cent would you
$73.00 = 6%. a^d to 6% ?
3. What per cent would you sub-
tract from 6% to find 4% ?
4. How do you find 2 % when you know 6 % ?
5. When you know 6%, what part of it must you find to
know 3%?
6. 8 % is how many per cent greater than 6 % ? 2 % is what
part of 6% ? What part of 6% then do you add to 6% to get
8%?
7. 9% is how many per cent more than 6% ? 3% is what
part of 6%? What part of 6% then do you add to 6% to get
9%?
8. How do you find interest at 3% the shortest way?
9. How do you find interest at 4% the shortest way? At
5%? At 7%? At 8%? At 9%? At 10%? At 4i%?
At 7i% ?
10. Find the interest on $486.50 for 1 yr. 2 mo. 24 da. at
6%. 7%. 9%.
11. Find the interest on $247.65 for 1 yr. 3 mo. 15 da. at
4%. 8%.
13. Find interest on $1,264.30 for 1 yr. 5 mo. 13 da. at 5%.
Find the interest of :
13. $798.81 for 1 yr. 1 mo. 1 d. at 8%.
14. $346.84 for 1 yr. 11 mo. 18 d. at 5%.
15. $816.24 for 1 yr. 7 mo. 6 da. at 7%.
le. $745.00 for 10 mo. 10 da. at 10%.
20 OJUl.
1. If 1 yd. of cloth cost $5i what will f yd. cost?
9. If 6 bbl, of beef cost $17i, what will li bbl. cost?
8. f of 15 is ^ of how many times 2 ?
4. f of 18 is f of how many times 7 ?
5. I of 24 is 1 J times what number?
6. If ^ of an acre of land is worth f 15 what are 10 J acres
worth ?
7. If you earn $i in a day and your brother $| in a day, how
much will you both ^am in 8 days ?
8. If Nellie has twice as many plums as Mary, and both
have 18 plums, how many has each?
0. 84 is V of how many times J of 25 ?
10. 7f are how many times 9 ?
H» Reduce to improper fractions : 6J, 9g, 7^, 5i, 4|,
18. What will J of a dozen of oranges cost at | of a cent
each?
13. What will 1 pt. of buckwheat coat if 3 pk. cost 48 cents?
14. Mr. C sold a farm for $1,200, which was f of its cost.
Find the loss,
15. How many cents will J of 100 oranges cost at J dime
each?
16. How many times 8 are \i of 26 ?
17. If 7 bbl. of cider cost $28, what will 6 bbl. cost?
18. If a boy can do a piece of work in 30 min., how many
hours will it take him to do 12 times as much work ?
19. If 8 yd. of cloth are worth 140, and butter is $3 a box,
how many boxes of butter will 9 yd. of cloth buy?
20. How much will 6 yd. of silk cost at $2^ a yard ?
31. Mr. Brown is 40 years old, and his son is g as old. How
old is the son ?
22. A man sold 2| yd. of velvet from a piece containing Sj
yd. How many yardft had he left?
28. What is the ratio of 8/ to 32/?
MEASUREMENTS. 21
1. What is the cost of carpeting a room 16^ ft. long^ 12 ft.
wide, with oil-cloth 1^ yd. wide, at 75/ a yard?
Note. — Have the least waste.
2. What will it cost to carpet a room 18 ft. long, 13 ft. wide,
with carpeting f yd. wide, at fl.25 a yard, breadths to run
lengthwise, and a waste on each breadth of 8 in. for matching
the figures ?
Note. — Sometimes, in order to have the flgurai In fchd carpet miitch, it is
necessary to make each breadth longer than the room. This is called a waste,
because it obliges the purchaser to buy more carpet than he needs for the room.
Add the waste on each breadth to the length of the room when the carpet runs
lengthwise, and to the width when it runs width wise, to find the length of each
breadth. Make a diagram to illustrate.
3. A church steeple is in the form of a pyramid. Its base
is a square 15 ft. on each side, and its slant height is 70 ft.
What is the cost of painting it at 30/ a square yard?
4. Find the difference in cost of painting this spire had the
base been a hexagon having the same dimensions.
5. What is the area of a semicircle whose radiuft is 12 ft. ?
6. Draw a circle with a radius of 5 in. This is a drawing
to represent a lot of land. If the scale is i in. tb a i^d, how
many square rods in the field ? How many acres ?
7. A rectangular cistern is 8 ft. long, 6 ft. 6 in. wide, and
4J ft. deep. Find the expense of lining the sides and bottom
with sheet lead weighing 9 lb. a square foot, at b)^ a pound.
8. A pile of wood contains 6^ cd. If the pile is 82 ft. long
and 6^^ ft. high, how wide is it ?
9. A room is 15 ft. 8 in. long and 12 ft. 6 in. wide. The
carpet is f of a yard wide. There is a loss on each strip of 4
in. for matching. Which way should the breadths run to use
the least number of yards of carpeting ? *
10. Find the area of a rectangle 16 yd. long, and 46 ft. wide.
11. Find the area of a rectangle 150 ft. long, 4 rd. widd<
22 TRAPEZOID,
1. What figure is this?
s^ 3. Cut out of paper a trau
\ pezoid, twice as large as this
\ figure.
\ 3. Fold so that the two
\ parallel edges will coincide.
Crease. Cut on the crease.
Place the two pieces so that what were the two parallel lines
shall form one continuous line.
4. What new form have you ?
5. How do you find the area of a parallelogram ?
6. The base of the parallelogram is the sum of what two
lines in the tr?ipezoid ?
7. The height of the parallelogram is what part of the height
of the trapezoid ^
8. Formulate a rule for finding the area of a trapezoid.
0. Instead of multiplying the sum of the two parallel sides
by one-half of the altitude, can you find another way for finding
the area of a trapezoid ?
Note. — Into what figures can you divide the trapezoid ?
10. Find the area of a trapezoid whose parallel sides measure
11 ft. and 16 ft., and the perpendicular distance between them
60 ft.
11. Find the area of a trapezoid whose parallel sides are 60
ft. and 130 ft., and altitude 40 ft.
12. One parallel side of a field in the shape of a trapezoid is
150 yd. The other is* 200 yd. How many square yards in the
field, the perpendicular distance between the sides being 60
yards ?
REVIEW OF DECIMALS, 23
(See pages 271-276.)
1. Divide five thousand fifty and five tenths by five hun-
dredths, and subtract twenty-five hundredths from the quotient.
2. Multiply seventy-three hundred-thousandths by one thou-
sand.
.035 X .0056 ^
^- .00007 ""•
4. Add fifteen thousandths, eighty-one ten-thousandths,
fifty-six millionths, seventeen ten-millionths, two hundred five
hundred-thousandths.
5. Change to decimals and add : 1^, 4}, 5i, 2f, Ixiy-
6. Multiply .032 by .005, and then divide .0512 by your
product
7. Take 27 and 28 thousandths from 97 and 7 tenths.
8. Reduce to common fractions : .055, .008.
9. Reduce to decimals : j\, ^^^ tthttt, 13Ji.
10. Multiply 12 thousandths by 12 hundredths, and from the
product take 12 millionths.
11. Multiply 160 by .016, and divide the product by .0026.
12. Add 8.3, 2.576, 3.42, 1.5, 6.279, .003, 1.417.
13. Multiply .084 by .0036.
14. Find the cost of 18,755 ft. of lumber at $24.75 per M.
15. Divide .005,232 by .016.
16. Multiply the difference between 4.4 and .00027 by the
product of 2.1 and .005.
17. Divide fifty and five thousandths by five millionths.
18. Multiply twelve thousandths by fifteen hundredths, and
divide the product by five tenths.
19. Divide two tenths by five ten-millionths.
20. Add: 243 thousandths, 203 ten-thousandths, 546 mil-
lionths, 12 and 1,234 hundred-thousandths, 116 ten-millionths.
21. Divide 15.625 by 31^.
22. Divide .08 by .0016. 670.08 by .016.
24 LEAST COMMON MULTIPLE.
1. Name a number that is divisible by 4.
2. A number that is divisible by another number is a mul-
tiple of it.
8. Name some numbers that are divisible by both 4 and 6.
4. A common multiple of two or more numbers is a number
that is divisible by all of the given numbers.
5. What is the least common multiple of 3, 4, and 6 ?
6. The least common multiple of two or more numbers is
the least number that is divisible by the given numbers.
7. A multiple of a number contains all the prime factors of
that number,
8. A common multiple of two or more numbers contains all
the factor 9 of each of the numbers.
9. Find the least common multiple of 8, 12, 20, and 30.
Arranging the numbers in a line, we notice
that 2 is a factor of all the numbers, hence (8) it
must be a factor of their multiple. Kemove the
factor 2 by division. We notice again that 2 is
a factor of 4, 6, and 10. It must then be a factor
of any number that can be divided by 4, 6, and
"o T T T 10. As at first, remove the factor 2 by dividing,
and since 2 is not a factor of 15, place the 15 in
2x2x2x3x5 = 120 the next line with the quotients. In the same
way continue to remove any factor of two or
more of the numbers. The continued product of these divisors and the last
quotients will be the least common multiple.
10. Find the least common multiple of 8, 16, 30, 48, 60, and
75.
11. Of 28, 30, 40, 56, and 60.
12. Of 42, 68, 84, 91, and 98,
18. Of 2, 3, 4, 5, 6, T, 8, and 9.
14. Of 14, 18, 22, and 24.
15. Of 15, 27, 35, 42, and 70.
16. Of 27, 36, 45, 90.
17. Of 18, 24, 27, 45.
2)8
12
20
30
2)4
6
10
15
3)2
3
5
15
6)2
1
6
6
ORAL PERCENTAGE, 25
1. Findl26%of 12; 20; 80; 120; 240.
3. Find .665 of 27. Find 663% of 27.
3. Express in common fractions in the lowest terms: 10%;
75%; 150%; 37i%; 116§%.
4. Find 50% of $72; $i ; $12§; 40 books; .4; 50%; $500.
5. A man gained $26 by buying flour at $5 a barrel, and
selling it at a gain of 20 % . How many barrels did he sell ?
6. What is 100 % of $10 ? 4 oranges ? 79 m.?
7. A regiment of 400 men went into battle, where 25% of
them were killed. How many men were not killed ?
8. A merchant paid $5.75 for an article, and sold it at a
profit of 20%. How much did he gain? How much did he
sell it for?
9. What part of 12 is 6 ? 16 is 4 ? 25 is 5 ? 12 is 2 ?
10. Substitute per cent for part in example 9, and give answer.
11. What part and what per cent of 10 is 5? 100 is 25?
500 is 50? 20 is 20? 120 is 40? 75 is 25? 40 is 32? 240
is 20?
12. A herd of 300 cattle was driven through a town. If
the farmers bought 50 cows from the herd, what per cent of the
herd did they buy ?
13. I invested $540 and lost $90. What per cent did I lose?
14. I bought 5 dozen oranges, but threw away 6 because they
were poor. What per cent did I throw away ?
15. Of what number is 40, 20% ? 12, 6% ? 9, 100% ?
16. 12 is J more than what number?
17. 12 is 133i%> of what number?
18. 20 is \ more than what number ?
19. 20 is 125% of what number?
20. 20 \B2b^o more than what number ?
21. How many inches in 50% of a yard?
22. A grocer paid 60/ a pound for tea, and sold it so as to
gain 25%. Find the gain.
26 FRACTIONS.
(For Summary of Processes in Fractions, see pages 26^270.)
Note. — In addition and subtraction of fractions find the least common
multiple of the denominators. This can often be found by inspection.
1. Add: 20§, 128?, 4H, SlJ.
2. Add: 71 J 96i, IT/g, 92, 44H.
3. Add: 136J, 7f, 36 A, TlJ.
4. From 145§ take 76 J.
6. From 717i take 196^-
6. From 3871 take 214|.
7. Multiply: 875i by 234.
8. Multiply: 645| by 412.
9. Multiply: 215 by 41 5.
10. Multiply 675 by 69?.
11. Find § of I of S of §.
12. Find I of if of 4|,
18. (A + 5) X (A - O-
14. Multiply 136 J by 41 f.
16. Multiply 61 2 J by 428.
16. Divide 6,346 g by 16.
17. Divide 683 J by 43|.
18. Simplify
10. Simplify
18i ^ m a off
16i -16g* iof2r
5§-3i' 68| +U '
20. Simplify:! x i- . ^X^.
i tV I «
81. Divide: 347g by 15. 692 by 19?.
22. Divide: 19| by 16§. 786 J by 30j.
38. Add: 2t^, 8J, 27§, 9J.
24. Add: 8ft, 37|, 28|, 9«?^, 19i.
26. From 27^;u take 18?. 12i - 91 = ?
26. How many miles an hour does a man walk who walks
21| miles in 4^ hours ?
BOARD MEASURE. 27
All kinds of lumber are measured by board feet. A board
foot is 1 ft, long, 1 ft. wide, and 1 in. thick. Boards less than
one inch in thickness are reckoned as one inch thick. In this
book, when no thickness is mentioned, one inch is understood.
1. How many feet of lumber are there in 24 boards, each 12
ft. long, 10 in. wide, and 1 in. thick?
If this board was 1 ft. wide, it would contain
10 in. = f ft. as many board feet as it has feet in lengtli. Since
12 ft. X I = 10 ft. it is only 10 in. (f ft.) wide, it will contain only
10 ft X 24 = 240 ft ^ ** many feet, or 10 ft. And 24 boards will con-
* * tain 24 times 10 ft., or 240 ft.
24 X 12 X 5
or? A ^' How many feet of lumber in a
^ - , ., , plank 9 ft. long, 8 in. wide, and 2 in.
Cancel when possible. ly^- i^o
9x2x2 Since a board is only 1 in. thick, a plank 2 in. thick will
q ' make 2 boards. Hence find the number of feet in one board
as aboye, and then multiply by the thickness in inches.
Find the number of board feet in each of the following :
3. A board 16 ft. long, 12 in. wide, 1 in. thick.
4. A plank 16 ft. long, 9 in. wide, 2 in. thick.
5. 14 ft. long, 6 in. wide, 1 in. thick.
6. 14 ft. long, 6 in. wide, 3 in. thick.
7. 12 ft. long, 8 in. wide, 1 in. thick.
8. 18 ft. long, 9 in. wide, 3 in. thick.
9. 12 ft. long, 3 in. wide, 3 in. thick.
10. 14 ft. long, 12 in. wide, 2 in. thick.
11. 14 ft. long, 18 in. wide, 1 in. thick.
13. 15 ft. long, 8 in. wide, 1 in. thick.
18. 16 ft. long, 9 in. wide, 2 in. thick.
14. 18 ft. long, 18 in. wide, 3 in. thick.
15. 12 ft. long, 9 in. wide, 2 in. thick.
le. 9 ft. long, 10 in. wide, 2 in. thick.
17. 10 ft. long, 6 in. wide, 3 in. thick.
28 BOARD MEASURE.
1. If I purchase 50 boards, each 12 ft. long and TJ in. wide,
for how many feet of lumber must I pay?
Note. — Fractions of an inch in the width of boards are never counted.
Call it the nearest inch.
2. Find the number of feet of lumber required to floor a
barn 36 ft. long, 17 ft. 6 in. wide, the planks being 2^^ in. thick.
3. At $18 per M., what will be the cost of the boards to
build a fence 4 boards high round a field 160 yd. long, 120 yd.
wide, if each board is 6 in. wide ?
4. How many feet of 6-in. boards are required to build 20
rd. of fence 4 boards high ?
5. How many feet of lumber are required for 140 ft. of tight
board fence 5J ft. high ?
6. How many feet of boards will be necessary to cover both
gables of a building 34 ft. wide, if the gable is 8^ ft. high ?
7. How many feet of boards will be needed to cover both
gables of a building 27 ft. wide, if the height of the gable is 9 ft. ?
8. A floor is 12 ft. by 16 ft. How many feet of lumber will
be needed for this floor, if each plank is 2 in. thick ?
9. In the 7th example, which way should the planks run,
if each plank is 16 ft. long ?
10. Find how many feet of planks will be needed if 12-foot
planks of the same thickness are used ?
11. At 116 per M., what is the cost of 42 16-f t. fence boards ?
Fence boards are 6 in. wide.
12. At $28 per M., find the cost of 750 boards, each 14 ft.
long, 12 in. wide, and 1 in. thick.
13. What is the cost of 46,250 ft. of pine lumber at $28.40
per M. ?
14. Find the number of board feet in 478 joists, each 24 ft.
long, 10 in. wide, and 3 in. thick.
15. Bought 20 joists, each 18 ft. long, 5 in. wide, and 8 in.
thick, at $30 a M. What did they cost me ?
STATEMENTS. 29
(See Part L, page 136.)
1. The divisor is 675, the quotient 488, and the remain-
der 548.
2. A man owns a rectangular garden plot 320 ft. long, 210
ft. wide. Around the outside is a walk 6 ft. wide.
3. Two men purchased a lot of wood for $31. In dividing
the wood one man took 6^ cords, and the other 9 cords.
4. A man had $12000. He lost in business the first year
12^% of it and 15% of the remainder the second year.
5. A square field is 72i rd. on each side. The fence that
incloses it cost $1.85 a rod.
6. A grocer bought 1152 gal. of molasses. After 12%
leaked out, the remainder was sold at 65/ a gallon.
7. The largest circle possible was drawn on a sheet of paper
12 in. long and 10 in. wide.
8. The sum of two numbers is 528, and one of them is 11
times the other.
9. $1,763.25 was on interest 1 yr. 3 mo. 14 da. at 5%.
10. I of a ton of hay cost $12.30. I bought 4 T. 500 lb.
11. A man owned J of a ship, and sold J of his share for
$16,800.
12. A room measures 18^ ft. by 15 ft. The carpet is | of a
yard wide. The breadths run lengthwise.
13. A man spent $646 for board and expenses. This was
84% of his salary.
14. Having lost 28% of my money, I have $17,640 left.
15. Goods that cost $764 were sold at a loss of 17^%.
16. The quotient is 71, the divisor 42, and the remainder 15.
17. The radius of a circle is 4 J ft.
18. A piece of land is 120 ft. wide, and 150 ft. long.
19. A man bought a horse for $72, and sold it for 25% more
than it cost.
30 ORAL,
1. If one pipe will fill a cistern in 4 hours, and another in
6 hours, how long will it take to fill it when both pipes are
ninning ?
2. If water is running out of the second pipe while running
in the first, how long then will it take to fill the cistern ?
3. A man sold a watch for $90, and gained 50 % . What did
it cost?
4. A man bought a hat for $5, and sold it for |6. What
per cent did he gain ?
5. John lost f of his money, and spent J of the remainder,
and then had only 10 cents. How much money had he at first?
6. How old are you if % of 80 is 4 times your age ?
7. How long will it take a man to save $60, if he earns $15
a week and spends $9 ?
8. If J of a yard of cloth cost 63 cents, what will | of a
yard cost?
9. A man sold a watch for $120, which was J of what it
cost him. How much did it cost ?
10. If 4 men can do a piece of work in 12 days, how long
will it take 6 men to do it?
11. If 4 pipes will fill a cistern in 40 min., how many pipes
will fill it in 10 minutes ?
12. A house was insured for $3,200 at 1%. What was the
premium ?
18. An agent sold $300 worth of property, and charged 5%
for so doing. Find his commission.
14. Given the cost and loss per cent, what can be found ?
15. A watch costing $80 was sold at a loss of 10%. For
how much was it sold ?
16. What number increased by 12 equals 16 ?
17. 6 added to a number equals 14. Find the number,
18. What number diminished by 5 equals 9 ?
19. What is the ratio of 108 to 12 ? Of 60 to 12 ?
REVIEW OF PERCENTAGE, 31
(For definitions and review, see pages 285-289.)
1. A house which cost $1,200 was sold at a gain of 25%,
What was the selling-price ?
2. Land which cost $42 an acre was sold at a gain of 8^%.
Find the selling-price.
3. A fruit-dealer bought apples at $1.76 a barrel, and sold
them at $2.25 a barrel. What per cent did he gain?
4. A man bought a chair for $2.34, and afterwards sold it
for $2.73. What per cent did he gain?
5. What does a bill for $2,478 become after a reduction
of5%?
6. What is the cost of insuring 640 bbl. of flour, worth $4
a barrel, if the cost of insurance is i% of the value of the
flour?
7. A horse was bought for $175. At what price must he
be sold to gain 12% ?
8. A merchant bought apples at $1.20 a bari-el, and sold
them at a gain of 26%. Find the selling-price a barrel. How
many barrels did he sell if he received altogether $187.50?
9. The population of a certain city was 48,000 this year.
If it increases at the rate of 2J% each year, find its population
1 and 2 years hence.
10. A person gave $1,600 for a piece of land, and sold it at
a gain of 20%. Find the selling-price.
11. A man built a house for $3,500, and rented it for $400 a
year. For what per cent of its value did he rent it ?
12. Turn to your geographies, and find the area of the basins
of the principal rivers, and then find what per cent of the area
of North America is drained by each river.
18. A quantity of goods was sold at an advance of 12i%. If
the gain was $34, what was the cost ?
14. When the cost is $8,000 and the selling-price $7,400,
what is the rate of the loss ?
32 MISCELLANEOUS REVIEW.
1. Find the interest on $800 for 2 yr. 6 mo. 15 da. at 6%.
2. Find the interest on $346.50 for 1 yr. 8 mo. 2 da. at 6%.
3. Find the interest on $750 for 3 yr. 3 mo. 3 da. at 6%.
4. A boy sold a sled for $1.40, and by doing so lost 12^%.
What did the sled cost?
5. A man paid $270 for a horse, which was 10% less than
his carriage cost. Find the cost of both.
6. A grocer bought 535 lb. of sugar at 6J/ a pound, and
sold it at a profit of 15 %. What did he receive for it?
7. A merchant sold some cloth at $3 a yard, and lost 25%.
What did it cost him ?
8. For what must a merchant sell a barrel of flour that cost
him $5.10 to gain 9i%?
9. A miller bought a consignment of wheat, and ground it
into flour. He sold 980 bbl. for $6.50 a barrel, making 25%.
How much did he pay for the wheat ?
10. Mr, Longley sold his house for $4,248, and by so doing
gained 20%. Find what the house cost Mr. Longley.
11. How much will it cost to plaster and paint the walls of
a room 28 ft. long, 20 ft, wide, and 12 ft. high, at 33 J/ a
square yard?
12. At $1.50 a yard, what will it cost to carpet a room 18
ft. long, 15| ft. wide, the carpet being J of a yard wide?
13. What is the area of a circle whose diameter is 100 ft.?
14. How many square inches of gold leaf will be required to
cover the convex surface of an equilateral triangular pyramid,
each side of whose base is 10 in., and whose slant height is 4 ft.?
15. The greater of two numbers is five times the less. If .
their sum is 72, what are the numbers?
16. A father is five times as old as his son, and the sum of
their ages is 36 years. How old is each?
17. If twice a number is added to four times the same num-
ber, the sum will be 66. What is the number ?
DIAGRAM.
33
3
m
a
This diagram represents a
garden plot, drawn to a scale
of J in. to 8 ft.
1. Find the distance round
the garden.
2. Find the area of the gar-
den.
3. If ml is extended to ji^ it
will cut off on the left Mary's
flower-bed. Find the area and
g perimeter of her bed.
4. If she sets out plants six
inches apart, how many plants
can she have in the garden?
5. Extend ih until it meets ef\ you have cut off on the
right Bessie's flower-bed. The area of Bessie's bed is what
per cent of the area of Mary's bed?
6. The perimeter of Bessie's bed is what per cent of the
perimeter of Mary's bed ?
7. If Bessie and Mary both start from a, and walk round
the garden in opposite directions with equal speed, where will
they meet ?
8. If nm and cd are extended until they meet, the part cut
off above is the vegetable garden. The area of Bessie's and
Mary's flower gardens will be what per cent of the area of the
vegetable garden ?
9. The perimeter of Bessie's garden is what per cent of the
perimeter of the vegetable garden ?
10. Find the area of the garden not belonging to Mary,
Bessie, or the vegetable garden.
11. How much will it cost, at $18 per M., to build a fence
4 boards high round the entire garden, if the boards are 8 ft
long and 6 in. wide ?
34 REVIEW OF FRACTIONS.
1. A man deposited in a bank $475^ Monday, fSTO^ Tues-
day, UNrOOy Wednesday, and drew out f563t Thursday, and
$145Tfiy Friday. How much had he in the bank Saturday ?
2. What is the least common multiple of 20, 24, 36 ?
3. A man had 490f bu. of grain, and bought 784^^ bu. more,
and then sold 900?. How many bushels had he left?
4. The area of an oblong is 24 square feet. What part of
its area is that of a square whose side is 2 ft. ?
5. Reduce 380$ to 72ds.
6. If a car runs 48i miles in an hour, how far will it run in
16 days, running 14i hours a day?
7. Find the cost of 15j cd. of wood at $6.37i a cord, and
7i cd. at $6i a cord.
8. (i + § + J) XG XSX16)=?
9. Find the cost of 19^ cd. of wood @ $6l.
10. A has $540, which is § of 3 J times as many as B has.
How much money has B ?
11. How many yards of cloth does a merchant buy if he
spends $1,200, of which $680 was spent for cloth at $5g a yard,
and the remainder at $4 J a yard?
12. Divide 2J by 2J, and multiply the quotient by the quo-
tient of 4J divided by 2^.
13. Find the cost of a house and lot when the house costs
$6,300, which is If times the cost of the lot.
14. From 20} take 16J.
15. The diameter of a circle is 15 ft. Find circumference.
16. The circumference is 33 rd. Find diameter.
17. The radius is 17 in. Find circumference.
18. The diameter of a circular pond is 16 rods ; what is the
area ?
19. Find the convex surface of a triangular pyramid, each
side of whose base measures 6 ft., and its slant height 24 ft.
20. Add 19S, 17#, 21^3^, and ISyV
ORAL. 35
1. If 3 apples are worth one peach, and 3 peaches are worth
one orange, how many oranges can be bought for 45 apples ?
2. Name two factors of each of the following numbers :
10
12
14
15
16
18
20
22
24
25
26
27
28
30
82
38
35
36
40
42
44
45
48
49
50
54
55
56
60
63
64
66
70
72
.77
81
88
96
108
121
132
144
3. What will 2 tons of iron cost if 1 lb. costs 10 cents ?
4. How many inches in i yd. ? In J yd. ? In J yd. ? In
iyd.? Iniyd.?
5. What will 7 quarts of wheat cost at 128/ a bushel?
6. If a coat and vest cost f 20, and a pair of trousers i as
much, how much will all cost?
7- If 1 pound of cheese costs 12 cents, what will 2f lb. cost?
8. Nellie has 46 cents, and Grace has f as many. How
many has Grace ?
9. If I buy an article for $60, and sell it for J of what it
cost, how much shall I gain ?
10. A man's salary is $120 a month. If he spends J of it
for a watch, J of it for a suit of clothes, and J of it for board,
how much of his salary remains ?
11. What will J of a gallon of molasses cost at 10/ a pint?
13. Bought a third of a barrel of sugar for $3. What will
2Sbbl. cost?
13. If iof a yard of cloth cost $2, what will 2^ yd. cost?
14. If I of a bushel of com cost 10 cents, what will 2 bu.
cost?
15. If f of a pound of spice cost 15 cents, what will 3j lb.
cost?
16. Find the interest on $600 for 19 da. at 6%.
17. In 672 eggs there are how many dozen eggs?
36 INTEREST. DIFFERENCE IN DATES.
1. Find the interest of 1396.16 from July 15, 1901, to Feb.
6, 1903.
From July 15, 1901, to July 16, 1902, 1 yr.
From July 15, 1902, to Jan. 15, 1903, 6 mo.
From Jan. 15, 1903, to Feb. 6, 1903, 22 da.
$ 3.9616 int. for 2 mo.
$ 3.9616 int. for 2 mo.
$35,654 int. for 1 yr. 6 mo. Iq^a- i I' 7 i a '
1 iQQ • 4. r ^o A f35.6o4 mt. for 1 yr. 6 mo.
1.188 mt. for 18 da. . «^^ . ^ . ^.-^ ,
.^Q . ^ . oj o^ 1-^2^ mt. for 20 da.
.198 mt. for 3 da.
.066 int. for 1 da.
.132 int. for 2 da.
$37,106 int. for 1 yr. 6 mo. 22 da.
$37,106
NoTB. — Use this method for finding the difference in dates. As soon sus
possible do the work mentally, and write only the results.
Find the interest at 6% of:
2. $649.21 from June 8, 1900, to Aug. 15, 1903.
3. $1460.78 from June 6, 1902, to April 23, 1904.
4. $284.30 from Feb. 23, 1902, to Aug. 5, 1903.
5. $366.44 from Jan. 5, 1903, to Jan. 27, 1905.
6. $491.73 from Nov. 16, 1901, to Nov. 28, 1903.
7. $91.36 from Aug. 12, 1900, to June 10, 1902.
8. $436.74 from March 25, 1902, to July "^9, 1904.
9. $589.76 from May 11, 1901, to Jan. 7, 1903.
10. $550 from May 8, 1900, to June 13, 1903.
11. $125.40 from Sept. 25, 1901, to March 16, 1902.
12. $679.08 from Feb. 10, 1902, to Dec. 7, 1903.
13. $137.65 from Oct. 14, 1902, to Dec. 29, 1904.
14. $146.35 from June 7, 1902, to Feb. 11, 1904.
15. $154.25 from Apr. 18, 1903, to Jan. 25, 1906.
16. $817.57 from Aug. 7, 1900, to Sept. 8, 1901.
17. $132.25 from Nov. 13, 1901, to May 2, 1903.
18. $446.50 from July 18, 1900, to Sept. 4, 1901.
19. $3166.49 from Aug. 16, 1901, to May 1, 1905.
REVIEW OF DECIMALS. 37
1. Add five, and three hundred eighty-two ten-thousandths ;
one thousand two hundred thirty-five hundred thousandths;
eight hundred ninetynsix, and fifty-one thousand three hundred
twenty-seven millionths.
2. Add nineteen, and forty-nine ten-thousandths ; seventy-
three, and one hundred fiftynsix millionths; thirty-four, and
eight hundred-thousandths ; five thousand eighty-two, and one
thousand nineteen hundred-thousandths,
3. Multiply .0000915 by .0056.
4. Multiply .58273 by 1000.
5. Multiply 2.4675 by 100.
6. Multiply 4.3982 by 500.
7. Divide .0009 by .003.
8. Divide .0002784 by .032.
9. Divide 10 by .001.
10. Divide .31 by .0005.
11. Divide 18.45 by 10. By 100.
12. Divide 436.457 by 100. By 1,000.
13. Divide 1464.25 by 100. By 1,000.
14. Divide 1867.8 by 4,000.
15. Divide 375.82 by 500.
16. Find cost of 825 bu. @ $1.66 J.
17. Find cost of 72 gal. @ 13.87^.
18. Add 8153 and 45 hundredths; 32 and 28 ten-thou-
sandths; 237 and 483 thousandths*; 5 and 165 hundred-thou-
sandths ; 6 hundredths.
19. Change to common fractions: .0419; .0048; .00625;
5.00125.
20. Change to decimals : A, A. Ai ih ih §1, 1%.
21. Divide 16 by 10,000.
82. Divide 1846 by 100. By 1,000,
88. Change to decimals: 6J. 82f, 46 A, 2^xhv
24. Find the difference between .406 and .62.
38 REVIEW IN PERCENTAGE.
1. A man owned 1,016 acres of land. He sold 12^% to
one customer, and 42^% of the remainder to another customer.
How many acres had he left ?
2. A man's income is $1,800 a year, of which he pays 12^%
for house-rent. What rent does he pay each month ?
8. What number increased by 40% of itself equals $1,694?
4. A book-keeper spends $600 a year, which is 24% of his
salary. Required his salary.
5. What per cent of 675 is 136?
6. A merchant bought 275 bbl. of flour. After losing 20%
of it, he sold 25% of the remainder. How many barrels re-
mained ? What per cent of the whole remained ?
7. I bought $820 worth of cloth, and sold it at a gain of
16%. What was the gain ? The selling-price ?
8. A dealer bought coal at $4.25, and sold it at 6 % advance.
What was his selling-price ?
9. Cost $7.50, profit 18%. Find the selling-price.
10. Cost $1,500, gain 16|%. Required the selling-price.
11. Cost $80, gain $36. Required the gain per cent.
12. Selling price $125, loss 20%. Find the cost.
13. Gain 25%, cost $5.60. Required the selling-price.
14. An agent sold 426 bales of cotton weighing 408 pounds
each, at 8i/ a pound. How much money did he receive ? He
kept 2|% of this as his commission. How much did he return
to his employer?
15. A regiment went into battle with 960 men, and came
out with 600 men. What per cent was lost?
16. A clerk's salary is $800 a year. He spends 10 % the first
quarter, 16% the second, 16% the third, and 14% the fourth.
How much does he save ?
17. My agent sold goods for $6,400 ; his commission was 2 J %,
and other charges $17.60. What amount should he send me?
18. Find 33i% of 729.
STATEMENTS. 39
1. $625.80 was on interest at 6% from Nov. 28, 1900, to
•Sept. 16, 1904.-
2. 18 men are at work on a piece of work that 24 men can
do in 9 days.
3. I bought 42 yd. of cloth. 35 yd. cost $12.25.
4. Goods that cost $3,072 were sold for $2,560.
5. A merchant withdrew $2,058 from a bank. This was
28% of his deposit.
6. A farmer sold 525 bu. of wheat at $1.12 a bushel, and
20% less of oats at 45/ a bushel.
7. A farmer owned a flock of 580 sheep, but lost 20% of
the flock in a snow-storm.
8. A plank is 16 ft. long, 1 ft. 3 in. wide, and 2 in. thick.
9. Wood cost me $4.75 a cord. I paid $28.50 for a pile.
It was 24 ft. long and 4 ft. wide.
10. A farmer paid 48/ a cubic yard for digging a ditch 22 ft.
9 in. long, 8 ft. 6 in. wide, and 8 ft. high.
11. Property worth $7,500 was insured at li%.
12. A man's salary is $2,500 a year. He spends 30% for
board, 12^^% for clothes, and 20% for other expenses.
13. A number diminished by 20% of itself is 936.
14. A merchant took a sum of money to market, and spent
24% of it. He brought back $760.
15. A farmer raised 55 bu. of potatoes. He sold 20% to
one man, and 25% of the remainder to another man.
16. A merchant bought 96 yds. of cloth at 40/ a yard, and
sold it at a gain of 33 J%.
17. A man sold his library for $840, which was 16% less
than it cost.
18. A cylinder is 40 ft. long and 15 ft. in diameter.
19. A dealer sold 438 tons of coal at $4.75 a ton, and a
number of tons of another kind at $5.20 a ton. He received
for aU $4,254.10.'
40 ORAL.
!• A man bought a horse for 1200, and sold him so as to
gain 10% of his cost. Find the gain.
2. Mr. Jones invested $2,000 in business, and gained 20%
of his investment every year. How much did he gain in 1 year?
5 years ?
8. An agent sold a house for $1,200. His commission was
10%, How much did he keep as commission?
4. Plush cloth bought at $6 a yard was sold at $6 a yard.
What part of the cost was gained? What per cent was gained?
5. A man receiving a salary of $2,400 spends 33i% of it
for expenses. How many dollars does he spend?
6. Ten bushels out of 100 bushels is what per cent?
7. $1 out of every $10 is what part? Is what per cent?
8. $5 out of every $20 is what part?. Is how many hun-
dredths ? Is what per cent ?
9. 2i is J of what number?
10. 2i is 25% of what number?
11. Yesterday I worked J of the day, and the day before J of
a day. What part of a whole day did I work in all ?
12. Change to a whole number: {, V, V» V-
18. If I pay 8 cents for a pint of milk, what must I pay for
a gallon at the same rate ?
14. Make a problem to illustrate, Given two numbers to find
their difference.
15. 46 is I of what number? ^ of what number?
16. 81 is f of what number? f of what number?
17. If 8 is § of a number, what is the number? What is |
of it?
18. How many cubic feet in a box 4 ft. long, 3 ft. wide, 2 ft.
high?
19. How many yards square is a floor that is 12 ft. square?
How many square yards are there in the floor ?
20. What is the ratio of a mile to a rod ?
DIAGRAMS.
41
Starting at A, the boundary line of a garden runs east 5 rd.,
thence south 2 rd., thence west 2 rd., thence south 2 rd., thence
east 2 rd., thence south 1 rd., thence east 1 rd., thence south 1
rd., thence west 6 rd., thence north to A.
1. Make a drawing, scale 1 in. to 1 rd,
2. Find the perimeter of the garden.
8. Find the area of the garden.
4. Find the number of posts, placed 8i ft. apart, needed for
a fence.
6. Find the feet of lumber in the two scantling (rails) run-
ing round the lot, each 4 in. by 3 in.
6. Find the feet of lumber in a tight board fence 5 ft. high
all round it.
7. Find the cost of the posts at 25/ each.
8. Find the cost of scantling and boards at $16 per M.
9. Find the cost of painting the fence at 33 J/ a square yard.
10. This is a plan of a hall
in a public building, drawn
to a scale of \ in. to 12 ft.
Find the number of square
feet in the floor.
11. If the walls are 12 ft.
high, find the square feet in
the walls and ceiKng.
12. Beginning at a point
called A, the northern boun-
dary line of a park runs west
30 rd. to B ; thence south 25
rd. to C; thence east 40 rd.
to D ; thence to place of beginning. Draw a diagram, scale J
in. to 6 rd. Find the square rods in the park.
Ij3. How many square yards of cloth will it take to cover a
column that is 16 ft. high, and 18i ft. in circumference ?
42 MISCELLANEOUS REVIEW.
1. Find the difference between 100,100,100 and 90,090,090.
2. If a man buys 396 acres at $37 an acre, and sells his pur-
chase for $15,176, what will be his gain ?
3. Multiply 183,600 by 427,000.
4. Multiply 630,000 by 3,800.
5. If the product of two numbers is 143,186,076, and one
of the numbers is 82,871, what is the other number?
6. Divide 27,180,000 by 16,100.
7. A man had $2,013.42. He bought 370 bu. wheat at
$1.12 a bushel; 980 bu. corn (a) 64/; 636 bu. rye (a) 62/;
and invested the remainder of his money in flour at $6.26 a
barrel. How many barrels of flour did he buy ?
8. A merchant sold 12 bbl. of pork, averaging 200 lb. a
barrel, at 12/ a pound, and took in payment 160 hams weigh-
ing 10 lb. each. Find the price of the ham a pound, n
9. Reduce to improper fractions : 27^^, 42f .
10. Find the difference between 438^^ and 287|.
11. Multiply 76W by 64tV
12. Divide 816,^ by 16f.
13. A can finish a piece of work in 4 days, B in 6 days, and
C in 8 days. In what time can the work be completed if all
work together?
14. Find the cost of 619 lb. (a) 27^/.
15. What is the cost of 3^^ reams of paper at 12J/ a quire ?
16. A owns 16% of a business; B, 26% ; C, 28% ; and D,
the remainder. What is the value of A's share if D's share is
worth $17,232?
17. The greater of two numbers is four times the less. If
the difference is 36, what are the numbers?
18. Two trains leave New Haven at the same time, and move
in the same direction. At the end of one hour they are 20
miles apart. If one has gone two times as far as the other,
how far is each from New Haven?
MISCELLANEOUS REVIEW. 43
1. What will it cost to floor a room 17^ ft. long and 16 ft.
wide at fl.lO a square yard?
2. A man had a capital of $2,600. He put 25% of it into
business, 33 J % of it into a bank, and invested 28% of it in real
estate. How much had he left?
3. A groder bought 800 bags of coffee, each bag containing
491 lb., at 18/ a pound, and sold it at a profit of 16§%. How
much did he receive for the whole lot ?
4. I lost 10% by selling goods at 27/ a yard. How much
did I lose on 485 yd. ?
6. What will be the cost of 35 3-in. planks, 24 ft. long, 16
in. wide, at $16.75 per M?
6. A man sawed a pile of wood 40 ft. long, 4 ft. wide, and
5 J ft. high, for $1.25 a cord. How much did he earn?
7. If I of my share of a farm is worth $420, and I own J
of the farm, what is the value of the farm ?
8. Find the convex surface of a log whose circumference is
18 ft. and length 35 ft.
9. What is the area of a circle whose circumference is 160
yd.?
10. The parallel sides of a trapezoid are 25 yd. and 21 yd.,
and its altitude 16 yd. What is the area?
11. The length of a rhomboid is 17 ft., and the perpendicular
height 16 ft. What is the area?
12. How many acres in a field 800 rd. long and 128 rd. wide ?
13. Find the area of a triangle whose base is 49 yd., and
altitude is i its base.
14. If 4i tons of coal cost $18, what will 18 tons cost?
16. If 5 bu. 3 pk. of potatoes cost $4.60, what will 2 bu. 1
pk. cost?
16. Three boys, A, B, and C, together receive $81. A re-
ceives twice as much as B, and C three times as much as B.
How many dollars does each receive ?
44 MEASURBMEifTS,
1. Find the convex surface of a triangular prism whose
length is 12 ft. and each side of whose base is 2 J ft.
2. Required the number of square feet in the surface of a
square pyramidal roof, the length of whose sides is 20 ft., and
whose slant height is 18 ft.
3. What length of tire will it take to band a cart-wheel 5 ft.
in diameter?
4. What is the difference between the area of a floor 40 ft.
square, and that of two other floors, each 20 ft. square ?
5. The diameter of a circular grass-plot is 17.5 ft. What
is its circumference ?
6. If the circumference of a tree is 50 in., what is its diam-
eter?
7. How manj^ board ft. in a plank whose length is 20 ft.,
breadth 16 in., and thickness 3 in. ?
8. Required the area of a pasture in the form of a trapezoid
whose parallel sides are 786 and 473 ft., and altitude 986 ft.
9. What is the cost of sawing a pile of wood 20 ft. long, 4
ft. wide, and 6 ft. high, at 11.25 a cord?
10. A field contains 199^ sq. rd. It is 18j rd. long. How
wide is it?
11. Draw, name, and describe four kinds of quadrilaterals.
12. A certain flower-bed is in the form of a trapezoid. The
two parallel sides are 10 ft. and 12 ft, and the perpendicular
distance between them is 8 ft. Find the area.
13. What is the floor measurement of a house built in the
form of an octagon, whose side is 12 ft., and the perpendicular
distance from the center to the middle of each side is 8 ft. 6 in. ?
14. A wheel whose diameter is 3 ft. turns how many times
in going a half-mile ?
15. A cistern is in the form of a rectangular prism, 12 ft. 8
in. long, 8 ft. 6 in. wide, and 12 ft. deep. How many square
feet are there in the sides and bottom ?
ORAL. 46
1. If I can do a piece of work in 4 days, what part ci^i I
do in 2 days ?
2. t of 25 is H of how many?
3. f of 28 is J of how many ?
4. By selling land at $150 an acre I gained 25%. Find
cost.
5. A merchant sold goods for $500 at a loss of 20% . What
was the cost?
6. I sold my horse for $200, and by so doing lost 20%.
What was the value of the horse ?
7. Grace gave one-half of her oranges to her mother, and
one-third of them to h6r father. What per cent had she left ?
8. If you sell an article for 12 J cents that cost 10 cents,
what will be your rate of gain ?
9. If an article costs $600, what will be the gain at 1%?
Atiof 1%?
10. A man paid $500 for wheat, and sold the whole at a loss
of 6%. Find the loss. Find the selling-price.
11. One-half of a lot of goods cost $180. Find the loss at
10% on the whole lot.
12. i of 40 is what per cent of J of 20?
13. J of 30 is what per cent of J of 40?
14. How many barrels will it take to hold 25 bu., if 1 bbl.
holds 2i bushels ?
15. Divide 5j by §. 6| by h
16. A man paid $24 for f of an acre of land. If he sold i of
an acre for $15, how much did he gain on the part sold ?
IV. Find the cost of 12 primers at 124 each.
18. At the rate of $7i a ton, what will 12 tons cost?
19. What is the difference between J of two and f of 1 ?
20. If 4 yd. of cloth cost 72 cents what will I yd. cost?
21. I of 30 is I of what number?
82. What is the area of a plot of ground 80 ft. by 50 ft. ?
46 SUBTRACTION COMPOUND NUMBERS,
1. Subtract 8 bu. 3 pk. 7 qt. from 47 bu. 1 pk. 6 qt. 1 pt.
A^ _ ^ »_ 'i _ 1 ^^ pints from 1 pt. leaves 1 pt. 7 qt. from
^ ^ o _ IT __ A 6 qt. we eamiot take, so we take 1 pk. from
the column of pecks, which is equal to 8 qt.
"" "" "~ 8 qt. and 5 qt. are 13 qt. 7 qt. from 13 qt.
leaves 6 qt. 3 pk. from pk. we camiot take. 1 bu. equals
4 pk. 3 pk. from 4 pk. leaves 1 peck. 8 bu. from 46 bu.
leaves 38 bu.
KoTB. — Numbers of the same denomination should be written in the same
column.
2. From 46 gal. 1 qt. 1 pt. 2 gi. take 25 gal. 2 qt. 1 pt.
3gi.
8. From 8 bu. 1 pk. 6 qt. take 3 bu. 2 pk. 4 qt.
4. Take 160 T. 1,800 lb. 6 oz. from 175 T. 298 lb.
6. From 7 mi. take 5 mi. 315 rd. 3 yd. 1 ft. 3 in.
6. From 471 cu. yd. 16 cu. ft. 972 cu. in. take 115 cu. yd.
17 cu. ft. 1,710 cu. in.
7. From 19 yr. 5 mo. 17 da. take 12 jt. 9 mo. 14 da.
8. From 20 gal. Ij pt. take 3 qt. 1| pt.
9. From 6 mi. 220 rd. 1 ft. 8 in. take 4 mi. 261 rd. 1 yd.
10. From 275 mi. take 50 mi. 130 rd. 3 yd. 1 ft. 3 in.
11. From 1,845 yr. 9 mo. 18 da. 20 hi*, take 1,774 yr. 11 mo.
20 da. 224 hr.
12. From 73 bu. 2 pk. 5 qt. take 59 bu. 3 pk. 7 qt.
13. From 17 mi. 311 rd. 1 yd. 1 ft. 3 in. take 3 mi. 79 rd.
1 yd. 2 ft. 7 in.
14. From 6 mi. take 4 mi. 64 rd.
16. From 116 cd. 4 cd. ft. take 105 cd. 5 cd. ft.
16. Subtract 5 mi. 215 rd. 5 yd. from 8 mi. 216 rd. 3 yd.
17. From 17 bu. 2 pk. 6 qt. take 8 bu. 3 pk. 4 qt.
18. From 14 cu. yd. 6 cu. ft. 1,011 cu. in. take 9 cu. yd. 17
cu. ft. 1,108 cu. in.
19. From 17 mi. 58 rd. take 10 mi. 117 rd. 2 ft.
REVIEW OF PERCENTAGE. 47
1. A merchant by selling cloth at $2.50 a yard gained 25%.
Find the cost.
2. An agent bought 50 carriages for $140 each, and charged
me 3i% commission. He also paid $75 for freight, and $35
for cartage. What did the carriages cost me ?
3. Find the interest of $475.05 for 6 yr. 10 mo. 10 da. at
4. A man sold a cow for $37.50, and lost 25%. Find the
cost.
5. A man sold a wagon for $41.25, and gained 25%. Find
the gain.
6. I sold a horae for $240, and lost 20%. Find the cost.
7. A man owned 156 A. of land, and sold 75% of it for
$5,265. What was the price an acre ?
8. A man bought a farm 180 rd. long and 160 rd. wide, for
$6,750, and sold it at a gain of 20 % . How much did he receive
an acre ?
9. I purchased 417 bbl. of flour at $5.25 a barrel. For how
much must I sell the whole to gain 35% ?
10. If it costs a man who earns $90 a month $72 for expenses,
what per cent of his money can he save ?
11. A manufacturer made 5,280 barrels of flour. He sold
12j% to one man, 33j% of the remainder to another, and 50%
of what was then left to another. If he received $4,666.20 for
what was then left, how much was that a barrel?
12. Find the interest of $875 for 80 days at 7%.
13. $596.50 for 33 days at 4%.
14. $1,375 for 5 mo. 3 days at 6%.
16. $7,000 for 2 yr. 2 mo. 16 da. at 10%.
16. If 1 lost 10% by selling goods at 18/ a yard, what did
they cost?
17. A merchant sold tea at 60/ a pound, gaining 20%.
Find the cost.
48 REVIEW OF DECIMALS.
1. How far will light travel in 12 min. 45 sec, if it travels
186,000 miles in one second?
2. Divide 567,891 by 729. 900,972 by 843.
3. Divide 120,000,000 by 12,000,000.
4. Add: Twenty-six and fifteen thousandths; eighty-one
and one thousand nine hundred ten-thousandths; eleven and
twenty thousand seven hundred four hundred-millionths ; twelve
hundred and twelve hundred-thousandths.
6. Multiply 36.03 by .06006.
6. Divide 5.958 by .0009.
7. Divide 16.27704 by 14.664.
8. If a man can build .425 of a rod of fence in an hour,
how many rods can 12 men build in 6.5 days, working 8.25
hours a day?
9. Thirty-five hundredths of a cargo of 8,000 bushels of
wheat were destroyed by fire; What was the value of the part
left at $.875 a bushel?
10. A farmer exchanged wood for coal. If he bought 9.5
tons of coal at $4.25 a ton, how many cords of wood at $3.75 a
cord did he give in exchange ?
11. What is .15 of $47.65?
12. Divide 10.201 by 101.
13. Divide 1.125 by 937.5.
14. Find the cost of 8,724 roofing-slate at $5.75 per hundred.
15. The bricks in a schoolhouse cost $10,875. If the price
was $7.25 a thousand, how many bricks were used?
16. Find the cost of 17,250 lb. of hay at $15.75 a ton.
17. A ton of coal costs $2.75 to mine it, $.85 for freight, and
$.25 for delivery. A dealer sold 425,600 lb. at $5.50 a ton.
How much did he make ?
In buying coal at wholesale, a ton is 2240 pounds. This is
called a lonff ton.
18. Divide 16 ten-millionths by 25 thousandths.
MISCELLANEOUS REVIEW. 49
1. A can do a piece of work in 6 days, B in 7, and in 8.
In what time caii they do it working together?
2. How many casks (40 gal.) of water will a cylindrical
cistern hold, whose diameter is 9J ft., and depth 10 ft. ?
8. If the diameter of my carriage wheel is 4 J ft., how many
revolutions will it make in going 2 miles and back again?
4. The height of a cylinder is 6 ft., and the diameter of the
base is 2j ft. Find the entire surface and volume. *
6. A field in the form of a trapezoid contains 234 acres.
One of its parallel sides is 95 rd., and the other 65 rd. What
is its altitude ?
6. Write a promissory demand note. Find the interest on
it for 1 yr. 3 mo. 17 da. at 7%.
7. Find the interest at 7i% on $256.34 from Nov. 17, 1901,
to Aug. 24i 1903.
8. Find the commission on $46,912.60 at 1|%.
9. At $1.85 a yard, find the cost of carpet 1 yd. wide to
cover the floor of a room 22 ft. long, 19 ft. wide, strips to run
lengthwise.
10. How many 4-oz. bottles (J pt.) can be filled from 4 gal.
2 qt. 1 pt. 3 gi. of alcohol ?
11. Find the cost of 86 pieces of maple flooring, 3 in. wide,
16 ft. long^ at $38.60 per M. ; same number of pieces 4 in. wide,
15 ft. long, at $35 per M. ; 48 boards 10 in. wide, 18 ft. long,
@ $28.75 per M.
Find the interest of :
15. $248 for 90 days at 7%.
13. $636 for 1 yr. 5 mo. 10 da. at 5%.
14. $1^478 for 1 yr. 2 mo. 13 da. at 6%.
16. Divide 200 into three parts so that the second part shall
be three times the first, and the third part two times the second.
16. Divide $800 between A and B, giving B $3 as often as
you give A $5.
50 ORAL.
Find the cost of one pound or one yard in the following :
1. 9 lb. figs cost. $1.08. 3 lb. steak cost $.76.
2. 15 yd. ribbon cost $1.05. 14 lb. rice cost $1.12.
3. A newsboy bought 75 papers at 2^ each, and sold them
at 3/ each. How much less than $1.00 did he gain?
4. I have a number in my mind. If I take 15 from it, 35
will remain. What is the number?
6. When is the selling-price equal to the cost and something
"added?
6. When is the selling-price equal to the cost and something
subtracted ?
7. When is the selling-price more than the cost?
8. When is the selling-price less than the cost?
9. When is the selling-price equal to the cost?
10. The cost is always considered what per cent ?
11. When the gain is 20% what is the selling per cent?
12. When the loss is 10% what is the selling per cent?
13. When there is neither gain nor loss what is the selling
per cent?
14. Eggs are 38/ a dozen at one store, and 45/ a dozen at
another. I bought 9 doz. at the second store. How much
would I have saved had I bought at the first store ?
16. Make change from three quarters for a 69/ purchase.
16. Make change from a half-dollar for a 31/ purchase.
17. Find the change due from three dimes for a 27/ purchase.
18. A man worked for a farmer at 20^ an hoar, and received
4 bu. of potatoes at 50/ a bushel. How many hours did the
man work ? *
19. If a man receives $6.60 for 22 houra' work, how much
does he receive an hour ?
20. What part of a pound is 12 oz.^ 4 oz.?
21. Find the cost of 5 T. 400 lb. of coal at $5 a ton.
22. What is the volume of a prism 10 ft. by 5 ft. by 4 ft.?
TO FIND THE CONTENTS OF CYLINDERS.
51
Fiff. 1.
Fiff.2.
Note. — These drawings were made from a cut-up cylinder. One should
be in the hands of the teacher when this lesson is given.
1. Of what is Fig. 1 a drawing?
2. Of what is Fig. 2 nearly a drawing?
Note. — The more parts into which the cylinder is cut, the more nearly
will it approach a prism, when arranged as in Fig. 2.
3. How do you find the contents of the prism ?
4. How does the length of the prism compare with the
circumference of the cylinder ?
5. How does the width of the prism compare with the
diameter of the cylinder ?
6. How does the height of the cylinder and prism compare?
7. Every cylinder can be changed into a rectangular prism,
with i the circumference as the length, the radius as the width,
and the same height. Learn : To find the contents of a cylin-
der, multiply the area of the base by the height.
8. What shape is one end of the cylinder ?
9. What does this circle become when the cylinder is
changed into a prism ?
10. Prove that the way to find the contents of cylinders is
identical with that of finding the contents of prisms. Find the
number of cubic units that can be placed in one layer on the
base, and multiply by the number of layers.
11. Find the volume of a cylinder whose altitude is 7 ft.
4 in. and the diameter of the base 5 ft.
52 SURFACE AND VOLUME OF CYLINDERS.
1. Find the volurile of a cylinder whose altitude is 8 ft. 6 in.
and the diameter of the base 8 ft.
2. Find the solid contents of a cylinder whose altitude is
15 ft., and the radius of the base 1 ft. 3 in.
3. What is the entire surface of a cylinder 6 ft. 6 in. long,
and the radius of its base 4 ft.?
4. What are the contents of a cylinder whose length is 5 ft.
and diameter of the base 15 in. ?
5. How many gallons will a circular cistern hold that is 6 ft.
in diameter and 10 ft. deep ?
Note. — There are 231 cu. in. in a gallon. For practical purposes it is suf-
ficient to say, there are 7 J gal. in a cubic foot. Use the latter method unless
otherwise directed.
6. Find the entire surface of a cylinder 30 ft. long, and 30
in. in diameter.
7. How many gallons of water will a cylindrical vessel hold
liiat is 9 ft. deep and 3 ft. in diameter?
8. A cylindrical vessel 8 ft high and 6 ft. in diameter is
filled with potatoes. What is the value of the potatoes at 75/
a bushel ? Approximate measurement.
9. How many gallons of water are in a well 5 ft. in diam-
eter, if the water is 7 ft. deep ?
10. Find the entire surface and volume of a cylindrical col-
umn 3i £t. in diameter, and 28 ft. high.
11. At 32/ a cubic foot what is the value of a log 45 ft. long
and 2 ft. in diameter?
12. A cylindrical water-tank is 25 ft. high and its diameter
is 30 ft. How many gallons of water will it hold ?
13. A circular reservoir is 80 ft. in diameter, and 20 ft. deep.
How many gallons of water will it hold when full?
14. How much more gilding will it take to cover a 9-inch
cube, than to cover a cylinder whose height and diameter are
each 9 inches ?
REVIEW m Pit ACTIONS.
53
1. A boy sdld 3 bu. 8 pk. of pears at the rate of 8 for 5
centS; Each peck ayehtged 3J doz. pears. How much did he
receive iot all ?
2. If a number diminished by Jf of itself is 7,296, what is
the number?
8. A horse fend carriage are worth $763. The carriage is
worth § as much as the horse. What is the value of each ?
4. A house and lot cost $13,600. The cost of the lot was
-ft of the cost of the house. Find the cost of each.
6. A man divided his estate, giving his oldest son J of it
ahd his youngest son J of it. If $875 was the difference be-
tween the sons' shares, what was the value of the estate ?
6. A and B can do a piece of work in 6 days. A can do it
alone in 10 days. In what time can B do it?
7. I called for bids in a piece of work. A agreed to do it
in 1| mo. at $2.76 a day; B agreed to do it in 2 J mo., at $2.25
a day; and C in Si mo., at $1.50 a day. Each counted only 24
working days to a month. Which bid should I accept ?
8. If T% of A's money is equal to f of B's money, and B has
$8,000, how many dollars has A?
9. A drover bought cows at $27.40 ; if he had paid $28J,
tbey would haVe cost him $120.70 more. How many cows did
he buy?
10. When 9 hours is a day's work, and $1.50 is a day's
pay, find each man's pay in the following time sheet :
MON.
TUES.
Wed.
Thurs.
Fri.
Sat.
A
9
84
94
74
8
64
B
8
10
7
64
8
84
ei
8i
9
8
6
7
D
6i
5i
9
7i
6
6
E
8
9
7
8
9
8
11. 53^ is what part of 83i ?
64 STATEMENTS,
1. 2, 2, 6, and 7 are four of the five factors of 1,680.
2 A owns I of a ship, and sells f of his share for $3,600.
3. A pole stands i in the ground, | in the water, and 33
ft. above water.
4. A room is 40 ft. long, 31 J ft. wide, and 12 ft. high.
5. A man's salary is $1,500 a year. He spends 35% of it.
6. A merchant paid $10,050 for stock, and sold it an ad-
vance of 33^%. His expenses were $1,500.
7. A rectangular tank of water is 25 ft. long, 18 ft. wide,
and 16 ft. 9 in. deep.
8. A merchant sold a lot of goods for $129 at a loss of
33J%.
9. The same merchant sold another lot for $73.85 at a gain
of 16|%.
10. A stove-pipe is 16 ft. long and 7 in. in diameter.
11. A pile of wood contains 4^ cords. The pile is 7 ft. 6 in.
high and 4 ft. wide.
12. The circumference of a circular pond is 150 rd.
13. Bought a horse and carriage for $650, and sold them for
$806.
14. An article cost $90. It was sold at a gain of 12^%.
15. A farmer bought a twelve-acre field of wheat for $225.
He paid $1.45 an acre for cutting, and 5^ a bushel for thrash-
ing, and $1.75 a load (42 bu.) for teaming. The wheat yielded
28 bu. to the acre, and was sold for $1.1 2^ a bushel.
16. A train leaves New Haven at 8.45 A.M., and goes 27 J
miles an hour. Another train follows at 9 a.m., and goes 41
miles an hour.
17. The inside dimensions of a rectangular fort are 240 ft.
by 190 ft The wall surrounding this fort is 6 ft. thick and
15 ft. high.
18. Two men had each $420. One of them spent 15% and
the other 18^% of this sum.
ORAL. 55
1. A merchant bought 6 boxes of butter for $50, and sold
them so as to gain $10. What did he receive for each?
2. What number added to twice itself gives 18 ?
8. If one man can do a piece of work in 44 days, how
many men can do the same work in 4 days ?
4. At 80^ a bushel, what is a peck and a half of corn worth?
6. Five boys bought a ball for 85 cents, and sold it for 70
cents. How much did each boy lose if they divided the loss
equally ?
6. A man bought a calf for $12, and sold it to the butcher
so as to gain 5%. How much did he gain?
7. What is the amount of $60 for 60 days?
8. What is the interest for $60 for 1 yr. 6 mo. at 5% ?
9. What is the interest of $100 for 1 yr.? For 2 yr.?
1 yr. 6 mo.? 2 yr. 3 mo.?
10. $20 is 16 J % of what sum?
11. A grocer sold tea at 50/ a pound, and thereby gained
25%. What was the cost a pound ?
12. What is 8J% of 48 books?
18. If 3* bbl. of flour cost $20?, what will 6i bbl. cost?
14. If a family consume | barrels of flour in a month, how
long will 3| barrels last them ?
15. At it of a dollar a rod, what will it cost to build ^ of a
rod of fence ?
16. A boy had 36 hens, and sold | of them at 50/ each.
17. How many are J of J of 36 ?
18. In one room there are 32 pupils, f of these are § of the
number in the other. How many pupils are there in the second
room?
19. In a pile of wood there are 24 cd. Three-eighths of the
pile are worth $54. What is 1 cd. worth?
20. 54 are how many times J of 24 ?
21. What is the ratio of a day to 3 hours ?
56 REVIEW OF PERCENTAGE.
1. A man had $60,000. He put 87% of it in the bank,
17% in a store, and the remainder in a railroad. How many
dollars did he invest in the railroad ?
2. A man owes |8,496, but can pay only $6,372. What
per cent can he pay ?
3. I have invested $4,896 in business, which is 16% of all
my money. How much money have I ?
4. What number increased by 62i% of itself equals 3,942i?
6. The Mississippi River is 4,200 miles long, which is 5%
longer than the Nile, and that is 63% longer than the Amazon.
Find the length of the Nile and of the Amazon.
6. A drover sold 34% of his cattle, and had 990 left. How
many did he sell ?
7. A man bought 800 tons of coal at $8.50 a ton, and sold
it so as to gain. 45%. What did he receive for it?
8. A grocer sold flour at $3.50 a barrel, and lost 30%.
Find the cost.
9. A man sold an engine for $1,650, and gained 25%.
Find the gain.
10. The same man sold another engine for the same price,
and lost 25%. Find the loss. How does the loss compare
with the gain in example 9 ?
11. An agent collected $1,680 for me, but only sent me
$1,600. How much did he keep as commission ? What per
cent of what he collected did he keep?
12. Find the interest of $2,763 from Sept. 5, 1901, to Jan.
13, 1903, at 4i%.
13. Find the interest of $106.45 from Nov. 28, 1900, to June
6, 1903, at5i%.
14. Find the amount of $1,047.50 for 1 yr. 9 mo. 10 da., at
7%.
15. Find the interest of $750 from Sept. 8, 1901, to Aug. 8,
1904, at 6i%.
MISCELLANEOUS REVIEW. 57
1. A and B can do a piece of work in 8 days, A and C in
9 days, and A alone in 12 days. In how many days can B and
Cdoit?
2. How many board feet are there in 56 joists, each 20 ft.
long, and 9 in. by 3 in. ?
8.' Add i, §, i, 9.647, 8.93J, 59.7|, 4.07J.
4. At 75^ a square yard it cost $99 to pave a walk. If the
walk is 6 ft. wide, how long is it?
5. How much will it cost at 86/ a square yard, to plaster
the walls and ceiling of a room 18 ft. x 14 ft. x 9 ft., allowing
180 sq. ft. for openings.
6. A room is 24 ft. long and 19 ft. 8 in. wide. How many
yards of carpet 1 yd. wide will it take for this room if the
breadths run lengthwise, and there is a waste of 10 in. on each
breadth for matcjhing ? Would it take any less if the breadths
ran the other way.
7. Find the cost^f building a tight board fence 4 ft. 6 in.
high, on two sides of a lot 32 rd. long, and i as wide. The
boards are nailed to 2 scantlings each 2 by 4 in. The posts are
8 ft. apai-t, and cost $27 a hundred. The scantlings cost $16
per M., and the boards $18 per M.
8. Find the interest of $2,679.13 from June 6, 1901, to
July 6, 1903, at 7%.
9. Find the area of a triangle whose altitude is 16 ft. and
base 18 ft.
10. Find the area of a trapezoid when the parallel sides are
84 rd. and 66 rd., and the altitude 38 rd.
11. What must be the height of a pile of wood 32 ft. long
and 6 ft. wide, to contain 9 cords ?
12. A man bought a horse, cow, and pig for $160. If he
paid three times as much for the cow as for the pig, and four
times as much for the horse as for the cow, what was the price
of each?
68
RATIO.
1.
5.
7.
(See
Part II., pages 64,
66.)
What is the ratio of:
36 to 12?
12 to 4?
2 ft. to 4 in. ?
49 to 7?
4 to 40?
Ijyd. to 6 in.?
72 to 8?
16 to 7?
1 lb. to 8 oz.?
42 to 6?
8 to 56?
$4 to 50 cents?
What is the ratio of :
3to 6?
4 to 20?
8 to 24?
2to 4?
6to 8?
12 to 60?
2 to 10?
3 to 12?
18 to 54?
6 to 30?
16 to 48?
86 to 48 ?
Find the ratio of :
1 da. to 2 hr. 4
mo. to 1 yr.
2 wk. to 2 da.
1 ft. to 1 yd. 2
min. to 30 sec.
1 pt. to 1 qt.
1 gal. to 2 qt. 4
oz. to 1 lb.
2 pk. to 1 bu.
itoi. *to|.
itoi.
Find the ratio of :
•
8.7 to 2.9. .64 to .08. 3i to 4f .
6 mi. 15 rd. to 2 mi. 6 rd. 3 ft,
. 9 in. to 8 ft. 4 in.
6 lb. 4 oz. to 6 lb.
10 oz. 2J to 3i.
Find the ratio of :
1.26 : 37.5
2t:3i
2 mi. :640 ft.
6.25: 2.5
6|:6i
5 wk. : 4i da.
16.8 : 8.4
8i:li
5 tons : 500 lb.
Find the ratio of :
65: 15
225: 75
93 : 31.
25 : 625
275 : 550
48 : 72.
342 : 228
75 : 125
144 : 48.
Find the ratio of:
900 : 300
96: 72
144 : 60
200 : 600
63 : 108
33:88
150 : 450
66: 84
36:72
600 : 200
64: 18
24:16
PROPORTION. 69
\
1. In every ratio the first term is called the antecedent, and
the second term the consequent.
2. What is the ratio of 8:4? What is the ratio of 6:3?
These two ratios being equal may be written 8:4 = 6:3. This is a propor-
tion. Proportion is an equality of ratios. A proportion may be read in two
ways : The ratio of 8 to 4 equals the ratio of 6 to 3, or 8 is to 4 as 6 is to 3.
8. In the proportion above, how many ratios are there?
Name the antecedent of each ratio. Name the consequent of
each ratio.
4. In every proportion the first and fourth terms are called
extremes, the second and third means.
5. Name the extremes and means in the above proportion.
6. Multiply the extremes together; multiply the means to-
gether. What is true of your products ?
7. Since this is true, if any three terms of a proportion are
given, the other can be found.
8. In the proportion a; : 4 = 6 : 3, we know that 3x2?==
6x4. Hence, 3 a; = 24. a; = 8.
9. In the proportion 8 : a; = 6 : 3, we know that 8x3 =
a; X 6. Hence 6 a? = 24. a? = 4.
10. In the proportion 8 : 4 = a; : 3, we know that 8x3 =
4 X a;. Hence 4 a; = 24. a; = 6.
11. In the proportion 8 : 4 = 6 : x, we know that 8 x a; =
6x4. Hence 8 a; = 24. a? = 3.
Find the missing terms in the following proportions :
12. 4 : 6 = 8 : a:. a; : 8 = 10 : 40. 9 : a; = 7 : 21.
18. 6 : 3 == a; : 11. 8 : 4 = 12 : a:. a; : 5 = 8 : 20.
14. 9 : 5 = a; : 15. 8 : a; = 6 : 12. 10 : 6 = 6 : a:.
16. 140 : 8 = 70 : a;. 8 : a: = 5 : 120. 12 : a; = 42 : 63.
16. a; : 8 = 7 : 3. 64 : 9 = a; : 10. 63 : 7 = a; : 5.
17. 35: a; = 21: 3. 48 : a: = 12 : 2. 36 : 6 = 24 : a;.
18. I : a? = J : J. a; : | = J : i- J : f = a? : J.
19. 2:6= a;: 10. 3 : 4 = 9 : a;. a::3 = 12:36.
60 ORAL.
1. What is the ratio of 86 to 5? Of 6 to 18?
2. What is the ratio of 60 to 12? Of 8 to 16 ?
3. What is the ratio of 1 to i? Of i to 1 ?
4. What is the ratio of i to i ? Of i to i ?
Find the value of 2; : —
6. a::4 = 6:3 2: a;==6:12 3:5 = 2::10
4:3«8:a; x: 6 = 2:4 8:2: = 4:2
8:6 = a;:4 7:14 = 3:2; a::6 = 4:8
2:a; = 4:8 10: 5=2::3 6:2 = 9:a;
(For solving the following problems see Part II, page 59, note.)
6. If 4 oranges cost 12 cents, what will 8 oranges cost?
7. If 5 yd. of cloth cost $15, what will 6 yd. cost?
8. How many oranges can be bought for 50/, at the rate of
4 for 10 cents ?
9. What will 6 tons of coal cost, if 3 tons cost f 18 ?
10. If 4 men can mow 8 acres in a certain time, how many
acres can 8 men mow in the same time ?
11. If 7 men can build a wall 14 ft. long, how long a wall
will 4 men build ?
12. If 4 men can build a fence 12 rd. long, how long a fence
will 7 men build in the same time ?
13. If 5 men can dig a ditch 10 rods long in a day, how long
a ditch can 8 men dig?
14. What will 20 yd. of cloth cost, if 5 yds. cost $15 ?
15. If II 00 gains |6 in a year, what will $200 gain?
16. If 5 oranges cost 15/, what will 8 oranges cost at the
same rate?
17. If 3 tons of hay cost $40, what will 1^ tons cost at the
same rate ?
18. Required the cost of 21 bbl. of apples, at the rate of $6
for 3 barrels.
19. The ratio of the value of a purse to its contents is J. If
the purse is worth $2, how much money does it contain ?
PROPORTION. 61
1. If 6 tons of coal cost $28, how many tons can be bought
for $84?
(a) In examples like the above we have two ratios. $28 ia to be compared
with $84. 6 tons is to be compared with x tons. These two ratios will be
equal, for they bear the same relation to each other as cause to effect. The
sums of money are directly proportional to the tons of coal bought with the
money. $28 : $84 = 6 tons : x tons. 28 x = 504. x = 18.
(6) These problems may be solved without forming a proportion if desired.
The ratio of $84 to $28 is 3. Hence we can buy 3 times 6 tons of coal, or 18
tons.
2. If 6 men can build a wall in 24 days, how long will it
take 18 men to build the same wall ?
The first example is an illustration of direct proportion, where more dollars
will buy more tons. This example is an illustration of an inverse ratio, where
more men need less days; i.e., the days are not directly proportional to the
"^®^- 18 men : 6 men = 24 days : x days ?
18x = 144. x = 8days. Ans.
The following explanations may seem clearer to some, and
are therefore inserted.
Since we are seeking for tons, we
1. $28 : $84 = 6 tons : x tons. ^^^ q ^^^ ^^ ^j,^ ^hird place, be-
3 cause the third and fourth terms
3^ X 6 , Q form a ratio and must be of the same
= 18 tons. tind.
28
^^ We know that $28 will buy 6
tons, then $84 will buy more tons. This shows us that the consequent of our
second ratio is more than its antecedent ; therefore to have an equality of
ratios, the consequent of our first ratio must be larger than its antecedent.
Hence our ratio must be $28 : $84.
Cancelling, we have 18 tons.
o-io^ a ckA j\ J Since we are seekiner for
2. 18 men : 6 men = 24 days : x days. .„„„ „^ , o.i ^
J J days, we place 24 days as
^ our third term. Why? We
^ X ^^ ^ j^ . know that 6 men can build
—^— — o- ^f^' a wall in 24 days ; then 18
n men can build it in less days.
^ This shows us that our sec-
ond consequent is less than its antecedent ; hence to have an equality of ratios
we must make our first consequent less than its antecedent, and our ratio must
be 18 men to 6 men. Cancelling, we have 8 days.
62 PROPORTION.
Note. — Before comparing the quantities in the following examples, first
determine whether the terms are directly or inversely proportional.
1. If 45 sheep cost $225, what will 165 sheep cost?
2. If 35 acres of land cost $937.50, what will 175 acres
cost?
8. If 240 acres of land cost $1,500, how many acres can be
bought for 14,500 ?
4. If 15 yd. of cloth cost $24, what will 65 yd. cost?
6. If 24 bbl. of flour will last 160 men for 5 weeks, how
many barrels will last 180 men the same time?
6. If 8 men can dig a ditch in 4 days, how long will it take
7 men to do it?
7. If 9 men can build a wall in 15 days, how long will it
take 5 men to build it ?
8. If 14 men can mow 25 acres of grass in a day, how many
acres can 35 men mow ?
9. If a staff 3 ft. high casts a shadow 5 ft. long, how long
a shadow will be cast by a pole 120 ft. high at the same time ?
10. If 12 lb. of sugar cost $2, what will 30 lb. cost at the
same rate ?
11. A family of 6 persons pays $21 a week for board; at this
same rate what must a family of 8 persons pay ?
12. A tree 18 ft. high casts a shadow 45 ft. long; how high
must a steeple be to cast a shadow 135 ft. ?
18. If 16 men fcan build a house in 24 days, how long will
it take 12 men to build it ?
14. If 24 men can build a house in 15 days, how many men
can build it in 24 days ?
16. If 12 tons of hay cost $264, what will 19 tons cost?
16. If a train travels 220 mi. in 8 hr., how long will it be in
traveling 330 miles ?
17. If 16 horses eat a ton of hay in 12 days, how many
horses will eat a ton in 32 da3r8?
PROPORTION, OR ANALYSIS, 63
1. If f of a store is worth t5,200, how much is i of it worth?
^- ^ ^ Arrange the ratios as in
# : t = ^o,zOO : ^x whole numbers. Arrange for
$5,200 X t 5,200 X 7x9 ^0^0^ cancellation as usual. $5,200
~& = g g = f 0,iyU multiplied by I and divided
by J equals $6,200 multiplied
by I and multiplied by |. This clears the work of fractions.
Note. — To arrange for cancellation, place fractions in the second or
third term as for multiplication. Since we are to divide by the fraction in the .
first term, it will be the same to multiply by the same fraction inverted.
2. If it requires 42 yd. of carpeting | yd. wide to cover a
floor, how many yards J yd. wide will cover the same floor?
3. If 6i cd. of wood cost $19i, how many cords can be
bought for 178 ?
4. If 3i yd. of velvet cost $5 J, how much will 7i yd. cost?
6. If 4 J lb. of butter cost $1.35, what will 17i lb. cost?
e. If J of a yard of silk cost $2.10, what will 33j yd. cost?
7. If 9 weeks' board cost $94i, what will 12 weeks' board
cost?
8. If 7 men can lay 2 miles of water-pipe in 15 days, how
many days will 48 men require ?
9. If 5 of a yard cost $lf, what will f of a yard cost?
10. If a loaf of bread weighs 8 oz. when flour is worth $5 a
barrel, what should it weigh when flour is worth $6 a barrel?
11. If 2| yd. of cloth can be bought for $3.30, what will 155
yd. cost?
12. If 25 men can do a piece of work in 12 days, in how
many days can 10 men do the same work?
18. If a steeple 126 ft. high casts a shadow 93 ft. long, how
long a shadow will a steeple 168 ft. high cast at the same time?
14. If a steeple 216 ft. high casts a shadow 162 ft. long, how
long a shadow will be cast by a steeple 124 ft. high at the same
time?
16. 83J : 55f = X : 93}. Find x.
64 MEASUREMENTS.
1. Find the feet of lumber in the following boards : 2 boards
18 ft long, 8 in. wide, and 1 in. thick ; 3 boards 16 ft. long, 9
in. wide, f in. thick ; 1 board 12 ft. 8 in. long, 8 J in. wide, 1
in. thick ; 2 planks 16 ft. long, 10 in. wide at one end, 8 in.
wide at the other, and 15 in. thick.
Note. — Boaxds less than 1 in. in thickness are called 1 inch. Fractions
of an inch in width are omitted, and the nearest integer taken. When the
fraction is one-half the next integer is taken. When boards are not of uniform
width, the average width is taken.
2. How many square feet in the surface of a spire which is
in the form of an hexagonal pyramid, whose slant height is 80
ft and each side of its base 12 ft ?
3. How many bushels of com will a box contain which is
8 ft long, 3 ft wide, and 20 in. deep ? (AUow i j cu. ft. to l bu.)
4. What will it cost, at 13/ a square yard, to plaster the
walls and ceiling of a room 16 ft. long, 15^ ft. wide, and 9 ft
high, deducting 95 sq. ft. for openings ?
5. A rectangular monument of granite is 3 ft square at the
base, and 8 ft. high. How many cubic feet does it contain ?
6. At 19/ a square yard, what will it cost to paint the out-
side and the inside of a cylindrical tank 9 ft long, and 6 ft in
diameter, no attention being paid to thickness of the material ?
7. Find the cost of 1,860 ft of lumber at $24.50 a thousand. /
8. A house is 32 ft. 8 in. long. The rafters are 22 ft long.
How many shingles laid 4 inches to the weather will cover the
roof? (Allow 1000 shingles to 100 sq. ft.)
9. A cylinder is 3 ft 8 in. long, and has a diameter of Ij ft
How many gallons will it hold ? (Allow 71 gal. to l cu. ft.)
10. A rectangular lawn measures 15 yd. 2 ft in length, and
33 ft. 8 in. in width. At 18/ a square yard, find the cost of
sodding the lawn. Outside the lawn is a walk 3 ft. 8 in. wide.
Find the square feet in it. At 16/ a running foot, find the
cost of a fence just outside the walk.
ORAL. 65
Give the names of the following figures :
1, A figure having three sides.
2. A figure having three equal sides.
8. A figure having three sides, two of which are equal.
4. A figure having four equal sides and four right angles.
6. A four-sided figure, having opposite sides parallel, but
containing no right angle.
6. A figure having four equal sides and opposite sides paral-
lel, but containing no right angle.
7. A figure having four sides, with opposite sides parallel
and equal.
8. A figure having four sides with only two parallel sides.
9. Any figure having four sides.
10. A figure having four sides with no parallel sides.
11. A rose garden is 12 ft. long and 9 ft. wide. How many
bouquets can I gather, if 3 sq. ft. will furnish 1 bouquet?
12. 9 + 7-6 + 5-8 + 4-9 + 7-6-3 + 2=?
18. How many cubic feet in a wall 30 ft. long, 4 ft. high, 2
ft. thick?
14. How many one-foot cubes can be placed in a cubical box
one yard" long, one yard wide, and one yard high ?
16. A farmer sold his wheat for $267, and his oats for $234.
How much did he receive for both ?
16. At 45^ a pound, how many pounds can be bought for
$1.35?
17. Divide 18 lb. 12 oz. by 3.
18. Divide 30 da. 15 hr. 45 min. by 3.
19. What part of 2 gal. is 2 qt. 1 pt. ?
30. 15 is ? of J of what number?
21. If 5 of a stack of hay is worth $42, what will 2 stacks
be worth ?
dd. What is the least common multiple of 8, 12, and 24 ?
23. Divide 15,000,000 by 30,000.
66 REVIEW OF PERCENTAGE,
1. A merchant sold goods which cost him $4,768.75 at a
profit of 18%. Find the gain.
2. A man raised 1,640 bu. of grain, and sold 246 bu.
What per cent did he sell ?
3. A teacher's salary, having been decreased 83%, is now
$1,200. What was it at first ?
4. A grocer bought $400 worth of oranges. 26% spoiled
before he sold them. The remainder he sold 16§% above
cost. Did he gain or lose ? and how much ?
5. A farm which cost $3,400 was sold at a gain of 22%.
Find the selling-price.
6. By selling hay at $15 a ton a dealer loses 10%. Find
the cost.
7. What is the commission, at 3^%, on the sale of $4,769
worth of goods ?
8. An agent sells 276 bbl. of flour at $6.50 a barrel. His
commission was 2^%. How much money should he return to
me?
9. A ship was valued at $72,000, and insured for | of its
value at 2^%. Find the premium.
10. A man bought 6,200 bu. of grain at $1.50 a bushel. He
sold 20% of it at a 5% loss, 40% of it at a 10% gain, and the
remainder at cost. What was gained on the whole ?
11. Find the cost when $17.25 is the loss from selling an
article at 15 > below cost.
12. Find the interest on $1,500 for 1 yr. 3 mo. 27 da. at 9%.
13. Find the amount of $960 for 3 yr. 7 mo. 9 da. at 5%.
14. A farmer raised 2,480 bu. of grain. 36 % of it was rye,
24% of it oats, and the rest com. How many bushels of each
kind of grain did he raise ?
16. A man bought 120 acres of land at $60 an acre, and paid
25% of the cost of the land for repairs and building. For how
much must he sell to gain $2,000 ?
REVIEW IN FRACTIONS. 67
1. What are the prime factors of 5,075 ?
2. Reduce ^^ to lowest terms.
8. When 17f lb. cost $lli, how many pounds can be bought
for«^5i?
4. What will 94 J yd. of cloth cost, if 165J yd. cost $94 ?
6. If a boat sails 254 miles in 19| hours, what is the rate
an hour?
6. What will 117i yd. of cloth cost, if 378| yd. cost $1,515 ?
7. Whatwilll43j bu. of apples cost, if 584tbu. cost $1,022 J?
8. A owns f of a store, and B the remainder. If A owns
$465 more than B, what is the value of the store and of each
-one's share ?
9. How many marbles have two boys, when one owns f of
all, and has 60 marbles more than the other?
10. Eight-elevenths of 2 J 28 is H of what number?
11. A man gave 180 acres to his son, and had J of his farm
left. What is the value of the remainder at $39 J an acre ?
12. An estate was divided between two persons, so that A
received ^ of the whole, and B the remainder, or $6,400. Find
the value of the whole estate.
18. An estate was divided among 3 children, so that one
received f of it, the second % of it, and the third $4,840. Find
the value of the estate.
14. How much more is 4 J^ times 9^ than 2} times 5f ?
15. A owns f of a store, B f of it, and C the remainder, or
$31,000. What is the value of the store ?
16. A room is 15 ft. by 20 ft., with walls 12 ft. high. Find
how many square feet there are in the walls and ceiling. If
there are 3 windows 2^ ft. by 6 ft., and 2 doors 3 J ft. by 8 ft.,
find how many square feet there are in the doors and windows.
Find how many square feet there are in the walls and ceiling,
after taking out the doors and windows.
17. From two take two ten-millionths.
68 MISCELLANEOUS PROBLEMS,
1. What is the area of a circular garden whose circumfer-
ence is 180 rd. ?
2. Find the area of a trapezoid when the parallel sides are
120 in. and 96 in., and the altitude 86 in.
3. A and B together have 2,638 acres of land, and B has 5
times as much as A. How many acres has each.
4. A drover has 427 sheep and cows. If he has 125 more
sheep than cows, how many has he of each ?
5. If ISJ bu. of com cost $6i, what will 16i bu. cost?
6. If 25 oxen eat the grass from 36 acres in a month, for
how many oxen would 468 acres furnish feed for the same time ?
7. Find the interest of $23.75 for 6 yr. 7 mo. 21 da. at 5%.^
8. How long must 3 piles of wood be to contain 27 cd., if
each is 5^ ft. high, and 4 ft. wide ?
9. A can do a piece of work in 6 days, B in 8 days, and C
in 12 days. In how many days can all do it together ?
10. Divide .00017 by .034.
11. Add 94, .845, 7§, 56j, .65j, 59.34, 37f .
12. Reduce 1,345,165 seconds to higher denominations.
13. Find the cost of 2,400 qt. of onions at 18/ a pk.; 3,200
qt of milk (a) 24/ a gallon ; and 4,200 qt. sirup ^ 12/ a pint.
14. Find the gain in buying 2,480 gal. of vinegar at 30/ a
gallon, and selling it at 9/ a quart.
15. A man divided a field containing 16 acres into lots con-
taining 40 square rods each. He sold the lots at $175 each.
How much did he receive?
16. Find the cost of 17 bu. 5 qt. of oats at 2/ a quart, and
7 lb. 2 oz, of spice at 15/ an ounce.
17. A mill is worth $9,900, a house $3,000, and a farm ^rs
of the difference between the value of the house and mill.
Find the value of all.
18. In a school there are 495 pupils, and twice as many girls
as boys. How many girls are there ?
TO FIND CONTENTS OF CONES AND PYRAMIDS. 69
To illustrate this principle there should be in the room a hollow cone and
cylinder, each having the same base and altitude, also a hollow pyramid and
prism, each having the same base and altitude. These can easily be made
from cardboard. Using sand or sawdust, let each pupil determine, by measur-
ing, the ratio or relation of the contents of the cone to the contents of the
cylinder, and the contents of the pyramid to the contents of the prism. This
they will find to be one-third.
Note. — To find surface, the slant height must be known ; but to find vol-
ume or contents, the altitude must be known.
Learn : To find the contents of cones and pyramids, multi-
ply the area of the base by J of the altitude.
1. Find the volume of a square pyramid each side of whose
base is 4 ft. and altitude 18 ft.
2. Find the volume of a cone the circumference of whose
base is 9 ft., and whose altitude is 14 ft.
3. What is the volume of a square pyramid, the perimeter of
whose base is 8 ft., and whose altitude is 10 ft.?
4. What is the volume of a pyramid whose base is 6 ft.
square, and whose altitude is 21 ft.?
6. Find the volume of a cone whose diameter is 6 ft., and
altitude 18 ft.
6. Find the volume of a cone whose circumference is 31.416
feet and altitude 20 feet.
7. Find the volume of a square pyramid with a base 10 ft.
square, and an altitude 25 ft.
8. Find the volume of a cone, when the circumference of
its base is 48 ft., and its altitude 50 ft.
70 ORAL.
1. If a boy buys oranges at 40/ a dozen, how must he sell
them apiece so as to make 20 % on each orange ?
2. If a yard of silk costs a merchant 80/, for how much
must he sell it to gain 12 J % ?
8. What is the gain per cent on an article that is bought
for 40 cents and sold for 60 cents ?
4. I went out into the country and bought eggs at 25/ a
dozen, and brought them to the city and sold them for 30/ a
dozen. What was my gain per cent?
5. At the same time I bought potatoes at 60/ a bushel, and
sold them at 25/ a peck. What was my gain per cent ?
e. A lawyer agreed to collect for me at 5% commission.
He collected i of a debt of $1,200. How much commission
did he keep?
7. At 8%, find the interest on $50 for one year.
8. At 7%, find the interest on $40 for a year.
9. At 10%, find the interest on $60 for 60 days.
10. At 6%, find the interest on $200 for 33 days.
11. A watch cost $70 more than a chain, and together they
cost $160. How much did each cost?
12. A can do a piece of work in 3 days, B in 4 days, and C
in 5 days. How much can each do in a day ? How much can
all do in a day? How many days will it take, all working
together, to do the whole work?
18. Two men can do a piece of work in 3 days. If one man
can do it in 5 days, in how many days can the second man do it?
14. A is 40 yd. ahead of B. If B runs 5 yd. while A runs
4, how many yards must B run to overtake A ?
15. A is 30 ft. ahead of B, but B runs 5 ft. while A runs 2
ft. How many feet will B run to overtake A ?
16. Divide 64, 76, 96, 104, 112, 216 by 4.
17. If J a yard of cloth costs $4, what will Ij yd. cost?
18. J of 24 are how many eighths of 12?
MISCELLANEOUS REVIEW^ 71
1. The difference between two numbers is 2,001,005; the
larger number is 89,009,089. Find the smaller.
2. At $5.75 a barrel, how many barrels of flour can be
bought for $1,161.50?
8. The quotient is 4,769. What will the quotient be if the
dividend is multiplied by 9 ?
4. The quotient is 4,664. What will the quotient be if the
divisor is multiplied by 8 ?
5. The quotient is 805. What will the quotient be if the
divisor is J of what it is now ?
6. The quotient is 909. What will the quotient be if both
dividend and divisor are multiplied by 15 ?
7. The quotient is 478. What will the quotient be if both
divisor and dividend are divided by 12 ?
8. If the divisor were 8 times as large as it is, the quotient
would be 489. What is the quotient?
9. What number must be taken from 7,684 that it may be
exactly divisible by 33 ?
10. A book agent bought 112 books at $3.20 each. He sold
them at $4.90 each. If his expenses were $75, and he could
not collect the money for 4 books, how much did he gain or
lose?
11. If 37 is added 99 times to itself, the result will be how
much less than 3,750 ?
12. If 37 be added to a certain number, 85 can be subtracted
from it 113 times. Find the number.
18. Of what number is 463 both divisor and quotient?
14. If 593 is subtracted 347 times from a certain number,
the remainder is 287. Find the number.
16. A house and lot cost $9,600. The house cost 5 times as
much as the lot. Find the cost of each.
16. How much will a dealer gain by buying 2,464 bu. of
potatoes at 87i/ a bushel, and selling them at $1.12 J a bushel?
72 STATEMENTS.
1. The base of a cone is 82 in, in diameter, and its altitude
is 3 ft
'2. A rectangular prism has an altitude of 12 ft Its base
measures 18 in. by 14 in.
3. The area of a rhombus is 720 sq. in. Its altitude is 24 in.
4. The area of a triangle is 7 sq. yd., and its altitude is
21ft
5. A man traveled 65 days at the rate of 768 miles in 4
days.
6. An article that cost $75 was sold for $95.
7. $953 was on interest from Aug. 9, 1901, to April 1,
1902, at 5%.
8. A merchant insured his store for $7200, at 2^% a year
for three years.
9. A collector received $121.40 for collecting a debt at 5%.
10. By selling flour at $5.85 a barrel, a merchant lost 22%.
11. When the selling-price was $162.50, there was a profit
of 30%.
12. A boy earned $8.40, which is 15% of what he had
before.
13. A man lost $1770 out of a business of $2950.
14. 83 J % of a debt of $8400 has been paid.
15. A room measures 21 ft by 16 ft. The carpet is f yd.
wide, and costs $1.40 a yd. The breadths run crosswise, and
there is a loss on each breadth of 1 ft for matching.
16. From an acre of land there was sold a piece in the form
of a trapezoid 10 rd. long, 7 rd. wide at one end, and 5 rd. wide
at the other.
17. It is 75 ft. in a straight line across a circular pond.
18. The area of a parallelogram is 300 sq. yd. The dis-
tance between its parallel sides is 50 ft
19. Some 4-foot wood is piled 6 ft high. There are 24 cords
in the pile. • •
MISCELLANEOUS REVIEW, 73
1. If 16 cd. of wood cost $96, what will 25 cd. cost?
In solving problems like the above, it is a good plan to make statements
like the following :
1. I am asked to find the cost of 25 cd. of wood.
2. I know the cost of 16 cords.
3. I need to know the cost of 1 cord.
. _, 96 X 25 ^_^
4. Process — — — = $150.
lo
5. Is the result reasonable ? 25 cords are a little more than l\ times 16
cords. My result should be a little more than 1^ times $96.
Note. — If pupils were always obliged to take the 5th step we should
have fewer nonsensical answers.
Solve the following problems by this method :
2. If a young man earns $36 a month, in how many years
will he earn $5,616 ?
3. If 7 tons of coal cost $35, what will 4 tons cost ?
4. My sister Alice hires a bicycle for 30/ an hour. What
does she pay for its use if she uses it from 10 a.m. till 3.30 p.m. ?
5. A has J of a sum of money, B J of it. B has $30 more
than A. How much has each?
6. If a bushel of beans weighs 60 lb., and a barrel holds 3
bu., how many barrels will it take to hold 5 tons ?
7. If an ocean steamer uses 300 tons of coal in a day, how
•many pounds will it use in a month of 30 days ?
8. Four men cut a pile of wood. If the first man cut J of
it, the second J, and the third i, how much did the fourth cut?
0. I hired 4 men to work at $3 a day each. The first
worked J a day, the second ^, the third 2 J days, and the fourth
Ij days. How much must I pay them?
10. If 3 bbl. of oil cost $15.75, what will 18J bbl. cost?
11. If I buy a 42-gal. barrel of oil for $3.50, and retail it at
12^/ a gallon, how much do I make ?
12. My house cost me $5,000, and my tax is $16 on a thousand.
If I rent my house for $30 a month, what is my annual income ?
74 REVIEW OF FRACTIONS.
!• Reduce to whole or mixed numbers : ^^ , Y/ » W > ^$P,
2. Reduce to improper fractions: 616J, 84|, 134g, 125?,
160f, 106 A, 1135,63}.
8. Reduce i and | to 12ths. } and i to Sths. { and I to
24ths. f and { to SOths.
4. Reduce to equivalent fractions having a common denom-
inator: i, t, I, I, ^flff, 4i, i, A.
5. Add: 16J, 22J, 46i, 60A, 8i, lOf, 14?.
6. Add: A, H,l; i,§i,5l.
7. From 20} take 8^^ . From 47j take 19ft.
8. What is A of 8,218 ? ^ oi 3,002 ?
9. JofiofI? ioi^joi^f? ft of A of A?
10. Multiply 408 by 20. 122 by 6?.
11. Divide 231 by H. 66i by 18$. .
13. A man lost f of his money and had $411 remaining.
How much had he at first?
13. If i of a farm is valued at $2,253i, what is the value .of
i of it?
14. 96| -'^?
^ 16J
15. If 8 J tons of coal cost $308, how many tons can be
bought for |127| ?
16. A farmer raised 2,146 bbl. of apples. He sold | of them
at $1| a barrel, and the remainder at $2i a barrel. How mucl
did he receive ?
17. A boy bought 120 oranges at the rate of 5 for 2 cents.
He sold J of them at the rate of 3 for 1 cent and the remainder
at 3 for 2 cents. How much did he gain or lose ?
18. If Ij lb. of beef and Ig lb. of flour are allowed to each
man as a ration, how much will rations for 618 men cost if beef
is 11}/ a pound and flour 3 J/ a pound?
19. From 126i take 13i times 1|.
ORAL. 75
1. If 4 men can do a piece of work in 6 days, how long will
it take 12 men to do the same work ?
2. If 8 men can do some work in 6 days, how long will it
take 4 men to do it ?
8. If f of a man's age is 18 years, how old is he?
4. If I of a ton of coal is worth $3|, how much are 8J tons
worth?
5. A boy spent i of his money, then earned J as much as he
spent, and had $20. How much had he at first?
6. Five-sixths of 60 is § of how many times 6?
7. A farmer sold a cow worth $45 at a loss of 10%. What
did he receive for her?
8. Three-fifths of 80 is what per cent of i of 120 ?
0. A butcher buys pork at 6/ a pound, and sells it at 10/ a
pound. What is his gain per cent?
10. One-fourth of 80 is how many times 6 ?
11. Seventy is J of how many times 5 ?
13. A farm was sold for $1,800, which was f of its cost.
What was the loss ?
13. Goods bought for $120 must be sold for what price to
gain33i%?
14. From what number must % of 20 be taken 3 times to
leave 3?
15. Twenty-four is 60% of what number?
16. A can do a piece of work in 4 days; B can do it in 5
days. In what time can A and B do it if they work together ?
17. A rectangular field contains 1 acre, and is 40 rd. long.
What is its perimeter ?
18. A man spent 16§% of his month's salary. If he spent
$15, what was his month's salary?
10. The selling-price of a cow. is $60, the gain is 25%.
What two things can be found? Find them.
20. What part of 100 is 12i? 37 J? 87i?
76 REVIEW OF PERCENTAGE.
1. A man bought a house for $4,200, and sold it at a gain
ol 25%. Find the cost.
2. A man spent 30% of his income for family expenses,
and 25% of it for books and clothing, and saved the test. If
he saved $900, what was his income ?
3. Change ^% to a decimal. Change .005 to a per cent.
4. Of a lot of goods a man sold $528 worth, which was 5%
of the cost What was the cost ?
6. 75 is 15 per cent of what number? 24 is 40 per cent
of what number ?
6. 26 is 13 per cent of what number? 105 is 35 per cent
of what number ?
7. If an agent's salary was increased 30%, making it $2,600,
what was it before it was increased ?
8. A man bought a farm for $1,250, and sold it at a gain
of $250. Find the gain per cent.
9. What per cent of 800 is 48 ? Of 700 is 35 ? Of 450 is
45? Of 75 is 37.5?
10. If the ore is 35% pure, how much iron will 6,893 lb. of
ore produce ?
11. A man owned a part of a mill for which he paid $12,250.
He was obliged to sell at a loss of 15%. Find his loss.
12. Given the amount gained and the rate of gain. What
can you find ? Illustrate by an example.
13. Given the cost of an article and the rate of gain. What
can you find ? Illustrate by an example.
14. Given the cost of an article and the gain. What can
you find ? Illustrate by an example.
15. Given the cost and the selling-price. What can you
find? Illustrate.
16. A dealer lost 20% by selling a carriage for $120. What
was the cost?
17. Find 65% of $684. Find 6j% of 8j.
MEASUREMENTS. 77
(See general summary, pages 276-282.)
1. Find the contents of a rectangular prism 12^ yd. by 10
ft. by 16 in.
2. A rectangular prism is 20 i ft. long, 17 J ft. wide, and 6
in. deep. Find its contents.
3. At $3.00 a cord, find the cost of a pile of wood 24 ft.
long, 4 ft. wide, and Ti ft. high.
4. I bought 6 boards, each 16 ft. long and Ij in. thick.
Their width was as follows : 8 in., 10 in., 12 in., 13 in., 14 in.,
9 in. How many feet of lumber did I buy ?
5. The entire surface of a cube is 294 sq. in. How long is
the cube ?
6. Find the entire surface of a 9-in. cube.
7. Find the entire surface and the contents of a prism 20
ft. long, 14 ft, wide, and 10 ft. high.
8. What is the difference in volume between a square prism
4 in. wide and 25 in. long and the largest cylinder that can be
cut from the prism ?
0. A well is 32 ft. deep and 5 ft. in diameter. How many
cubic feet of water in it if it is § full ?
10. The area of a rectangular field is 12 acres. If its length
is 20 rd., what is its altitude ?
11. In digging a trench 3 ft. wide and 4 J ft. deep, 330 cu.
yd. of earth were removed. How long was the trench ?
12. From a lot of land 40 rd. square I sold 40 sq. rd. What
is the remainder worth at $120 an acre?
13. A railroad company fenced 8 miles of its road at 67^/ a
rod. Find the cost of the fence.
14. On a pond covering 1 acre the ice when removed was 15
in. thick. If a cubic foot weighs 57^ lb., how many tons were
cut?
15. What will it cost to polish the sides and top of a granite
shaft 6 ft. by 2 ft. by 22 in. at Ij/ a square inch?
78 MISCELLANEOUS REVIEW.
1. At $60 an acre, find the cost of a quadrangular piece of
land whose parallel sides are 26 rd. and 35 rd., and 60 rd. apart.
2. What will it cost to sod a yard 28f ft. long, and 241 ft.
wide, at 76/ a square yard ?
3. A and B can build a wall in 10 days ; A can build it in
18 days. How long will it take B to build it alone ?
4. What will 5 cd. ft. of wood cost at 13.76 per cord ?
5. What will it cost at 30/ a square yard, to plaster a room
27 ft. long, 18 ft. wide, and 9 ft. high, allowing for 2 windows
and 1 door, each 6 ft. by 2^ ft.?
6. 26% of 660 is 33 J % of what number?
7. If 6 acres of land cost $72, what will ^^ of 80 acres cost?
8. One-eighth of a certain number is 16 more than one-ninth
of it. What is the number ?
9. I sold an article to a man for J more than it cost me.
He sold it for $12, which was f less than it cost him. What
did it cost me ?
10. A pile of wood contains 200 cords. It is 8 ft wide and
8 ft. high. How long is it ?
11. What will it cost to dig a ditch 2 ft. deep and Ij ft.
wide around a lot 4 rd. long and 3j rd. wide, at 64/ a cubic
yard?
12. How many rods of fence will inclose a rectangular field
oif 20 acres, whose width is 40 rd. ?
13. If 27 bu. of apples cost $60|, how many bushels can be
bought for $461 J?
14. If a stick of timber 20 ft. long, 12 in. wide, 10 in. thick,
is sawed into boards 1 in. thick, how many board feet will there
be?
15. Four times a certain number added to three times the
same number gives 112. Find the number.
16. The sum of two numbers is 344, and the greater is 7
times the less. What are the two numbers ?
MISCELLANEOUS REVIEW. 79
1. Find the cost of carpet at $1.25 a yard, for 30 in. wide
carpet, for a room 18 ft. by 14 ft, if the strips run lengthwise.
2. Find the cost of papering a room 32 ft. long, 22 ft. wide,
13 ft. high, with paper 18 in. wide, 8 yd. in a roll, at 66/ a
roU, if 60 sq. yd. are allowed for openings.
3. How many board feet in twelve 4-in. planks 16 ft. long
and 10 in. wide ?
4. What will it cost to shingle a roof, each side of which is
30 ft. long, and 25 ft. wide, at $4.50 a square ?
5. What will it cost to build a wall 90 ft. long, 7i ft high,
2 ft thick, at $6 a cubic yard ?
6. Find the number of bushels of grain in a bin that is 6
ft. long, 5 ft. wide, 4 ft deep. (\^ cu. ft. in 1 bu.)
7. What is the breadth of a rectangular field containing 7 J
acres, if the length is 242 yards ?
8. Make out a bill that shall contain five debit and one
credit items.
9. A man sold 40 horses at $200 each. On one-half of
them he gained 25%, and on the rest he lost 20%. Find the
entire gain or loss.
10. Find the interest on $1250 from Nov. 15, 1902, to Mar.
1, 1904, at 5%.
11. If 2.45 tons of straw cost $29.40, how many tons can be
bought for $9 ?
12. Find the surface and volume of a prism measuring 4' 8"
X 3' 10" X 3' 6".
13. Three men. A, B, and C, earned $330. A earned four
times as much as B, and C as much as both A and B. How
much did each earn ?
14. In an orchard there are three times as many pear trees
as apple trees, and four times as many peach trees as pear trees.
In all there are 224 trees in the orchard. How many are there
of each kind?
80 ORAL,
1. For how much must silk that cost $1,20 a yard be sold
to gain 20%?
2. A dealer bought cloth at $4 a yard, and sold it at $6 a
yard. What per cent of profit did he make ?
3. A grocer bought flour at a profit of f 1.20 a barrel, which
was a gain of 25%. What was the cost a barrel?
4. What per cent was lost on a horse which cost $90 and
was sold for $76 ?
5. When a hat was sold for $2 there was a gain of 33 J%.
What was the cost?
6. A mowing-machine was sold for $36 at a loss of 25%.
7. A house was sold for $1,500 at a gain of 25%.
8. A house was sold for $1,800 at a loss of 25%.
9. A cow that cost $45 was sold for $40.
10. When goods are sold for § of tlieir cost what per cent is
lost?
11. When I of an article is sold for what the whole article
cost, what per cent is gained?
12. A grocer sold a pound of butter for 24 cents, by which
he gained 20% on its cost. What would have been his gain
had he sold it at 30 cents a pound ?
18. Mr. Smith sold a city lot for $700, losing $100. What
per cent did he lose ?
14. A gain of $5 on goods that sold for $25 is a gain of what
per cent?
15. How shall I mark cloth that cost 12/ a yard to make
25%?
16. What per cent of an acre is a rectangular lot 4 rd. by
5 rd.?
17. If one-fourth yard of cotton cloth cost one and one-haK
cents, what will 12 yards cost?
18. At 20/ a yard what will 42 ft. of ribbon cost?
19. What is the interest on $400 at 6% for 10 months?
GREATEST COMMON DIVISOR. 81
1. Name a divisor of 8.
2. A divisor is a number that will exactly divide another
number.
3. Name a common divisor of 8 and 12.
4. A common divisor is a number that will exactly divide
two or more numbers.
5. Name the gi-eatest common divisor or factor of 8 and 12.
6. The greatest common divisor or factor of two or more
numbers is the greatest number that will exactly divide them.
7. What is the greatest common divisor of 18, 30, and 48?
2) 18 30 48 By the definition we see that any factor or divisor must be
g\ 9 15 24 * divisor of all the numbers. We arrange the numbers in a
^ — - — » horizontal line, and by division remove all factors that are
/ common to all. The product of these factors must be the
greatest common factor. Removing 2, we find our quotients to be 9, 16, and
24. Any number, to be a part of our greatest common divisor, must be a part
or factor of these quotients. We remove three as common to them all. Our
quotients are 3, 5, and 8. Since these numbers are prime to each other, there
can be no more factors common to all.
2 X 3 or 6 then must be the greatest common divisor.
8. What is the greatest common divisor of 60, 72, 48, 84?
9. What is the greatest common divisor of 45, 75, 90, 135,
150, 180 ?
10. What is the greatest common divisor of 76, 300, 450 ?
11. Find the greatest common divisor of 108, 270, 432.
12. Find the greatest common divisor of 16, 20, 24.
13. Find the greatest common divisor of 44, 110, 164.
In the following examples find the greatest common measure of the numera-
tor and denominator of each fraction, and divide each term by it to reduce the
fraction to its lowest terms.
14. Reduce to lowest terms :
M M M M ii 41 if U
M If U tVit H U H I!
15. Reduce to lowest terms :
m m hu m m m m m
82 COMPOUND NUMBERS.
Note. — These topics have little value, and may be omitted without loss.
1. Multiply 7. bu. 3 pk. 5 qt. by 6.
7 bu. 3 pk. 5 qt. Multiply as in whole numbers. 6 times 6 qt.
6 are 30 qt. 6 times 3 pk. are 18 pk. 6 times 7
»o u — -io 1^ — QQ ^ bu. are 42 bu. Draw a line under this answer,
— ^ * — ^ * and change to the next higher denomination if
47 bu. 1 pk. 6 qt. possible. 30 qt. equal 3 pk. and 6 qt. Write
the 6 qt. 18 pk. and 3 pk. are 21 pk., which equal 6 bu. and 1 pk. Write
the 1 pk. 42 bu. and 5 bu. are 47 bu.
2. What is the weight of 9 loads of hay, each weighing 1
T. 645 1b. 12 oz.?
3. Multiply 217 rd. 4 yd. 2 ft 7 in. by 32.
4. Multiply 13 gal. 2 qt. 1 pt. 7 gi. by 17.
5. If it takes 4 d. 6 hr. 15 min. to build a machine, how
long will it take to build 24 machines ?
6. Multiply 46 T. 439 lb. 8 oz. by 7.
7. Multiply 14 m. 85 rd. 9 ft. 11 in. by 11.
8. Divide 50 bu. 3 pk. 2 qt. by 6.
8 17
6)50 bu. 3 pk. 2 qt.
48 bu. One-sixth of 60 bu. is 8 bu. (write it above bushels) and 2 bu.
2 bu. remaining. 2 bu. equals 8 pk., and 3 pk. make 11 pk. One-
^ sixth of 11 pk. equals 1 pk. and 5 pk. remaining. 5 pk. equal
o T^ 40 qt., and 2 qt. make 42 qt., one-sixth of 42 qt. equals 7 qt.
3 pk. Divide :
n pk. 9. 17 hr. 32 min. 24 sec. by 4. 23 sq. yd. 9
^ pk. sq. ft. 117 sq. in. by 9.
5 pk. 10. 35 wk..5 d. 15 hr. 12 min. 18 sec. by 6*.
_? 11. 21 cu. yd. 20 cu. ft. 17 cu. in. by 7. 89 bu.
^^ ^*- 1 pk. 7 qt. by 7.
-i "!*• 12. 23 cu. yd. 12 cu. in. by 4. 53 T. 176 lb.
S't ^y28.
— "^ • 13. 112 A. 8 sq. rd. by 9. 332 lb. 8 oz. by 19.
14. 125 cd. 7 cd. ft. 7 cu. ft. by 9.
COMPOUND NUMBERS. 83
Note. — See note on page 82.
1. Change H gal. to integers of lower denominations.
Since there are 4 qt. in one gai., in
1^ X 4 qt. == i^ qt. = 2} qt. ^j gaj. there are H times 4 qt., or 2f
I X 2 pt. = ^ pt. = 1 J pt. qt. Since there are 2 pt. in 1 qt., in
i X 4 gills = 2 gills. i qt. there are | times 2 pt. or IJ pt.
Ans. 2 qt. i pt. 2 gills. ^^^^^ '^^^® *^^ ^ «f^- ^^ ^ P*- ^^ * P**
^ ^ there are J times 4 gi., or 2 gi.
2. Change \^ gal. to lower denominations.
3. Change ^^ of a ton to lower denominations.
4. Change ^ bu. to quarts and pints.
6. Change .6875 of a gallon to integers of lower denomi-
nations.
The explanation is the same as for common fractions.
.6875 gal. See above.
4
ont'c.r. ^ 6. Change .85 lb. to integers of lower denomi-
2 nations.
j-gQ^ 7. Change .325 T. to integers of lower de-
4 nominations.
2^gi. ®' Change .0135 cd. to cubic feet.
9. Change .08J yd. to feet, etc.
1 9 • ^^* Express .09375 A. in square rods.
2 qt. 1 pt. 2 gi. ^^ Change .015625 bu. to pecks, etc.
12. Change 2 pk. 6 qt. to the decimal of a bushel.
8 )6 qt. Since 8 qt. make a peck, 6 qt. are equal to .76 pk., which,
4)2.75 pk. united with the 2 pk., make 2.76 pk. Since there are 4 pk.
— fiQ7K~u — ^ * bushel, 2.76 pk. are equal to .6875 bu.
13. Change 2 pk. 6 qt. to the fraction of a bushel.
6 qt. -!- 8 = I pk. = I pk.
23 ^u ji ^\. The explanation is the same as for chan-
* ^ * ~ -.^ ^ ' , ging to a decimal.
^ pk. -f. 4 = \^ bu. ^ ^
14. Change 4 yd. 2 ft 5.25. in. to the fraction of a rod.
15. Add J of a gallon and .375 gal.
16. Add 103.75 ft. and .845 mUes.
84 MISCELLANEOUS REVIEW.
1. Multiply 45 bu. 3 pk. 6 qt. 1 pt. by 16.
2. Divide 212 m. 26 rd. IJ yd. by 7.
3. Add 43 A. 32 sq. rd. 127 sq. ft. ; 240 A. 20 sq. rd. 200
sq. ft. ; 96 A. 26 sq. rd. 75 sq. ft. ; 12 A. 100 sq. ft. ; 137 sq. rd.
30 sq. ft
4. How many acres in a farm 225 rd. long and 175 rd.
wide?
6. How many cubic feet»in a room 18 ft. long, 17 ft. wide,
and 15 ft. high ?
6. In 3,538,944 cu. in. how many cubic yards ?
7. Reduce 4 mi. 213 rd. 16 ft to inches.
8. How much rice at 8/ a pound will pay for 5 bu. 3 pk.
of cherries at 9/ a quart ?
9. Find the least common multiple of 13, 39, 56, 63.
10. Find the greatest common divisor of 315, 945, and 63.
11. K § of an acre of land cost f66f, how much will 8f acres
cost ?
12. What is the value of a pile of wood 48 ft. long, 8| ft
high, and 4 ft. wide, at $6.50 a cord?
13. Multiply 27 millionths by 12 hundredths, and divide the
product by 324 thousandths.
14. A farmer had \ of his sheep in one pasture, J in another,
I in another, and the remainder, 26, in a fourth. How many
sheep had he ?
15. Take four hundred and twenty-five ten-thousandths from
ten thousand, and multiply the remainder by ten hundredths.
16. The sum of two numbers is 56. The larger is seven
times the smaller. Find the numbers.
17. A boy has a certain number of pears and four times as
many peaches. His pears and peaches together number 25.
How many of each has he ?
18. The difference of two numbers is 24. The larger is five
times the smaller. Find the numbers.
ORAL. 86
1. Find the cost of 1,000 cords of wood at $7.37i a cord.
2. A man earned $100 a month. If he spent $4 out of
every $10, how many dollars did he save every month?
3. How much did a grocer pay for a quart of milk, if he
sold a gallon for $.32, at a gain of 33 J%?
4. A rectangular field contains 1 acre. If it is 80 rd. long,
how wide is it ? What is its perimeter ?
5. I sold a dozen oranges costing 24 cents for 18 cents.
What per cent did I lose ?
6. I sold a dozen oranges costing 18 cents for 24 cents.
What per cent did I gain ?
7. What will 27 lb. of coffee cost at 33 J/ a pound?
8. What will 40 pt. of milk cost at 2^/ a pint?
9. What is the interest of $2,468 for 30 days at 6% ?
10. If 2 J qt. of beans cost 26 cents, what will 10 qt. cost?
11. Add all the prime numbers between 1 and 10.
12. 5 qt. is what decimal of a peck? What per cent of it?
13. A can do a piece of work in 4 days, B can do it in 6 days.
In what time can A and B do it working together?
14. I bought a table for $4. At what price must I sell it to
gainl2i%?
15. How many 2-in. cubes in a 10-in. cube?
16. If a dealer sells goods for double what they cost him,
what per cent does he make ?
17. I sold my watch for $18, which was 12i% more than it
cost. Find the cost.
18. What part of a dollar did I pay for 6j lb. of candy at
12/ a pound ?
10. How many hours is it from 2.45 p.m. to 5.16 P.M.?
20. A man bought a gallon of milk for 28 cents, and sold it
at 4/ a pint. How much, did he gain on a quart ?
21. How many cows at $25 a head can be bought for $260 ?
22. What part of $100 is 2 times $16§?
86 REVIEW OF INTEREST.
(For general sunmary, see pages 203, 204.)
What is the interest of :
1. $845 for 6 mo. 24 da. at 4% ?
2. $46.60 for 123 days at 6% ?
3. $74.60 for 1 yr. 5 mo. at 6% ? *
4. $2,463.76 for 11 mo. 23 da. at 4J% ?
5. $6,900 for 3 yr. 6 mo. 17 da. at 7% ?
6. $400.60 for 2 yr. 11 mo. 3 da. at 4% ?.
7. $10,000 for 63 days at 6% ?
8. $640.80 for 4 yr. 7 mo. 11 da. at 7% ?
9. $16,420 for 9 mo. 24 da. at 6i % ?
10. $734.75 for 3 yr. 9 mo. at 4% ?
11. $459.28 from Dec. 14, 1901, to May 5, 1902, at 4^% ?
12. $658.48 from Aug. 17, 1900, to Apr. 4, 1902, at 8% ?
13. $2,184 from Jan. 24, 1901, to Mar. 30, 1904, at 6% ?
14. $609.50 from Mar. 5, 1899, to Sept 14, 1901, at 4i% ?
15. $489.25 from May 5, 1901, to Aug. 11. 1902, at 5% ?
16. $625.57 from Aug. 15, 1900, to Dec. 29, 1905, at 3^% ?
17. $647.48 from Sept. 30, 1900, to May 5, 1902, at 7^% ?
18. $492 from Aug. 31, 1901, to Dec. 30, 1902, at 3j%?
19. $1,827 from Jan. 16, 1901, to Oct. 11, 1902, at 4% ?
20. $945.96 from June 4, 1901, to Sept. 10, 1903, at 4^% ?
21. $2,846 for 8 yr. 4 mo. 12 da. at 6j% ?
22. $862 for 4 yr. 7 mo. 22 da. at 8% ?
23. $8,624 for 1 yr. 2 mo. 17 da. at 5% ?
24. $946.25 for 89 days at 4j% ?
25. $3,010 for 2 yr. 7 mo. 7 da. at 8% ?
26. $480 for 3 yr. 1 mo. 24 da. at 7% ?
27. $847.25 for 1 yr. 8 mo. 7 da. at 4% ?
28. $756.75 for 2 yr. 2 mo. 5 da. at 7^ % ?
29. $1,050 for 3 yr. 5 mo. 3 da. at 6% ?
30. $2,500 for 4 yr. 11 mo. 9 da. at 6% ?
31. $800 for 115 days at 3j% ?
PROBLEMS IN INTEREST. 87
Note. — This page may be omitted without loss.
1. What principal at 6% will yield $225 int in 2 yr. 6 mo. ?
What is the interest of $1
$.15 = int. of «1 for 2 yr. 6 mo. ^^^ ^ yr. 6 mo V What is
^ooK « 1K «^ <ifci ^^^ mterest of U as stated
^ZZa . ^.10 = ^x : ;tt>l. jj^ j.jjg example? What is
.15a: = $225. the ratio of $225 to $.16?
X = $1500. What is the ratio of $a; to
$1 ? Since these two ratios
are equal, write them as a proportion. Find the value of x.
2. In what time will $940 at 6% gain $432.40?
$56.40 = int. of $940 for 1 yr. at 6%. j, JestVlSo
$432.40 : $56.40 = x yr. : 1 y r. . for i yr. at 6% ?
56.40a; = 432.40. What is the in-
X = 7^ yr. terest stated in
x=7 yr. 8 mo. ^^® ^^^P^® ^^^
•^ X years at 6% ?
Form the time and interest into a proportion, and find the value of x.
3. At what rate will $900 gain $231 in 3 yr. 8 mo.?
_^^^ . ^/N^^ r « o ^^ What is the inter-
$198 = mt. on $900 for 3 yr. 8 mo. at 6%. est on $900 at 1%
$33 = int. on $900 for 3 yr. 8 mo. at 1% for 3 yr. 8 mo. ?
$231 : $33 = aj% : 1%. What is the interest
33a; = 231. x = 7. ^^ ^^^ *^ *% *»
stated in the ex-
ample ? How does the interest given at x% compare with the interest you get
at 1% ? Form the two ratios into a proportion, and find the value of x,
4. What principal on interest at 6% will gain $15 in 2 yr. ?
5. At 5% what principal will gain $20 in 4 yr.?
6. What principal at 7 % will yield $360 in 1 yr. ?
7. In what time will $142,64 gain $13,105 at 4% ?
8. In what time will $500 gain $5.00 at 4% ?
9. In what time will $900 gain $13.50 at 6% ?
10. At what rate will $426 gain $63.90 in 2 yr. 6 mo. ?
11. At what rate will $62.75 gain $8,785 in 2 yr. 4 mo. ?
88 PROFIT AND LOSS.
1. Profit is the excess of the selling-price over the cost.
2. Loss is the excess of the cost over the selling-price.
3. Selling-price is always the cost plus the profit, or cost less
the loss.
4. The gain or loss is always reckoned as a percentage of
the cost.
6. I bought f 640 worth of goods, and sold them at a gain
of 12%. Find the selling-price.
6. An agent buys §660 worth of goods at 40% off, and sells
at a gain of 25% on the cost. For what does he sell?
7. George bought a building-lot for $850, and sold it to
Henry at a gain of 25%. Henry sold it at a loss of 20% ;
what did Henry receive for it ?*
8. A grocer lost 15% by selling eggs at 17 cents a dozen.
Find the cost.
9. A dealer sold a horse at 12J% loss, and lost $25. Find
the cost.
10. If j of an article is sold for J of its cost, what per cent
is lost ?
11. A dealer lost 16% by selling goods for $4200. Find the
cost.
12. If I sell goods that cost me $.84 a yard for $.63, what is
my loss per cent?
13. If I sell a horse for $175, and gain 5%, what per cent
should I have gained if I had sold him for $200 ?
14. Sold some goods for $200, and thereby gained 25%.
What per cent should I have gained had I sold them for $220?
15. A lady spent $64.60 for jewelry and dress goods, paying
15% more for dress goods than jewelry. How much did she
pay for each ?
Note. — 100% = jewelry. 116% = dress goods. 216% = both.
16. A and B together have $1,680, and A has 25% less
money than B. How much has each ?
PROFIT AND LOSS. 89
1. I sold a bicycle for $45 at a loss of 10% ; for what should
it be sold to gain 10% ?
_ ,^ ^ , ^ , ^ The first three statements need no
1. S. P. at 1st loss, = $45. explanation.
2. S. % at 1st loss, = 90. what is the ratio or comparison of
3. S. % at a; gain, = 110. 110% (which we wish to find) to 90%
110 • 90 = ' ^ ® = ^J^ (which we know) ? It is as 110 : 90 or
ji 'i %AK 2^55 W or i^L. If it is ^^ of the second
V- 01 ^*o ~ ^oo. statement, it must be ^ of the first
statement, because the first and second are equal, both representing the Selling.
Price at first loss.
2. If by selling a farm for $1,425 a fanner gains 14%, what
per cent would he have gained by selling it for $1,600 ?
1. S. P. at Ist gain, 11425. 1600 : 1425 = HS» = *f
2. S. % at Ist gain, 114. ft of 114% = 128%.
3. S. P. at X gain, 1600. 128% - 100% = 28%.
(See explanation above.)
3. Sold a farm for 18,128, and made 27 per cent on the
cost. What per cent should I have gained had I sold it for
19,600?
4. A pair of horses were sold for $297, at a gain of 35%.
What would have been the gain per cent if sold for $253 ?
5. A horse was sold for $144, at a gain of 20%. At what
price would it have been sold if there had been a loss of 20% ?
6. A piano was sold for $252, which was at a gain of 12%.
At what price would it have been sold to gain 25% ?
7. A boy sold a pair of skates for 92 cents, and gained
15%. At what should he have sold them to gain 18|% ?
8. Sold a lot of hay for $644, at a gain of 15%. At what
price should it 'have been sold to gain 20% ?
9. A merchant's income is $5760. This is a gain of 18f %
on the capital invested. His income last year was 25% of the
capital. Find his income of last year.
10. I sold a farm for $5000, and made 25%. What per cent
should I have gained or lost if I had sold it for $3,500?
90 ORAL,
1. What is the loss per cent on goods costing 15/ a yard,
if sold at 12/?
2. What per cent of a score is a dozen ?
• 8. What per cent of an acre is a lot that is 5 rd. by 16 rd.?
4. If 3.5 acres of land cost $35, what will a farm of 40
acres cost?
6. A farmer sold a calf for $4, and lost 30%. What did
it cost ?
6. A boy earned $20, which was 20 % of what he had before.
How much has he now ?
7. A man sold a wagon for $16 less than it cost him, and
lost 20%. What did the wagon cost?
8. An article that cost $4.50 was sold for $6. What was
the gain per cent ?
9. An article sold for $32 at a loss of 20% . What was the
first cost ?
10. A man gained 12 J % by selling a wagon for $8 more than
it cost him. What did it cost ?
11. Find the cost of a hat that was sold for $1.60 at a loss
of 20%.
12. By selling a watch for $60 I gained 20% . Find its cost.
18. Flour sold at $6 a barrel yields a profit of 20%. Find
the cost of 10 barrels.
14. Sold a cow for $5 less than cost, and lost 12j%. Find
the cost and selling-price.
16. I sold a harness for $48, which was 20% below cost.
What did it cost?
16. An article that cost $40 was sold for $60. What was
the gain per cent?
17. If a merchant buys shoes at $5 a pair, at what price must
he sell them to gain 25% ?
18. How much will a 40 qt. can of milk cost at 24/ a gallon?
19. If I yd. of cloth cost 24/, what will IJ yd. cost?
STATEMENTS. 91
1. My furniture, worth $1,800, is insured for f of its value
ati%.
2. A grocer bought flour at $4.35 a barrel, paying for it
$1,148.40, and sold it at $5.15 a barrel.
3. In a school 174 pupils are present, and 6 are absent.
4. Bought 175 bu. of wheat for $315, and sold it for $2 a
bushel.
5. On a commission of 2^% I sold 400 bales of cotton, each
weighing 480 pounds, at 32^ a pound.
6. Selling-price, $125.46; profit, 8i%.
7. Cost $88.60, loss 7^%.
8. A house which cost $4,800 rents for $24 a month. The
expenses on it are $48 annually.
9. Land which cost $5,600 was sold at a profit of 25%.
10. A line of wire 1,560 ft. long is supported by 13 posts,
placed the same distance apart
11. 17 gal. 3 qt. 1 pt. 2 gi. were sold from a cask containing
25 gal. 2 qt. 1 gi.
12. I lost 12% by selling some goods for $215.60.
13. I bought 320 lb. of sugar at 5j^ a pound. I lost ^%
by drying, and sold the rest at 6^ a pound.
14. A man lost J of his money, and then J of what was left
He then had $12.60.
15. In a school the girls are 52% of the whole number, and
there are 240 boys.
16. James spent \ of his money, then f of what remained,
then f of what still remained. He then ha-d $15.
17. A lawyer collected a note of $2375. His commission
was 5%.
18. Twelve pairs of shoes were bought at $3.25 a pair, with
a discount of 6%.
19. 22^ yd. of ribbon, costing $6.75, were sold at a gain of
20%.
92 MISCELLANEOUS BEVIEW,
1. Divide $336 between two persons so that one may have
J as much as the other.
2. A tank 18 ft. long and 15 ft. wide requires 96 sq. yd.
of lead to line its sides and bottom. How deep is it?
3. A room 20 ft. long, 17 ft. 6 in. wide, will require how
many yards of carpet 2 ft. 6 in. wide to covqr it, if no allowance
be made for waste ?
4. Find the area of a gravel walk 6 ft. wide just inside a
fence surrounding a lot 320 ft. long^ 210 ft. wide.
Note. — Find area of one rectangle 320 by 210 ft., and of another 308 by
198 ft. Find tiieir difference.
6. What is the difference between 66 divided by .65, and
.65 divided by 65 ?
6. By selling a carriage for $117 a dealer lost 10%. For
how much should he have sold it to gain 10% ?
7. A man gave § of his money to his wife, § of the re-
mainder to his oldest daughter, and the remainder, $5,000, he
divided equally between his two younger daughters. How
much was the man worth ?
8. Add seven hundredths, thirty-six ten-thousandths, and
seventy-two millionths.
9. An agent charged $25.50 as commission at 2J% for sell-
ing 200 bbl. of flour. At what price a barrel was the flour
sold?
10. I paid $25 for an insurance policy on my house. If the
rate is i%, for how much is my house insured?
11. I collected 80% of a debt of $5,600, and charged 4J%
commission. How much ought I to return to my employer ?
12. A man spent /y of his money, and invested y\ in busi-
ness. He had the rest, $1,850, in a bank. How much was he
worth at first?
13. Find the interest on $769.74 for 3 yr. 9 mo. 17 da. at 4%.
14. $2,300 is 15% more than what?
^ INSURANCE. 93
1. Insurance is security against financial loss on account of
the destruction of property, or by the injury or death of a person.
2. The three common forms of insurance are Fire, Accident,
and Life.
3. Premium is the sum paid for insurance. In Fire insur-
ance it is estimated at a certain per cent of the amount in-
sured. In life insurance it is estimated at a certain amount a
year for each thousand dollars of insurance, and varies with the
age of th6 person insured.
4. A man insures his life for $2,500 at the rate of $22.50
for every $1,000. What is his annual premium ?
5. A man insures his life for $3,000, paying $14.24 semi-
annually for every $1,000. If he dies in twelve years, how
much more than his premiums will his heirs receive ?
6. A mill was insured for | of its value. The premium
paid was $350. The rate of insurance 5%. Find the value of
the mill.
7. A factory valued at $50,000 is insured for J of its value.
The premium is $500. What is the rate of the insurance ?
8. How large an insurance can I plac6 on my house by pay-
ing a premium of $122.50, if the rate of insurance is 1|% ?
9. A ship worth $52,000 was insured for f of its value at
2^ %. The cargo worth $8,640, was insured for f of its value
at 8%. Find the premium.
10. A house worth $12,000 is insured for f of its value.
Find the pi-emium at f %.
11. I insure my house, worth $6,000, at i% on f of its value,
and $1,800 worth of funiiture on J of its value at f %. How
much premium must I pay ?
12. Mr. J. paid $250 for insurance on his stock of goods.
The face of the policy is $10,000. Find the rate.
13. Find the amount insured, when $68.24 is the premium
at J%.
94 COMMISSION,
1. Commis9ion is compensation paid by one person to an-
other for transacting some business.
2. These business transactions are usually buying or selling
property or collecting bills.
3. The person doing business for another is called Agent,
Factor, Broker, Commission Merchant.
4. The person for whom the business is done is called Prin-
cipal, Employer.
5. The services performed by agents are of two kinds:
a. Where they receive money for selling or collecting, to be
remitted to their principal ; 6, Where money is sent them to
be expended for their principal.
6. The agent's commission is usually some per cent of the
amount collected or expended.
7. The Net Proceeds is the sum of money due the Princi-
pal after the commission and other charges have been deducted.
8. The Entire Cost of a purchase is the price paid, plus
the commission and all other expenses.
9. My agent bought 70 bbl. of flour at 15.25 a barrel. His
commission was 3%. Find the agent's commission.
10. A commission merchant sold a lot of goods for f 1,480,
charging 2J% commission. What should he send his employer?
11. A broker bought for me some goods for which he paid
12,146. His commission for buying was i%. What did the
goods cost nie ?
12. My collector charges me 3% for collecting bills. In one
month I paid him $345. How much did he collect?
13. My agent sold 23 wagons ad $85 each. What were the
net proceeds if his commission was 6% ?
14. I collected 95% of a debt of $2,148. My commission
was 3^%. What sum must I send my employer?
15. The commission for selling some property was $985. If
the rate was 5%, find the value of the property sold.
OBAL PERCENTAGE. 95
1. What will it cost to get a house insured for $4,000 for
10 years, at J% a year?
2. A house valued at $3,500 was insured at 1%. What
was the premium ?
3. If a book-agent received $50 for selling $150 worth of
books, what was his' rate of commission ?
4. An auctioneer sold $500 worth of goods at a commission
of 4%. What was his commission?
5. Can you lose more than 100^ in selling an article ?
6. Give a case where you lose 100%.
7. Can a gain be greater than 100% ? If so, make an
example to illustrate.
8. If not greater than 100%, tell why not. Make an ex-
ample where the gain is just 100%.
9. A man sold a horse for $90, and gained 20%. Find the
cost of the horse.
10. Read the 9th example, using a common fraction instead
of per cent.
11. When cloth that cost $6 a yard is sold for $4 a yard,
what is the loss per cent ?
12. If you buy an article for $80, and sell it at 25% profit,
what will be your selling-price ?
13. A man paid $160 for ahorse, which was 20% less than
the cost of a carriage. Find the cost of both.
14. What per cent of a number is f of it?
15. What per cent of a number is S of it?
16. What per cent of a number is § of it?
17. A horse that cost $84 was sold at a gain of 25%. Find
the selling-price.
18. If 25% of the cost of a bicycle is $9, what is the cost of
the bicycle?
19. Cloth which cost $.75 a yard was sold at a loss of 33^%.
Find the selling-price.
96 ALGEBRAIC PROBLEMS.
1. Charles and Henry together had 72 cents. If Charles
had 3 times as many as Henry, how many did each have ?
2. Three men, A, B, and C, formed a company with a capi-
tal of $8,000. B put in 4 times as much as A, and C three
times as much as A. How many dollars did each put in ?
8. The sum of two numbera is 99, and the greater is twice
the less. What are the numbera ?
4. Three times a certain number added to two times the
same number gives 75. Find the number?
6. The sum of two numbers Ls 366, and the greater is 5
times the less. What are the two numbers ?
6. A man bequeathed $48,000 to his wife, son, and daugh-
ter. The will provided that the son should receive twice as
much as the daughter, and the wife 3 times as much as the
daughter. What was the share of each?
7. A man divided 80 cents between two children. If he
gave the second three times as much as the first, what did he
give to each?
8. Divide the number 88 into three parts, so that the second
part shall be three times the first, and the third four times the
first.
9. Divide the number 120 into three parts so that the sec-
ond part shall be two times the first, and the third part as much
as the sum of the first and second.
10. Four men have together $480. B has 4 times as much
as A, C has 6 times as much as B, and D has J as much as C.
How many dollars has each?
11. A boy, being asked how many marbles he had, said that
if he had 5 times as many more he should have 240. How
many had he?
12. In a school there are 495 pupils, and twice as many boys
as girls. How many boys are there ?
13. 4a; — a; =7 -f 5. Find a;.
' MEASUREMENTS. 97
Make accurate drawings to represent the examples in this lesson. Find the
answers by measuring.
1. Two men, Brooks and Scott, start from A. Brooks walks
directly north 36 miles, and Scott directly ea^t 48 miles ; how
far apart are they in the shortest line ? Scale, ^ in. to a mile.
2. A house is 18 ft. wide. From the attic-floor to the ridge-
pole it is 12 ft. How long rafters must be used to project 6 in.
over the wall of the house ? Scale, i in. to 3 ft.
8. If the length of this roof is 32 ft., how many boards 16 ft.
long and 6 in. wide will it take to cover it ?
4. At a point, A, draw a line east 6 in. to B ; south 2 in.
to C ; west 2 in. to D ; south 1 in. to E ; west 1 in. to H ;
south 4 in. to I ; west to a point directly south of A to J. Con-
nect A and J. How far is it round this field ? Scale, i in. to
a rod.
6. From B to H is what part of the whole distance ?
6. 4 of the distance round this figure is how many yards?
7. A house is 40 ft. long, 26 ft. wide, and has 18 ft posts.
The gable is 13 ft. high, and there is no allowance for doors and
windows. What will it cost at 22/? a square yard to paint the
outside ?
8. Three boys were standing by a tree. William remained
there ; George and Henry walked east 24 ft. to another tree.
Geoi^e stopped there, and Henry walked north 32 ft. to another
tree and stopped. How far is Henry from William? Scale,
1 in. to 8 ft.
9. Beginning at the north-east corner, the boundary-line of
my lot runs as follows : west 40 rd. to a tree, called B ; south
16 rd. to a point, C; east 24 rd. to another tree, called D; then
north-east to the point of beginning. How many square rods
in the lot? How many rods round the field? Scale, 1 in. to
8rd.
10. Find the area of a quadrilateral, 55 yd. by 16 ft.
98 MISCELLANEOUS REVIEW,
1. Sold 96 yd. of carpeting at $1.87^ a yard, and thereby
gained $38.40. How much did it cost me a yard ?
2. What would you gain by selling at 6i^ a pound, 1,522
lb. of rice costing |4.25 a hundred, and 636 lb. of barley cost-
ing $6.60 a hundred?
3. Multiply seventy-eight ten-thousandths by five hun-
dredths ; divide the product by thirteen thousandths, and re-
duce the quotient to a common fraction.
4. A commission merchant sold 600 pieces of cloth for $130
a piece, and paid his employer $65,260. What was the rate of
his commission ?
5. What will be the net proceeds of a sale of 626 bbl. of
beef (a) $18.26, allowing 3% commission, and 6/ a barrel for
storing ?
6. How much is 5^ tons of coal worth if 17f tons are worth
$100?
7. If the numerator of a common fraction is divided by 3,
what is the effect upon the value of the fraction ? Illustrate it.
8. Two men start from two towns 105 miles apart, and walk
toward each other. They meet at the end of 15 hours. If the
first traveled 3 miles an hour, how many miles did the second
travel an hour?
9. What is the area in acres of a triangle whose base is 56
rd. and altitude 63 rd. ?
10. How many farms containing 90 acres each can be formed
in the town of Gi-anby, if it is in the form of a rectangle 6 miles
long and 4 J miles wide ?
11. What is a fraction? Explain addition of fractions, and
give the reason for every step.
12. What is interest? Explain your method of finding the
interest on any sum for any time at any rate.
13. From four billion take two billion, one hundred five
million, two hundred fifty thousand, forty-seven.
FRACTIONS. 99
1. If % yd. of cloth cost I ft^, what will A of a yard cost?
2. Make out the following bill, supplying names and dates :
32 lb. soap, @ lb\ff \ 35 lb. starch, @ 5|^ ; 85 lb. sugar, @
8J/ ; 62^ gal. of vinegar, @ 25>f ; 28 lb. coffee, @ 23^; 112 lb.
butter, @ 33]^ ;f. Discounts, 10% and 5%.
8. A merchant bought 76.75 yd. of cloth for fll5|, and
sold J of it at an advance of f i a yard. How much did he
receive for the part sold ?
4. What will 76 lemons cost if three dozen cost $li ?
5. What will 8 lb. 12 oz. of butter cost, if IJ lb. cost
30 cents?
6. When 2 J tons of hay cost |33, what will 18? tons cost?
7. If 3i yd. of -cloth cost *10, what will 2§ yd, cost?
8. A man sold a horse for $125i and gained $26 i. How
much did the horse cost ?
9. A man divided 6.3 bu. of potatoes among his workmen,
giving each j^ of a bushel.
10. What is a complex fraction? Write one. Change it to
a simple fi-action.
11. What is a mixed number? Write one. To what other
form can you change it ? Do so. Have you changed its value?
13. What are the terms of a fraction called ? Why are they
so called ?
13. Show what effect it has upon the value of a fraction to
(1) multiply its numerator by 2 ; (2) to divide its denominator
by 2; (3) to multiply both numerator and denominator by 2;
(Ay to multiply its numerator by 2, and divide its denominator
by 2.
14. At 7i)^ a pound, how many pounds of sugar can you buy
for $4.80 ?
15. How many feet in 5 J rd. and 4J yd. ?
16. What fraction of 4i is 6§ ?
17. What part of 5i is 2J ? Of 14§ is 4f .
100 OBAL.
1. K a certain number diminished by J of itself is 15, what
is the number ?
2. Nellie has 18 buttons, and 5 of her buttons equals i of
Mary's buttons. How many buttons has Mary?
8. What is ^ of § ? I of J ? *i of ? ?
4. Having lost } of his money, Harry found J of what he
lost, and then had 70 cents. How much had he at first?
5. If a man can walk 3 J miles in 2 hours, how far can he
walk in 8 hours?
6. How far apart are two places, when 7 times 3f miles is
5i times the distance between them?
7. If cloth costs $5 a yard, how much can you buy for $2i?
8. If your brother can earn Jj6 a week, how long will it
take him to earn $4i ?
9. If 5 pints of milk cost 12 cents, what will 25 pints cost?
10. If 4 men can do a piece of work in 4 J days, how long
will it take 12 men to do the same work?
11. A girl gave 3 J apples to each of her girl friends. If she
gave away 14 apples, to how many friends did she give her
apples ?
12. A watch cost $40, and a chain cost $12. What per cent
of the cost of the watch is the cost of the chain ?
13. A cow cost $24, and | of the cost of the cow is J of the
cost of the horse. Find the cost of the horse.
14. Five-sixths of 72 is how many times J of 16 ?
15. A horse cost $150, and J of this is three times the cost
of a sleigh, and the sleigh cost twice as much as a harness.
Find the cost of the sleigh and the harness.
16. A vessel was sunk in 9 fathoms of water. How many
feet deep was the water? A fathom is 6 ft.
17. The distance round the school-yard is 160 paces. How
many feet is it? How many yards is it?
18. 8 is one factor of 24. What is the other?
COMMERCIAL DISCOUNT. 101
(For summary, see page 291.)
Commercial discount is a reduction from the nominal price of
anything.
1. Find the net amount of a bill of $1,440, with 25%, 10%,
and 5 % off.
^ $1,440 is the gross amount of
25% of 11,440 = $ 360.00 the bill.
$1,440 -$360= 1080.00 a.!^,^'^.'^::^^^.
10% of $1080 = 108.00 Find 10 fo of this amount and de-
$1080 -$108 = 972.00 duct it. The result is $972. Find
c ^ t jbrvTo Ao £ifk 5% of this amount and deduct It.
5% of $972 = 48.60 ^^ ^^^^ j^ ^^23.40. This is the
$972 -$48.60= 923.40. Ans. net amount of the bill.
2. Find the net amount of a bill of $1,920 with 26% and
7}% off.
Find the net amount of a bill of :
- 8. $1,275 with 20% and 15% off.
4. $562 with 35% and 15% off.
5. $1,088 with 50% and 10% off, and an additional 5% off
for cash.
6. 8 doz. bolts, at $3.00 with 40%, 5%, and 25% off.
7. 11 gross screws, at $2.25, with §, and 30% off.
8. 6 doz. handles, at $1.50, with 40%, 5%, 25%, and 17i%
off.
9. 480 lb. tea, at 62i^, with 37i%, and 15% off.
10. 560 articles at 87i/, with 25%, 165%, and 10% otf.
11. 25 lb. crushed sugar, at 10^ ; 40 lb. maple sugar, at 12^ ;
6 lb. cheese, at 13/; 8 lb. butter, at 28,^; 4 lb. raisins, at
13/; 2 lb. cream tartar, at 35/; with 15%, 20%, and 5% off
on the whole.
12. 20 oxen, at $53.50, with 45%, 15%, and 5% off.
18. 15 cows, at $23.25^ with 25%, 15%, and 10% off.
14. 15 reams note paper, at $1.25, and 25 reams letter paper,
at $1.76 with 30%, 22i%, and 12i% off.
102 PERCENTAGE.
1. I bought books listed at $12 a dozen for ^ and 10^ off.
I sold them at 11.00 each. What per cent did I gain ?
3. A farmer sold 1,000 bu. of com at $1 a bushel, and esti-
mated his loss at 5%. What per cent would he have gained
had he sold at $1.20 a bushel?
3. A dealer sold skates at $3.60 a pair, and made 20%.
What per cent would he have made if he had sold them at $4.00
a pair?
4. Find the net gain on two houses sold for $2,100 each, if
on one there is a gain of 16S% and on the other a loss of 12^%.
6. A house was insured for } of its value at |%. The
premium was $13.60. What was the value of the house ?
6. A real-estate dealer sold for me 75 lots of land at $275
a lot. If he charged me 2% commission for selling, and $5 a
lot for recording the deed, how much will I receive for all my
land ?
7. A merchant sold 73,680 ft. of lumber at $20 per M., and
gained $294.72. What was his per cent of gain ?
8. By selling a piece of land for $160, I lost 25%. At
what price should I have sold it to have made 20% ?
9. I bought a house for $2,250, and sold it for $2,700.
What per cent did I gain ?
10. A dealer bought 56 house-lots for $256.00 each, and
sold them at an advance of 9J%. How much did he receive
for all the lots?
11. A merchant paid 18|/ a yard for cloth, and exchanged
12f yd. for 16 doz. eggs at 25^ a dozen. What per cent of
profit did he make ?
12. By selling a piece of land for 16 J % profit I cleared $150.
What did it cost?
18. If I pay $150 for insuring $8,000 worth of goods, what
is the rate ?
14. 32 J is 6J% of what number?
MEASUREMENTS, 103
1. What wUl it cost to dig a cellar 70 ft. long, 35 ft. wide,
5 ft. deep, at 62^^/ a cubic yard ?
2. At $4.25 a cord, what is the value of a pile of wood
72 ft. long, 4 ft. wide, and 12 ft. high?
8. At $18 per M. find the cost of planks for flooring a barn
40 ft. by 32 ft., if each plank is 16 ft. long, 15 in. wide, 2 in.
thick?
4. The width of a gable is 30 ft. and its perpendicular
height 20 ft. What will be the cost of boarding two gables at
$16 per M. ?
5. A bin is 8.5 ft. long, 4.25 ft. wide, and 3.75 ft. deep.
How many bushels of oats will it hold ? li cu. ft. in 1 bu.
6. How many gallons of water will a tank hold 4 ft. by 3i
ft. by 2 ft. 4 in. ? 7i gal. in 1 cu. ft.
7. A rectangular garden is 200 ft. long and 150 ft. wide.
There is a walk 4 ft. wide running all round it, and also through
the center in both directions. What part of the whole area of
the garden is taken up with the walk ?
8. A section of land is one mile square. At 60^ a rod for
fencing, and $1| an acre for plowing, find the sum paid out for
a quarter-section of land.
9. A room is 18 ft. by 24 ft. A carpet is } of a yard wide,
and runs lengthwise. There is a waste of 9 in. on each breadth
for matching. At $1.25 a yard find the cost of the carpet.
10. What will it cost to build a fence round a square 660 ft.
on a side, if the posts are placed 6 ft. apart, and cost 18/ each?
The pickets are 2 in. wide, and placed 2 in. apart, and cost $3.25
a hundred. The two rails are 4 in. by 4 in., costing $12 per M.
board feet. The labor is $125.
11. At $72 an acre, a farm is worth $12,240. What will
it cost to fence it at $1.25 a rod if it is in the form of a rect-
angle 160 rd. wide?
12. Find area of a rectangle 33 rd.. by 34 rd. 2 yd.
104 MISCELLANEOUS REVIEW,
1. A man paid $87 for insuring his house, worth $7,250.
What was the rate of insurance ?
2. K I pay $27 for insuring property *t i% premium, what
is the value of the property ?
3. Find the net amount of a bill of $1,875, the discounts
being 20% and 5%.
4. A merchant imported 175 chests of tea. Each chest con-
tained 45 lb., valued at 48^ a pound. He sold it at a gain of
25%. Find the selling-price.
5. An elevator in Minneapolis is valued at $24,000, and the
grain in it at $25,000. The elevator is insured for | of its
value at 1%, and the grain is insured for f of its value at ^%.
Find the entire premium.
6. The perimeter of a rectangle is 42 inches. The horizon-
tal sides are twice as long as the vertical sides. How long is
each side ?
7. A earns 18% more in a week than B, and the sum of
their wages is $76.30. How much does each earn?
8. A merchant paid $1.50 for a book, and marked it to be
sold for $2.00. He discounted 12^% from his price. How
much did he gain ?
9. An agent purchased 4^ tons of raw sugar at 3^/ a pound.
What was his commission at 2^% ?
10. The net proceeds of a sale were $1,368. The commission
was $57. What was the rate of commission ?
11. Find the interest of $2,862 for 93 days ^6%.
12. Divide 96,496 by 592 ; 76,368 by 516.
13. Add .96, 7.3004, 8010, .00093, 1.24650.
14. A flag-pole 140 feet high was broken off, so that the
part broken off was 6 times as long as the part left standing.
How long was the part broken off ?
15. The sum of two numbers is 49. The larger is 6 times
the smaller. Find the numbers.
ORAL, 105
1. If 6 bbl. of flour cost $33, what will 11 bbl. cost?
2. If 9 tons of coal cost $54, how many cords of wood at $4
a cord will cost as much as 5 tons of coal ?
3. 27 + 15+18 + 25 + 9 = ?
4. If 4 lb. of cheese cost 36 cents, how much cheese can be
bought for 3 cents ?
6. If 2 bu. of cider apples cost 40 cents, what will 3 pk.
cost ?
6. 25% of $24 is what per cent of $200?
7. f of 35 are g of how many times 12 ?
8. How many square inches on the surface of a 6-in. cube ?
9. At 6% a year what is the interest of $1,000 for 48 days ?
For 33 days?
10. What will 5 pints of molasses cost at $.80 a gallon ?
11. How many 4-in* squares can be cut from a 20-in. square ?
13. If I sold a horse for $105, and gained 16§%, what did
the horse cost?
18. How many times 4 are f of j of 18 ?
14. S of 40 are i of what number ?
15. What per cent is lost on goods sold at | of their cost?
At g of their cost? At J of their cost? At half price ?
16. What per cent of 27 is 20% of 45?
17. What is the ratio of 18 : 9? 35 : 7 ? 16 : 4? 5 : 15 ?
9:12?
18. 48 is the antecedent, and 8 is the consequent ; what is
the ratio ?
19. 12 is the consequent, and 5 is the ratio; what is the
antecedent ?
80. To what sum will $100 amount when on interest at 6%
for 2 yr. 6 mo. ?
21. At 5%, for how much can I insure my store if I pay
a premium of $50 ?
22. How much will 8 tons of coal cost at $5.25 a ton?
106 STUDY OF LINES.
1. Place a cube on your desk. How many dimensions has
it? How many faces has it?
2. These faces are called its surfaces. A sur&ce is a bound-
ary of a solid. Define surface.
8. How many dimensions has each surface ?
4. Surfaces are bounded by edges called lines. A line is
the limit of a surface, or it is the path traced by a point as it
moves from one position to another. To read a line we usually
use two letters, naming the starting-point first.
6. How many dimensions has a line ?
6. How are the lines of the cube limited ?
7. A point is the limit of a line and has no extent, only
position. Define point.
8. In the cube how many faces meet to form a line ?
9. Each face is bounded by how many lines ? If the cube
has six faces, and each face has four lines, how many lines has
the cube ? Why is not the number 24 ?
10. How many lines meet at each point? If the cube has 12
lines, and each line has two points, how many points has the
cube? Why not twenty-four?
11. In a square prism, how many surfaces, lines, and points
are there?
13. By how many lines is each surface bounded ? How many
surfaces meet in each line? How many lines meet at each
point? Are the surfaces the same shape? The same size?
18. Examine in the same way a triangular prism and an hex-
agonal prism.
14. What kind of lines have you found on these solids ?
15. A straight line is a line which has the same direction
throughout its entire length.
16. Define a straight line. Draw one.
17. Look at a cylinder. How many edges or lines has it?
Are these lines straight? What are they?
8TUD7 OF LINES. 107
1. Define a curved line. Write : A curved line is a line
that constantly changes its direction.
2. Fasten a weight to one end of a cord. Hold the other
end at rest in the hand. This is a plumb line, and is said to be
vertical.
3. Define a vertical line. Draw one.
4. A horizontal line is a line which has the direction of any
line in the surface of still water. Practically it is a line that
points towards the horizon.
5. Lines neither vertical nor horizontal are called inclined
lines or oblique lines.
6. How are horizonal lines represented on paper ? Verti-
cal lines ?
7. Hold your ruler vertically, horizontally, inclined. Draw
on paper lines to represent these three positions.
8. Draw a vertical line, and through it two horizontal lines.
9. Draw two lines which have the same direction, that is,
do not meet, however far extended. These lines are said to be
parallel. Define parallel lines.
10. Draw two parallel straight lines. Two parallel curved
lines.
11. Draw two parallel horizontal lines. Two parallel verti-
cal lines.
12. Draw two lines not parallel. Prolong them till they
meet.
13. This point of meeting is called their intersection. De-
fine intersection.
14. Hold two pencils parallel. Hold them so they would
intersect.
16. Hold two rulers parallel ; not parallel.
16. Are two vertical lines always parallel to each other?
17. Can two horizontal lines ever intersect each other?
18. Give examples of vertical and horizontal lines.
108 CONSTRUCTION
1. By means of a triangle and a ruler draw through a point
outside of a given line, a parallel to that line.
2. Draw several parallel lines freehand. Test and correct
them with ruler and triangle.
3. Write : When one line meets another line so as to make
the adjacent angles equal, the lines are said to be perpendicular
to each other.
4. Draw two lines so aa to form equal adjacent angles. De-
fine perpendicular lines.
5. Draw a line perpendicular to a vertical line. Draw one
perpendicular to an inclined line.
6. Draw three lines : a) All parallel ; b) Two parallel, one
perpendicular to them ; c) No two parallel, all intersecting at
one point ; d) No two parallel, and not all meeting in a point.
7. In case d)^ in how many points do the lines intersect ?
8. Draw freehand a horizontal line of any length. Draw a
vertical line of equal length. Test your work.
9. Draw two lines whose ratio shall bel:2; 1:4; 1:6.
10. Draw two lines whose ratio shall be as 2 : 3 ; 3:4; 2:6.
11. What is meant by drawing to a scale?
12. How long would you draw a line to represent 20 in.,
using a scale 1:8?
13. How many centers can a circle have? How many
circles can have the same center?
14. Can a plane surface and a curved surface be parallel?
Draw a straight line and a curved line that shall be parallel.
15. If I use a scale of J in. to a foot, what ratio do I use?
16. What is the standard unit of length in this country?
17. Draw to a convenient scale lines representing 130 ft ;
250 yd.; 75 rd.
18. What scale did you use ?
19. How many lines parallel to a given line ean be drawn
through a point outside of the line ?
MISCELLANEOUS REVIEW. 109
1. A tank can be filled by one pipe in 16 min., and by
another pipe in 30 min. In what time can it be filled by both
together?
2. Suppose water runs in through the first pipe and out
through the other. In what time then will the tank be filled "^
8. A cistern can be filled by one pipe in half an hour, by a
second pipe in 45 min., and by a third pipe in an hour. In what
time will the cistern be filled if all run together?
4. Suppose water runs out through the second pipe and in
through the other two. In what time then will it be filled ?
5. The population of a certain town was 35,416 in 1890.
If it increases 50% in 10 years, what will it be in 1900?
6. Find the gain or loss per cent :
Cost, $20 ; selling-price, $25.
Cost, $2.00; selling-price, $2.12i.
Cost, $12.40 ; selling-price, $10.23.
Cost, $74.00 ; selling-price, $70.30.
7. A house is sold for $400, and 25% is made. How much
profit would be made by selling for $336 ?
8. By selling a house for $7,590 a man gained 10%. What
per cent would he have lost if he had sold it for $6,210 ?
9. A draper bought 960 yd, of silk at $2.00 a yard. He
sold J at a gain of 25%, J at a gain of 20%, and the remainder
at a loss of 15%. For what was it sold?
What is paid for goods marked :
10. $600, with a discount of 33i% ?
11. $1,200, with a discount of 16§%?
12. $1,000, with a discount of 27% and 10%?
13. $600, with discounts of 20%, 10%, and 1% ?
14. $2,500, with discounts of 20%, 5%, and li%?
15. Which is cheaper, to buy goods at a discount of 80%
and 5%, or with S3|% off? How much cheaper on a bill of
$600?
110 ORAL.
1. Buy oranges at 15/ a dozen, and sell them at 2 for 5
cents, and tell me your gain per cent.
2. Buy a horse for $300, and sell it for $360, and tell me
your gain per cent.
3. A boy bought some marbles for 12 cents, and sold them
for 15 cents. Find the gain per cent.
4. A merchant bought muslin at 10/ a yard, and sold it at
a gain of 20^. Find his gain on 25 yards.
6. Eggs cost me 20 cents a dozen, but in selling them I
gained 25%. How many dozen eggs must I sell in order to
receive $1.00 ?
6. $7 is 16§% more than what?
7. A farmer had 60 sheep, and bought 20% more. ' How
many had he then ?
8. If ^ of an acre of land is worth $60, how much is | of
an acre worth ?
9. If a boy lost | of his money, and had 8 cents left, how
much had he at first ?
10. A boy lost J of his marbles and sold i of them, and then
had 21 left. How many had he at first ?
11. If f of a yard of cloth cost 30 cents, how much will J of
a yard cost ?
13. If 5 yd. of cloth are worth 80 cents, how much is f of a
yard worth?
18. $30 is f of the cost of a cow. Find the cost of two cows
at the same rate.
14. If 3 bu. of com cost $3}, how many bushels can I buy
for $2.50 ?
15. If a yard of cloth cost $|, how many yards can be bought
for $10?
16. A man exchanged 5 sheep at $9 each, and 2 cows at $30
each, for pigs at $5 each. How many pigs did he get?
17. If A of A's money is $50, what is his money?
MEASUREMENTS, HI
1. What is the circumference of the largest circle that can
be drawn on a 9 ft. square ?
a. If it requires 440 ft. of lumber to boaid up the gable ends
of a barn 40 ft. wide, how high is the ridge above the eaves ?
3. The area of a triangle is 36 sq, yd., and the base is 36 ft.
What is the altitude ?
4. Find the convex surface of an equilateral triangular pyr-
amid, the sides of whose base are each 8 ft., and whose slant
lieight is 24 ft.
5. How long a band of iron wiU it take to surround a cylin-
drical tank 15 ft. 8 in. in diameter?
6. What is the area of a semicircle whose radius is 24
inches ?
7. At il.25 a square yard, how much will it cost to pave a
triangular space, one of whose sides is 80 yd., and the perpen-
dicular distance from the opposite vertex to that side 180 feet?
8. The diagonal of a trapezium is 18 ft., and the perpendicu-
lars from the opposite vertices are 9 ft. and 8 ft. respectively.
What is the area?
9. Find the convex surface of a cone, the radius of whose
base is 16 in., and whose slant height is 8 ft.
10. At 25^ a square yard, it costs $18.75 to paint a triangu-
lar surface. If the base is 60 ft., what is the altitude ?
11. How many acres are there in a field in the form of a
trapezoid, if the parallel sides are 24 rd. and 16 rd., and the dis-
tance between them 18 rd.?
12. What is the surface of a sphere whose circumference is
24 ft. ?
13. If a bin is 8 ft. square, how deep must it be to hold 256
bu. of apples ?
14. A field in the form of a trapezoid contains 11 J acres. • If
the parallel sides are 60 and 40 rd., how far apart are they?.
15. How many feet in a board 15 ft. long by 16 in. wide?
112 MISCELLANEOUS REVIEW.
1. Make out a correct bill, supplying dates and items.
2. Make out a correct monthly statement.
8. A dealer bought 13 head of young cattle for $325. He
kept them for 4 months at an expense of $2 a head a month,
and then sold them at $32 each. Did he gain or lose? and how
much ?
4. It is 40 rd. round a field. At $22.65 a thousand find
the cost of rails for the fence. Eacli rail is 11 ft. long, and
the fence 6 rails high.
• 5. A farmer raised 8,526 bu. of wheat. He had it ground
into flour. If 1 bu. made 40 lb. of flour, how many barrels did
he receive?
6. A horse and carriage are valued at $420 ; J of the value
of the horse is equal to \ of the value of the carriage. Find the
value of each.
7. A bought pears at the rate of 6 for 5 cents, and B bought
peaches at the rate of 3 for 4 cents. How many peaches should
B give to A for 120 pears?
8. If telegraph poles cost 25/ each, and wire % of a cent a
yard, how much will the material cost for 2 miles of telegraph
line consisting of 6 wires, if the poles are 80 ft. apart?
9. A rectangular field containing 27 A. is 40 rd. wide.
What will it cost to fence it at 35/ a yard ?
10. A wagon upon which 4-ft. wood was piled was 12 ft.
long. How high was the wood, if there were 2 J cd. ?
11. Find the number of board feet in 14 planks, 8^ ft. long,
16 in. wide, and 3i in. thick.
12. What will it cost to plaster a room 32 ft. long, 18 ft.
wide, and 13 ft. high, at 15/ a square yard, allowing 148 sq. ft.
for openings.
13. At what rate will $652 gain $440.10 in 15 years?
14. What sum of money on interest at 4J% will jdeld. an
annual interest of $1,200? *
MISCELLANEOUS REVIEW. 113
1. A company charges f 30.37 i for insuring $1,850 worth
of property. What was the rate of insurance ?
2. A mill was insured for $5,000 in one company at 1\%
and for $6,000 in another company at lj%. What was the
total premium paid?
8. A bankrupt pays 42 i/ on a dollar. How much will a
creditor lose whose bill is $1,460 ?
4. A bankrupt's liabilities are $30,000 and his assets $8,000.
How much can he pay on a dollar?
5. A house is insured for f of its value at li%. The
premium is $81.00. What is the value of the house?
6. A merchant fails in business, owing $7,200. His assets
are $3,000. How much will a man receive who is creditor to
the amount of $600 ?
7. My horse and buggy together are worth $300, and the
horse is worth 4 times as much as the buggy. What is each
worth ?
8. If the divisor were one-third what it is, the quotient
would be 948. What is the quotient?
9. If 63 be added to a certain number it will contain forty-
two 246 times. What is the number ?
10. How many times must 720 be added to 522 to make
987,642 ?
11. A man bought an equal number of lemons and oranges
for $6.25. For the lemons he paid 2^ each, and for the oranges
3/ each. How many of each did he buy ?
12. One train left Boston at 1 p.m. on the B. & A. R. R.
A second train left at 3 p.m. The first goes 30 miles an hour,
and the second 40 miles an hour. When will the second over-
take the first ? and how many miles from Boston ?
18. What fraction divided by § of 12 will give t for a quotient?
14. If .3 of a farm is worth $963, what is the value ol § ol
the farm?
114 STATEMENTS,
1. A house worth $8,600 is insured for f of its value at |^.
2. A man owns f of a store, and sells f of his share for $492.
3. I insured my barn for | of its value at lj%, and paid a
premium of $15.
4. I lost $1,280 on 160 acres of land, which I sold for $34
an acre.
6. Goods marked $64 were sold at 6J% discount and b%
for cash.
6. A tree 45 ft. high was broken at such a point that the
part broken off was 4 times the length of the part left standing.
7. After deducting his commission of 5%, an agent returned
to his employer $1,436.40.
8. A commission merchant sells goods for $5,728, and
sends to his principal $5,649.24.
9. A company insured a mill and its machinery for $117,-
944, the machinery being worth 15% of the value of the mill.
The owner paid 2% on the mill, and lj% on the machinery.
10. Three sevenths of a certain number exceeds \ of the
same number by 25.
11. A certain number multiplied by three thousandths will
give 3645.
12. The premium at 3% is $756.
13. My agent sold for me some property for $1080, and
charged me $81 commission.
14. A man built \} of a wall in 5J days.
15. It cost 55/ a cubic yard to dig a well 6 ft in diameter,
and 30 ft. deep.
16. Forty cents were divided among three boys, A, B, and C,
in such a way that B and C each had twice as much as A.
17. A farmer sold a horse and cow for $240. He sold the
horse for five times as much as the cow.
18. $206.25 was the premium paid for insuring a factory
at 1|%.
OBAL. 115
1. t of 20 are § of what number? | of what? f of what?
3. When oranges cost 25/ a dozen, how many can be
bought for 66§% of $2.25? For 60% of 11.25? For 3 times
16S cents?
3. Charles is 12 years old. § of Charles's age is f of Bes-
sie's. S of Charles's age is § of George's age. How old are
Bessie and George ?
4. What is the cost of 54 yd. of cloth at $0.16§ a yard?
5. What per cent of 100 is 25% of 80? 66§% of 75?
6. What per cent of 80 is 25% of 100 ? 66§% of 75 ?
7. Find the cost of tea a pound when 10% is gained by
selling it at 55/ a pound ?
8. A teacher said, " I should have 50 pupils in my room,
but 5% are absent.'* How do you know she made a mistake?
9. At 6% find the interest for 12 days :
$3000 $840 $30 $ 1500
3500 620 10 2400
1200 150 15 10000
10. A ship worth $60,000 was insured at 4%. What was
the premium ?
11. A bill of goods amounting to $500 I bought at a dis-
count of 10%. What did I pay?
13. If 36 is the antecedent and 3 is the ratio, what is the
consequent ?
13. What is the area of a triangle whose base is 24 ft. and
altitude 8 ft. ?
14. If the area of a triangle is 48 sq. ft. and the altitude is
8 ft., what is the base ?
15. If the area of a rectangular field is 77 sq. rd. and the
base is 11 rods, what is the altitude ?
16. How many board feet in a plank 20 ft. long, 15 in. wide,
and 2 in. thick ?
17. What wiU 8 boxes of oranges cost if f of a box cost $9 ?
116 ALGEBRAIC PROBLEMS,
1. The greater of two numbers is twice the less, and the
sum of the numbers is 135. What are the numbers ?
2. The sum of the ages of a mother and daughter is 36
years, and the age of the mother is eight times that of the
daughter. What is the age of each ?
3. A man traveled 320 miles in three days. If he traveled
three times as far the first day as he did the third, and four
times as far the second day as the third, how far did he go each
day?
4. Divide 36 into three parts so that the first part shall be
three times the second, and the second two times the third.
5. A farmer bought a horse, a cow, and a calf for $104.
The cow cost three times as much as the calf, and the horse
three times as much as the cow. What was the cost of each ?
6. One boy has three times as many marbles as his com-
panion. If both boys have 28 marbles, how many marbles has
each?
7. Divide 126 into two such parts that one part may be
five times as large as the other part.
8. Divide 96 cents between two children so that one child
shall have three times as many cents as the other child.
9. If from 8 times a certain number three times the num-
ber is taken, the remainder will be 45. What is the number ?
10. Five times a certain number and eight times the same
number is 156. What is the number?
11. Three men, A, B, and C, had 270 acres of land. B had
3 times as much as A, and C had five times as much as A.
How many acres had each ?
12. Divide the number 264 into two parts so that one part
will be 5 times the other.
13. Three men. A, B, and C, earned $330. A earned 4
times as much as B, and C as much as both A and B. How
much did each earn ?
ALGEBRA. 117
1. The product of 4 and 8 is written 4x8, but the product
of a and b is not written a x 6, but ab. The product of 5, x
and y is written bxy.
2. Write the product of 8 and c; 4, a, and d\ 7, a;, y,
and z.
3. In the expression 5a;y, 6, x^ and y are the factors of 5a:y.
When one of the factors of an expression is a numerical quan-
tity, it is called the coefficient of the remaining factors.
4. The index has the same meaning as in arithmetic. 3a and
a' are not alike. 8a means a + a + a. c? means a x a x a.
If a = 4, find the value of 8a and of a*. Be careful to dis-
tinguish between coefficient and index.
6. Express in the form of a fraction 15 -?- 5. In the same
way express Sx -i- 2 ; 4:X -i- 5 ; a -^ b ; 2 -^ c.
6. If a = 4, J = 3, (? = 1, 2; = 2, y = 5, 2 = 6, find the
value of :
4a, 8c 4/, a^ 2b\ 5a?
Iz. (?. 3^, ^, 4y\ Ix,
2aby Bbc, 4«r, 5xy, 6y2, 2aV.
7. If a = 4, 6 = 1, <? = 8, a; = 6, y = 7, 2 = 0, find the
value of :
Sx + 5z — lb; 3y — 42 + 7c ; 3a — 5J + c.
5c— 8z + Sa; 4x— 2y — 8b; 2x — 8y + 5a.
^ a?-8a^ + 2<?\ b"" - 2i J^ 3a^; xz - zb - 2cz.
8. Express four increased by five ; a increased by b.
9. Express 9 diminished by 6 ; a diminished by b.
10. If the dividend is 12, and the quotient 4, express the
divisor.
11. How much does b lack of being 15?
12. If a man walks 3 miles an hour, how many miles will he
walk in x hours ?
13. If a man walks x miles an hour, how many miles will he
walk in c hours ?
118 MISCELLANEOUS REVIEW,
1. Reduce if to a fraction whose denominator is 500.
2. Find the value of a mill if # of | of it is worth $4,690.
8. A man owned f of a large factory. He sold f of his
share for $9,900.90. What is the value of the factory?
4. 4 of f of 378 is I of 1^5 of what number?
5. The product of three numbers is 79. Two of them are
8i and &^. What is the other?
6. What is the cost of 8,170 lb. of iron at $9j a ton, and
10,160 lb. at |6i a ton?
7. A bought 140 oranges at the rate of 2 for a cent, and
160 at the rate of three for a cent, and sold them all at the
rate of 6 for two cents. Did he gain or lose ? and how much ?
8. A rectangle is 27 ft. wide, and contains 945 square feet.
What is its length?
9. If 15f bbl. of flour cost $68, what wiU 10.4 bbl. cost?
10. At 7J%, what is the interest of $864 for 2 yr. 11 mo.
13 days?
11. A and B are together worth $102,375. How much does
each own, if A owns \^ as much as B ?
12. A man bought f of a mill, and sold % of what he bought
to one man, and the remainder to another man for $4,000.
What was the value of the mill ?
18. How much did a lawyer receive for collecting his bills,
one of $225 at 8i%, and the other of $789 at 9j% ?
14. A man invested J of his money in real estate, J in a mill,
\ in bank-stock. The remainder, $2,300, he kept in cash.
How much was he worth?
15. A bam is 40 ft. long and 20 ft. wide, with 16*ft. posts, and
gables 8 ft. high. Find the cost of painting the bam, if the work
costs 80^ a square, and it takes 20 gal. of paint at $1.50 a gallon.
16. An agent remitted his employer $8,775 for the sale of
some land. If his commission was 2^%, what was the value
of the land sold?
MISCELLANEOUS REVIEW. 119
1. What is the difference on a bill of $1,275 between a dis-
count of 40% and a* discount of 30% and 10% ?
2. What per cent is gained by selling articles at 21/ each
that cost $3.36 a dozen less 25% and 14f % ?
3. If you buy a bicycle at a discount of 25% from the list
price, and sell at list price, what is your gain per cent?
4. A lawyer, collecting a note at a commission of 5%, re-
ceived $9.75. What was the face of the note ?
6. Find the area of a trapezoid when the parallel sides are
84 rd. and 66 rd. and the altitude 38 rd.
6. What must be the height of a pile of wood 32 ft. long
and 6 ft. wide, to contain 9 cords ?
7. William has a certain number of marbles ; Charles has
five times as many as William ; Henry has as many as twice
William's subtracted from Charles's ; and Henry's added to
William's are equal to 40. How many has each ?
8. What was the amount of risk if $25.20 was paid for in-
surance at 70/ on $100 ?
9. Find the per cent of loss on a bill of goods bought for
$3,360 and sold for $2,520.
10. What number decreased by 25% is $342.60?
11. Find the number of feet, board measure, in a plank 24 ft.
long, 9 in. wide, and 3 in. thick.
12. Scale, J in. to a rod. Draw a plan of Mr. Gordy's farm,
whose boundary runs as follows : l^'rom A east to B, 30 rd. ;
from B south to C, 12 rd. ; from C east to D, 24 rd. ; from D
south to E, 36 rd. ; from E west to F, 54 rd. ; from F north to
A. Find the whole area of his farm, and the length of his
boundary fence. Connect A and C with E, and find the area
of each of the three fields.
13. A man after drawing out 20% of his money, and then
10% of the remainder, found that he had in the bank $1,512.
How much had he in the bank at first?
120 ORAL.
1. A house valued at $800 was insured for three years at
li%. What was the premium ?
2. A man sold a horse and carriage at a loss of 20%. If he
received f 240, what was the cost ?
8. A dealer sold a set of books for $25, and gained 25%.
What per cent would he have gained or lost if he had sold for
$21?
4. By selling a carriage for $180, a man gained 20%. Find
the cost.
5. At $1.25 a rod, find the cost of building a wall one mile
long.
6. A man sold a cow for $60, which was | of the cost. How
much did he lose ?
7. Make an example like the 6 th, using per cent.
8. If your steps are each 2 ft. long, how many steps will
you take in walking 2 rd. 1 yd.?
9. How many surface feet in a rectangular piece of marble
4 ft. long, 3 ft. wide, and 2 ft. thick?
10. Find the commission that an agent receives for selling 40
bbl. of flour at $6 a barrel, if he receives 3%.
11. A bookseller bought a book for $3.75, and sold it for
$4.50. What per cent did he make?
12. How many acres in a field 80 rd. long and 40 rd. wide ?
13. How many cords in a pile of wood, 32 ft. long, 4 ft. high,
and 4 ft. wide ?
14. At $1.25 a cubic foot, find the cost of a block of granite
4 ft. long, 3 ft. wide, and 2.5 ft. thick.
15. At $1 a hundred board feet, find the cost of 6 boards,
each 12 ft. long, 10 in. wide.
16. At 20/ a square yard, what will it cost to paint a ceiling
18 ft. by 24 ft.?
17. Jf § of a yard of cloth cost $8, what will 2i yd. cost?
18. What will 4 tons of hay cost at $18.75 a ton ?
RULES FOR PRACTICAL MEASUREMENTS. 121
Note. — These rules are for reference, and need not be memorized.
1. In painting and plastering, it is customary to deduct from
the whole area of the room one-half of the area of all doors, win-
dows, or openings. This rule is not always observed.
2. Papering. American wall-paper is usually 1^ ft. wide and
24 ft. long for a single roll, 48 ft. long for a double roll.
There are various rules : —
(a) Pind the perimeter of the room in feet, and divide by \\ ft. (width of
paper); the quotient equals the number of strips of paper required. Divide the
length of a roll by the height of the room to find the number of strips in a roll.
. Divide the strips in the room by the strips in a roll to find the rolls required.
In the first and third division, if there is a fraction, take the next higher
integer; in the second division, take the next lo9\ r integer.
(&) Same as a, except from perimeter of room, deduct the width of doors
and windows.
Use this method unless otherwise directed
(c) Find the area as for plastering, divide the square feet in the area by 36
(the square feet in one roll of paper); this will give the number of rolls.
3. Shingling. Shingles are put up in bunches consisting of
25 layers on each side, 20 in. wide. Every 4 in. is reckoned as 1
shingle. Four bunches make 1000.
In practice, allowing for waste, a thousand will cover 100 sq.
ft. when laid 4 in. to the weather.
Find the number of squares, then compute the number of
shingles.
4. Laths. A lath is 4 ft. long, and 1\ in. wide, usually nailed
J of an inch apart. There are 50 or 100 laths in a bunch. In this
work bunches of 50 estimated to cover, allowing for waste, 3 sq.
yds. are reckoned.
Find the number of square yds. and divide by 3 to find the num-
ber of bunches of laths required.
5. Clapboards. A clapboard is usually 4 ft. long, 6 in. wide,
and 25 are put in a bundle. They are usually laid 3^ in. to the
weather. 1 bunch will cover 25 sq. ft. allowing for waste.
Find square feet in area, and divide by 25 to find the number of
bunches required.
Note. — Laths and clapboards are sold only by the bunch. One-half of
openings is usually deducted in making estimates.
122 RULES FOR PRACTICAL MEASUREMENTS.
(See note on page 121.)
1. Stone Masonry. All stone work is usually reckoned by
the cubic foot.
In measuring for cellars and buildings the distance round the
outside of the walls is taken for the length, thus measuring each
corner twice. This is considered an offset for the greater labor in
constructing the corners. For the same reason no allowance is made
for an opening unless it is a large one ; then deduct one-half.
2. Brick Work. In measuring, the same rule applies as in
stone work.
(a) Find the number of square feet in the surface, and multiply by 7 if the
wall is one brick in thickness ; by 14 if 2 bricks in thickness ; by 21 if 3 bricks
in thickness.
(6) Find the number of cubic feet in the wall, and multiply by 22 ; for 22
bricks including mortar fill 1 cubic foot.
3. To find capacity of bins. A bushel contains 2150.4 cu. in.
This is nearly IJ times a cubic foot.
(a) For practical purposes take f of the number of cubic feet in the bin for
the required number of bushels.
(6) For accurate purposes divide the number of cubic inches by 2150.4.
Note. — Use the first method unless the second is asked for.
Note. — In measuring bulky fruits and vegetables, as apples and potatoes,
a bushel will fill li cu. ft.* Then find § of the number of cubic feet for the
number of bushels. In small fruit, as berries, or in grain, li cu. it. are used.
4. To find the number of gallons in a cistern. A gallon contains
231 cu. in. Hence 7^ gal. equals 1 cu. ft.,* and 1 bbl. equals 4J cu. ft.
1 gal. of water weighs 8 J lb. 1 cu. ft. of water weighs 62^ lb.*
(a) Multiply the cubic feet in the cistern by 7i, to find number of gallons.
(6) Divide the cubic inches in cistern by 231, to find the number of gallons.
5. To measure coal. A short ton of hard or anthracite coal
measures about 36 cu. ft. A short ton of soft or bituminous coal
measures about 42 cu. ft.
Divide the cubic feet in the bin by 36 or 42 as the case demands.
6. To gauge or find the volume of a barrel or cask. Find mean
diameter. This is the head diameter plus two-thirds of the differ-
ence between the head and bung diameters.
(a) Square the mean diameter (multiply it by itself) ; multiply by the
length of the cask in inches and that by .0034 ; the result will be the number
of gallons.
(6) When the cask is not full, multiply the square of i of the sum of the
head, mean, and bung diameters in inches by the depth of the liquid in inches,
and this by .0034.
* Approximately.
APPLICATIONS OF THE PRACTICAL BULE8. 123
1. At 14.25 per M., how much will the shingles cost for a
double roof, rafters 22 ft. long, house 84 ft» long? The shingles
are laid 4 in. to the weather.
2. At $2.75 per M., how much will the laths cost for a ceil-
ing 18 ft. by 24 ft.?
3. How many barrels of water will a cistern hold that is
6 ft by 7 ft. by 8 ft?
4. A cistern 6 ft wide and 10 ft long holds 40 bbl. of
water. How deep is the water ? Approximate measurement.
5. How many barrels of water in a cistern 6 ft in diameter,
if the water is 6 ft. deep ?
6. How many barrels of apples will a crib contain that is
8 ft wide, 9 ft high, and 30 ft long ?
7. A bin 6 ft. wide and 10 ft long holds 150 bu. of pota-
toes. How high is the bin ?
8. A house is to be built 40 ft by 30 ft If the wall be-
neath it is to be 6 ft high and 16 in. thick, how many cubic
feet of stone are required ?
9. How many bricks are required' for the 8-in. walls and
bottom of a rectangular cistern, the outside dimensions of which
are: length 8 ft, depth 7 ft., width 6 f t ?
10. A room is 16J ft by 15 ft. Carpet is I yd. wide. If
the breadths run widthwise of the room, and there is a waste
of 6 in. on each breadth for matching, how many yards of car-
pet will it take to carpet the room?
11. A room 18 J ft. long and 15 ft. wide is carpeted with
Brussels carpeting worth $1.1 2i a yard. Find the cost of car-
peting the room if the breadths run lengthwise, with an allow-
ance of 8 in. on each breadth for matching.
12. A circular cistern is 5 ft. in diameter. If the water in
it is 8 ft deep, how many gallons are there ?
13. A cubical cistern is 6 ft. deep. How many gallons of
water will it hold ? Approximate measurement
124 APPLICATIONB OF THE PRACTICAL RULES.
1. At $4.00 per M., find the cost of shingles for a roof 50
ft. long, and each of the two sides 23^ ft wide, if you allow
1000 shingles for every 126 sq. ft.
2. How far must a man walk in plowing lengthwise a field
16 rd. long, 8 rd. 12 ft. wide, if each furrow is 18 in. wide?
3.^ A roof is 86 ft. long, and each side 24 ft. wide. Slated
are 16 in. by 12 in., and lap one-half. How many slates will it
take to cover the roof?
4. At 25/ a cut, find the cost of sawing into 4 pieces a pile
of wood 56 ft. long, 4 ft. high, and 4 ft. wide.
6. At 18/ a pound, 4^ lb. to a square foot, find the cost of
lead to line a tank 6 ft. by 5 ft, and 4i ft deep.
6. A house is 38 ft long, 24 ft. wide, and 20 ft high, with
a gable 8 ft. high. . How many bundles of clapboards will it
take to cover the house ?
7. A haU is 18 ft by 6i ft, and 7i ft high. There is 1
door and 1 window, each 3i ft. wide. At 24/ a roll, find the
cost of the paper.
8. At $4.1 2 J a cord, find the cost of a pile of wood 8 ft
long, 4 ft wide, and 3 ft. high.
9. A's lot is 400 ft. by 25 ft, and B's is 100 ft square.
Which has the larger area ? Which man will pay the more for
fencing, and how much more at 32/ a foot?
10. A building-lot containing J of an acre is 36 ft. wide.
How deep is it?
11. What length must be cut off an inch board 8 in. wide to
obtain 3 board feet?
12. If a man can walk 1 mile, 15 rd. in 20 min., how many
hours will it take him to walk 84 mi. 175 rd. ?
13. The walls of a hall are 120 ft long, 75 ft wide, 30 ft
high, and 21 in. thick. At $3 per M., find the cost of brick,
deducting one-half for openings ; 12 windows, 3 ft by 6 ft. ;
4 doors, 6 ft. by 10 ft
MEASUBEMENTS. 126
1. What is the cost of digging, a cifitem in the form of a
cylinder whose diameter is 6 ft., and whose depth is 18 ft., at
62/ a cubic yard ? How many gallons of water will it hold?
2. A grocer placed in his window a pyramid of oranges 6 ft.
high, and 6 ft. square at the base. How many dozen oranges
did he use, if 9 oranges fill the space of a cubic foot?
8. A grain-box has a base 8 ft. long and 6 ft. wide. The
height of the box is 11 ft. How many bushels of grain will fill
the box ?
4. A cylindrical shaped bin has a diameter at the base of
16 ft., and the height of the cylinder is 24 ft. This cylinder
has a cone-like base, with a depth of 9 ft. How many bushels
of grain will the bin hold ?
5. A lot was bounded as follows : beginning at the north-
west comer, the line ran east 35 ft. to B ; thence south 90 ft. to
C ; thence west 60 ft. to D ; thence to A. How many loads
of gravel will it take to raise this lot 4 ft. ?
6. The following represents a cellar 6 ft. deep. Scale, 1 in.
to 10 ft. Draw, starting from the northeast comer. A, south
1 in., west 1 in., south 1 in., east 1 in., south J in., east J in.,
south \l in., west 1^ in., north \l in., west 2 in., north 2^} in.,
east 3 in. Find the cost of excavating this cellar at 46/ a
cubic yard.
7. Scale, 1 in. to 8 rd. Find the cost, at $75 an acre, of a
field whose boundary runs as follows : beginning at A, the north-
west comer, running eastward 26 rd. to B ; thence to the south-
east to a point C, 12 rd. east of D, a point 24 rd. directly south
of B ; thence southwesterly to E, 24 rd. south of D ; thence
west 26 rd. to F; thence northwest to G, which is 12 rd. west
of H, a point 24 rd. north of F ; thence to A.
8. How many board feet in 100 boards, each 8 in. wide and
12 ft. long?
9. Find the area of a circle whose diameter is 77 ft.
126 APPLICATIONS OF PERCENTAOE.
1. By selling goods for $47.50 a man lost 6%. What
would he have gained if he had sold them for $67 ?
2. A man sold his house and lot for $8,304, at a profit of
15^%. If the lot cost f as much as the house, find the cost
of each.
3. Find an agent's per cent of commission, when he re-
ceived $340.20 on a sale, the net proceeds of which were
$9,379.80.
4. If you buy oranges at 30/ a doz., and sell them at $2.80
a hundred, what will be your gain per cent?
5. The premium on an insurance of $7,440 is $44.64.
What is the rate ?
6. If by selling an organ for $30, I lose 26%, at what price
should I have sold it to gain $16 ?
7. Find the cost of flour a barrel, when a commission mer-
chant receives $323.00 for selling 1,360 bbl. at 5%.
8. A man bought 2 houses for $3,460, paying 30% more
for one than he did for the other. Find the cost of each.
9. Find a man's income when 46% of $1,800 is 18% of i
of his income.
10. A man having $9,600 in the bank, di-ew out 14% of it
at one time, and then deposited a sum equal to 150% of what
he had drawn out. How much had he in the bank then ?
11. 66 S% of the cost of my horse is 66% 'of the cost of my
carriage. If the horse cost $166, find the cost of both.
12. A boy sold two balls for 35 cents. This is a loss of
12i%. Find the cost of a ball.
13. Mr. Williams sold a piano to Mr. George at a gain of
14f %. Mr. George sold it to Mr. Bruce for $320, thus losing
20%. What did Mr. Williams pay for the piano ?
14. A man received $24 as 20% of the price of a bicycle,
sold at a gain of 20%. What did the wheel cost in the first
place ?
FRACTIONS. 127
1. A man lost J of his money in business one year, and
gained the next year % of what he had left. How much had he
at first, if at the end of the second year he had $70,000 ?
2* How many marbles has a boy if the difference between
f and J of the number he has is 135 ?
3. How many 2j-in. cubes can be cut from a large cube
2 J ft. on a side?
4. One man can do a piece of work in 7 J days ; his brother
can do it in 6 days. In how many days can both do it working
together?
5. What part of 10 A. 140 sq. rd. is 7 A. 80 sq. rd.?
6. Change .0033J, .0875, and .066S to common fractions.
7. A can do a piece of work in 12 days, A and C in 9 days,
and A and B in 6 days. In how many days can A, B, and C
do it ? How many days will it take B and C ?
8. A man owned J of an acre. How much has he left after
selling 28| square rods ?
. 9. A man had $4,200 of a fortune left after having put \ of
it in a bank, and spending J and J of it.
10. If I of a yard of silk cost $1.20, how many yards will
cost $51.60?
11. it is what part of /^ ?
12. What is the cost of 64 J bushels of seed at $2j a bushel ?
18. A man had 13^ acres of land, which he divided into
building-lots each containing ^ acres. How many lots did he
have ?
14. Change .968| to a common fraction.
15. Change J* to 225ths ; iV& to l,309ths.
16. If 8i tons of coal cost $37t, what will 27t tons cost?
17. George lost f of his marbles, then bought J as many
as he had at first. If he then had 84« how many had he at
first?
18. Change \^ of a gallon to pints.
128 MISCELLANEOUS REVIEW.
1. The surface of a lake is 4i sq. mi. How many gallons
of water will it take to raise the surface i in. ?
2. How many cubes whose sides are 4 in. are equal in
volume to a cube whose side is 2i ft. ?
3. If George has 33^% more marbles than his brother, what
per cent less than George has his brother ?
4. If by selling land at 180 an acre I lose 25%, how must
I sell it to gain 40% ?
5. A man sold | of a barrel of flour for what f cost, and
the rest of the barrel for what i cost. What was the gain or
loss per cent on the whole barrel ?
6. How many blocks 7i in. by 4i in. will it require to
pave a lot 100 ft. by 12| ft. ?
7. If W of an acre of land cost $37.75, what will 43 J acres
cost?
8. What is the expense for carpeting a room 17 ft. 6 in.
long, 14 ft. wide, with carpet J yd. wide, at f 1.25 a yard ;
breadths to run lengthwise ?
9. The ratio is 3 J and the consequent 12. What is the
antecedent ?
10. A room is 20 ft. square and 10 ft. high. If each side of
the room were 10 ft. longer, how much greater would the entire
surface of the room be ?
11. A merchant marked his goods at 75% above cost, and
sold them at 38^% below his marked price, deducting 10% for
cash. What per cent of profit did he make ?
12. Divide two thousandths by eight raillionths.
13. At 16/ a square foot, what will it cost to paint the ceil-
ing of a room 16 ft. 6 in. long, and 12 ft. 6 in. wide?
14. What is the acreage of a rectangular field whose length
is 284 rd. and whose breadth is 165 rd. ?
15. If three times a certain number is added to the number
the sum will be 12. What is the number ?
MISCELLANEOUS REVIEW. 129
1. If the sod 2^ in. thick is removed from a field contain-
ing i of an acre, how many cubic yards are taken ?
2. A garden whose breadth is 4 rd., and length 2i times its
breadth, has a wall 2 ft. thick and 3 ft. high around it, outside
of the line. Find the cost of the wall at 5/ a cubic foot
(exact measurement).
8. Outside of the wall in example 2 is a ditch 3 ft. wide
and 4 ft. deep. What did it cost to dig it at li/ a cubic foot?
4. How many bricks will be required to build a house 36 ft.
long, 28 ft. wide, and 20 ft. high? The wall is IJ ft. thick,
and has four doors 4 ft. by 8 ft., 32 windows, 3 ft. by 6 ft.
5. If it takes 3 days to dig a cellar that measures 8 ft. each
way, how long will it take to dig one of the same depth, but
the other dimensions li times as large ?
6. What is the area of a circular pond which contains 15
times as much area as one 25 rd. in diameter ?
7. What is the value of a lot of land 4i rd. long, 50 ft.
wide in front, 42 ft. wide in the rear, at 65/ a square foot?
8. How many board feet in 15 sticks of timber 27 ft. 9 in.
long, and the other dimensions 8 in. and 10 in.?
9. A built a square house 40 ft. on each side. B built a
house containing the same area, but 80 ft. long. The perimeter
of A's house is what per cent of the perimeter of B's house ?
10. A general placed 4800 men in three regiments so that
the 2d regiment had twice as many as the 1st regiment, and the
3d regiment had as many as both the others. How many were
placed in each regiment?
11. Three times, eight times, and four times a number is
360. What is the number?
12. In what time will $165 amount to $179.85 at 6% ?
18. If 15% is lost by selling an estate for $10,200, for what
must it be sold to gain 20% ?
14. Reduce 563,147 inches to feet, etc.
130 ORAL FEBCENTAGE.
1. Find 20% of 50 men. 70% of 120 yr. 25% of 120 bu.
62i% of 64 days.
2. Find 60% of 600 yd. 8i% of 72 bu. 40% of 80 tons.
6% of 140.
3. A farmer raised 4,200 bu. of grain, and sold 20% of it.
How much did he sell ?
4. A house was bought for $4,400, and sold at a gain of
26%. Find the selling-price.
5. 40 is 25% of what number? 60 is 10% of what num-
ber? 80 is 40%? 12isl6S%?
6. 64 is 20% of what number? 125 is 50%? 70 is 33j% ?
30 is 12i% ? 200 is 66§% ?
7. A farmer owns 420 acres of land, which is 25% of what
his neighbor owns.
8. What per cent of 75 is 15 ? Of 80 is 40 ? Of 80 is
32? Of 120 is 30? Of 72 is 27?
9. What per cent of 120 is 90 ? Of 400 is 160 ? Of 200
is 120? Of 64 is 40? Of 90 is 30?
10. What per cent of 63 is 27 ? Of 56 is 49 ? Of 48 is € ?
Of 90 is 15? Of 160 is 10?
11. If you buy an article for f 210, and sell it for $42 g^in,
what per cent do you gain ?
12. A regiment entered battle with 960 men, and came out
with 912 men ; what per cent were missing?
18. A merchant buys lead-pencils at i^ each, and sells them
at 3/ each ; what per cent does he gain ?
14. If a merchant buys goods for f of his selling-price, what
per cent does he gain ?
15. A man is 72 years old, and 12^% of his age is 26% of
his son's age. What is the son's age ?
16. My house is insured for $4,000 at 1|% premium. Find
the premium.
17. How many inches in 26% of one foot?
ANGLES, 131
1. Take the dividers, and open the points one inch.
2. Do the legs of the dividers now point in the same direc-
tion or in different directions ?
3. Draw lines to represent the legs of the dividers.
4. These lines are said to make an angle.
5. What is an angle ? An angle is the difference in direc-
tion of two lines.
6. In this figure the lines ah and ac are
^y called the sides of the angle. The point where
y^ these lines meet, as at a, is the vertex.
7. Define vertex.
8. The angle in 6 is named by reading the angle d, or the
angle c ah.
9. Draw two lines perpendicular to each other. The angle
you have formed is called a right angle.
10. Define a right angle.
11. Draw two lines that shall meet but not be perpendicular
to each other. The angle formed is an oblique angle.
12. Define an oblique angle.
13. Is the angle you have formed less than a right angle?
If so, it is an acute angle.
14. Define an acute angle.
16. Is the angle you have formed greater than a right angle ?
If so, it is an obtuse angle.
16. Define an obtuse angle.
17. At 2 o'clock what angle do the hands of a watch make ?
18. At 9 o'clock what angle do the hands make ?
19. What kind of an angle does a vertical line make with a
horizontal one ?
20. Starting at 12 o'clock, in what time will the hands of a
clock be at right angles to each other ?
21. What kind of angles do you find on a cube? On a
square prism ? On a triangular prism ?
132
MEASUREMENT OF ANGLES.
1. For convenience, every circle is Bupposed to be divided
into 360 equal parts called degrees.
2. By means of an instrument called a protractor any angle
can be measured.
8. From cardboard cut out a figure like this, and mark it in
the same way. This is a protractor sufficiently accurate for all
practical purposes.
4. Draw a circle and two diameters at right angles to each
other. If there are 360 degrees in a circle, how many are there
in J of a circle ? In J of it ?
5. Using your protractor, meas-
ure each of the angles in this figure.
What is their sum ?
6. In the wheel of my carriage
there are 12 spokes. Haw many
degrees between two spokes ? Be-
'0 ^f tween the first and fifth?
7. Between the first and eighth? The third and seventh?
8. When the hour hand of the clock is at 12, where must the
minute hand be that the two hands may be 30° apart? 16°?
150°? 75°? 45°? 60°? 120°? 90°? 180°?
9. Using your protractor make an angle of 86°.
CONSTRUCTION. 133
1. Draw several angles, and measure them by using a pro-
tractor. First estimate their size before measuring.
2. Using your protractor, make an angle of 46® ; 60° ; 90° ;
30° ; 150° ; 135° ; 100° ; 40° ; 20°.
3. Draw several angles, and by means of the protractor
make other angles equal to each of them.
4. Draw an angle. Using protractor, make another angle
twice the size of the given angle. Do the same, using dividers.
5. Draw an angle. Make another angle of \ the size ; of
four times the size.
6. Draw a horizontal line, AB, At a point C in the line,
draw an oblique line that shall not cut AB, Measure the angles.
Add their result. What is the sum ?
7. Do the same after drawing one vertical and two oblique
lines.
8. Draw three lines cutting each other at one point. How
many angles are formed ? Measure them. Add results. What
is the sum ? What ought it to be ?
9. Draw a four-sided figure. Estimate the angles of the
figure. Record your estimate in one column; the true value,
found with a protractor, in another ; and the error in a third
' column.
10. Substituting a five-sided figure, repeat 9.
11. Substituting a six-sided figure, repeat 9.
12. Using a protractor, erect a perpendicular at each extrem-
ity of a horizontal line ; of an oblique line ; of a vertical line.
13. Draw a vertical line. Using protractor and ruler, con-
struct a square upon this line.
14. How many degrees are passed over by the hour hand of
a clock in an hour? in 3 hours? In how many hours will the
hour hand pass over 90° ?
15. Make any angle. Using your protractor, make an angle
three times as large.
134 DUTIES OB CUSTOMS.
NoTB. — A study of imports should precede this topic.
1. Have you ever been in Washington, D.C. ?
2. What can be seen there ?
8. Name some of the persons engaged in making and exe-
cuting our laws,
4. Who pays these men for their work?
6. Where does the Government get all its money for all
expenses ?
Ana, I. From an Internal Revenue, — a tax on the right to make or sell
liquors, tobacco, etc. II. From Customs or Duties, — taxes on imported goods.
6. What are imported goods ?
7. What goods do we import ?
8. Make a list of the more important things that are im-
ported, and the name of the country from which they are
imported.
9. Who determines the amount of Duties to be paid?
Ana, The Government, by an Act of Congress, usually called, " The Tariff
Act.'* This is a list of goods on Which duties must be paid, with the rate of
duty assessed on each.
10. What is Tariff?
11. Where are these duties collected?
Ana. At Custom Houses, — buildings owned by the United States, where
the Collector and other officers do business.
12. The Government designates certain places called Ports
of Entry, where Custom Houses are built.
13. What are Ports of Entry ?
14. Are Ports of Entry ever found except on the sea-coast ?
Why?
15. What is smuggling ?
16. Duties, are either specific or ad valorem. Specific duties
are based on the number, quantity, weight of the merchandise.
Ad Valorem duties are based on the value of the merchandise.
17. What are Specific Duties?
18. What are Ad Valorem duties?
CUSTOM-HOUSE BUSINESS, 135
' 1. Importers are required to submit to the collector an in-
voice. This is a description of the goods, and their cost in the
country from which they are imported.
2. What is an invoice ?
3. Allowances, called Tare, Leakage, and Breakage, are
deducted before estimating duties.
Tare is a deduction from the gross weight hecause of the weight of the
box, etc. Leakage is an allowance made on liquids in casks or barrels.
Breakage is an allowance made on liquids in bottles.
4. Invoices are made out in the i^oney of the country from
which the goods are imj^rted. When changed to United
States money the duty is computed on the nearest dollar.
5. What is the duty at 35% on 75 pieces of satin, each piece
containing 47 yd. at $1.65 a yard?
6. What is the duty on 20 casks of wine containing 40 gal.
each, invoiced at $1.12^ a gallon, at 28%, leakage, 6% ?
7. A grocer imports 360 bags of cocoa, gross weight 145 lb.
each, tare 3i%, invoiced at 13/ a pdund. What was the duty
at 2i/ a pound?
8. What is the duty upon merchandise invoiced at 120 Kra,
allowing 7^% for breakage; rate of duty, 27%? A lira is
equal to $.193.
9. If the rate of duty is 50%, and tare 2%, find the duty
on merchandise invoiced at 4,670 guilders. A guilder is equal
to $.402.
10. Find the duty on 1,160 gal. of brandy, leakage 2%, rate
of duty, $1.75 a gallon.
11. At 35% ad valorem^ what is the duty on 150 doz. pairs
of kid gloves invoiced at 68 francs a dozen ? A franc is equal
to $.198.
12. What is the duty at 2^/ a pound on 500 sacks of cocoa,
each containing 85 lb., the tare being 1^% and 600 sacks each
containing 75 pounds ?
136 DUTIES AND CUSTOMS.
1. What is the duty at 30/ a dozen, and 15% ad valorem^
on 750 doz. linen collars valued at 88 J/ a dozen ?
2. If velvet cloaks cost 600 franca each in Paris, and the
duty is 50%, what will be the duty on a dozen cloaks?
3. By the last Tariff Act the duty on varnish is 35%, and
$1.32 a gal. Find the duty on 6 bbl. of varnish, 31 i gal. to a
barrel, invoiced at $6 a gallon, leakage 10%.
4. If the duty on Brussels carpets is 44/ a square yard, and
40% ad valorem, find the total cost to me of 300 yd., J yd.
wide, invoiced at Qs. a y^rd. A shilling is $.243.
6. The duty on an importation of lace at 60% was $624.
How many yards were there if the lace was invoiced at 80/
a yard ?
6. What is the duty at 20% on 25 oil-paintings, averaging
$1,375, and on 15 pieces of statuaiy averaging $978 each?
7. What is the duty at 2/ a pound on 375 boxes of figs,
weighing 138 lb. each, tare 16 lb. on each box?
8. What is the duty (Jn 12 casks of molasses, 68 gal. each,
at 3/ a gallon, leakage 15 gal.?
9. The duty on plate glass is 35/ a square foot. Find the
duty on 416 plates, each plate 9i ft. by 12^ ft.
10. Find the duty at $.65 a cubic foot on a block of Italian
marble 2i by 3J by 8 ft., invoiced at 3,450 lira.
11. Merchandise invoiced at 7,689 florins pays a duty of
35%. A florin is equal to $.359.
12. At 44/ a square yd., and 40% ad valorem, find the duty
on 2,468 yd. Brussels carpet 27 in. wide, invoiced at 3i shillings.
13. A merchant imported 26 hhd. of sugar, invoiced at
1,045 lb. each. What will be the duty, tare being 12^%, and
the rate 2/ a pound ?
14. What is the duty on 40 bales of peanuts invoice^ at
80 lb. each, and 25 bales invoiced at 70 lb each, tare being 6%,
and the duty i/ a pound ?
MISCELLANEOUS REVIEW. 137
1. A butcher bought 7i doz. turkeys for $108, and found
20 % of them spoiled. How must each one of the remainder be
sold to gain 83^% on the lot?
Note. — Most merchants choose some word of 10 letters as their private key,
thus : ?r???I*^**^'?« Now, if a merchant wishes to mark an article 26^, he
uses the letters r a.
2. A merchant's private key for marking goods is " abridg-
ment." If he buys goods at hd a yard, how must he mark them
so as to gain 20 % ?
3. Using the same key, how must shoes that cost %h.dt be
marked to gain 60% ?
4. What is the duty on 224 chests of tea, each weighing
67 lb., tare being 4 lb. to a chest, at fl.50 a hundred weight?
6. $475 Rolyoke, April 1, 1903.
Three months after date^ we jointly and severally
promise to pay J, A. Dickinson^ or order, Four
Hundred seventy-five Dollars, with interest. Value
received. John French.
James Fiske.
Is this a negotiable note? What is the meaning of jointly
and severally ? Find the interest due when the note is legally
due.
6. A grocer borrowed |400 at 6% interest, and bought flour
at $4 a barrel. He kept the flour 1 yr. 3 mo., when he sold it
all at an advance of 25%. After paying his note, how much
had he gained ?
7. What will it cost to build a half mile of road at $4.75 a
rod?
8. How many screws in 5 J gross and 4 J doz. ?
9. What will be the cost of 8 bu. 3 pk. 6 qt. of nuts at
13.20 a bushel?
10. How many feet in 7 J rd., llj^ yd. ?
138 EQUATIONS.
(Review page 117.)
1. If one part of 12 is 7, what is the other part?
2. If one part of 15 is x^ what is the other part ?
3. If one factor of 25 is 5, what is the other factor ?
4. If one factor of 18 is Xj what is the other factor ?
6. If a pear costs 2x cents, and a peach Zx cents, what will
represent the cost of both ?
6. What will 8 yards of cloth cost at 2a: cents a yard ?
7. Draw a rectangle. Call the length x in. and the width
y in. Express the perimeter of the rectangle. Express one-
half of the perimeter. Express the difference between the
length and the breadth.
8. Express the area of the rectangle ; \ of the area.
9. Express the square of x ; the cube of y ; the fourth
power of a.
10. If X and y represent two numbers of which x is the
greater, what will represent their sum ? their difference ? their
product ? their quotient ?
11. The expressions 6 4- 4 = 10, or 6a -f- 4a = 10a are called
equations. The parts at the left and right of the sign = are
called members, or sides. They are distinguished as first mem-
ber, or left side, and second member, or right side.
12. In the equation 6 + 4 = 10, add 2 to the first side.
What must you do to the other side to preserve the equa-
tion?
13. Learn : If anji;hing is added to one side of an equation,
an equal amount must be added to the other side.
Note. — A self-evident statement like the above is called an Axiom.
14. If Zx = 9, what does ^x + b equal?
15. If a; — 4 = 8, what does x equal.
16. If a; = 2, how can you change the equation so that the
right side shall be 6 ?
Note. — The last three examples are illustrations of Axiom 1. Prove it.
EQUATIONS. 139
(Review pages 117, 138.)
1. In the equation 6 + 4 = 10, subtract 4 from the left
side. What must you do to the other side to preserve the
equation ?
Axiom 2. — If anything is subtracted from one side of an equation, an
equal amount must be subtracted from' the other.
2. If a; = 7, what will x —Z equal ?
3. If a; -f- 4 = 6, what does x equal ?
4. If a; = 9, how can you change the equation so that the
right side shall be 5 ?
6. In the equation 6 = 6, multiply the left side by 3. What
must you do to the right side to preserve the equation ?
6. Write the statement as Axiom 3.
7. If a; = 4, what will 7a: equal ?
4a:
8. If -^ = 20, what does 4a: equal ?
9. In the equation 6=6, divide the left side by 2. What
must you do to the right side to preserve the equation ?
10. Write the statement as Axiom 4.
11. If 4a: = 12, what does x equal?
Express in the form of equations the following statements :
12. a is equal to b added to c.
13. 25 exceeds 19 by 6.
14. The excess of 17 over 8 is 9.
15. The excess of x over y is 2.
16. Write three times the expression three plus four.
17. Write two times the expression x minus y.
18. What number is less than 16 by 4?
19. Learn : We may add the same number to both members
of an equation ; subtract the same number from both members ;
multiply both members by the same number, or divide both
members by the same number, and not affect the value of the
equation.
140 ORAL.
1. How many cubic feet in a rectangular block 2 ft. square
at the end and 6 ft. long ?
9, How many times larger would a block be that was twice
as long, twice as wide, and twice as thick ?
3. How many blocks J of an inch on a side can be sawed
from a 2-in. cube ?
4. When a number is used twice as a factor, or multiplied
by itself, the product is called the square of a number.
5. Name the squares of the numbers from 1 to 10.
6. When the number is used three times as a factor, the
product is called the cube of the number.
7. Name the cubes of the numbers from 1 to 10.
8. When 100 shares of bank-stock are sold for ^^1 7,650,
what is the price per share ?
9. Divide 24,584 by 10 ; by 100 ; by 1,000.
10. Divide 16,485 by 10,000 ; by 1,000 ; by 100 ; by 10.
11. At $7.50 each how much will 100 trunks cost?
12. At $2.50 each, how many chairs can be bought for $50 ?
13. Separate $20 into 5 equal piles. What is the answer?
Since you are asked to find J part of $20, examples like this are
called Partitive Division.
14. Separate $20 in piles of $4 each. What is the answer ?
Since here the size of the pile is given, and we are asked to
measure the larger pile by it to find the number of piles, this is
called Measuring Division.
15. Name the terms used in Division.
16. In the partitive form of division, which of these terms are
alike?
17. In the measuring form of division which of the terms are
alike ?
18. Make an example to illustrate Partitive Division,
19. Make an example to illustrate Measuring Division.
20. Define quotient. Divisor. Dividend.
STATEMENTS. 141
1. A man sold some property at a profit of 20%, and with
the proceeds bought some more property, which he sold for
$4,860, at a loss of 65%.
2. A man sold a bicycle for $93.10, thereby losing 5%.
3. A teacher's salary after being increased 20% was $3,000.
4. I received $1,642.60 for some property which my agent
sold for $1,720.
5. As agent, working on 4^% commission, I received
$129.51.
6. I bought a barrel of flour for $5.94. The dealer gained
44 cents.
7. I sold a house for $638 less than I paid for it I sold it
for 92|% of its cost.
8. I sold goods on 2% commission, and remitted to my
employer $4,777.50.
9. A man bought 20 sheep for $250, and 20 cows for
$1,060. He sold the sheep at a gain of 12i%, and the oxen at
a gain of 7i%.
10. A man left by will $4,500 to his wife, and the re-
mainder of his property to be equally divided among his three
children. It was found that the share of each child was one-
flfth of the whole property.
11. A bill of goods amounted to $5,650. It was offered for
sale with 20% and 5% discount and 10% off for cash. The
offer was accepted.
12. On a bill of $600 a dealer offered me a discount of 33J %.
13. A lawn-tennis court is 78 ft. long and 27 ft. wide. At
each end there is a margin of grass 12 ft. wide, and at each side
a margin 6 ft. wide. It cost 50/ a square yard to turf the
mai-gin, and 85/ a square yard to gravel the court.
14. A rectangle 150 ft. by 120 ft. has in the center a rect-
angular grass-plot 80 ft. by 60 ft. Cover the rest with gravel
8 in, deep at a cost of 62/ a cubic yard.
142 MISCELLANEOUS REVIEW.
1. How many cubic inches in a sugar-loaf in the form of a
cone, the diameter of the base being 8 in., and the height 18
in.?
2. How many cubic feet in a cylinder 60 ft. long, and 8 ft.
in diameter?
8. Find the contents of a pyramid whose base is 9 ft. square,
and whose altitude is 79 ft.
4. How many board feet in 15 2-in. planks 12 ft. long, 18
in. wide at one end, and 12 in. wide at the other end?
6. How many acres in a field in the form of a triangle whose
base is 965 rd., and altitude 576 rd. ?
6. A man placed $4,200 insurance on his house, $2,400 on
his furniture, and $700 on his library, for 8 yr., paying a pre-
mium of $109.50. What was the rate per annum?
7. I paid $50.12i as premium for insuring my house at 2^% .
What was the value of the house ?
8. A store valued at $10,000, and a stock of goods valued
at $15,000, were insured for 75% of their value at 3%. If there
were a total loss by fire, what would be the owner's loss ? Com-
pany's loss ?
9. A man bought a horse, carriage, and harness for $240.
He gave three times as much for the carriage as for the harness,
and as much for the horse as he did for both the carriage and
the harness. How much did he give for each ?
10. A farmer, when asked how many cows he had, replied
that if he had twice as many more he would have 60. How
many had he ?
11. If the interest on $960 at 6% is $54.40, what is the
time?
12. A gave his note Aug. 6, 1902, for $670, interest at 7%.
He paid the note and interest May 17, 1903. How much did
he pay ?
13. Find the interest on $9 for 9 yr. 9 mo. 9 da. at 9%.
MISCELLANEOUS REVIEW, 143
1. The product of three numbers is 2,090 ; one is 28i and
another 22. What is the third number ?
2. Find the sum, difference, product, and quotient of f , |,
using the larger fraction as minuend and dividend.
3. Divide the least common multiple by the greatest com-
mon divisor of 18, 48, 72, 66.
4. If 61i lb. of tea cost $55/^, what is the cost of iVl^js lb.
5. If an agent receives 2i% for collecting, and is paid $739,
how much will his employer receive ?
6. A sold some goods to B at a profit of 10%. B sold them
to C at a profit of 10%. C sold them to me at a profit of 20%.
Now, if I paid f 1,452, what did they cost A ? Had A sold
directly to me at the price I paid C, what per cent would he
have made ?
7. If a broker sells goods that cost $4,800 at a profit of 6%,
and charges 5% on the amount received as commission, how
much does the owner of the goods receive as profit ?
8. How many cubic yards of earth will be thrown out in
digging a ditch 3 ft deep and 4 ft. wide just within the boun-
dary of a rectangular field 12 rd. long and 8 rd. wide?
9. Express .035, .625, .12288 as common fractions in their
lowest terms.
10. If a piece of cloth contains 246 yd. 1 ft. 6 in., how many
times can you cut from it 14 yd. 1 ft. 6 in, ?
11. Reduce |i, t^^, and ^JHtt to decimals.
12. If 7 men build 6^ rd. of wall in 15^ days, in how many
days can 12 men do as much ?
13. A father is 30 yeara older than his son. The sum of
their ages is 42 years. How old is each ?
14. If eight times a number exceeds five by as much as 17
exceeds three times the number, what is the number ?
16. A horse and cart cost 1115. The horse cost $5 less than
twice the cost of a cart. Find the cost of each.
144 TAXES.
1. Mention several things for which a city or town needs
money.
2. Where does the money for schools, roads, sewers, lights,
police, etc., come from ? Ans, From taxes.
3. What is a tax ? Arts, A tax is a sum of money assessed
on persons and owners of property to meet the expenses of a
town, city, county, or State.
4. Find out what need a county has for money.
6. Find out what need a State has for money.
6. Poll means head. What, then, is a poll-tax ?
7. Who pays a poll-tax?
8. Does a man need to own any property in order to be
assessed a poll-tax ?
9. How many kinds of property are there? Ans. Ileal
Estate and Personal Property.
10. Real estate consists of lands, houses, or, in general, im-
movable property, and is taxed in the place where it is situated.
11. Personal property consists of horses, money, merchandise,
etc., or, in general, movable property, and is taxed in the place
where the owner lives.
12. Any tax upon real estate or personal property is called
. a property tax.
13. Assessors are persons appointed to make an inventory of
all taxable property and estimate its value.
14. How large a poll-tax is assessed in your town or city ?
16. How large a tax on property is assessed in your town or
city?
Taxes are usually assessed and collected as follows :
16. The State determines the amount to be expended for State
purposes, and divides that amount among the counties according
to their valuation, previously determined.
17. The county adds to this sum the amount it will need for
county expenses, and divides the total among the towns of the
county according to their valuation.
TAXES. 145
1. Each town adds to this amount whatever it needs for
schools, police, roads, salaries, etc., and thus finds its total tax,
or tax levy,
2. The assessors now find the number who must pay a poll-
tax, and multiply this by the tax on one poll. This is the poll-
taxy and is subtracted from the whole tax. The ampunt of the
tax that is left must be assessed on property, and is called the
Property Tax.
3. The assessors now find the entire valuation of the town
by adding all the Real Estate and Personal Property. This is
called the Total Valuation.
4. If the tax that is to be raised on the property is now
divided by the property, what will the result be ? This tax on
$1 is called the tax-rate.
5. The assessors had to find the amount of property each
man owned when they found the valuation of the town. If the
sum of money that each man owns is multiplied by the tax-
rate, what will be the result?
6. In some States the county and State tax are found sepa-
rately, and are not united with the town tax.
7. Write the different steps taken in computing taxes, and
tell the reason for each step.
8. Do you see the reason why all persons are not taxed the
same amount ?
9. Sometimes there is assessed an Income tax ; that is, a
tax. on a man's income.
10. Sometimes, to provide for abatements and uncollected
taxes, the property tax is inci-eased by a small percentage.
11. There are 4,120 persons, each of whom pays $1.50 in
the city of B. The total valuation is 15,864,528, and the total
tax 194,147.92. What must Mr. Philips pay, who owns a mill
valued at $4,500, and who owns personal property valued at
$1,240?
146 TAXES,
1. The property valuation of a town is $1^600,000, and the
tax levy is f 12,500. There are 250 male adults, each paying
$2. Mr. Dunbar's real estate is valued at $7,600, and his
personal property at $4,500. Find his tax.
2. In the town of C there are 2,500 polls, each taxed $2.
The tax leyy is $245,000, and the taxable property $12,000,000.
Find Mr. J.'s tax on property worth $125,000.
3. A tax of $50,000 is levied in a town, valued at $3,200,-
000. There are 1,000 persons who pay a poll-tax of $2. What
is my tax if my property is valued at $9,000 ?
4. In the city of H there are 1,200 male adults, each poll
taxed $1. The taxable property is $30,000,000, and the tax
levy $151,200. What tax does Mr. Sims pay, who is assessed
$12,000 for real estate, and $2,500 for personal property?
6. The real estate of a city is $3,099,500 ; personal prop-
erty, $1,487,280 ; tax levy, $66,023.92. 1,340 polls are each
assessed $1.35. What does Mr. A pay, whose property is
valued at $18,000?
Note. — In all these examples a poll-tax is included in finding the whole
amount of a man^s tax.
6. What is the rate of taxation in a city when $125 is the
tax on a house assessed at $6,000 ?
7. How much tax will a man pay on $6,250, if the rates
are lj% for a city tax, and |% for State and county tax ?
8. If the tax-rate is 21 mills on a dollar, what is the as-
sessed value of a property that pays $62,622 tax ?
9. The valuation of the town of H is $2,432,500, and the
tax-lexy is $48,650. Mr. Smith owns $150,000 worth of prop-
erty, assessed at f of its value. What is his whole tax ?
10. If a tax-rate 'of 2^/ on a dollar produces $130,000, what
is the assessed valuation of the property ?
11. What is the rate of taxation when $2,285.10 is the tax
upon $152,340 ?
MISCELLANEOUS REVIEW. 147
1. Find the aica of a walk, 9 ft. wide, round the outside of
a park that is 9 rd. square.
2. What is the duty, at 15%, on 640 yd. of silk invoiced
at $2.26 a yard?
8. What is the duty, at 12^ a cwt., on 36 bags of salt, each
containing 116 lb., tare 2 lb. a bag?
4. How much tax will a person pay whose property is
assessed $246,600, if he pays If % city tax, 1% school tax, |%
county tax ?
5. Real estate of a town, $1,106,843; personal property,
$249,031 ; tax to be raised, $21,268.11. There are 860 polls,
each assessed $1.10. What tax will Mr. Martin pay if his
property in the town is assessed for $8,660, and he, not being
a resident of the town, pays no poll-tax ?
6. An insurance company took a risk of $876,000 at t%.
It reinsured $176,000 in another company at 1%. How much
premiuin did it clear above what it paid ?
7. When the rate of insurance is 3i^% and the premium
$14.63, what is the value of the property insured?
8. The premium for insuring a house at | of its value was
$64.60. If the rate was If %, for what amount was the house
insured?
9. A note for $1,460 was given, to be paid in 1 yr. 7 mo.
21 da., with interest at 6%. What was the amount due at the
expiration of the time?
10. What number increased by 22% of itself is 23,646 ?
11. A horse and carriage were sold at a profit of 20%, which
was a gain of $100. The horse cost 60% more than the car-
riage. What was the cost of each?
12. I sold 76 bu. of wheat at $1.06 a bushel. This was
more than I paid for it. How much did I gain?
13. What part of 6 rd. is 14 yd. 2 ft.?
14. Write a promissory demand note.
148 MEASUREMENTS.
1» Find the area of a paralellogram 15 in. by 9 in.
2. Find the area of a rhomboid whose base is 42 ft. and
whose altitude is 16 ft.
3. Find the area of a trapezoid whose parallel sides are 14
ft. and 10 feet) and whose altitude is 12 ft.
4. Find the cost of gilding a ball 50 in. in diameter at 5/
a square inch*
5« Find the volume of a cone if its height is 18 in. and the
diameter of its base 20 in.
6. A bin 24 ft. long, 6 ft. wide, and 2 ft. deep is 62 J % full
of oats. How many bushels are in the bin ?
7. Find the volume of a cone whose base contains 57 sq. in.,
and whose height is 28 in.
8. Find the number of bushels of grain required to fill a
bin 14i' X 8V X 4'.
NoTB. — The Signs ' and '' are sometimes used In place of feet and inches.
9. Find the number of gallons of water in a well 4^ ft. in
diameter if the water is 9 ft. deep.
10. How many feet board measure in 20 beams 8" x 10" and
24' long?
11. How many feet board measure in 48 boards J in. thick,
4 in. wide, and 15 ft. long?
12. What is the height of a pile of wood containing 60 cords,
if it is 400 ft. long and 4 ft. wide ?
13. A canal was dug 300 ft. long, 15 ft. wide, and 10 feet
deep. How many cubic yards of earth were removed ?
14. Estimate the number of bricks required for the walls of
a building 80' x 50' x 22', if the walls are Ij ft. thick, and if
500 cu. ft. are allowed for doors and windows.
15. The rafters of a house are 21 ft. in length. If the house
is 36 ft. long, how many shingles, laid 4 in. to the weather, will
be needed?
16. Find the area of a rectangle 24 ft. by 86 ft.
COMMON FRACTIONS, 149
1. Reduce to whole or mixed numbers : W ; W ; W ;
8. Reduce to improper fractions : 47f; 96H; ISA; 41tVv.
8. Find product of : Y X 549 ; U x 108 ; Jf x 384;
^ X 500.
4. Find product of; iJ x H ; i of A of V; 22 x 6^^.
5. Find the value of : | of 3^ of ift^ -^ 5^.
e. 97i is H oi what number?
7. 89g is il of what number?
8. 99 is J more than what number?
9. If If of a cord of wood is worth $6, find the cost of
9J cd.
10. Find the sum of 1*5, t'jj, f, t^.
11. Find the sum of 26H, 37 A, 16i, 48^
12. From 4181 take 11 f.
13. What number must be added to 18# to make 24} ?
14. A can do a piece of work in 16 days, B in 18 days, and
C in 12 days. How many days will it take the three together
to do the work?
15. If a man walks 3| miles an hour, how many hours will it
take him to walk 187 J miles?
10. A man raised 187^ bu. of barley on 7| acres of land.
How many bushels an acre did he raise ? -
17. From 120 acres of land, 44j A. were sold to one man, and
i of the remainder to another. Find the value of the unsold
acres at $65 an acre.
18. f of § of the value of a pair of horses is $210. What is
the value of the pair ?
19. A can do a piece of work in 6 days, B in 8 days, and C
in 10 days. How much of it can they do in 2 days, working
together ?
20. At 16i/ a yard, what will 415§ yd. of cloth oost?
21. Divide 12i x 16§ by 6§ x 6i.
150 ORAL.
!• What is the result of an example in subtraction called ?
2. (a) A man had $25, and lost f 10. (J) You have f 25, and
I have $10. In (a) is the answer a difference or remainder?
In (6)?
8. What kind of numbers can be subtracted?
4. When numbers are unlike, what must be done before
subtracting?
5. When fractions are unlike, what must be done before
subtracting?
6. What are the terms in subtraction called ?
In the following four examples, give the answer in terms of
both the minuend and subtrahend :
7. 3 lb. - 32 oz.
4hr.
-180 min.
4 yr. — 86 mo.
8. 5 yd. - 12 ft.
5 ft.
- 48 in.
9 ft. - 2 yd.
9. $12 - 350/.
2 T.
- 2000 lb.
5 dimes — 20 ct.
.0. 4 bu. - 8 pk.
6pk.
-24qt.
10 qt. - 8 pt.
.1. Find the missing
term:
Minuend, 45
X
43
53 X 61
X 95 74
Subtrahend, x
36
27
X 34 29
64 X 39
Remainder, 24 16 x 11 42 x 19 32 x
12. In the morning I had $2, during the day I spent 35 cents,
a half-dollar, a quarter, and a dime. How much money had I
at night?
18. What day of what month is the 75th day of every year
not a leap year ? What change does leap year make ?
14. What number is 12 less than 76 - 38 ?
15. What kind of numbers can be compared ?
16. What are the terms of a ratio called?
17. When do you have a fraction for your ratio ? When an
integer?
18. What effect on an integer is made by annexing a cipher?
By dropping a cipher?
19. Name the terms used in multiplication.
INTEREST. 151
1. Find the interest of $4,378 for 3 yr. 8 mo. at 5%.
2. Find the interest of $426.25 for 20 yr. 9 mo. 24 da. at
6i%.
3. Find the interest of $417.16 for 4 yr. 11 mo. 17 da. at
4%.
4. A has $16,000, and loans 12j% of it to B for 2yr. 3 mo.
7 da. at 4i%. How much does B owe him at the expiration of
the time ?
5. Find the interest of $900 for 93 days at 7 %.
6. A man bought 43 cows at $38.50 each, and paid half
cash and the remainder in a note for 2 yr. 4 mo. at 2J%. What
was due on the note at maturity ?
7. Find the amount of a note for $916.84, given Aug. 16,
1902, at 6i%, and due Mar. 11, 1905.
8. A man bought a horse Oct. 3, 1902, for $186, and gave
in payment his note at 7%. On June 13, 1903, he sold the
horse for $192.50, and paid his note in full. How much did he
gain or lose ?
9. What principal on interest at 6% will gain $105 interest
in 3 yr. 6 mo. ?
10. Interest $21.69§, time 2 yr. 4 mo. 9 da., rate 5%. Find
the principal.
11. At what rate will $3,470 give $1,408.82 interest in 6 yr.
9 mo. 6 da. ?
12. A man gave his note Jan. 11, 1903, for $6,000 at 5%.
Sometime afterward he canceled the note by paying $9,326.66.
What was the date of cancellation ?
13. Write a note supplying all the data.
14. John James paid Amos White, of Salem, Mass., Sept. 1,
1902, $112.24, amount due for meat. Write the receipt in full.
15. I loaned a friend $8,000 June 16, 1903. If money is
worth 6J%, how much is due me Sept. 1, 1906?
16. Find the interest of $267 for 2 yr. 10 mo. at 4%.
162 PERCENTAGE.
1. What is 138% of 64.6?
d. 210 is 16 1 % more than what number?
3. $174.72 is 28% of what sum ?
4. A miarohant dropped $4.65 from his price on a chamber-
set by taking off 15% from his price. What was the price
asked ?
5. Find the net amount of a bill of $642 with i and 10%
off.
6. What is paid for an article listed at $2.40, and sold at
25% and 10% off?
7. A man offered for sale a piece of land that cost him $660
so as to gain 40%. He discounted his price 16|%. How much
did he receive for it?
8. A man built two houses at a cost of $2,785 each. He
sold one at a gain of 16%, and the other at a loss of 5%.
What was his gain ?
9. A commission merchant sold 7,500 bu. of wheat at 65/,
charging 2i% commission. How much money should he remit
to his principal or employer ?
10. Find the cost of insuring a stock valued at $4,685, risk
being taken at li% on J of its value.
11. What would be the tax on $1 if $5,838 must be raised
on $486,500 ?
12. Find the duty at 20% on 348 doz. of olives, invoiced in
Italy at 4 liras per dozen. QLira « $.193.)
18. Find the interest on $476.50 for 3 yr. 7 mo. 4 da. at 8%.
14. Find the amount of $726.50 for 3 yr. 4 mo. at 4%, com-
pound interest.
15. An agent paid $58.50, at 2%, for insuring 1,000 bbl. of
flour. If the flour was insured for 75% of its cost, what was
the cost?
16. An agent sells buggies for $76 each. If the rate of his
commission is 15%, how many must he sell to earn $1,026 ?
TO FIND THE SURFACE OF A SPHERE. 153
1st Illustration. —Draw a line 6 in. long, and divide it into 12 equal parts. Make 9
dots at each extremity of the line. Take the compass, and with a radius of five in. draw
arcs intersecting as in the figure. Cut out, and fold into a sphere. This is not quite exact.
2d Illustration. — Take a wooden hemisphere, and drive a tack in the center of Its
ounred surface. Commencing at the tack, carefully wind a cord about the curved surface,
as a boy winds a top. When the surface is covered, cut the cord. Drive a tack into the
center of the base of the hemisphere, and wind the cord tightly about the tack. When the
surface is covered, just ^ of the string will have been used. This proves that the hemi-
sphere is two times greater than the circle in surface. Hence the whole surface of the
sphere is four times greater than the circle. How do the circumference and diameter of
the circle compare with those of the sphere ? How do you find the area of the circle? By
what would you multiply this to get the area of the sphere ?
If you multiply ^ the circumference by | of the diameter, and then multiply by 4, you
will find the area of the sphere. But since } times } times 4 equals 1, we see that the area
of the sphere is found by multiplying the circumference by the diameter.
3d Illustbation. — Place a sphere on the table. Holding it between the palms of
your hands, roll it over once. How long a space has been passed over ? Continue until tlie
pupil sees that it is a space as long as the circumference of the sphere. Let him next
determine how wide a space has been passed over. Continue until he discovers that it is
a space as wide as the distance between the hands, or the diameter of the sphere.
Lead the pupils to see that you have covered a rectangle, as long as the circumference
of the sphere and as wide as ttie diameter. With the area of this form all are familiar.
Find the surfaces of the spheres, when the following dimen-
sions are known :
1. Radius 6 in. 6. Radius 6 in.
2. Diameter 8 in. 7. Diameter 12 ft.
3. Circumference 31.416 ft. 8. Circumference 62.832 in.
4. Diameter 2 ft. 6 in. 9. Radius 5j ft.
6. Diameter 4 J ft.
10. How much leather will it take to cover a base-ball if its
diameter is 8 J inches?
11. At 15/ a square foot, what will it cost to ^Id a sphere
14 in. in diameter ?
12. Find the surface of a sphere 4 ft. 6 in. in diameter.
154 MISCELLANEOUS REVIEW,
1. I asked $184 for my horse. This was a gain of 15%. I
sold him for $150. What per cent did I lose ?
2. A speculator sold two farms at $4,000 each. On the one
he gained 20%, and on the other he lost 20%. Did he gain or
lose ? and how much ?
3. An agent sold goods to the value of $13,656, and received
as his commission $307.26. What was the rate of commission ?
4. What tax must I pay if my property is valued at $5,600
in a town where a tax of $3,240 is to be raised? There are
398 polls, each paying $1.50, and the taxable property is valued
at $440,500.
5. A house worth $16,000 was insured for 5 of its value at
li%. If it was totally destroyed by fire, find the loss to the
owner and to the company.
6. Find the interest on $1,248 from Mar. 4, 1901, to Aug.
12, 1904, at 7%.
7. Find the interest on $3,020 from Apr. 14, 1902, to June
10, 1905, at 5i%.
8. A town whose valuation is $4,900,000 raises by taxation
$63,210. There are 980 poll-tax payers, each taxed $2. What
is Mr. Brown's tax, who owms a farm valued at $3,200, personal
property amounting to $1,800, and who pays one poll-tax ?
9. A man imported from France 12 doz. pairs of gloves, in-
voiced at 6 francs a pair. The duty was $2.25 a dozen and 50%
ad valorem. What did the gloves cost him? A franc is $.193.
10. Find the duty on 1,000 boxes of cigars weighing 2,480
lb., invoiced at $3.50 a box ; tare, 8 oz. a box ; duty, $2.50 a
pound and 25% ad valorem.
11. Eight times a certain number minus 11 equals five times
the number plus 25. Find the number.
12. My sister is 4 years younger than I am, and the sum of
our ages is 26 years. What are our ages ?
13. 2a:-^3 = 13. Find a:.
MISCELLANEOUS REVIEW. 155
1. What is the duty, at 22/ a gallon, on 600 qt-bottles of
oil, breakage 6% ?
a. A merchant imported 600 yd. of dress goods, invoiced at
8 francs a yard. What is the duty at 40 % ad valorem ?
8. What is the duty at 18/ a sq. yd., and 30 % ad valorem^
on 1,000 yd. of carpet | yard wide, invoiced at 10 francs a yard?
4. If a man travel 720 miles in 30 days of 7 hr. each, how
far will he travel in 7 days of 10 hr. each?
5. Brooks raised five times as many bushels of potatoes as
Avery, and Maxfield 10 bushels more than both of the others.
If all together they raised 328 bushels, how many did each
raise?
6. Find the entire surface of a cylinder whose altitude is
8 ft. and the radius of the base 4 ft. Find contents.
7. The slant height of a cone is 50 ft., and the diameter of
the base 18 ft. Find its convex surface.
8. A room measures 18' x 12' x 10'. At 37/ a square
yard, find the cost of plastering it, allowing 80 sq. ft. for doors
and windows. At $1.1 2i a yard, find the cost of Brussels car-
pet laid lengthwise, allowing 8 in. loss on each breadth for
matching.
9. What is the cost of 10 girders each 40 ft. long, 14 in.
wide, and 12 in. thick, at $28 per M. ?
10. A bam is 80 ft. long, 50 ft. wide, with 20-ft. posts. The
gable is 10 ft. high, and the rafters are 28 ft. long. How many
boards 16 ft. long and 15 in. wide will it take to roof and
board it?
11. An irregular stone was thrown into a cylinder 3 ft. 4 in.
in diameter, which was partly filled with water. After the
stone was thrown in, the water in the cylinder rose 10 in. Find
the contents of the stone.
12. A bin is 11 ft. 6 in. long, 4 ft. wide, 6 ft. 3 in. deep. How
many bushels of wheat will it hold ? How many of apples ?
156 SIMPLE EQUATIONS.
1. re — 4 a= 6 el) 1° ^^^^ equation we desire to get rid
of — 4 in the first member. This we can
do by adding 4 to both members. What
difference do you notice between the first
and third equation ?
X- 4 = 6
(1)
+ 4 = 4
(2)
x= 6 + 4
(3)
3! =10
(4)
2x+ 6= 14
(1)
- 6=-6
(2)
2a; = 14 -6
(3)
2x= 8
x= 4
2. 2 a? 4- 6 = 14 Cl^ ^^ *^^* example we desire to elimi-
nate + 6 in the first member. Why do
we add — 6 to both members ? What
difference do you notice between the
first and third equations ?
Note. — In the first and second ex-
amples we notice in each case that the
terms that we wish to eliminate from one member appear in the other mem-
ber with their sign changed. This operation is called transposition. Accord-
ing to this any term can he removed from one side^ provided we put it on the
other side with the oppoaite aign.
Find the value of a? in the following :
3. a:-7=4 62;-i-2 = 12.
4. 2a; + 6=8 ^z + 5 ^18.
5. 3a; ~ 7 = 11 2a;- 9 = 21.
6. 2a;- 5 = 7 8a?--7 = 2a
7. J a; = a; + 5 (1) j^ ^^^^ example we must eliminate x
— X = — X (2) in the second member. Why do we add
2x a; = 5 C^V ~~**'® ^^^ members ? Compare the first
x = 5
and third equations.
8. 2a;+ 7 = 15 7 a; — 70 = 5x — 20,
9. 3a;-18= 7-2a; 8a: + 12 = 2a; + 18.
10. 7 a; — 13 = 7 - 6 a; 16 a; -f 14 = 8 a; + 64.
11. 8a; — 16 = 6a; -3 9a; + 12 = 3x + 80.
12. 6a;- 4= 6+3a; 3a;— 6 = 2a;— 2. .
^ __ Q /^i \ Compare the first and second equations. What
4 ^^ change has taken place ? Why did we multiply
. both sides by 4? How is the third equation
!L£ = 12 (2^ formed from the second ?
^ Note. — This operation is called clearing the
a; = 12 (S^ equation of fractions.
^^- ^ + 2 + 4 = ^- 2 + 3='^"^-
ALGEBRAIC EQUATIONS, 157
1. Find two numbers whose sum is 180, and whose differ-
ence is 40.
2. Divide the number 133 into two parts such that one part
is 15 more than the other.
8. Three men together have 168,400. If the second has
18,000 more than the first, and the third $2,400 more than the
second, how much has each ?
4. If to three times a number I add 38, I shall obtain 98.
What is the number ?
6. A man and his two sons sawed 25 cd. of wood. The
elder son sawed 5 cd. less than three times as many as the
younger son, and the father sawed twice as many as the elder
son. How many cords did each saw ?
6. A man sold three houses of equal value, and a barn for
$24,400. If the barn brought $1,600 less than a house, what
was the price of each ?
7. One number is 4 times another, and their difference is
30. Find the numbers.
8. John is two times as old as Henry, and the sum of their
ages is 18 years. What is the age of each ?
9. Alice had some money, and earned twice as much. After
spending 9 cents, she had 21 cents left. How much money did
she earn?
10. The sum of two numbers is 155, and the greater is 4
times the less. What are the numbers ?
11. The difference of two numbers is 132, and the greater is
4 times the less. What are the numbers ?
12. Samuel has a certain number of marbles, and John has
15 more than Samuel. If together they have 103 marbles, how
many has each ?
13. The sum of two numbers is 58, and the greater is 7 less
than 4 times the smaller. What are the numbers?
14. 3a? + 4 = a; + 10. Find the value of x.
158 CONSTRUCTION.
1. Draw a horizontal line, AB. Draw two lines, CD and JEF^
perpendicular to AB. Where will CD and J?^meet? Why?
2. Draw a horizontal line, AB^ and at each extremity draw
an- oblique line, making with AB an angle of 45°.
3. Draw a vertical line, AB, At each extremity draw two
oblique lines, each making an angle of 75° with AB. Where
will the oblique lines meet? Why?
4. Draw an oblique line, AB. Draw two lines, CD and EF^
making with AB an angle of 65°. Will these oblique lines
meet?
5. Draw a horizontal line, AB. Draw twa lines, CD and
EF^ making respectively angles of 45° and 75°. If you pro-
duce the two oblique lines, will they meet ?
6. Draw a horizontal line, AB. Draw an oblique line, (72),
making with AB an angle of 60°. Draw EF parallel to AB^
and cutting CD. Draw OH parallel to (7i>, and cutting AB.
7. What is the ratio of one rt. Z to 4 rt. ^ ?
8. What is the i-atio of one-fourth of a rt. Z to 2 rt. ^?
9. How can you test whether two intersecting lines are
perpendicular to each other?
10. Make an angle. Make another angle 3 times as large.
Make another \ as large.
11. An angle formed by a vertical line meeting a horizontal
line is an angle of degrees ?
12. Draw an angle that is equal to one-half of a right angle.
13 Draw a right angle. Cut from it an angle of 25 degrees.
How many degrees are there in the remaining angle ?
14. Draw a quadrilateral that is not a parallelogram.
15. Draw an equilateral rectangular parallelogram.
16. Draw a rhombus having an angle of 75°. Write the
number of degrees in each of the other angles.
17. Using protractor, draw two lines perpendicular to each
other.
REVIEW OF PERCENTAGE. 159
1. What per cent of 12 bu. is 6 pk. ?
2. A horse was sold for $226 at a gain of 12 J%. What
did it cost ?
8. Two boys have 567 apples. One has 83 J % more than
the other. How many has each ?
4. What fraction increased by 20% of itself equals § ?
5. One farmer has 306 sheep; this is 66% less than B has.
How many have both ?
6. If I lose 13% by selling an article for 52 cents less than
cost, how should it be sold to gain 12% ?
7. I gained 23% by selling 24 bbl. of apples for $8.28 more
than cost. What was the cost a barrel ?
8. A commission merchant charged $17.28 for selling 320
bu. of potatoes at 60/ a bushel. What was the rate of his com-
mission ?
9. Find the premium paid for insuring buildings for $3,500
at li%, and furniture worth $3,000 at i%.
10. If I pay $468 to insure property worth $10,400, what is
the rate?
11. The valuation of a town is $175,600. There are 276
polls, each assessed $1. The town wishes to raise $2,910. What
tax will a man pay who owns a house valued at $3,000 ^
12. What is the interest on $1,785 from Aug, 16, 1898, to
Mar. 28,1901, at7%?
13. A and B received $3,159 as the interest on their money
invested for 6 yr. 6 mo. at 6%. If B's money equals i of A's,
how much money has each ?
14. A furniture dealer sold a parlor-set for $70, and gained
16 §% by so doing. What per cent would he have gained had
he sold it for $72 ?
16. A bookseller sold a set of encyclopedias for $26, at a
profit of 8J%. What per cent would he have made by selling
the set for $28 ?
160 ORAL.
1. In 840 in. how many feet?
2. At 2/ a foot, find the cost of a rope 720 in. in length.
3. At 1/ an inch, find the cost of three pieces of ribbon.
In one piece there are 27 in., in another I yd., and in the third
li yards.
4. If you earn $25, and then spend $17.26 for a suit of
clothes, $3.25 for a pair of shoes, how much of your money will
you have left ?
6. If two desks are worth $li, what are two dozen desks
worth ?
* 6. At 40/ a peck, what will a farmer receive for 96 qt. of
beans?
7. Find the cost of 4i lb. of beefsteak at 16/ a pound, and
four chickens at $li each.
8. If a box of butter contains 26 lb., what will 4 boxes cost
at 25/ a pound ?
9. Find the cost of an ounce of tea when ^ lb. costs $.82.
10. If a man's salary for 8 months is $840, what will be his
salary for 6 months ?
11. 250 H- ? of 25 = ?
12. How many times can you sell a quart of chestnuts if you
seU3bu.?
13. The divisor is 9, the quotient 16. What is the divi-
dend?
14. If I have $3.50 at first, how many peanuts can I buy at
5/ a quart and have 30 cents left ?
16. Find the cost of 4^ gal. oil at 8/ a gallon.
16. Find the cost of 1 lb. of candy if 2 oz. cost 12/.
17. What is the seventh part of 574 ?
18. If 2 bu. of apples cost $2.40, what will a peck cost?
19. How many sevens must be added together to make 224 ?
80. $18 pays for how much insurance at i% premium ?
21. What will 6 acres of land cost at $60i an acre ?
COMPOUND INTEREST. 161
(Review pa^s 11 to 19.)
1. What is a savings bank ?
2. How many of you have money in a savings bank ?
3. Why is it a good plan to put money in a bank?
4. Suppose you each put $200 in a savings bank Jan. 1,
1902, and the bank pays 4% interest How much interest will
be^ due Jan. 1, 1903 ?
5. To whom does this f8 belong?
6. If you go to the bank Jan. 1, 1903, and the bank pays
you f 8, how much money will you have in the bank for the
next year ?
7. If you do not call for the $8, what will be done with it
by the bank officials ?
8. If the $8 is added to your $200, how much money will
you have on interest the second year ?
9. What will be the interest of $208 for the second year?
10. When the interest is added to the principal, so as to find
interest on principal and interest, it is called Compound Interest.
11. Interest at savings banks is usually compounded semi-
annually. Sometimes quarterly. Compound annually unless
otherwise directed.
12. In the 9th question what will be done with the $8.32 ?
13. How much will you have on interest then ?
14. How much will the interest be the third year?
16. What will be the new principal for the fourth year?
16. This example is worked like the following :
$200.00 let Principal.
8.00 Interest for 1st year.
$208.00 Principal for 2d year.
8.32 Interest for 2d year.
$216.32 Principal for 8d year.
8.653 Interest for 3d year.
$224,973 Amount in bank at end of 4tli year.
200.00 Ist Principal.
$24,973 Compound Interest for 3 years.
162 COMPOUND INTEREST,
1. When there is a part of a period remaining, find the
intei-est for the part period, aild add it as for the whole period.
2. What is the compound interest of $450 for 4 yr. 3 mo.
15 da. at4%?
3. What is the difference between the simple and compound
interest of $500 for 6 yr. 6 mo. 18 da. at 6% ?
Find the compound interest of the following :
4. $4,500 for 2 yr. 6 mo. at 8i%.
5. $1,278 for 3 yr. 9 mo. at 4%.
6. $2,576 for 4 yr. 2 mo. 12 da. at 5%.
7. $1,563 for 5 yr. 3 mo. 21 da. at 4i%.
8. $6,793 for 3 yr. 6 mo. 15 da. at 4%,
9. $728 for 2 yr. 9 mo. 24 da. at 3%.
10. $1,560 for 6 yr. 6 mo. 6 da. at 6%.
11. Find the compound interest of $600 for 1 yr. 9 mo. at
4% annually, compounded semi-annually.
Note. — 4^ is the annual rate. The rate for 6 mo. would be what ?
12. Find the compound interest of $550 for 1 yr. 4 mo. 12 da.
if compounded quarterly at 1 % a quarter.
Find the difference between the simple and compound in-
terest of:
13. $1,678 for 2 yr. 5 mo. 19 da. at 6%.
14. $2,768 for 3 yr. 3 mo. 3 da. at 3%.
15. $1,248 for 4 yr. 4 mo. 4 da. at 4%.
16. $678 for 3 yr. 11 mo. 21 da. at \\%.
17. $624 for 5 yr. 7 mo. 21 da. at 5%.
18. $4,635 for 2 yr. 4 mo. 15 da. at 3i%.
19. $6,745 for 3 yr, 9 mo. 17 da. at 4%.
20. $6,476 for 4 yr. 5 mo. 13 da. at 5%.
21. $4,124 for 3 yr. 3 mo. 3 da. at 6%.
22. $2,146 for 4 yr. 4 mo. 4 da. at 4%.
23. $1,486 for 5 yr. 5 mo. 5 da. at 5%.
84. $4,238 for 3 yr. 8 mo. 18 da. at 6%.
MISCELLANEOUS REVIEW. 163
1. At $1.75 a cubic foot, find the value of a block of marble
in the shape of a square pyramid, whose height is 18 ft., and
the sides of whose base are each IJ ft.
2. At 8/ a square foot, what will it cost to paint the gable-
ends of a house 26 ft. wide, if the height of the gable is 8^ ft. ?
3. A plank contains 40 J board feet. If it is 20' 2" long,
and 3" thick, how wide is it ?
4. What must I pay for 6 pieces of broadcloth, 150 yd.
each, at 80/ a yard, 10 ^{j off for cash?
5. Bought a bill of goods for $640, but by paying cash I
was allowed a discount of 10% and 5%. What was the dis-
count? and what did I pay?
6. What is the compound interest of $3,460 for 4 yr. at 6 % ?
7. A man placed a certain sum of money at simple interest,
at 6%, when his son was bom, and when the son became of
age the amount was $2,260. What was the sum?
8. I borrowed $500 May 6, 1902, at 6% interest. I returned
it when it amounted to $600 ; when was the money returned ?
9. A house valued $3,600 rents for $20 a month ; what
rate per cent does it yield ?
10. What is the interest of $877.50 for 50 d. at 8% ?
11. What is the interest of $1,250 for 54 d. at 6% ?
12. What is the interest of $650.30 from May 10, 1901, to
July 16, 1904, at 7% ?
18. What is the duty on 60 hhd. of 63 gal. each, and 26 gal.
of syrup, worth 70/ a gallon, leakage 3%, duty 12i%.
14. Mr. Snow owns property, valued at $16,480, in a town
whose valuation is $960,000. What will be his share of a tax
of $33,260?
15. If the cost of insuring property at 2j% is $121.50, what
is the value of the property?
16. The area of a triangular lot is i A. If the base is 200
ft., what is the altitude ?
164 MISCELLANEOUS REVIEW.
1. A farmer raised in four fields the following quantities
of com : 66 bu. 3 pk. 2 qt. ; 98 bu. 1 pk. 3 qt. ; 110 bu. 5 qt. ;
176 bu. 3 pk. 6 qt How much did he raise in the four fields ?
2. If a man travels 97 m. 120 rd. every day for 15 days,
how much does he lack of having traveled 1,600 miles?
3. At 12/ a pound, how much sugar can be bought for
$1,128.96?
4. Reduce 25,480 m. to weeks, etc.
6. At $6i a cord, find the cost of a pile of wood 20 ft.
long, 4 ft wide, 6 ft. high.
6. Bought 8 A. of land at $175 an acre, and sold it at 22/
a sq. ft. What was the gain or loss ?
7. What is the value of 16,840 lb. of wheat at 98;? a
bushel? A bushel of oats weighs 60 lb.
8. Find the value of a load of straw weighing 1,860 lb. at
$18 a ton, and a load of hay weighing 2,520 lb. at $18.50 a
ton.
9. Divide 4.7151 by .604^.
10. If 5| yd. of carpet cost $7^, how much will 12i yd. cost?
11. A man's income in 3 years was $4,020. If his income
the second year was 15% more than the first, and his income
the third year was 20% more than the first, what was his
income each year?
12. Find the volume of a cone whose altitude is 36 ft. and
the radius of the base 15 ft.
13. At $12.75 a sq. ft., what will it cost to gild the hemi-
spherical dome of an observatory 40 ft. in diameter?
14. At each comer of a square 50 ft. on a side, with a radius
of 25 ft., segments of circles are drawn. Find the area within
the square not included in the segments.
15. Three boys had 85 marbles. The second boy had 4 more
than 3 times as many as the first, and the third had 6 times as
many as the first How many had each ?
TO FIND THE CONTENTS OF A SPHERE. 165
TTse the wooden sphere to illustrate this point, or take a sphere (a round
potato or apple) and cut it into several small pieces, each shaped like a
pyramid. To do this the cut must in each case reach the center of the
sphere.
Examine each piece. The base of each piece is a part of what ? The sum
of all the pieces is what ?
The altitude of each piece is what of the sphere ?
How do you find the volume of each piece ?
State the rule for finding the volume of a sphere.
If we multiply the surface of a sphere (the sum of all the bases of the pyra-
mids) by J of the radius (the height of the pyramids), what shall we have?
Learn : To find the volume of a sphere, multiply its surface
by J of its radius.
Can you tell why you multiply by } of the radius ?
1. Find the volume of a sphere whose radius is 5 in.
2. If the diameter of a sphere is four inches, what is its
volume ?
3. The radius of a sphere is 15 in. Find its surface ; its
volume.
4. How does the volume of a sphere 4 in. in diameter com-
pare with the volume of a cylinder 4 in. in diameter and 4 in.
in altitude ?
5. In the fourth example, if the sphere was cut out of the
cylinder, what part would be cut away ?
6. A man cut a cylinder of the largest possible size out of
a cubical block of wood measuring 2 ft. What part of the cube
did he cut away ?
7. If a sphere is cut out of a cube, what part of the cube
must be cut away?
8. How does the volume of the sphere compare with the
volume of the cube ?
9. Find the volume of a sphere whose diameter is 4 ft.
10. From a sphere 15 in. in diameter two inches in thickness
were cut off. How many cubic inches were cut off ?
11. Find the volume of a sphere whose radius is 6 in.
166 REVIEW OF PERCENTAGE.
1. If at 2J% premium, I pay fl77.60 for insurance on my
property, what is the value of the property ?
2. I paid a lawyer $115.65 for collecting a bill at 4i%.
What was the amount of the bill ?
3. An agent sold goods to the amount of $6,680.00. He
paid $19.40 for storage, $60 for freight, and returned $6,300
to me. What was the rate per cent of his commission ?
4. An auctioneer sells goods to the amount of $1,920, and
charges $48. What per cent does he receive as commission?
6. An agent sold 75 bales of cotton, each bale weighing
850 lb., at 12i^ a pound, on a commission of 2|%. What was
his commission ?
6. A dealer sold goods at $1.84 a yard, and gained 15%.
At what should he have sold a yard to gain 18|% ?
7. Sold two houses at $4,968 each, gaining on one 8%, and
losing 8% on the other. What did I gain or lose ?
8. By selling goods at $237.60 less than cost I lose 44%.
At what should I sell them to gain 16% ?
9. A merchant having $10,000 worth of goods, lost 25% by
fire, and sold the remainder at a gain of 40%. What was the
gain or loss per cent?
10. If the cost of coal at the mine is $2.60 a short ton, and
the freight $1.30 a ton, at what price must it be sold to gain
30%?
11. Bought a house for $3,300 when property was depre-
ciated 40% in value. What was the value of the house ?
12. The number of the pupils enrolled in the schools of a
city is 5,805, which is 35% more than the number in atten-
dance. What is the number in attendance ?
18. Find the amount of duty on the following invoice :
80 hhd. molasses, 63 gal. each ^ 26/ ; 38 hhd. sugar, 380 lb.
each (a) 4i/ ; 250 boxes raisins, 20 lb. each (a) 7^/. The duties
are : molasses, 20% ; sugar, 30% : raisins, 10%.
MEASUREMENTS. 167
1. Two of the boundary lines of a field run north and
south, and are 60 rd. and 48 rd. in length. The distance be-
tween them is 36 rd. Find the area of the field.
2. A cylinder is 8 in. in diameter and 28 in. in length.
Find the volume of the largest cone that can be cut from it.
How many cubic inches of the cylinder must be cut away ?
3. Find the volume of a square prism the perimeter of
whose base is 44 ft., and whose altitude is 42 ft.
4. At $1.87i, what will it cost to carpet a room 18 ft. long,
15 ft. wide, with carpet 27 in. wide, if the breadths run length-
wise?
6. At $30 per M., what shall I pay for 3 boards, each 12 ft.
long, 16 in. wide at one end, and 10 in. at the other?
6. What will a pile of wood cost at f 7.50 a cord, if it is
16 ft. 8 in. long, 4 ft. wide, and 6 ft. 3 in high?
7. A piece of land 35 rd. long and 7 rd. wide is divided
into 5 square lots of equal size. What will be the cost of
boundary and cross fences at $2.1 2^ a rod ?
8. My house is on a corner lot, — 150 ft. on one street, and
60 ft on the other. The sidewalk is 6 ft. wide. How many
cubic feet of snow do I shovel in clearing my walk after a
15-in. snowstorm?
9. How many gallons in a tub having a base of 3J sq. ft.
and a depth of 15 in. ?
10. How many bricks are necessary for the 12-in. walls of a
house 40 ft. long, 28 ft wide, 22 ft. high, deducting 66 cu. ft.
for openings ?
11. How many feet (board measure) in 48 joists, each 18 ft.
long, 10 in. wide, 2^ in. thick ?
12. How many bunches of shingles laid 4 in. to the weather
will cover a roof, each half of which is 40 ft. x 20 ft.?
13. What will it cost, at 25/ a cut for every cord, to saw
into four pieces 4-ft. wood, piled 60 ft. long, and 6 ft. high ?
168 SIMPLE EQUATIONS,
1. -
2.
8.
4.
1- 8=^-4
2 4
0)
*/ 82 - ^'^ 16
2 4
(2)
2a; -32 = a; -16
2a;- a; = 32 -16
a; = 16
I + * = T + ''
V%T=f + ^
2« + «/-2/_24
4 4
7 a; a; a; 31
2 ■^4'^8~ 2
3a; 2a;
4 ^- 8 +^
2a; a; . x x
3 ''■4~12~6
+ 7
i-|+2=s
^ + 16-1 + 1 + 17
« + i-^» = I«
Compare the second equation
with the first. Why did we mul-
tiply by 4 instead of 2 ? What is
the least common denominator?
Carefully note and explain each
step in the solution of this prob-
lem.
Note. — In the following ex-
amples multiply by the least com-
mon denominator to clear the
equation of fractions.
2a;
-2 =
X
^.
4
3
1x
-5 =
5x
-3.
8
3
6a;
10a: ,
< A
- = tt.
4
8
X
38
35
X
4~
■ 2 ~
2 ■
~4-
2a;
6a;
^ 6
-8
= 9,
1-8+1 = 6-8.
XX X _ 5
4 "*" 8 " 6 ^ 12*
, X X rj
^ + 2 + 4 = ^-
10. a; + ^-40 = ^ f+f + f = 28.
2 ' 4 ' 8
2a; , .„ 4a; . ^ „ 3a;
X
11. t^ + 12=Y + 6 3a; + Y + 15 = | + 41.
12 ^4- 4-10- i^ if 2-q ^^
18. 2+4-10-— __2-3-— .
IS. 3a; - 3 = 7a; - 15. 5a; - 10 = 3a; + 4.
CONSTRUCTION. — TRIANGLES. 169
(Review page 168.)
1. Draw a horizontal line, AB^ 2 in. long. At point A
make an angle of 60° by drawing AC 8 in. long. Join BC.
2. What is the figure ABO called ?
3. How many sides and angles has it?
4. What kind of angles are A, B, and C?
6. Since all the angles are acute, the figure is called an
acute triangle.
6. Draw the line AB. At A make an angle of 110° by
drawing AC. Connect BO.
7. This figure is called an obtuse triangle. Why ? What
is an obtuse triangle ?
8. Draw AB. At A make a right angle by drawing AO.
Connect BO.
9. What name is given this figure ? Why ?
10. What is a right triangle ?
11. In this right triangle the line AB is the base, — the
line on which it rests ; the line AO is the
perpendicular; the line BO is the hypotenuse.
12. The hypotenuse is always opposite
what angle?
18. Define base, perpendicular, hypotenuse.
14. How many right angles can a triangle have? Obtuse
angles?
15. Draw two lines, AB and A (7, each 3 in. long, and form-
ing an angle at A. Join BO.
16. When two sides of a triangle are equal, the triangle is
called an isosceles triangle.
17. Define an isosceles triangle.
18. Draw an isosceles triangle that shall contain an obtuse
angle.
19. Draw an isosceles triangle that shall contain a right
angle.
170 ORAL.
1. Two towns are 160 miles apart. If the railway fare is
$3.00, what is the rate a mile ?
3. What is the cost of 6 suits of clothes at f 15 each, and
•1 hats at $1.50 each?
8. At 3/ an ounce, what is the cost of 2 lb. of pepper?
4. What is the cost of 5 J lb. of cheese at 18/ a pound?
5. If 16 sheep cost $96, what will 30 sheep cost ?
6. How many yards long is a piece of cloth that is 720 in,
long ?
7. Find the cost of Ij bu. of potatoes at 15/ a half-peck.
8. How many quarter-inch squares can be cut from a 2-iiu
square ?
9. Give answers:
8,100 - 100. 66,000 -8- 22,000. 18 x 800.
9,300 -r- 100. 89,000 ^ 13,000. 24 x 400.
11,200 - 100. 66,000 -- 12,000. 16 x 200.
10. At 32/ a pound, find the cost of 2 lb. 5 oz. of butter.
11. What is the total cost of 2 lb. tea at 75/ a pound, and
i bbl. flour at $6 a barrel?
12. At 80/ a pound, what will 6 oz. of tea cost ?
18. A boy had 64 stamps. How many had he after he had
sold 28 and bought 16?
14. A grocer sold 16 lb. 8 oz. of tea on one day, and 17 lb.
8 oz. on another day. How much tea did he sell on both days ?
15. If you have 75 cents, how much will be left after paying
for J lb. of 80-cent tea, and a pound of 35-cent coffee ?
16. A man paid $50 for a parlornset, and $30 for a bedroom-
set. He paid \ in cash, and tlie rest in 6 monthly payments.
17. Find the cost, at 18/ a pound, of two hams, one weigh-
ing 4 lb. 8 oz., and the other 6 lb. 8 oz.
18. When apples are worth a half-dollar a bushel, how many
bushels can be bought for $16 ?
19. How many quarts in 25% of a peck?
PARTIAL PAYMENTS. 171
$690. New ffaven, Conn., Feb. 3, 1901.
On demand^ for value received^ I promise to pay
to Charles W. Ashley^ or order^ Six Hundred Ninety
Dollars^ tvith interest at 6%, M. E. Martin.
1. In the above note who is the Maker? The Payee?
What is the Face, or Principal ?
2. Promissory notes can be bought and sold like other
forms of property, when they are negotiable.
8. What words in the above note gives Charles Ashley a
right to sell this note ?
4. If Charles Ashley should sell this note, he would write
his name on the back as an Indorser, and thus be held responsi-
ble for its payment to the holder in case the maker fails to pay
it when due.
5. Is this a demand or a time note ?
6. Substitute "three months after date" for the words
"on demand." What kind of a note is it now ?
7. Sometimes two persons sign the note, when it would
read, "we, jointly and severally, promise to pay," or "we, or
either of us, promise," etc.
S\ If the words " with interest " are in the note, it draws
interest from date to payment. If these words are omitted from
a demand note, the note will not bear interest ; if omitted from
a time note, the note will not draw interest until after it
becomes due.
9. Write a non-negotiable demand note.
10. Write a time, joint and several, negotiable note.
11. Substitute "bearer" for "order." Does this change
the meaning?
12. In each of the notes that you have written, name the
maker and payee.
13. Write a time, interest-bearing, negotiable note.
172 PARTIAL PAYMENTS.
1. Instead of paying the note in full, it frequently happens
that part payments are made at different times.
2. A record of each partial payment, with date of payment,
is made on the back, and called an Indorsement.
3. These indorsements were made on the foregoing note :
Dec. 8, 1901, $40; April 3, 1902, $60; Dec. 3, 1903, $150.
4. If Martin, on Dec. 3, makes a payment, how long will
he have used Ashley's money ? How much will he owe as in-
terest? How much will be due Dec. 3, 1901? Ana. $724.50.
5. If Martin then pays $40, how much will he still owe ?
6. If Martin comes again April 3, 1902, how long will he
have used $684.50? How much interest will he then owe?
What is the interest of $684.50 for 4 mo. ? What is done with
this interest? How much will Martin owe Ashley, April 3,
1902? If he pays him $60, what balance will still be due ?
7. On Dec. 3, 1903, how long has $638.19 been on interest ?
To what has it amounted ? What payment was then made ?
What balance was left due ?
8. If the note was paid Feb. 27, 1904, what sum was paid ?
9. The following is a good form to use :
Original principal $690.00
Interest from Feb. 3, 1901, to Dec. 8, 1901 (10 months) . 84.50
Amount due Dec. 8, 1901 $724.60
First payment 40.00
Balance due Dec. 3, 1901, or Second Principal . . . $684.50
Interest from Dec. 3, 1901, to April 8, 1902 (4 months) . 13.69
Amount due April 3, 1902 $698.19
Second payment 60.00
Balance due April 8, 1902, or Third Principal . . . $688.19
Interest from April 8, 1902, to Dec. 8, 1903 (20 months) . 63.82
Amount due Dec. 1908 $702.01
Third payment • . . . 150.00
Balance due Dec. 8, 1908 $662.01
Interest from Dec. 8, 1903, to Feb. 27, 1904 7.72
Amount due at settlement $559.73
PARTIAL PAYMENTS. ITS
1. Write a composition on Partial Payments. Carefully
describe each step taken in working a problem. Write a rule
for another's guidance.
2. $1,000. Springfield, Dec. 13, 1901.
For value received^ I promise to pay Clarence
Rogers, One Thousand Dollars, with interest at
6%, G. A. Morse.
Indorsements: $231, June 19, 1902; 1350, Oct. 1, 1902;
$125, Feb. 19, 1903. How much was due June 25, 1903?
3. $2,600. Boston, Mass., Dec. 14, 1900.
On demand, for value received, I promise to pay
Charles Conway, or order. Two Thousand Five
Hundred Dollars, mth interest.
Martin 0. Sikes,
Payments : March 26, 1901, $50 ; Nov. 1, 1901, $500 ; Dec.
19, 1902, $1250. What is due Dec. 31, 1903 ?
4. $850.75. Worcester, Mass., Jan. 1, 1900.
For value received, I promise to pay Flora Jor-
dan, or order. Eight Hundred Fifty -^j^ Dollars,
with interest at 6%. E. S, Smith.
Indorsements: July 16, 1900, $150.00; July 80, 1902,
$450.00 ; April 9, 1903, $342.39. If this note is settled Dec.
17, 1903, what amount will pay it?
6. Face, $480.50. Date, June 15, 1900. Indorsements:
Nov. 30, 1900, $175.75 ; Sept. 2, 1901, $140.00 ; Oct. 9, 1902,
$85.00 ; May 18, 1903, $90.00. What is due June 1, 1904?
6. Face, $1600. Date, Sept. 16, 1900. Indorsements:
June 18, 1901, $400 ; Oct. 15, 1902, $500 ; Jan. 15, 1903, $300.
With interest at 5%, what is due May 10, 1903?
7. Supplying names, write the 5th example as a note.
174 PARTIAL PAYMENTS.
1. 9336. BogUm, March 26, 1901.
On demand^ we promise to pay J. C. Stephens^
or bearer. Three Hundred Thirty-six Dollars, with
interest. Value received. Rogers ^ Brown.
IndorsemeDts: July 20, 1901, $66; April 7, 1902, *8,
Sept 26, 1902, f6.00; Jan. 7, 1903, «160. What is due
May 1,1903?
Principal 9336.00
Interest to July 20, 1901 (3 months, 24 days) .... 6.38
Amount due July 20, 1901 f342.38
First payment 55.00
Second Principal 1287.38
Note. — We find that the interest due at tho next payment is $12.36. As
the payment was less than the interest, we make no use of it on that date, but
consider it as paid Sept. 26, 190^. We find that the interest from July 20,
1901, to Sept. 26, 1902, is ^0.40. As the united payments only amount to
f 14.00, we must again consider that no payment was made on the principal
until Jan 7, 1903.
Interest from July 20, 1901, to Jan. 7, 1903 $25.29
Amount due Jan. 7, 1903 $312.67
Payments ($8 + $6 +f 160) 174.00
Balance due, or Third Principal . $138.67
Interest from Jan. 7, 1903, to May 1, 1903 2.63
Amount due May 1, 1903 $141.30
Note. — Unless the payment and the interest are very nearly equal, you
can mentally calculate whether the payment exceeds the interest or not.
2. Face, $1,650. Date, May 12, 1900. Indorsements:
Jan. 24, 1901, $140.50; Dec. 6, 1901, $20.10; Aug. 15, 1902,
$136.87; Dec. 6, 1902, $75. What was due April 24, 1903?
3. Face, $165. Date, April 15, 1900. Indorsements :
May 24, 1901, $24.18 ; July 18, 1902, $5.25 ; Sept. 6, 1902,
$45.00 ; Jan. 24, 1903, $40,00. What was due April 15, 1904 ?
4. A note of $720, dated Aug. 14, 1897, has the following
indorsements: Dec. 26, 1898, 200; Sept. 14, 1901, $175;
Dec. 31, 1902, $400. Settled Dec. 31, 1903. Fmd the sum
paid at time of settlement.
6. Write the rule for Partial Payments.
PARTIAL PAYMENTS. 175
Mebchant^s Rule.
This rule is usually used when settlement is made within a year.
A note of $850 was dated Jan. 2, 1903. The indorsements were : March
18, $200 ; May 2, $160 ; Aug. 18, $300. What was due Dec. 2, 1903 ?
Amount of $850 from Jan. 2, 1903, to Dec. 2, 1903 . . . $896.75
Amount of $200 from March 18 to Dec. 2 . . . $208.47
Amount of $160 from May 2 to Dec. 2 . . . . 166.26
Amount of $800 from Aug. 18 to Dec. 2 . . . . 305.20 668.92
$227.83
1. What is the face of this note ? For how long a time
was it on interest? To what does it amount Dec. 2, 1903?
2. What was the first payment? On what date was it
paid ? For how long a time did the payee have the use of
this $200 ? To what did it amount?
3. Answer the same questions for each of the other pay-
ments.
4. To what do all the payments amount?
6. What is the difference between the amount of the note
and the amount of the payments ?
6. Write a clear analysis of an example in Partial Pay-
ments performed by the Merchant's Rule.
7. A note of $1,250.60, dated July 6, 1901, with interest
at 7%, was indorsed as follows: Sept. 21, 1901, $260; Nov.
22, 1901, $325 ; March 6, 1902, $120 ; May 17, 1902, $250.
What was due at settlement, July 6, 1902?
8. A note of $2,000, dated Jan. 20, 1902, had the follow-
ing indorsements: May 20, 1902, $100; July 20, 1902, $175;
Dec. 23, 1902, $250. Find the balance due Jan. 15, 1903.
9. A note for $3,000, dated Jan. 1, 1902, had indorsements
as follows: March 1, 1902, $300; Oct. 1, 1902, $150; Nov.
1, 1902, $1,500. What is due Jan. 1, 1903?
10. Note, $460. Date, May 9, 1902. Settled, Feb. 24,
1903. Payments: July 1, 1902, $120 ; Sept. 16, 1902, $150;
Jan. 2, 1903, $100.
176 REVIEW OF PERCENTAGE,
1. A note of $1,200, dated Aug. 15, 1902, was indorsed as
foUows : Dec. 16, 1902, f 20 ; Sept. 15, 1903, $150. Find the
amount due Aug. 15, 1904.
2. By selling a farm for $4,400 I lost 8i%. What had I
paid for the farm ?
3. A farmer offered a cow for sale for $40. He sold her
at 10% discount, and yet made 25%. What was the cost of
the cow?
4. I bought a chair for $8, 20% off, and sold it for $10,
15% and 6% off. How much did I gain?
5. A dealer sold two horses for $300 ; on one he gained
12J%, and on the second he lost 20%. Did he gain or lose if
for the second horse he received § as much as for the first?
6. A furniture dealer had 800 chairs, worth $6 each. A
fire destroyed 30% of them, and he sold the remainder at $8.60
each. How much did he lose ?
7. If a man starts in business with $8,000, and each year
gains 12 J % of his capital, what will he have at the end of three
years?
8. Find the interest of $651 for 16 days at 9%.
9. A debtor owed me $1,560. A lawyer collected 76% of
the debt, and charged 5% commission. How much did I
receive ?
10. A man who owned 75% of a ship, sold 40% of his inte-
rest for $30,000. At that rate, what was the value of the
whole ship?
11. What is the value of 75% of a farm, if | of it is worth
$4,000?
12. Bought 67^ yd. of carpet at $1.87i, receiving a discount
of 15%. What was my bill?
13. 1,320 is 12% less than what number?
14. Mr. S. has a salary of $1,800, and pays $378 rent.
What per cent of his salary does he pay for rent?
REVIEW OF PERCENTAGE. 177
1. If a merchant's gain at retail is 35%, and he sells at
wholesale for 10% below his retail price, what is his gain at
wholesale ?
2. Find the per cent of lighting surface to floor surface in
a room 28' x 82', with 8 windows, each 3' 6" x 8'.
3. The owner of 66f % of a ship sold 50% of his interest for
132,000. Find the value of the whole ship at the same rate.
4. A man bought a horse for 1400, which was 20% less
than its real value, and sold it at 20% above its real value.
Find selling-price.
5. What is 62i% of a sum of money, if 76% of it is «1,200
more than 66§% of it?
6. What premium must a man pay on furniture worth
$1,800, insured at 87^% of its value, at lf% premium?
7. By selling an article at $6.65 a man lost 5%. For how
much must he sell it in order to gain 5% ?
8. A man sold a paper-mill, receiving 45% of the price in
cash. He invested J of the sum received in a farm worth
$2,160. For how much was the mill sold ?
9. A man, who had been paying $25 a month rent, borrowed
$4,000 at 5%, and bought a house. Instead of rent he now
pays interest on the borrowed money, $50 a year taxes, $8
wateivtax, $12 insurance, and $25 for repairs. Find his gain
or loss a year.
10. Leaving 87^% of my money at home, I spent 5% of the
rest for butter at 29/ a pound. I bought 40 lb. of butter.
How much money had I at first ?
11. If a lawyer retained $9.08 for collecting $181.60, at the
same rate, what would he need to collect to receive $20,000 a
year?
12. What is the compound interest of $236 for 1 yr. 6 mo.
at 8% per year, interest compounded semi-annually?
13. What is the duty on 46,080 pencils at 2/ a gross?
178 MISCELLANEOUS REVIEW.
1. A physician, whose charges are $1.50 a visit, made an
average of 6 visits a day in the year 1903. He collected 65%
of his charges, and saved 40% of the sum collected. At that
rate how much could he save in 3 yr. 6 mo. ?
2. Owing to a deficiency in the appropriation bill, the sal-
aries of the clerks in the post-office were reduced 16% for the
last quarter of the fiscal year. How much did a clerk who was
paid $336 for the last quarter receive during the whole year?
3. The cost of insuring a store at 1^% is $108 a year, and
the cost of insuring its contents at 2J% is $175.50. What is
the whole amount of insurance ?
4. The ice on a circular pond is 18 in. thick. If the pond
is 800 ft. in circumference, how many cubic feet of ice does it
contain ?
5. Two merchants offer the same quality of goods at the
same list-price. The first offers a discount of 10% and 5%,
and the second offers a discount of 15%. Of whom will it be
more advantageous to buy ? and how much will be saved on a
bill, the list^price being $1,050?
6. A gross amount of a bill is $570.35, and the discounts
are 10%, 10%, and 5%. What net cash will pay the bill?
7. Find the cost of papering the walls and ceiling of a hall
36 ft. long, 24 ft. wide, and 18 ft. high, if 64 sq. yd. are allowed
for openings. The paper costs 37 J/ a roll.
8. Subtract 23 ten-millionths from 2 hundredths of 6
thousandths.
9. Multiply four hundred thousandths by four hundred-
thousandths, and divide the product by four tenths.
10. A man's farm is 120 rd. wide. He sells 12 A. off one
end. How much shorter is his farm than it was before ?
11. A suit of clothes cost $17. The trousers cost $1 less
than 3 times as much as the vest cost, and the coat cost 2
times as much as the trousers. Find the cost of each.
FRACTIONS, 179
1. Charles rode 6 hours on his bicycle, going 11 f miles the
first hour, Ti^^j miles the second, 9| the third, 7 J the fourth, and
7f the fifth. How many miles did he ride?
2. A man owning a vessel gave | of it to his son, and sold
25% of the remainder for $2,000. What was the value of the
vessel ?
8. Mr. S. started to walk 21j miles. After walking 5 h.
at the rate of 3 J miles an hour, how many miles of his journey
remained ?
4. Change to common fractions: 2.00375, 76.88, 15.0125.
5. 3 men reap i of a field of wheat in li days. How many
dajrs will it take one man to reap the whole field ?
6. From 2 orchards 120 bbl. of apples were picked. If one
orchard produced § as many barrels as the other, how many
barrels were picked from each field?
7. A cubic foot of water weighs 62^ lb. If copper is Sf
times as heavy as water, what is the weight of a cubic foot of
copper?
8. If a man can row 4| miles an hour in still water, how
many miles can he row in 3 J h. up a river that flows at the rate
of li miles an hour?
0. How many miles can he row in 2§ h. down the same
river ?
10. How many hours will it take him to row 15 miles down
the river ? 15 miles up the river?
11. If a train runs 40 i miles in an hour, what part of a mile
does it travel in a minute ?
12. If a man's debts amount to $10,500, and his property
is worth $4,650, how many cents on a dollar can he pay ?
18. A can build a wall in 12^ days, and A and B together
can build J of the wall in a day. In how many days can B
build it alone ?
14. The divisor is 7^, the quotient If, what is the divi-
dend?
180 ORAL.
1. If 3 men can do a piece of work in 4 days, how long
will it take 24 men to do it ?
2. At what rate will $400 gain $40 in 1 yr. 8 mo.?
8. Seven is three-eighths of what number?
4. Sold a cow for $24, losing thereby 40%. Had I sold
her for 20% advance on the cost, what would I have received
for her ?
6. What is the effect on the value of a decimal, if the deci-
mal point is moved two places to the right "^
6. What is the effect of multiplying the numerator and
denominator of a fraction by 4 ? Why ?
7. Divide .006 by 100. Multiply the same numbers.
8. If the denominator of a fraction is divided by 3, what is
the effect upon the value of the fraction ?
0. In every fraction, what is shown by the denominator?
By the numerator ?
10. What is meant by a decimal fraction?
11. What is a factor ?
12. If a cipher is added at the right of a decimal, what
effect has it on the value of the decimal ?
13. Reduce if to a decimal fraction.
14. What is meant by the ratio of one quantity to another ?
16. What is meant by minuend? By quotient? By mul-
tiplicand ?
16. How many board feet in a plank 14' long, 6" wide, and
3" thick?
17. What is the volume of a square pyramid whose altitude
is 15 in., and each side of the base 10 in. ?
18. A room is f as wide as it is long. Its length is 16 ft
Find the square feet in the floor.
10. What will 5 of a yard cost, if 5 yd. cost 90 cents?
20. How many days from May 18 to July 4 ?
21. How many hours in 83^% of a day?
COMMISSION. 181
To find the sum to be invested, after deducting the per cent
commission from the amount remitted.
1. A merchant sent $9,180 to his agent in Chicago with
which to buy wheat. If the agent charges 2% for buying, how
many bushels of wheat can he buy at 90/ a bushel ?
(a) If an agent is expending money for another, on what has he a right to
take a commiuion ?
(6) Did he spend all of the $9,180 for his employer? Has he a right to
take a 2 % commission on that sum ?
(c) Does this $9,180 include the agents commission?
(d) Does it include the sum spent for wheat ?
(e) What per cent of any number is the number itself ?
(/) If $9,180 includes the agent's commission, 2%, and the sum spent for
wheat, 100%, what per cent is it of the sum spent for wheat ?
(g) If $9,180 is 102% of the sum spent for wheat, what is 100%, or the sum
•pent for wheat ?
102% = $9,180
l%=t^^of $9,180=$90
100% = 100 X $90 = $9,000
(A) At 90^ a bushel how many bushels of wheat can be bought for $9,000 ?
$9,000 -r 90^ = 10,000 times, i.e. 10,000 bu.
(i) 2d Explanation. — If the agent keeps 2%, 2,<z^ on a dollar, how much
money must be sent him to allow him to buy $1 worth of wheat ? If he buys
$1 worth of wheat with every $1.02 sent him, how many dollars* worth will he
buy with $9,180 sent him ?
8. If $10,250 includes the amount expended for wool and
2^% commission to the agent, how much money does the agent
spend in wool ?
8. If $3,549 are remitted to an agent to buy cotton, after
deducting 4% commission how n^uch will be invested in cotton?
4. How many baiTels of flour at $5 each can be bought
with a remittance of $2,575, after deducting 3% commission?
5. A country merchant forwarded 800 bbl. of apples to be
sold at $1.25 a barrel, the agent to receive a commission of 3%
for selling. After paying $5.75 for cartage, and deducting his
commission of li% for investing, he invested the proceeds in
sugar at $9i a bhd. How many hogsheads did he buy?
182 COMMISSION.
1. When $9,823 are sent an agent, whose commission is
4J%, how much is spent for goods?
2. An agent is paid 14% for purchasing goods. What
amount does he purchase from a remittance of $1,258.60 ?
3. An agent is paid 6% for buying goods, what amount can
he buy with $2,660, after deducting his commission?
4. A merchant remitted to an agent $1,412.46, with in
structions to buy apples at $2.16j a barrel after deducting his
commission of 2^%. How many barrels did he buy?
5. A merchant shipped 240 bbl. of flour to be sold at $6 J
a barrel at 3% commission. After paying $15.90 for cartage,
he buys hay at $18 a ton, commission 2j%. How many tons
of hay does he buy?
6. I remit to my agent in Chicago $169,302 to purchase
flour. After deducting his commission of \\% and $48 for
other expenses, how many barrels of flour at $4 a barrel, will
the money purchase ?
7. A dealer shipped $40,000 worth of goods to his agent
with instructions to buy groceries with the proceeds. The
agent charged 2i% for selling and 2% for buying. What sum
did the agent receive as commission ?
8. A commission merchant sold 1,000 bbl. of apples at
$2.60 a barrel at 3j% commission, and invested the net pro-
ceeds in cloth at 26/ a yard. How many yards did he buy,
commission 6%?
0. An agent sold goods for a merchant to the amount of
$1,200, and invested the net proceeds in apples less a commis-
sion of 2i% in both cases. What was his whole commission?
10. I sent $12,300 to my agent, with which to purchase
flour at $5 a barrel, after deducting his commission of 2^%.
11. A grain dealer in Chicago received $2,460 with direc-
tions to purchase com at 60/ a bushel, after deducting his com-
mission of 2i%. How many bushels of com did he purchase?
MISCELLANEOUS REVIEW. 183
1. A and B have together Jfl,05S. If | of A's money is
equal to | of B's, how much has each?
2. The driving-wheels of a locomotive are 15 ft. 9 in. in
circumference. How many revolutions will they make in a
mile? If the wheels revolve 2 J times a second, what is the
rate of speed a mile?
8. How much will it cost to fence 3^ miles of railroad at
the rate of 62i/arod?
4. How many bushels of grain will a bin 6 ft. long, 4 ft.
wide, and 5i ft. high, hold?
5. From a pile of wood 16 ft. long, 4 ft. wide, and 8 ft.
high, there were sold at one time 2\ cd., and at another time
12 cd. ft. What is the remainder worth at $5.75 a cord?
6. A fruit-dealer bought 202i crates of peaches for $225,
but was obliged to sell them at a loss of 20%. For what were
they sold a crate ?
7. How many posts 8 ft. 6 in. apart will it take to inclose
a rectangular field 23 rd. lOi ft. by 179 ft. 6 in. ?
8. A cubic foot of water weighs 62i lb. If pine wood is
60% as heavy as oak wood, and 40% as heavy as water, how
much will a cord of oak wood weigh ?
0. If oranges cost me 20/ a dozen, which is the better offer
and what per cent better : 3 cents each, or 20 % profit ?
10. A clerk who received $100 a month, paid for living
expenses $800 a year. When his salary was increased 25% he
increased his expenses 30%. Did he save more or less than
before his increase? and how much?
11. Make out and receipt a bill of goods sold to-day to
Mary R. Sullivan, as follows: 17i yd. cloth @ 15|/; 2 pr.
shoes @ $3.75 ; If yd. silk @ $2.25.
12. A bushel of com weighs 56 lb., and a bushel of wheat
60 lb. How many bushels of wheat will weigh as much as
446 bu. of com?
184 MISCELLANEOUS REVIEW.
1. A rectangular field contained 40 acres. Each comer
was cut off, forming a triangular lot 60 rd. by 20 rd. What
per cent of the field remained ?
2. I paid |37i for a carpet at $1.26 a square yard. The
width of the floor was 16 ft. What was its length?
3. At what rate will $800 gain $62.60 in 1 yr. 3 mo. ?
4. Sold a span of horses at 80% gain, and with the money
bought another span, which I sold for $364 and lost 12i%.
What did each span cost?
5. A man paid % of his money for a farm ; had he paid $76
more he would have paid f of his money. Find the cost of the
farm.
6. A horse was sold for $184 at an advance of 16%, What
would it have brought at a gain of 20 % ?
7. 75% of a farm is cultivated; 80% of the remainder is
pasture ; and the remainder, 2 A. 80 sq. rd., m woodland. What
is the area of the farm ?
8. What is the difference between specific and ad valorem
duties ?
0. Why does moving the decimal point to the left two
places give two months' interest at 6%.
10. A dealer obtained $360 for a piano on the list-price of
which he had discounted 60%. He still made a profit of 20%.
Find the cost and list-price of the piano.
11. Carpeting J yd. wide is used for a room 18 ft. square.
The waste in matching is 8 in. to a strip. What is the cost at
$1.37jayard?
12. The last reading of my gas-meter was 64,700 cu. ft.
The previous reading was 47,900. At $1.50 a thousand, with
a discount of 12^%, what was the amount of my gas-bill?
13. If the valuation of a town is $6,400,000, and my property
is assessed at $11,200, how much of a tax of $40,000 ought I
to pay?
BANK DISCOUNT. 185
1. What is a bank? A bank is an institution chartered by
the Government ; i.e., given permission to do business.
2. What business is done by banks? They furnish a safe
place of deposit for money, they exchange money, issue notes
for circulation, borrow and lend money, and collect money on
notes and drafts.
3. Suppose Mr. R. J. Bartlett has some money on deposit
in the Home National Bank, but is owing $100 to C. R. Hooker
of New York. Instead of sending the money, Mr. Bartlett fills
out a blank check like the following :
Hobjoke, Mass 19 2fo ^
Home National Bai^k.
Pay to the order qf. ^
Dollars,
This check is sent to Mr. Hooker, who takes it to any national bank in
New York, and they will pay it or collect it for him. All banks have dealings
with one another, so that through a ** Clearing House ^' the check comes back
to the Home National Bank, and the amount is placed on the book against
Mr. Bartlett. In this way banks help in exchanging money.
4. Fill out a blank check.
5. We have learned also that banks can issue notes or bills
for circulation. Examine carefully some bank-notes, and see
how they read.
Note. — Before a bank can issue bills of Its own it must deposit with the
Treasury Department in Washington Government Bonds equal in amount to
the bills issued. The bills are printed by the department and the bonds are
held in trust for the security of the bill holders.
6. Who are the stockholders of a bank? They are men
who own all the property, and, like partners in other kinds of
business, share in the gains and losses.
7. How does the Government try to protect those who de-
posit their money in the banks ? By having the banks exam-
ined at stated intervals by Bank Examiners.
8. From what does a bank derive its income? From
loaning money, discounting notes, etc.
9. How may one get a note discounted at a bank?
186 BANK DISCOUNT.
A. C. Bardwell has the following note for $500.00, taken in bofiEiness,
which he wishes to get discounted at a bank.
f600. Holyoke, Mass., Sept. 10, 19
8iai^ days^^^^^^^-^^^^^^^-^^^^^^^^^^^^^-^"^^^-'^^ — after date I promise to pay
to the order of A. C. Bardwell,-
^'^'-'^''--^^'^^-^-'^^'"^ — Five Hundred
payable at Home National Bank.
Value received,
(a) A. C. Bardwell must write his name across the back; i.e., indorse it,/
and thus become responsible for its payment, if Mr. Jones should fail to pay it
when due.
(6) Mr. Bardwell can now take the note to the bank, and if the officials
are satisfied that the note is good, they may accept it and loan the money.
(c) The time when the note is due will then be ascertained by adding the
time specified in the note to the date, which will make it due Noy. 0th.
In some States three days of grace are allowed, in which case the above note
will be due Noy. 12th.
Note. — Days of grace have been abolished in many States. In Massa-
chusetts and some other states they are still in force on sight drafts. Count
them or not according to your location.
(d) As Mr. Bardwell presented the note to the bank Sept. 10, the bank
finds the interest on $600 for 60 days, which is $5.00, and keeping this as their
discount, gives Mr. Bardwell the rest, $495, called Proceeds or Avails.
(e) If Mr. Bardwell had not taken the note to the bank until a later day,
say Oct. 1, the bank would have found the i&terest on $500 from Oct. 1 to Nov.
9, or for 39 days, which would be $3.25, and the proceeds $496.76.
1. Bank Discount is the interest retained by a bank for ad-
vancing money on notes before they become due.
2. The Proceeds, or Avails, is the amount received by the
borrower, and is equal to the Face of the note less the Dis-
count.
3. The Term of Discount is the time a note has to run
from the date of discount to the date of Maturity ; i.e., the day
when the note is due.
4. Notes for discount are usually without interest. Some-
times a man may receive an interest-bearing note. At the time
he receives it, or at any time before it is due, he may wish to
obtain the money on it. If this note is discounted, the amount
at maturity, and not its face, will be the sum discounted.
BANK DISCOUNT. 187
1. In the note on Page 186, who is the maker ? The payee ?
What is the face ? What is the date ? Is it a demand or a
time note ? When is it due ?
2. Who is the Indorser of this note ? By indorsing it he
makes himself liable to what? Under what circumstances will
Mr. Bardwell be called upon to pay the note ?
3. If the note is not paid by Mr. Jones on Nov. 9, the note
is said to have gone to protest, and a notary public notifies the
indorsed. This notice must be made within 24 hours after the
note is due.
4. Give a good reason why a business man is very careful
never to allow his note to be protested.
5. How much money belonging to the bank did Mr. B. use
for 60 days ?
6. What is the interest of $495 for 60 days ?
7. Why does Mr. B. pay the bank $6 interest instead of f 4 .95 ?
8. What is the difference between simple interest and bank
discount?
Practice varies in estimating the time of maturity and term of discount.
Some banks reckon the time in exact number of days, others in months and
days, others the exact muuber of days when the time is leSs than two months,
but in months and days when the time is more than two months. It is best
to conform to the custom of your own locality. In this book the exact state-
ment, whether months or days, is used in finding the date of maturity. In
finding the term of discount the exact number of days is found. Days of
Grace will not be used.
0. $595^. Boston, Feb. 10, 1903.
Three months after date, I promise to pay to the
order of James McKemie Five Hundred Ninety-
Jive and VW Dollars at the City National Bank.
Value received. William Kenny.
Discounted at date at 6%. Find proceeds.
Find the bank discount and proceeds of the following notes :
Facb.
Date.
Time.
Day
OF Discount.
Rate
10. % 846.
Feb. 7.
60 da.
Feb. 22.
6%.
11. «1,450.50.
Mar. 6.
2 mo.
Apr. 2.
5%.
12. ^ 376.40.
Apr. 9.
3 mo.
May 6.
6%.
13. $ 248 60.
May 12.
4 mo.
June 2.
5%.
188
BANK DISCOUNT.
Find the bank discount and the proceeds in the following :
Facb.
Datb.
Time.
Day of Discount. Bate.
1.
$1,234.
Sept. 10.
60 da.
Sept. 80.
6%. •
s.
$2,846.
Nov. 13.
90 da.
Dec. 10.
5%.
3.
$3,456.
Aug. 11.
45 da.
Aug. 17.
4%.
4.
$4,667.
Jan. 5.
75 da.
Feb. 6.
6%.
B.
$6,678.
July 7.
60 da.
July 14.
4%.
6.
$6,789.
Nov. 21.
2 mo.
Nov. 21.
6%.
7.
$7,890.
Oct. 2.
3 mo.
Nov. 1.
6%.
8.
$8,901.
Feb. 21.
4 mo.
March 11.
6%.
9.
$9,012.
May 20.
100 da.
June 80.
8%.
10.
$9,876.
July 15.
96 da.
Aug. 1.
6%.
11.
$3,766.
Dec. 24.
80 da.
Jan. 2.
6%.
12.
$7,664.
Nov. 18.
75 da.
Dec. 6.
6%.
18.
$6,648.
Feb. 4.
70 da.
March 17.
7%.
14.
$5,432.
March 6.
«Oda.
March 6.
6%.
16.
$4,321.
April 17.
90 da.
May 5.
6%.
16.
$3,210.
June 16.
8 mo.
June 16.
6%.
17.
$2,109.
Sept 13.
4 mo.
Oct 13.
8%.
18.
$1,098.
Dec. 80.
1 mo.
Jan. 2.
6%.
19.
$276.60
June 19.
63 da.
July 6.
7%.
20.
§796.70.
March 20.
76 da.
March 29.
6%.
21.
$548.30.
July 21.
3 mo.
July 21.
5%.
22.
$274.
July 22.
8 mo.
Aug. 11.
6%.
23.
$382.
Aug. 20.
4 mo.
Oct 13.
4%.
24.
$496.
Sept 18.
2 mo.
Sept 30.
5%.
20.
$618.
Oct 16.
30 da.
Oct. 16.
1%.
26.
$736.
Dec. 12.
45 da.
Jan. 6.
6%.
27.
$448.
Jan. 10.
75 da.
Feb. 7.
7i%.
2&
$569.
Feb. 8.
90 da.
March 13.
6%.
29.
$224.
Sept 5.
2 mo.
Oct 1.
6%.
30.
Making yourself the payee, and
your teacher the maker,
write notes, using
; data given
in the first ten examples
REVIEW OF PERCENTAGE. 189
1. After taking out his commission of 4% and $80.80 for
other charges, an agent remitted to his employer §1,820, the
amount due him on wheat sold at $.C0 a bushel.
2. A man put 16% of his money in the bank, and spent
40% of the remainder. If he had 11,008 left, how much did
he have at first?
3. A man willed 30% of his money to his wife, 20% of the
remainder to his children, 12^% of what was left to the city
library, and the remainder, $4,900, to benevolent institutions.
How much was the whole property and each share ?
4. A father bequeathed $5,580 to his son. This sum was
26% of what the son already had. How much did the son
have after receiving his father's bequest?
6. A man spent $100 a year for*3 years in repairs on his
house, and then sold it for $100 less than its first cost, and his
entire loss was 4% of its cost. Find its cost.
6. S owned a half interest in a manufacturing industry.
He sold 12% of his share for $4,500. At that rate what is the
value of my share if I own 12 J % of the other half?
7. A widow received 36% of her husband's estate, each of
two daughters 22% of it, and the son the remainder. If the
widow received $12,600 less than all the children, what was
the share of each ?
8. B bought some land for $2,500, and sold it immediately
for $3,000, taking in exchange a six months' note without in-
terest. How much did he make if he had the note discounted
at a bank at 6 % ?
9. Find the value of x :
Cost.
Selling-I^rtce.
Gain or Loss.
OAnc OB Loss %.
f 20.00
$ 16.00
X
X
10.
f 40.00
$ 44.00
X
X
11.
$400.00
$ X
X
10
12.
$ 2.60
X
$ 1.00
X
190 ORAL PERCENTAGE,
1. If I pay $10.50 for having my house msured at |%, for
what amount do I get it insured?
2. I paid $9.76 for a load of coal at $6.00 a ton. How
much did it weigh?
8. A commission merchant received $25.00 for selling butter
at 2i%. How much did the butter bring?
4. I bought a bill of goods amounting to $26, with a trade
discount of 20%, and 6% off for cash. What was the net
amount of the bill?
5. A broker sold cotton to the amount of $620 at 2^%
commission. How much did he receive for his services?
6. A sleigh that cost $28 was sold at a loss of $4. What
per cent was lost?
7. Smith & Murray ^ell lace curtains at $10 a pair, and
thereby gain 25%. What did the curtains cost them?
8. When flour is selling at $4.60 a barrel, a merchant loses
10%. What would be his gain per cent if he sold at $5.50
a barrel?
9. A grocer bought eggs at 36/ a doz., and sold them at the
rate of 6 eggs for 21 cents. What per cent did he make ?
10. A sleigh which cost $50 was sold for $40. What per
cent was lost?
11. A farmer sold a cow for $40, which was 80% of the cost.
What was his loss ?
12. If dress goods sell at 60/ a yard, a gain of 20% is made.
How much is the gain per cent when sold for 70/ a yard?
18. A grocer, by selling flour at $6.25 a barrel, gains 25%.
What did the flour cost him a barrel?
14. A dealer made $20 on a buggy by selling at an advance
of 20%. For what did the buggy sell ?
16. If 20% was gained by selling a parlor chair for $6.00,
what per cent would be gained by selling it for $7.00 ?
16. How many sheets of paper in 75% of a quire?
REVIEW OF PERCENTAGE. 191
1. A merchant bought, Aug. 2, 1902, 12 bales of cloth (16
pieces in a bale, 45 yd. in a piece), at 7/ a yd., for which he
gave his not^ on interest at 6%. On Dec. 8, 1902, he sold 3
bales at 12 J/ a yard, and gave the money in part payment of
his note. On the 20th of June, 1903, he sold 2 bales at 14/,
and paid it all as part payment on his note. Oct. 26, 1903, he
sold the remainder at 15/ a yard, and settled the note. How
much did he gain ?
2. A person takes a 3-months' note for $217.80 in payment
for a horse. On getting the note discounted at a bank at 6%,
he finds that he has lost 20% of what the horse originally cost
him. Find the cost.
3. What is the compound interest on a note for f 600, dated
Sept. 18, 1902, and paid Dec. 2, 1903, interest at 6% per
annum, payable semi-annually?
4. A mill was insured for J its value at |%. If the pre-
mium was 1123.76, what was the value of the mill?
6. Find the proceeds of a note of $1,440, dated Oct 14,
payable in 90 days, and discounted Nov. 21.
6. I bought a farm of 80 acres of land for $3,000, and
sold it at a profit of $7.60 an acre. What was the gain per
cent?
7. A car-load of peaches was bought at 80/ a basket, and
sold at a loss of 12J%. If the loss was $43.20, how many
baskets were in the car ?
8. A grocer mixed 30 lb. of 26/ tea, with 20 lb. of 60/ tea,
and sold the mixture at 52^ a pound. Did he gain or lose?
and what per cent ?
9. A merchant has policies of insurance on his goods as
follows: $6,000, at i% ; $11,800, at t% ; $16,200, at 1%;
$8,000 at 1%. The goods cost him $48,000. If a fire should
totally destroy the goods, what would be his loss, including the
sum paid for insurance ?
192 ALGEBRAIC PROBLEMS.
1. Three boys have together 160 marbles. The first has 50
less than the second, and 40 more than the third. How many
has each ?
8. Edith is 6 years more than \ the age of her brother. If
their united ages amount to 42 years, how old is each ?
3. Divide 80 cents between two girls so that one shall have
§ as many as the other,
4. Divide 164 into three parts, such that the first shall be
12 greater than the second, and the second 16 greater than the
third.
5. What number multiplied by 8, and then diminished by
13, is equal to 11 ?
6. Find two numbers which differ by 14, and one is three
times the other.
7. A horse and wagon are worth $300, and the horse is
worth 3 times as much as the wagon. What is each worth?
8. Divide $500 among A, B, and C so that B and C may
each have twice as much as A. How many dollars will each
have ?
9. The sum of three numbers is 140. The second is four
times the first, and the third is J the second. What are the
numbers ?
10. If to a certain number, itself, one-fifth of itself, and 5 be
added, the sum will be 104, What is the number?
11. A farmer sold two-fifths of his farm to one man, one-
third to another, and had 20 acres left. How large was his
farm at first?
12. Three-eighths of what number is 30 less than the num-
ber itself ?
13. A and B went into business together with a cash capital
of $2,400. If A put in four times as much as B, how much
did each invest?
14. 5x — 4 = 3x + 2. Find value of x.
PARENTHESIS. 193
1 a^ 8 4- r3 4- 2^ a) To 8 add 3, and to this sum add 2.
* A\ e /Q 9^ W^^t, IS the result ? Add the two numbers
0) o -f- {^o — ^) within the parenthesis. Add their sum to 8.
Do you get the same result as at first ?
6) To 8 add 3, and from their sum subtract 2. What is the result ?
Subtract the numbers within the parenthesis, and add the difference to 8.
Do you get the same result as at first ?
2. If an expression within a parenthesis is preceded by the
sign +, the parenthesis can be removed without any change.
8. a^ 8 — (S 2^ ^) From 8 take 3, and from their differ-
j{ Q \n ON ®°c® ^^^® 2. What is the result ? Subtract
^) ^ \*^ i ^) the numbers within the parenthesis, and take
their difference from 8. Do you get the same result ? If a is written 8 — 3 +
2, would you get the same result as at first ?
6) Add the numbers within the parenthesis, and subtract the sum from 8.
What is the result ? If 6 is written 8 — 3 — 2, would you get the same
result?
4. If an expression within a parenthesis is preceded by the
sign — , the parenthesis can be removed, provided the sign before
each term within the parenthesis is changed, the sign + to — ,
and the sign — to + .
5. Numbers are grouped by using different forms of the
bracket, ( )? [ ]^ { }^ and the vinculum. The line also between
the numerator and denominator of a fraction acts as a vinculum.
Remove the parenthesis and unite the terms :
6. 16 + (7 - 5). 15 - (7 - 2). 17 - (4 + 3).
7. 21 - (11 - 7). 18 + (7 - 4). 15 - (9 + 2).
8. 12 - (7 + 5). 11 _ (6 - 2). 14 + (7 - 5).
If a = 4, J = 3, (? = 2, find the value of :
9. a - (6 + (?). 2 J - (c? + a). S c — (a - J).
10. 3 a - (2 6 - c?). 3 a - (6c - a), ib - {c^ + 2 a).
Find the value of x :
11. 2 a; - (3 + 4 ic - 3 a: + 5) = 4.
12. (2 a; - 5) - (a: ~ 4) + (a; _ 3) = aj - 4.
13. 7 a; - 5 - (6 - 8 x) + 2 = 3 a; - 7 + 94.
14. 15x-(6x + 3) = 304-(ox-hl'^).
194 CONSTRUCTION— TRIANGLES,
1. Draw a triangle that shall contain three acute angles.
2. Draw a triangle that shall contain a right angle. How
many acute angles will it contain ?
3. Can you draw a triangle that shall contain an acute, an
obtuse, and a right angle ?
4. Draw a triangle. With your protractor m<)asure each
angle. Add the sum. What is the result ?
6. Cut a triangle out of paper. Cut off the three angles,
and place them so as to show their sum. Fold to show the
same result.
6. Write : The sum of three angles of a triangle is equal to
two right angles or 180°.
7. Draw the line AB. At A make an angle of 60°, and at
B an angle of 60°. Prolong the lines until they meet at .0.
How many degrees in the angle at (7?
8. Substitute 80° and 40° in 7. How many degrees in Z CI
9. Draw a triangle having two angles of 40° each. Are
the sides equal ? What kind of a triangle is it ?
10. How do the angles at the base of an isosceles triangle
compare ?
11. Draw a triangle having two angles of 60° each. How
many degrees in the third angle ? How do the sides of this
triangle compare ?
12. In 11 what kind of a triangle was drawn?
13. If all its sides were equal, it was an equilateral triangle.
14. Define an equilateral triangle.
15. What is true of the angles of an equilateral triangle ?
16. Write : An equilateral triangle is equiangular.
17. Each angle of an equilateral triangle is of what magni-
tude?
18. Why could it never be other than 60° ?
19. Draw a triangle, no two of its sides to be equal. This
is called a scalene triangle.
CONSTR UCTION — TRIANGLES. 195
1. Cut from paper a scalene triangle. Compare its angles
by folding.
2. Is it equiangular? Can it be so? Are any of its
angles equal ? Can they be ?
3. Can a scalene triangle be a right A ?. If so, draw one.
4. Can it be an isosceles triangle ? If so, draw one.
6. Show why it cannot be equiangular.
Find the third angle of a triangle when two angles are :
6. 80^ 18' and 20° 45' ; 40° 20' 80" and 34° 45' 50".
7. 65° 41' 35" and 74° 16' 44".
8. 79° 10' 30" and 11° 44' 12".
9. Find the other angles when one angle of a right triangle
is 35° ; 40° ; 62° ; 30° 30' ; 20° 20' 20".
Take the proposition : " The sum of the angles of a triangle
is equal to two right angles." Fill the blanks in the following :
10. The acute angles of a right triangle are angles.
11. Each angle of an equilateral triangle must be .
12. In a triangle there can be but one .
13. In an isosceles triangle the angle at the vertex is 40° ;
find the angles at the base.
14. One acute angle of a right triangle is 40° ; what is the
other acute angle?
16. The angle at the base of an isosceles triangle is 60° 30'.
Find the angle at the vertex.
16. The angle at the vertex of an isosceles triangle is 50°
40'. Find the angle at the base.
17. Make five similar questions for class to solve.
18. What true statement can you make about a triangle,
knowing that its three sides are equal?
19. When the three angles of a triangle are equal, what is
true of the sides ?
20. The two acute angles of a right triangle are equal.
How many degrees are there in each angle ?
196 STOCKS AND BONDS.
Note. — If possible show the pupils a stock certificate or a bond.
1. Many business undertakings are so large that many pei>
sons must unite to provide the money necessary to cany on the
business. If these individuals secure a charter, and elect such
officers as a president, secretary, treasurer, and board of direc-
toi-s, the association is called a Corporation, or Stock Company.
2. Banks, railroads, insurance companies, and many manu-
facturing companies are illustrations of Corporations.
3. The Charter is the certificate given to the corporation,
usually by the Legislature of the State, stating its name, object,
amount of capital, etc.
4. The amount of money and other property owned by a
corporation is called its Capital, or Capital Stock.
5. The capital is divided into equal shares, usually of flOO
each.
6. A person who owns one or more shares is called a Stock-
holder.
7. Each stockholder receives a certificate of stock, giving
the number and value of his shares.
8. A stockholder cannot demand of the corporation the
return of his money, but he may sell his shares.
9. If the company is prosperous, these shares will sell for
more than they originally cost; that is, above par or at a
premium.
10. If the company is not prosperous, the shares will sell for
less than cost ; that is, below par or at a discount.
11. The profits of the company are called dividends, and are
usually distributed annually, semi-annually, or quarterly among
the stockholders.
12. Losses are in like manner divided among the stockholders.
13. The dividends or assessments of a stockholder do not
depend on the price at which the shares were boughti but on
their par value.
STOCKS AND BONDS. 197
1. When Corporations, or National, State, or City Govern-
ments, borrow large sums of money, they give bonds ; that is,
interest-bearing promissory notes.
2. Coupons are certificates of interest attached to the bonds.
There are as many coupons attached as there are payments to
be made. These coupons are detached, and presented to the
corporation when due.
a. Shares of stock are bought and sold in the market like
other property.
4. Persons who make a business of purchasing and selling
stocks and bonds are called Brokei-s.
5. Brokers are really commission agents, and are entitled
to a commission called Brokerage. This brokerage is always
on the par value of stock.
6. Bonds are usually named according to their rate of in-
terest and date of maturity. Thus Mass. 5's '05 means Mass.
bonds bearing 5% interest, and payable in 1905.
7. What is the cost of 25 railroad shares at 92, brokerage i%?
192 + U = cost of one share. ^uoh statements as " at 92 " mean
the agent pays $92 for a share, and asks $i a share for his work, what will one
share cost you ?
8. What is the income from the above stock, if it yields an
annual dividend of Ci% ?
flOO X 26 = $2,500. Par value. Why do we multiply $100
5i% of 12,500 = $137.50. Ana. ^y 25 ? See 13, page 196.
9. A man invests 5^36,000 in bank-stock at 90. It yields
3i% semi-annual dividends. Find his annual income.
What is the 90 ? If you
$36,000 -^ $90 = 400. No. of shares. divide the amount invested'
$100 X 400 = $40,000. Par value. by the cost of a share, what
7 % of $40,000 = $2,800. Ana. j"^ \^^ the result ? Where
'*^ ^ ' ' does the 7 % come from ?
198 STOCKS AND BONDS.
1. What must be the price of stock, in order that f 18,400
stock may be bought for $16,928 ?
$18,400 stock costs $16,928.
$1 stock costs T^i^x) of $16,928 = $.92.
The price will be 92.
2. How much 4% stock must be bought to give an income
of $1,280?
$1,280 : .04 = a; : $1. At 4%, what income will be derived from
ftQQ AAA ^^ stock ? The income at $z is to the in-
8. U. S. 4's are bought at 133J. What is the rate of
income ?
Income of $100 stock at 4% = $4. Cost of $100 stock = $133i. $4 is
what per cent of $133i ?
4. What amount of bonds at 97i can be bought for $7,790 ?
5. I have $19,971 invested in U. S. 4's. What is my in-
come if I paid 118i ?
6. How much shall I receive from the sale of 85 shares of
New England Central at lOOj, brokerage i% ?
7. I sold 8,000 U. S. 4's at 112?, brokerage i%. What did
I receive for them ?
8. How many shares at 110| can be bought for $12,265.50,
brokerage i% ?
9. What annual income shall I receive by investing $5,765.-
50 in 6% stock bought at 11 Of, brokerage i% ?
10. A man paid $21,978 for Boston and Maine 6's at llOf,
brokerage i%. What was his income ?
11. A man's income is $1,428. What amount did he invest
in 4i% stock at 103|, brokerage i%?
12. What is my rate of income if I buy 7% stock at 139 J,
brokerage i% ?
18. If you should sell $14,400 worth of U. S. 5's at par, and
invest the proceeds in New York Central 7's at 120, what
change would you make in your income?
STOCKS AND BONDS. 199
NoTK. — Give many examples from stock quotations in daily papers.
1. What sum must be invested in stock at 112, whicli pays
10% annually, to obtain an income of $5,500 ?
2. When bonds with a face value of $10,000 sell for $9,250,
at what per cent below par are they selling ?
8. Which will give you the larger income, one share of 7%
stock bought at 108, or a 6% stock bought at 105 ?
4. How much will be gained on 1,000 shares of Pacific
Railroad bought at 54 J, and sold at 67t, brokerage i for each
transaction?
5. Find the cost of 10,000 Atchison, Topeka, and Santa F6
R. R. 4% bonds quoted at 99i, 8,000 Central Pacific R. R. 5%
bonds quoted at 101}, and 6,000 Erie R. R. 6's quoted at 76,
brokerage J% in each case.
6. Find the income derived from the bonds in the fifth
example.
7. A has a farm of 120 acres, which yields him an annual
income of $2.62^^ an acre. A real-estate agent sells the farm
for $75 an acre, charging him 3% commission. A invests the
net proceeds in 3J% R. R. stock at 87, brokerage ^%. How
much did he increase his income ?
8. How much will you receive for 75 shares of stock in a
silver mine, if sold at 53|, brokerage J% ?
9. I own some 4i^% bonds. They yield me annually $1,800.
What is their par value? If my rate of income is 3%, what
did they cost me ?
10. What is the market-price of railroad stock when $11,600
stock costs $14,529, including brokerage J % ?
11. I bought 118 shares of stock at 52, and sold it at 64,
payi^ i brokerage for each transaction. What was my
gain?
12. What is the price of stock when $8,729 will purchase
$11,600 worth of stock ?
200 ORAL,
1. What is the duty on 400 lb. of coffee, at 4^' a pound ?
2. What is the duty at 25 % on a bill of goods, invoiced at
$4,000?
3. In the first question, is the duty specific or ad valorem ?
In the second question?
4. If a man owns $4,500, what will be his tax, if the rate
is 2%?
5. What premium must you pay at 2 % for insuring goods
worth $750 ?
6. If a man insures his life for $4,000 at 2i% per annum,
what will be his annual premium ?
7. In what time will $500 double itself at 4% ?
8. In what time will $400 gain $48 at 6% ?
9. At what per cent will $300 gain $72 in 4 years ?
10. A man sold a cow for $33, and gained 10% ; what did
she cost ?
11. A man bought a sleigh for $35, and sold it so as to gain
20%. Find the selling-price.
12. A man bought a bicycle for $100, but sold it at a loss of
163%. For what did he sell it?
13. If it costs a manufacturer $50 to make a bicycle that he
sells for $75, what is his per cent of profit?
14. At \ of 1%, what is the discount on a bill of $1,000?
15. Find the cost of 3^ yd. of cloth at $3J a yard.
Note. — Call the integral part of the cost 1 more. Multiply by 3 and
add i.
16. Find the cost of 6^ yd. of cloth at 6J/ a yard.
17. What will 12i doz. eggs cost at 12j/ a dozen?
18. What will 9i lb. nails cost at 9J^ a pound?
19. What will 2\ oz. of candy cost at 2 J/ an ounce?
20. At $8J a yard, find the cost of 8^ yd. of silk.
21. At 7i^ a yard, find the cost of 7i yd. of gingham.
22. Find the quantity of which $5 is 12j%.
REVIEW OF PERCENTAGE. 201
1. What is the discount on a piano, list-price $800, at
88i% off for cash?
2. A note dated May 5, for $764.48 payable in 3 mo. ^ith
interest at 8%, was discounted June 17, at 6%. Find the
proceeds. See 4, page 186.
8. Face of note, $2,160 ; date, July 12 ; time, 6 mo. ; dis-
counted Sept. 9. Find the proceeds.
4. Find the interest on $647.40 for 1 yr. 7 mo. at 7%.
5. If $48.76 is the premium paid for insuring $3,500, what
is the rate of insurance ?
6. My agent purchased for me 9,400 bu. of wheat. His
commission on the purchase at 2% amounted to $141. What
did he pay a bushel ?
7. I sent an agent $2,550 with which to purchase wheat,
after reserving his commission at 6J%. How much will he
invest in wheat?
8. A 90-day8' note for $125 was dated March 5, and dis-
counted March 21. Find the proceeds.
9. A man bought 75 shares of bank-stock at 108^, received
a dividend of 5|%, and then sold the stock for 107. How
much did he gain ?
10. An agent received $1,507.50 to purchase cloth, after de-
ducting i% commission. How many yards did he buy at $.62^
a yard?
11. A man borrowed the money at 7%, and bought $1,875 bu.
of wheat at 75/ a bushel, Sept. 5, 1902. On June 15, 1903,
he sold the lot for 87^/ a bushel. After paying back the money
he had borrowed, and the interest, how much had he left?
12. A man bought stock at 15% below par, and sold it at
10% above par. How much did he make on 115 shares?
13. I sold goods at 25% gain, and bought other goods \vith
the proceeds, and sold them at 20% loss. Did I gain or lose
by the operation ? and what per cent ?
202 REVIEW OF MEASUREMENTS,
1. Find the cost of painting, at 35/ a square yard, a church
spire whose base is a hexagon 6 ft on a side, and whose slant
. height is 65 ft
2. The area of a triangle is 270 sq. yd., and the perpendicu-
lar is 45 ft Find the base.
3. The distance round a circular park is IJ miles. How
many acres does the park contain ?
4. One of the side walls of a brick building measures 2 rd.
long, 22 ft high, 18 in. thick. How many bricks did it take to
build it?
6. If the building above was in the form of a rectangle
whose width was one-half of the given length, how many bricks
were required for the whole building ?
6. A rectangular field is 60 rd. long, and its width is 60%
of its length. How many boards 12 ft. long and 8 in. wide
will it take to inclose it with a fence 5 ft. high ? The boards
are placed 4 in. apart and 4 in. from the ground.
7. Find the surface of a sphere 22 in. in diameter.
8. The perimeter of one square field is 400 ft. and of an-
other 320 ft. How many square feet in a field equal in area
to both square fields ?
9. How many square yards in the sides of a square pyramid
whose slant height is 100 ft, and the perimeter of whose base
is 54 ft. ?
10. How many cubic feet in a stone 7^ ft. long, 5j ft wide,
and 4^ ft thick ? How many square feet in its surface ?
11. A circular field is 60 rd. in diameter. How many acres
does it contain ?
12. At 16.50 a cord, a pile of 4-ft wood 32 ft long cost
|35i. How high was the pile ?
13. If every person needs on an average 28 cu. ft of air an
hour, how many hours will the air in a room 18' x 14' x 9i'
last 9 men?
MISCELLANEOUS REVIEW. 203
1. The duty on 625 yd. of silk, at 40% ad valorem^ is $660.
For how much a yard must the importer sell the silk to clear
16%?
2. The amount of tax to be assessed in a certain city is
$44,382 ; the taxable property is $2,860,800 ; the number of
polls, each assessed $1.50, is 1,080. What is the rate of tax-
ation ?
8. A block of buildings worth $186,000 is insured for J of
its value in three companies. The first company takes \ of the
risk at 3% premium; the second, J of the remainder at J%
premium ; and the third, the remainder at 1 % premium. Find
the entire premium.
4. If the above block is damaged by fire to the amount of
$80,000, find the amount that each company will be obliged to
pay.
5. A company with a capital of $250,000 declares a
dividend of 3% with a surplus of $6,750. What were the net
earnings of the company ?
6. A commission merchant in Savannah received $23,648,
with which to purchase cotton after deducting his commission
of lj%. Find his commission and the amount expended for
cotton.
7. What number less 168% of itself equals 1,017.90 ?
8. A merchant sold 8% of a piece of cloth. If 128.34 yd.
were left, how many yards were there in the piece at first ?
9. How many yards of carpeting J yd. wide will carpet a
room 18§ ft. long, and 16J ft. wide, if the strips run lengthwise,
and there is a loss of 7 in. on each breadth for matching ?
10. How many bushels of wheat will fill a bin 6 ft. long, 3 J
ft. wide, and 3 ft. 8 in. deep? Approximate measurement.
11. How many cubic feet in a round timber 8 ft. long, and
2 ft. in diameter?
12. What is the ratio of 6 rd. to 3 yd. ?
204 PARTNERSHIP,
(Review Ratio and Proportion.)
1. David Jones is in business with a capital of $2,000. He
takes Joseph Smith into the business with him with a capital
of f 2,000. Such an association of two or more men is called
a partnership. The association is called a firm, or company.
The persons so associated are called partners.
2. What is partnership? What are the persons associated
in business called ?
3. In the firm of Jones and Smith, what is the whole capi-
tal? What part of the capital does each furnish?
4. Suppose the firm gains 1800 the first year, how ought
this gain to be divided among the partners ? Why should each
receive one-half of it?
5. Suppose that for the second year Oscar Brown is admitted
into the company with a capital of $4,000. What will the en-
tire capital be now? How much of the $8,000 will be fur-
nished by Jones? By Smith? By Brown? What part will
be furnished by each?
6. If Jones furnishes \ of the capital, what part of a gain of
$1,200 will he receive ? How many dollars will he receive ?
7. How do you divide the gain among the partners?
8. Write : Take the same share of the gain or loss as each
partner's capital is of the whole capital.
9. Partnership is also called Distributive Proportion, and
the examples can be performed by Proportion.
Note. — The wiiole capital is to a partner's share of the capital as the
whole gain is to the partner's share of the gain.
$8,000 (whole capital) : $2,000 (Jones's capital) = $1,200 (whole gain) :
X (Jones's gain).
The capital is the cause ; the gain the effect.
10. Three men purchase a store paying as follows: A,
$2,000; B, $4,000; C, $3,000, They gain $6,000. How
much does each gain?
11. Three men buy a house for $5,000. A pays $1,000;
B, $2,400 i C, $1,600. They rent it for $600. What is each
one's share of the rent?
PARTNERSHIP. 205
NoTS. — Be sure that the preceding page is understood,
Note. — It sometimes happens that the capital of the different partners is
invested for periods of time of unequal lengths. In this case the profit of each
partner depends on two elements, the amount of his capital and the time it is
employed. The element of time must be eliminated before the principals of
partnership can be applied.
1. A and B enter into partnership ; A furnishes $300 for
2 mo., and B $200 for 6 mo. They gain $150. What is each
one's share of the profit ?
$800 X 2 = $ 600 It is obvious that $300 in business for
$200 X 6 = $1,200 2 mo. is the same as $600 in business for
$1 800 ^ month. And $200 for 6 mo. is the same
«1 QAA «AAA «1 \f\ ^^ ^^'^^ ^^^ ^ month. $600 and $1,200 can
^1,0UU : iB)OUU = ilMOU : x ^^^ ^^ considered as the respective capitals,
600 X 150 ^-^ and then proceed as on Page 204.
1,800 "" *^^
2. A, B, and C engaged in business together. A put in
$6,000, B $2,000 more than A, and C $2,000 less than B.
The profits were $6,000. What was each partner's share ?
3. A, B, and C entered into partnership. A put in $4,000
for one year, B $3,000 for 9 mo., and C $2,500 for 6 mo.
Their profits were $1,612.50. What was each partner's share?
4. A and B hired a pasture together for $50. A put in 60
cows for 6 mo., and B put in 90 cows for 4 mo. What should
each pay ?
6. A, B, and C, enter into partnership. A puts in $714
for 5 mo., B, $742 for 7 mo., and C, $308 for 11 mo., and
they gain $694.40. How much is each one's share?
a. Three contractors agree to dig a canal for $1,010. A fur-
nishes 30 men for 2 days ; B, 20 men for 10 days ; and C, 16 men
for 9 days. Of the sum how much should A, B, and C receive ?
7. A, B, and C agree to build an embankment for $3,200.
A is to furnish 14 men for 30 days; B, 10 men for 40 days;
and C, 12 men for 32 days. How much should each receive
after pajdng expenses of $190 ?
206 REVIEW OF PERCENTAGE,
1. Find the bank discount and proceeds : Face, $1,500 ;
date, Jan. 5 ; time, 60 days ; day of discount, Feb. 1 ; rate of
discount, 6%.
2. If goods are bought at 20% below list-price, with 5% off
for cash, and sold at 14% above list-price, what per cent is
gained ? *
3. On a note of $800 at 6%, and dated March 4, 1903, are
the following indorsements : April 12, 1903, $75 ; July 9, 1903,
$150; Sept 5, 1903, $90 ; Dec. 8, 1903, $200. What was due
Jan. 1,1904?
4. If a man receives $318.75 as a dividend on $5,000 of
stock, what is the per cent of the dividend ?
6. Including J% brokerage, what is the cost of 23 shares
of stock bought at 8i % discount ?
6. A man invested $7,570 in 5% bonds bought at 5S%
discount, brokerage i%. What is his annual income there-
from ?
7. The assessed valuation of the property of a town is
$2,496,000. The estimate of expenses includes $4,500 for
schools, $4,800 for streets, $3,600 for salaries, and $2,076 for
contingent fund. What tax will be required of A, whose real
estate is assessed at $9,000 and personal property at $650 ?
8. An insurance company asks $120 as premium on property
insured for $16,000. At the same rate, what premium will
they ask for insuring $40,000 ?
9. A and B engaged in business Jan. 1, 1902. A invested
$5,000, and B $8,000. On Aug. 1 they took in C as a third
partner, who invested $7,000. On Jan. 1, 1903, their net gain
was $7,640. What was the share of each partner?
10. How large an investment in Holyoke City 5's at 105 will
give an income of $1,500 ?
11. If a man should invest $16,428 in 4% bonds at 74, what
annual income would he receive ?
MISCELLANEOUS REVIEW. 207
1. What are the proceeds of a 90-day8' note for $789.96,
discounted at 7% ?
2. K the rate of discount at a bank is oi%, what will be
the proceeds of a 3-months' note of $570, dated May 23, and
discounted June 14 ?
3. Jan. 1, 1902, A and B engaged in business, each contrib-
uting 16,000. April 1, 1902, C invested $8,000 ; Oct. 1, 1902,
D invested $9,000. The gain was $14,580. Find the share
of each.
4. Which yields the greater percentage on the investment,
4% bonds at 80, or 6% bonds at 110?
6. A note for $1,200 was given Jan. 1, 1900. . On the 16th
of May, 1900, $360 was paid ; and on the 1st of October, 1902,
$480 was paid. How much was due Dec. 1, 1903, interest at
7%?
6. If the net profits of a mill in 2 years are $8,118, and the
profits of the second year are 20% more than the first year, how
much were the profits the first year ?
7. 20% of a shipment of potatoes, originally 5,000 bu.,
were frozen. What per cent will be gained on the lot by sell-
ing the remainder at $1 a bushel, if the cost was 62i/ a bushel?
8. How many shares of railroad stock can be bought for
$33,293.75 at 945, brokerage i% ?
9. How many cubic inches are there in a grindstone 5 ft. in
diameter, 3 in. thick, and having a hole at the center 4 in.
square?
10. Divide $4.08 among 4 boys in the proportion of 3, 6, 7,
and 9. How many cents will each boy receive ?
11. A, B, and C formed a partnership. A put in $2,000 for
10 mo. ; B, $1,800 for 8 mo. ; and C, $3,000 for 6 mo. If they
gain $2,620, what is each partner's share ?
12. The sum of two numbers is 21 and their difference is 3.
What are the numbers ?
208 MISCELLANEOUS REVIEW.
1. A horse, wagon, and harness cost altogether §130. The
horse cost twice as much as the wagon, and the wagon 4 times
88 much as the harness. Find the cost of each.
2. Mr. J. is worth $6,560 more than S., and they are worth
together 135,978. How much is each worth ?
3. I gained |2,100 by selling § of my property for what J
of it cost. At the same rate of profit, what ought I to receive
for I of the remainder ?
4. Mr. Jencks owns six U. S. 4% bonds of $500 each. His
brother has $2,000 invested in a business that brings him in the
same annual income. What rate per cent does the brother
receive ?
5. One side of a rhombus measures 16 ft., and the distance
between its parallel sides is 10 ft. What is its area?
6. Find the proceeds of a note with the following data:
Face, $870 ; date. May 9 ; time, 60 days ; date of discount,
June 3; rate, 6%.
7. What is the interest on $465.82 from May 15, 1901, to
Jan. 6, 1903, at 6§% ?
8. If an insurance company takes a risk of $12,000 at 1^%,
and reinsures § of it in another company at 1|%, how much
does the first company make, if no loss occurs ?
9. What is the value of a stock of goods, if $420 is paid
for insurance on % of its value at 1 J % ?
10. A merchant buys goods at discounts of 20%, 10%, and
5%. At what discount from the list- price must he sell to gain
25%?
11. A commission merchant sold 6,375 yd. of caUco at 4|/
a yard, 3,790 yd. of gingham at 6f / a yard, 8,780 yd. of ging-
ham at 7|/ a yard. Find his commission at 2^%.
12. A floor 14 ft. 6 in. by 12 ft. 8 in. is covered with carpet
a yard wide, laid crosswise of the floor, at a cost of 87^/ a
yard. Find the cost
CONSTR UCTION — TRIANGLES. 209
1. One angle of a triangle is 100°. How many degrees in
each of the other angles if tliey are equal ?
2. Find the size of each angle of an isosceles triangle, when
one of the equal angles is 50° ?
3. Draw a triangle. Measure two of the angles. What
must the other measure ?
4. Cut a paper isosceles triangle. Can you fold it so as to
show that the perpendicular from the vertex divides an isosceles
triangle into two equal right triangles ?
Draw :
5. An acute isosceles triangle. An obtuse isosceles triangle.
6. A right isosceles triangle. An acute scalene triangle.
7. An obtuse scalene trangle. A right scalene triangle.
8. Define each of the above triangles.
9. Can you draw a triangle that contains two right angles ?
If not, why not ?
10. Can you draw a triangle that contains a right and an
obtuse angle ? If not, why not ?
11. Can you draw a triangle that contains a right and an
acute angle? If not, why not?
12. Can you draw a triangle that contains three obtuse
angles ? If not, why not ?
13. Can you draw a triangle that contains two obtuse angles ?
If not, why not?
14. Can you draw a triangle that contains one obtuse angle ?
Are the others acute ?
15. Two angles of a triangle are : a. 46° and 34°. I. 24°
10' and 32° 15'. c, 72° and 21°. d. 47° 12' 20'' and 62° 14'
21". Find the third angle in each case.
16. Draw a triangle whose sides are 4 in., 6 in., and 8 in.
What kind of a triangle is it? Measure its angles.
17. The sum of two angles of a triangle is 122° 45'. The
third angle is how large ?
210 ORAL.
1. A and B form a partnership with a capital of $4,000, of
which A puts in f 3,000, and B $1,000. How shall they divide
a gain of $1,200 ?
2. Two boys, Charles and Henry, invest $25 in a business,
of which Charles invests $16, and Henry $10. How ought a
gain of $30 to be divided between them ?
3. A, B, and C form a partnership. A puts in $3,000 ; B,
$2,000 ; and C, $4,000. If they gain $3,600, how ought it to be
divided ?
4. How many feet in 10 boards, each 16 ft. long and 6 in.
wide?
6. Find the interest of $3,000 for 20 days at 8%.
6. List-price, $675. Find the cost at 33 J % off.
7. How many times is % in. contained in 2 J inches?
8. The divisor is §, the quotient is f . What is the dividend ?
9. Mr. Brown purchased a horse, harness, and sleigh. The
horse cost | and the harness tV of the entire cost, and the price
of the sleigh was $30. Find the entire cost.
10. If 24 is J of a number, what is I of the number?
11. A man lost $230. If he had J of his money left, how
much had he at first ?
12. Gain 54 cents, rate of gain 27%. Find the cost and
selling-price.
13. Cost, 70 cents; rate of gain, 7|%. Find the gain and
selling-price.
14. Selling-price, $56 ; rate of loss, 20%. Find the cost and
loss.
16. i of J of a dollar is 20% of what?
16. If Carrie's money is 25% less than her brother's, his
money is what per cent more than hers ?
17. I of a number exceeds i of it by 9. What is the
number?
18. What is the ratio of 25% of a bushel to 50% of a peck?
INVOLUTION AND EVOLUTION. 211
1. What is the product of the two equal factors 4 and 4 ?
2. What is the product of the three equal factors, 4, 4,
and 4?
3. A power is the product of equal factors. The product
of two' equal factoids is a second power, or square. The product
of three equal factors is a third power, or cube. The product of
four equal factors is a fourth power, etc.
4. Each one of the equal factors used in producing the
power is a root. If there are two equal factors, each is a second
root, or square root. If there are three equal factors, each is a
third root, or cube root, etc.
6. 4* is read "the third power of 4," or "4 to iihe third
power," or " the cube of 4." The figure at the right and a little
above the root (in this case 3) is the exponent. It always tells
the power desired.
6. Find the powers as indicated:
54^ 67^ .14^ 1.42«, 37^ 78», 45*, 1.01', 361
2.08', .004S 28«, M\ 2.02«, 1.05*, .7^
7. A root is indicated by the sign Vi which is called the
radical sign. If any other root than the square root is desired,
a figure called the index is placed above the sign.
8. Find the roots indicated :
V16, ^64, a/125, ^J/81, a/64, Wt,
a/216, a/256, VMO, a/8000-
9. Square 86. 36 = 30 + 6
36
36
36 6 X 6 = (units)2
180 6 X 30 = units X tens.
180 30 X 6 = tens X units.
900 30 X 30 = tens2.
30 + 6
30"
SS5
900
30x6
=
180
6x30
=
180
6"
=
36
1,296 36 X 36 = tens^ + 2 (tens X
units) + units^.
30^ + 2 (30 X 6) + 6^ = 1,296
212
POWERS AND ROOTS.
I. lu example e, on page 211, is there any difference
between 30 x 6 and 6 x 30 ? Is using them both the same as
2 times the product of the tens by the unite?
36 What 19 the square of the tens? What is
36 two times the product of the tens by the units */
AAA _^ ^f What is the square of the units ? In this pro-
360 aa 2tu ^^^ ^^ squaring, how many partial products are
36 = M« ' obtained?
Note. — This is the algebraic method of squar-
l,296^t^ + 2tu + u\ ing numbers.
:d. Use this method in squaring the following: 25, 44, 62,
66, 75, 83, 91, 14, 28, 35, 42, 69, 67.
3. To square numbers geometrically.
Square 36. Let the line AB be 30
units long, and BE 6 units long. Con-
struct on AB a square. What will be its
area? Construct now on AE a square.
How many additions must you make to
your first square? Prove from the fig-
ure that the square of 36 will consist of
302 + 2 (30 X 6) + 62.
4. Draw a figure, and square 18, 26, 32,
80X«
B
B
48, 63, 84, 73, 98.
6. Find the square root of 1,296.
3 6
12 96
9 00 =c e«
607
96
60
2tu + u^
2tu
36 = u^
3 6
12 96
9
6r39
36
36
Divide 1,296 in periods of two figures
each. lu our root we shall have as many
figures as there are periods or fraction of a
period. In example 1, what three parts
did we have in the algebraic formula ?
What is the greatest square less than 12 in
the tens period ? What is its root ? Place
it above the tens period. Taking t^ away,
what two parts remain ? If tis 30, what
is 2 1 ? This is a trial divisor. How many
times is it contained in the remainder?
Place 6 above the units period. Find the
value of 2 tu. Subtract it. What remains ?
Take away u^. What then is the square
root of 1,296?
The short form is the same, only un*
necessary ciphers are omitted, and only
one figure brought down at a time.
36
Learn all the perfect powers between 1 and 100.
POWERS AND ROOTS.
213
30X6
a
b
30«-
900
8
X
c
d
Note. •^Page 212 ahoald be thoroughly unddrstood before attempting this
pa«e.
1. Square Root illustrated geometrically :
This diagram represents a square
surface whose area is 1,296. What is
the area of the square a 6 c d ? Tak-
ing this square away, how large an area
is left ? This remainder consists of how
many equal rectangles, and how many
squares ? Think of these as placed end
to end, forming a rectangle. What is
its approximate length ? If 896 is the
area of a rectangle, and 60 is its approx-
imate length, how wide is it ? What is the area of each rectangle ? Taking
away the area of both rectangles, how large an area is left ? How does the
side of the little square compare with the width of the rectangles ? Find the
area of the little square, and take it away. What is left ?
Find the square root of :
2. 3364, 3. 6241, 4. 5929, 6. 5329, 6. 8025, 7. 7569,
8. 1369, 9. 2809, 10. 5184, ii. 4356, 12. 8886, 13. 4489.
Note 1. — If there are more than two figures in the root, double the root
already found for a trial divisor, and proceed as at first.
Note 2. — When a cipher occurs in the root, place the cipher above its first
period, and bring down the other figure of that period and the first figure of
the next period.
Note 3. — When a number is not a perfect square, annex periods of
ciphers and continue.
Note 4. — When a number contains a decimal, begin at the decixoal point,
and mark toward the left and right to form the periods.
Find the square root of :
14.
12,544.
16.
15,625.
16.
87,025.
17.
93,025.
18.
77,841.
19.
81,225.
20.
15,376.
81.
38,416.
S3.
27,225.
23.
29,241.
24.
617,796.
26.
334,084.
26.
638,756.
87.
890,625.
28.
288,369.
89.
278,784.
30.
214,369.
31.
948,676.
82.
143,641.
83.
823.69.
34.
285.61.
36.
6.7081.
36.
32.7184.
87.
1866.24.
Note. — If a perfect square is retolved into its prime factors, the Square
root will consist of one-half of the number of each different factor.
. 1,296 = 2x2x2x2x3x3x3x3. .-.2x2x3x3 = 36. Ana.
214
APPLICATION OF SQUARE ROOT.
1. Draw a right triangle with a base 4 in. and a perpen-
dicular of 3 in. On each side draw a square. Divide each
square into square inches.
2. How many square
inches in each square?
3. Add the square inches
in the square of the base and
perpendicular. How does the
sum compare with the square
inches in the square of the
hypotenuse ? -
4. If there are 25 sq. in.
in the square of the hypote-
nuse, what is the length of
the hypotenuse ?
5. Subtract the square of
the base from the square of
the hypotenuse. How many
square inches ? How does it compare with the square of the
perpendicular ?
6. If there are 16 sq. in. in the square of the perpendicular,
what is the length of the perpendicular ?
7. Subtract the square of the perpendicular from the square
of the hypotenuse. Extract the square root of the remainder.
How does the result compare with the length of the base ?
8. How many sides of a right triangle must be known in
order to find the remainder ? Formulate a rule for finding each
side.
9. The base of a right triangle is 12 and the perpendicular
16. What is the hypotenuse ?
10. A ladder is 39 ft. in length. How far from the base of
a building must the foot of the ladder be placed, in order that
the top may reach a window 36 ft. above the base ?
Vv^
\x
X
B
APPLICATION OF SQUARE ROOT. 215
1. If the area of a square is 196 sq. ft., what is the length
of one side ?
2. How much will it cost at $1.75 a rod to build a fence
round a square field whose area is 10 acres?
3. A rectangular lot contains 1,875 sq. yd. If it is 3 times
as long as it is wide, what are its dimensions ?
Note. — Into how many equal squares can this rectangle be divided ? How
many square yards are there in each square ?
4. How much more will it cost to fence a rectangular field
312 rd. long and 78 rd. wide, than a square field of the same
area, at $2.50 a rod ?
6. Find the dimensions of a cubical box, if the area of its
faces is 12,696 square inches.
6. When 2,255 men were arranged in the form of the largest
square possible, 46 men were left over. How many men were
in rank and file ?
7. A park in the form of a rectangle is 80 rd. long and 72
rd. wide. What is the length in rods of a walk between the
opposite comers ?
8. Two vessels sail from the same port. One sails due
north 8 miles an hour, and the other due west 6 miles an hour.
What is their distance from each other at the end of 5 hours ?
9. What is the length of a field twice as long as wide, con-
taining 70 A. 50 sq. rd. ?
10. A field contains 63 A. 60 sq. rd. Its length is to its
breadth as 5 to 3. What will it cost to fence it at 72/ a
rod?
11. A field four times as long as it is wide contains 15f
acres. Find the dimensions.
12. Find how long the rafters must be for a house 82 ft.
wide, if the ridgepole is 12 ft. above the attic floor, and the
eaves project 2 ft. beyond the walls.
13. The product of what two equal numbers is 5184?
216 APPLICATION OF SQUARE ROOT.
1« A room measures 18 ft. by 24 ft, and is 12 ft. high.
Find the distance from one comer of the floor to the comer of
the ceiling diagonally opposite.
NoTB. — Find the diagonal of the floor. Uae this diagonal as the base,
and the height a« the p^rpendioular. See if you oan discover a shorter way.
2. Find the side of a square that shall contain as many
square feet as an oblong measuring 210 ft by 52^ ft.
3. Find the square root of 66.1249.
4. Find the square root of tI^Iv*
NoTB — Take the square root of eaoh term separately when possible. If
not possible, change to a decimal.
5. Find tixe square root of ^|.
6. How many square yards of canvas will make a conical
tent 10 ft. in diameter and 15 ft. high ? First find the slant
height
7. What is the largest square that can be cut from a circu-
lar cardboard 82 in. in diameter ?
8. At 87i/ a rod, how much more will it cost to fence
25 A. 96 sq. rd. in the form of a rectangle whose length is to
its width as 16 to 25, than to fence the same area in the form
of a square ?
9. A square field, containing 10 A., has round the outside
a driveway. If the driveway contains ^y of the whole square,
how wide is it ?
10. The foot of a ladder, 52 ft. long, is 20 ft. from the base
of a building. How high a window can the top of the ladder
reach?
11. What was the height of a tree, standing 25 ft from a
building, the top of which in falling struck the building 30 ft.
from the ground ?
12. Hypotenuse, 25 ; base, 15 ; find the perpendicular.
13. The length of an oblong is three times its width. It
contains 768 square feet. How wide is it?
MISCELLANEOUS REVIEW. 217
1. Extract the square root of 5,669,161.
2. A block of granite is 38 ft. long, and 9i ft. square at the
ends. How many cubic feet must be cut away to leave a per-
fectly cylindrical pillar ?
3. A farmer fed J of the corn he raised to his horses, \ he
sold, i he saved for seed, and the remainder, 520 bu., he has in
his granary. How many bushels did he raise ?
4. What sum must I invest in Massachusetts 5's, purchased
at 97i, to get an annual income of $1200?
6. Find the annual income from investing $14,229 in New
York 6's at 104i, brokerage J%.
6. A 4-months' note for $564.50 was dated April 9, and
bore interest at 6%. If it was discounted May 6, what were
the proceeds ?
7. On May 24, 1903, Mr. B. borrowed 11,200. If this sum
remained on interest at 7j^% until Oct. 7, 1904, what amount
would Mr. B. then owe ?
8. A lawyer collected 87i% of a bill of $2,400, and charged
6%. How much did he remit to his client?
9. A merchant sold goods to the amount of $31,378 this
year. If this was 8t% more than he sold last, what was the
amount of his sales last year ?
10. A man owned 600 acres of woodland. He sold 25% of
it to one man, and 33^% of the remainder to another. What
part of the land remained unsold? and what is it worth at $75
an acre ?
11. A lot containing 2 A. 120 sq. rd. 186 sq. ft. was sold
for $80. At the same rate, how much land can be bought for
$1,400?
12. Find the inner and outer circumference of a walk 6 J ft.
wide, running round a circular grass plot that measures 90 ft.
in diameter.
13. Find the area of the walk in example 12.
218 MISCELLANEOUS REVIEW.
1. Uow deep must a bin 12 ft. square be made to hold
864 bu. ? Approximate measurement.
2. Find the area of a right triangle, whose base is 25 ft.,
and hypotenuse 60 ft.
3. If the circumferenee of the base of a cone is multiplied
by i its slant height, what is obtained?
4. The slant height of a square pyramid is 15 inches, and
one side of the base 24 inches. Find its contents.
5. Two poles are 40 ft. apart. One is 60 ft high and the
other 80 ft. How long a line will connect their tops ?
6. A house is 24 ft. wide. The ridge-pole is 9 ft. higher
than the plate. How long are the rafters if they project 1 ft. ?
7. V2033.1081. V3444736.
8. A's share of the gain is ^} of the whole gain. B's
capital is $8,500. What is A's capital ?
9. Divide 256 hundredths by 16 hundred-thousandths.
10. Find two important facts from the following data:
Amount retained by an agent for purchasing wheat, ^72 ; rate
of commission, 2% ; cost of wheat a bushel, 90 cents.
11. What will be the rate of income of a 4% bond, bought
at 114}, and f % brokerage?
12. The capital stock of a company is $200,000. There is
a debt this year of $10,000. If I own 40 shares, how much
must I pay of the assessment levied?
13. I bought some railroad stock at 60% premium, paying
$19,200. How many shares did I get?
14. A eO-days' note for $429 was dated Feb. 21, and dis-
coimted Mar. 11, at 4?,%. Find the proceeds.
15. If your father places $600 in the Savings Bank, when
the rate of interest is 4% per annum payable semi-annually,
how much can he withdraw at the end of 1 yr. 6 mo. ?
16. I received from my agent $7,720, the net proceeds of a
sale of flour at 3j<y^ commission. Find the gross proceeds.
MISCELLANEOUS REVIEW. 219
1. A square field contains 22 A. 80 sq. rd. At the rate of
a mile in 8 min., how long will it take a boy to ride his bicycle
round the boundary of the field ?
2. At $1.10 a square yard, it will cost $495 to carpet a
room whose length is double its breadth. Find the length.
3. A man obtains an income of $60 from an investment of
$1,560 in 5's. What was the market price of the bonds ?
4. After spending i of his income, then \ of the remainder,
J of the second remainder, and \ of the third remainder, a man
had $1,200 left. What was his income ?
6. A man invested $76,800 in 2?% l)onds at 95i%. How
much stock at 109} can he buy with his first semi-annual inter-
est; brokerage, J% in each transaction?
6. The nearest of the fixed stars is estimated to be twenty
trillion miles distant. If light travels 190,000 miles a second,
how long does it take the light of the star to reach the earth ?
7. Find the trade discoimt on a bill of goods for $2,920,
with 15% and 7% off.
8. What is the cost of concreting the bottom of a circular
fountain 70 ft. in diameter at $1.75 a square yard ?
9. A farmer sold somC" sheep for $950, and took in payment
a 3-mo. interest-bearing note dated Jan. 6, rate 5i%. On March
1 he had the note discounted at 5%. How much did he re-
ceive for his sheep ?
10. A man, after deducting $2,000 from his income, pays
$85 income tax on the remainder. If the $2,000 had not been
deducted, the tax would have been $125. Find his income.
11. The outer diameter of a spherical iron shell is 10 in.,
and the inner diameter is 6 in. Find the cubic inches of iron
in the shell.
12. The breadth of a room is twice its height, and the length
is 3 times the height. It cost $86.40 to paint the walls at 6/
a square foot. Find the dimensions of the room.
220, ORAL PERCENTAGE.
1. A man's assets are $3,000, and his liabilities $4,000.
How much can he pay on the dollar?
2. A man's resources are $2,400, and his liabilities $3,600.
How much can he pay on the dollar?
8. A merchant bought a bankrupt stock at 40 cents on the
dollar, and sold it at 20% below the original price. How much
per cent did he gain ?
4. Find the cost of insuring a cargo of goods for $16,000 at
i%.
5. What will it cost to insure a house for $4,200 at 2J % ?
6. How much stock will $6,400 buy at 80 ?
7. How much stock wiU $87,500 buy at 75 ?
8. If I buy oranges at the rate of 60/ a score, at how much
a dozen must I sell to gain 33^% ?
9. If by selling an article for $9.50 I lose 6%, for how much
should I sell it so as to gain 5% ?
10. 10% of a number is what per cent of J of the number?
11. What fraction of 96 is 12? What per cent?
12. The rent of a house is $860, which is 12% of its value.
What is its value ?
18. Find 10% of $428.
14. Find 16J% of $792.
16. Find 33 J % of $624.
16. A horse traveled 5i miles in 88 minutes. What was his
average time a mile ?
17. Make six different numbers with the figures 6, 4, and 8.
18. What number must be subtracted from J to leave .15 ?
19. Find the number of board feet in a board 12 ft. long,
10 in. wide, and 1 inch thick.
20. Bought a typewriter for $120, and sold it for $100.
What per cent did I lose? What per cent should I have
gained if I had sold it for $160 ?
21. 250 is what per cent of 5000 ?
EQUATION. 221
Find the value of « :
1. 7 (a; + 3) - 4 (Zx - 16) = 45. 2. 3a: - 20 = - (a; + 4).
3. 4a: + 12 = 2a; - (a; - 21). 4. 9(a; +1) =12(a; - 2).
6. 2 (a; - 6) + 3 (2 a; + 5) = 3 (3a: - 2) - 1.
6. 2(a;-l) -2 (2a; -19) = 3 (a; -3).
7 ^_2^+i = 2 4-£±-§ 8 6a;-l 3-4a: _4 a:.
5 "^3''4 6 38
^ a:+3 , 7a;-2 6a;-l , 5a;+4
^. 3_^-l_2^+l ,, 60-a: 3a:-5 _ 3a:
4 3 14 7 4
1 o , x — \ » , X ,. 11 — 62: 9 — 7a; bx-b
l8.,+3+-^=7+g. 13.,-^ 2 6—
,^ 3x-l x-1 2a;-31 ,, 2^-6,- 2a;-3 „
"•^[0 4- = -^— "-^-^^ 3- = -'-
16 8 2
4 2 4
80.^ + ^ = 20-^. 8i.2a:-5±i+15=12£+26
2 o 2 3 5
x+l ^+2^ ^^+3^ ^-l_^-2_^-3^_
2^3 4 2 3 4
o..7a;-8, c 4a; + 4 a;-2
84.^_ + «_5 g —.
X x — 2 a; , 13 _^ , 5a: , 6a: ^^^
a7.^ + ^<yi^ = 3(x + 6)-20.
28. 3 (X -l)-2 (X -3) + (x -2)-5 = 0.
222 ALGEBRAIC PROBLEMS.
1. A number is as much larger thau 10 as 10 is larger than
J of the number. What is the number ?
2. A certain number and two-thirds of the number equals
15. What is the number?
3. What number is 6 6 larger than 10 6? 6 a; larger than
-3a? 12 larger than - 10 ?
4. A drover bought the same number of sheep and cows.
For the sheep he paid f4 each, and for the cows S32. If he
paid $288 for all, how many did he buy of each?
6. 86 divided by a certain number gives 2 less than 48
divided by the same number. What is the number?
6. Divide 36 into three parts so that the first will equal \
the third, and the second will equal the sum of the first and
third.
7. To what number can 3 be added making J of the sum
equal to i of the number?
8. i of a given number added to 2 equals f of itself plus 1.
Find the number.
9. If from i of a number you take 6, the remainder will
equal 2 less than \ of the number. Fin I the number.
10. From § of a number take onensixth of it, and the re-
mainder will equal 6. Find the number.
11. f of a number added to 6 gives the same result as 8 plus
^ of the same number. Find the number.
12. To f of a certain number 8 was added, and the sum was
12 more than J of the number. Find the number.
13. If from J of a number you take 1, and to i of the num-
ber add 2, the results will be equal. Find the number.
14. George has 12 more than i as many cents as his sister
Mary. If together they have $1.74, how many cents has
each ?
15. The sum of two numbers is 25, and the larger is 3 less
than three times the smaller. What are the numbers ?
REVIEW OF PERCENTAGE. 223
1. A farm is taxed for $81.60. The rate of taxation is
$13.60 on a thousand, and the assessed valuation is | of the
real value. Find the real value of the farm.
2. An insolvent debtor has liabilities of $40,000, and assets
worth $15,000. How much will a creditor obtain to whom he
owes $4,280?
3. If 3% more be gained by selling a horse for $133.20
than by selling for $129.60, what was the original cost of the
horse?
4. What per cent of 3 h. 45 m. are 5 m.?
6. A lady paid $27 for a cloak. § of the cost of the cloak
was 90% of the sum paid for other clothing. How much did
she pay out in all ?
6. At a forced sale, a bankrupt sold a suit of clothes for $8,
which was 20% less than its real value. If the suit had been
sold for $12, what per cent above its real value would it have
brought?
7. A man paid $5,000 for a farm, and then spent a sum
equal to 80% of this amount for a new house. He then sold
the whole for $12,000. What per cent did he make ?
8. A carriage was sold for $185, at an advance of 15% on
its cost. What would have been the gain per cent if it had
been sold for $222?
9. Ten sheep were sold for $69, at a gain of 15%. For
how much a head on the average should they have been sold to
gain 10% ?
10. A commission merchant received $450 as his commission
at 2i% for purchasing 3,600 bbl. of flour. What was the price
paid a barrel ?
11. In a city of 3,000 polls, each paying $1.50, the sum of
$166,500 is to be raised by taxation. The property is assessed
at $13,500,000. What is the tax of a man who pays for one
poll and tax on property assessed at $16,470 ?
A
224 REVIEW OF MEASUREMENTS.
1. One side of a square field of 22i A. abuts on a road.
This side is divided into building-lots 110 ft. deep, having a
frontage along the road of 90 ft. each. The building-lots are
sold at $60 each, and the rest of the field at $75 an acre. What
is the total received ?
2. What is the least amount of carpet f yd. wide that is re-
quired for a floor 24 ft. by 21 ft. ?
3. If a cubic foot of ice weighs 62i lb., how many tons of
ice can be stored in an ice-house which is 175 ft. long, 36 ft.
high, and 18 ft. wide?
4. The hot-air register in our schoolroom is 2 ft. 4 in. by
1 ft. 8 in., and J of the area is taken up by the grating. How
much air a minute must pass through each square foot of the
opening of this register into the room to supply each of 48
pupils with 4 cu. ft. of fresh air every minute ?
6. How many pounds of lead will be required to line an
open cistern, whose dimensions are 5i ft. long, 3J ft. wide, and
2i ft. deep, if the lead weighs 31 lb. to the square foot ?
6. If each person on an average breathes 28 cu. ft. of air
in an hour, how many hours will the air in a room 15 ft. long,
12 ft. wide, and 8 ft. high, last 9 persons, supposing the air can
be breathed only once ?
7. If a man takes 110 steps a minute, and the average
length of his step is 30 in., how far can he walk in 2 hours ?
8. How high must wood be piled on a car, which is 28 ft.
long and 8 ft. wide, to contain 14 cords ?
9. Find the cost of papering a room 36 ft. long, 24 ft. wide,
14 ft. high, with paper 18 inches wide. The paper costs 42^
a roll, and 64 sq. yd. are deducted for openings.
10. A room is 32 ft. long, 20 ft. wide, and 14 ft. high. Al-
lowing for 3 doors, each 8 ft. by 4 ft., 4 windows, each 7 ft. by
3i ft., and a wainscot 26 in. high, find the cost of plastering
the room at 12^ cents a square yard.
BEVIEW OF FRACTIONS. 225
1. A contractor found that it would take 8 men 12 weeks
to do a piece of work. He wished to complete the work in 8
days. How many men must he employ?
2. How many loads of gravel wiU it require to cover to a
depth of 3 in. a path 150 yd. long and 4 ft. wide ?
3. Divide .00625 by 250 ; and 62.5 by .025.
4. If a merchant sells J of an article for J of its cost, what
per cent does he gain ?
6. A started to walk a distance of 80 miles at the rate of 5
miles an hour, stopping to rest 20 min. at the end of every two
hours. Two hours and a half after A started, B started to drive
the same distance at the rate of 8 miles an hour. If B stopped
3i hours to rest his horse, how many hours ahead of A would
he reach his destination?
6. From a piece of cloth measuring 36^ yd., a merchant
sold 8i yd. at $1.25 a yard; 12| yd. at $1.12^; and the re-
mainder at 95/ a yard, throwing in i of a yai-d that was dam-
aged. How much did he make on the whole, if it cost him
87i>^ayard?
7. A cistern has 3 pipes ; the first will fill it in 2 hr., the
second in 3 hr., and the third in 4 hr. In what time will they
together fill the cistern ? Suppose the water flows out of the
first pipe, and in through the second and third, in what time
will the cistern be filled ?
8. If a bird can fly 8| miles in 20 min., how far can it fly
in 2 hr. 30 min. ?
9. A can do a piece of work in 12 days ; A and C can do
it in 9 days ; A and B can do it in 8 days. In how many days
can B and C together do it ?
10. A and B can finish a piece of work in 25 days. They
work together for 15 days, and then A finishes it by himself in
20 days. How long will it take A and B, working separately,
to do it ?
226 LONGITUDE AND SOLAR TIME.
1. Turn to a map of the hemispheres in your geographies, or,
better, study the globe. What are the lines called that extend
from the North to the South Pole ?
2. Meridian means the line of midday ; i.e., all places sit-
uated on the same line have midday or noon at the same
time.
3. We usually call the meridian that passes through Green-
wich the first or prime meridian.
4. Longitude is distance east or west from this meridian.
5. In which direction does the earth revolve? In which
direction, then, does the sun appear to move ?
6. Do places east or west of us see the sun rise first ?
7. If the places east of us begin the day earlier, will they
have earlier or later time than we have ?
8. Into how many degrees is any circle divided ? Draw a
circle with a 4-inch radius, and divide it into degrees.
9. How many hours are there in one day? Draw a
circle with a 3^ in. radius, and divide it into 24 equal
parts.
10. Fasten these two circles by an eyelet at the center, so
that one can revolve upon the other. You have now a mechan-
ical contrivance for solving all examples in Longitude and
Time.
11. Since 360 degrees of the earth's surface passes under the
sun in 24 hours, we say 24 hours of time corresponds to 360
degrees of longitude. One hour of time corresponds to how
many degrees of longitude ?
12. If 1 hour of time corresponds to 15 degrees of longitude,
what does 1 minute of time correspond to? One second of
time?
13. What is the difference in longitude between two places,
if the difference in time is 1 hour ? 8 hours ? 4 J hours ? 30
minutes ? 4 minutes ? 2 hours, 15 minutes ?
LONGITUDE AND TIME. 227
1. Formulate a rule for finding the difference in longitude
when the difference of time is known.
2. What is the difference in time of two places, if the dif-
ference in longitude is 15 degrees? 60°? 6°? 16'? 45'?
80° 30' ?
3. Formulate a rule for finding the difference in time when
the difference in longitude is known.
4. When it is noon here, what time is it at a place 15 de-
grees east of here ? 30° west ? 120° west? 90° east ? 45° 15' west ?
15' 30" east? 45' 15" west?
5. It is 0° on the meridian of Greenwich. In what direc-
tion, and how many degrees distant, is a place whose time is
2 hours earlier ? 1 hr. 10 min. earlier ? 3 hr. 20 min. later ?
2 hr. 15 min. later ?
6.
16° 10° 6° 0° 6° 10° 15° 20°
h i h k ^ h nr-^
Wb8t. m->- -<-« East.
How far apart are A and D? How found? B and F? How
found ? A and C ? How found ? C and G ? How found ? B
and C ? How found ? D and G ? How found ?
7. From the illustration above, formulate a rule for finding
the difference in longitude between two places when both are
in east longitude ? When both are in west longitude ? When
one is east and the other west longitude ?
8. Table of longitude :
New York, 74° 0' 3'' W. Boston, 7P 3' 30" W.
San Francisco, 122'=> ?6' 48" W. Canton, 113° 14' 0" E.
Paris, 2° 20' 22" E. Calcutta, 88° 19' 2" E.
Constantinople, 28° 69' E. St. Louis, 90° 15' 16" W.
Rome, 12° 27' 14'' E. Chicago, 87° 36' 0" W.
Washington, 77° 0' 15" W. New Orleans, 90° 3' 28" W.
228 LONGITUDE AND TIME.
1. Find the difference in longitude between Boston and
each of the other places given in the table on page 227.
2. When it is 2 P.M. at A, 60° west longitude, what time is
it at B, 90'' east longitude?
60° -I- 90° = 150° How do you find the difference in
150° corresponds to 10 hr. longitude between two plaxjes ? How
o .^/\^_ in -j-i.^ do vou Change difference in longitude to
2 P.M. + 10 hr. = 12 midnight ^.g^^^^ ,/y^^y ^ ^ ^^f^^ ^^^
of A ? In going east will you find earlier or later time ? If you wish to find
later time, do you add the difference to the given time or subtract it ?
3. Directions for using the chart : Turn the hour dial until
2, the hour at A, coincides with 60° W., the longitude of A.
Find 90° E., the longitude of B, and on the corresponding dial
read its time.
4. When it is 9 A.M. at C, 120° W., it is 12 m. at D. Find
the longitude of D.
12 hr. — 9 hr. = 3 hr. How do you find the difference in time be-
3 hr. corresponds to 45° ^^®®" '^^ P^«^®« ^ ^^^ ^^ ^^^ change differ-
1 90° _ J.^° — 7^5° A ®°°® ^" ^^"^® ^^ difference in longitude ? Does
' D have earlier or later time ? If later time, is
it east or west of C ? If C is 120° W., and D 45° east of it, how do you find
the longitude of D ?
5. Directions for using the chart : Turn the hour dial until
9 A.M. coincides with 120°. Find 12, and read the longitude
of the corresponding dial.
6. When it is 11 A.M. at Boston, what time is it at Paris ?
7. When it is 3 p.m. at New York, what time is it at New
Orleans ?
8. When it is noon at Canton, what time is it at
Washington ?
9. When it is 6 a.m. at St. Louis, what time is it at Rome?
10. When it is 4 p.m. at Chicago, what time is it at Boston ?
Find the missing term in the following :
11. A. Longitude 20° 15' 20" E. Time 6.30 a.m.
B. Longitude 15° 10' 20" W. Time?
STANDARD TIME. 229
1. This difference in time caused such inconvenience to the
railroads and persons traveling that in November, 1883, the
principal cities and railroads in the United States adopted
what is called " Standard Time,^^
2. Four meridians were chosen, 15 degrees apart, as cen-
tral meridians; these are the 75th, 90th, 105th, 120th. All
places 7i degrees on either side of each of these meridians
form a belt. Thus from 7J degrees east of the 75th meridian
to 7i degrees west of it is called the Eastern Belt, and the solar
time of the 75th meridian is the standard time of all places in
that belt. The other belts are called the Central Belt, the
Mountain Belt, the Pacific Belt.
3. Trace these Central Meridians on the map. Locate each
belt. Learn approximately the boundary of each, so as to teU
what States lie mostly in each belt.
4. Since these central meridians are just 15 degrees apart,
what is the difference in time between the Eastern and Central
Belt? The Eastern and Pacific Belt?
5. When it is 2 p.m. at Philadelphia, what time is it at
Detroit?
6. When it is 10 a.m. at Denver, what time is it at New
Haven ?
7. How should a watch be changed in going from Connec-
ticut to Minnesota ? From California to Ohio ?
8. What is the difference between the true time and the
standard time at a place whose longitude is 85° W.? 125° W. ?
Give the difference in standard time, and tell which place
has earlier time : —
9. Boston and St. Louis. New York and St. Louis.
10. Washington and New Orleans. Portland, Me., and Seattle.
11. Baltimore and Omaha. Philadelphia and Chicago.
12. How should a watch be changed in going from Boston
to St. Louis ? From Chicago to Washington ?
230 OEAL,
1. A piece of cloth containing 12 yd. was sold for $60, at
a gain of 25%. What was the gain on each yard?
2. By selling 6 yd. of cloth for $20 a merchant gained i of
what the cloth cost. What did it cost a yard?
3. What will 5 gal. of molasses cost, if 6 pt. cost 45 cents ?
4. What will 1 quart of seed cost, if 2 pk. cost $3.20 ?
6. 5^ times 5 is ^ of what number?
6. If 9 lb. of oatmeal cost $.54, what will 27 lb. cost?
7. If a pole 8 ft. long cast a shadow 4j ft., what will be
the length at the same time of day of the shadow of a pole
which is 15 ft. long ?
8. 1 J are how many times A ?
9. How many times is 2} contained in 8§ ?
10. A man after spending } of his money foimd that f of
what remained equaled $12. How much money had he ?
11. What part of 2 is i of J?
13. What part of 3 is ^ of i?
18. .} of 16 is J of how many times | of 21 ?
14. If 15 cords of wood are worth $45, how much are Sf cd.
worth?
16. Divide the number 45 into two parts which shall be to
each other as 7 to 2.
16. Divide the number 14 into two parts which shall be to
each other as 4 to 3 ?
17. Two men hired a pasture for $40. One put in 5 horses
and the other 3 horses. What ought each to pay?
18. Two boys have 41 marbles. If one has 5 marbles more
than the other, how many marbles has each ?
19. If 9 times a certain number is 24 more than 6 times the
same number, what is the number?
20. What number is that, to which if i of itself be added,
the sum will be 21 ?
21. 12c - 7a; = 25. Find x.
MISCELLANEOUS llEVIEW. 231
1. If it cost $270 to inclose a rectangular field 60 rd. long
and 40 rd. wide, how much will it cost to inclose a square field
of the same area with the same kind of a fence ?
a. The longitude of Boston is 71° 3' 30" west, and that of
San Francisco 122° 26' 15" west. When it is 4 a.m. at Boston,
what is the time at San Francisco?
3. The longitude of Cincinnati is 84° 26' west, and that of
Berlin 13° 23' 45" east. When it is 16 min. 20 sec. past
10 A.M. at Cincinnati, what is the time at Berlin-^
4. When it is 10 o'clock a.m. at Philadelphia it is 10 min.
past 3 P.M. at Paris. What is the longitude of Paris, if that of
Philadelphia is 76° 10' west?
6. The perimeter of a square and the circumference of a
circle are each 16 rd. Which has the greater area? and how
much greater?
6. The parallel sides of a trapezoid are 62 yd. and 48 yd.
and the altitude 68 ft. What is its area?
7. At 66^ a square yard, it cost $9.90 to pave a triangular
space the base of which was 18 ft. What was the altitude?
8. A ladder 60 ft. long stands 15 ft. from a building, and
the upper end rests against the building 4 ft. from the top.
How high is the building?
9. If I buy 5 % bank stock on which there is a semi-annual
dividend of $400, what will it cost at 1125 a share ?
10. A board fence 5 ft. high is built round a square piece of
land measuring 48 ft. on a side. At $16.50 per M., find the
cost of the boards. At 15/ a running yard, find the cost of
building.
11. Find one of the two equal factors of 811,801.
13. Ij is one of 4 equal factors of what number?
18. Find the square root of 2.5 raised to the fourth power.
14. How many tiles, 4 in. by 2\ in., are required to make a
hearth 6 ft. 4 in. long, 3 ft. 9 in. wide.
232 MISCELLANEOUS REVIEW,
1. Posts are placed 8 ft apart, round a square field of 40
acres, and round a rectangular field of 60 acres whose width is
J of its length. How many posts are needed for both fields ?
2. A lot 88 ft wide contains Ij acres. Find the length of
the field.
3. As a wholesale grain-dealer you sell, March 1, to James
Merrick, 38 bbl. flour (a) |5.80, 136 bu. com (a) 35A 220 bu.
wheat (a) 66/, 410 bu. oats (a) 35/, with 5% discount for cash.
Make out his bill and note in proper form, supposing he pays
one-half cash, and gives his note for 3 mo. for the balance.
4. The difference between two numbers is 144. ^ of the
greater number equals § of the smaller number. Find the two
numbers.
6. A grocer -bought 150 boxes of oranges at f 2.50 a box.
He found 16% spoiled, but sold the rest at $3 a box. Did he
gain or lose? and how much?
6. The base of a triangular lot is 24 ft, the altitude is
45 ft Find the area.
7. A room is 18 ft by 20 ft How many yards of carpet-
ing 27 in. wide are required, if we allow a waste of 9 in. on
each breadth for matching figures ?
8. A hall measures 40 ft by 25 ft. by 14 ft At 36/ a
square yard, find the cost of plastering the room, allowing ^ for
doors and windows and 1 ft. all round for the base-board.
There are 10 windows 5 ft. by 10 ft., and 5 doors 5 ft. by 8 ft
9. Find the cost of 45 bu. 3 pk. 6 qt of wheat at 75/ a
bushel.
10. Find the cost of a load of lumber consisting of 40 planks,
16 ft. long, 8 in. wide, and 2^ in. thick, at $16 per M.
11. From 7 times a certain number subtract 5 times itself
and 10 more. The remainder is 6. What is the number?
12. A horse and wagon cost $115. The horae cost $5 less
than twice the cost of the wagon. Find the cost of each.
MISCELLANEOUS REVIEW. 233
1. D commenced business with #1,800 stock; 2 months
later he took in E with a capital of $1,500, and a month later
F with a capital of $2,400. At the end of the year the firm
had gained $1,164. Find the share of each.
2. A, B, and C are partners with $4,040 capital. A's gain
for the year is $492, B's $450, and C's $270, What capital
did each furnish ?
3. A man spent | of his money, and then f of the re-
mainder. If he spent $91 more than he had left, how much
had he at flrst ?
4. At $3J a rod it costs $420 to fence a field in the form
of a square. What will it cost to fence a rectangular field of
equal area whose sides are to each other as 2 to 4^ ?
6. Bought a horse for $120, which is | of 5^ times the
cost of a buggy. Find the cost of the buggy.
6. A and B sailed together from New York. A kept his
watch by New York time, and B set his by the sun every day.
In a few days the watches were 2 h. 15 min. apart. Whose
watch was the faster? In what longitude were they?
7. In grading a lot 162 ft. long and 40 ft. wide, it was
found necessary to raise it 15 inches. How many loads
(1 cu. yd.) of earth were needed ?
8. How many square feet of tin in 140 feet of furnace
pipes 8 in. in diameter?
9. A circular park is 60 rd. in diameter. At $1.85 a rod>
what will it cost to fence the park ?
10. A lot 6 times as long as it is wide contains 18,481^
square feet. What are its dimensions ?
U. VlLSB + V2:56 = ?
12. The square of a number divided by the number equals
84. What is the number ?
13. The cube of tlie fifth power is what power?
14. Cube the square of 9.
234 METRIC SYSTEM.
(For tables, see page 306.)
1. Examine carefully a meter stick. Into how many equal
parts is it divided ?
d. Call each part a decimeter ; i.e., a tenth of a meter.
3. Into how many parts is each decimeter divided? Call
each part a centimeter ; i.e., a hundredth of a meter.
4. Into how many parts is each centimeter divided? Call
each part a millimeter ; i.e., a thousandth of a meter.
Note. — The prefixes "deci," "centi," and "milli" come from Latin
words, and mean a tenth, a hundredth, and a thousandth.
5. A millimeter is what part of a centimeter?
6. A centimeter is what part of a decimeter?
7. A millimeter is what part of a decimeter ?
8. Draw several lines of different lengths. Estimate their
length. Test your estimate.
9. The multiples of the meter are designated by the Greek
prefixes "deka" (10), "hecto" (100), "kilo" (1,000), and
"myria" (10,000).
10. FiU in the blanks, and learn the abbreviations in the
following :
millimeters (mm.) = 1 centimeter (cm.)
centimeters ■= 1 decimeter (dm.)
decimeters ^ 1 meter (m.)
meters ■= 1 dekameter (Dm.)
dekameters = 1 hectometer (Hm.)
hectometers «= 1 kilometer (Km.)
kilometers ■= 1 myriameter (Mm.)
lyoTE. — The word "meter" means measure. The standard meter is a
bar of platinum carefully preserved in Paris. It was intended that the meter
should he one ten-millionth of the distance from the equator to the poles. It
is 89.37 in. in length.
11. Measure off 10 meters on a string with knots to indicate
the meters. Find the length and breadth of the school-yard
in dekameters.
METRIC SYSTEM -^ aqU ABE MEASURE. 235
1. Draw a square decimeter. Divide the sides of this
square decimeter into centimeters. Connect the points. Into
how many little squares is the square decimeter divided ?
2. How many square centimeters, then, in a square deci-
meter?
3. Take the upper right-hand square centimeter, and divide
its sides into milimeters. Connect the points. Into how many
parts is the square centimeter divided ?
4. How many square milimeters in a square decimeter?
6. Write the table for Square Measure as follows :
square milimeters (sq. mm.) 1 square centimeter (sq. cm.).
6. How many square centimeters in a square meter ?
7. Find the area of a square whose side is : 72 m. 4.25 m.
545 dm.
8. When the area of a square is as follows, find its side :
625 sq. m. 515.29 sq. m. 63.2025 sq. m. 54.76 sq. Dk.
9. A square park contains 81 sq. Hm. How many trees
20 m. apart can be set out round it?
10. Find the area of a rectangle when the sides measure :
40 m. X 36 m. 7.5 Km. x 60 m. 45 Dm. x 14.24 Dm.
11. Find the other dimension of a rectangle when the area
and one side are as follows: 72 sq. m. and 30 m. 11.2 a. and
16 m. 100 ha. and | Km. 7.2 ha. and 0.25 Km.
Note. — The square dekameter is usually called an Ar (a.), and the square
hectometer a Hectar (ha.), and the square meter a Centar (ca.). They are
employed chiefly in measuring land.
12. How many ars in a field 64.5 m. by 42.8 m. ?
13. The perimeter of a rectangle is 36 m. If the length is
13.8 m., find the breadth and the area.
14. If the area of a rectangle is 1,500 sq. m., and its length
48m., what is its breadth ?
15. The area of a rectangle is 288 sq. m., and the breadth
8 m. Find the length and the perimeter.
236 METRIC SYSTEM-- SQUARE MEASURE.
1. The base of a triangle is .6 Dm*, and the altitude 40 dm.
Find the base of a triangle with the same altitude and area
twice as large.
2. Two triangles have the same altitude, and their bases are
800 cm. and 2.4 Dm. Find the ratio of their areas.
8. The perimeter of a rectangle is 120 m. If the length is
twice the breadth, find the length, breadth, and area.
4. The roof of a tower is in the form of a pyramid with
a square base. If a side of the base is 4.6 m., and its slant
height 6.2 m., how many square meters of tin are required to
cover the roof?
6. Find the area of a trapezoid, if the parallel sides are 4 m.
and 800 cm., and the altitude 1.8 m.
6. The area of a trapezoid is 369 sq. m. If the parallel
sides are 1,600 cm., and 2.5 Dm., find the altitude.
7. The area of a trapezoid is 12,480 sq. dm., the altitude
.64 Dm., and one of the parallel sides 128 dm. Find the other
side.
8. One man has a garden in the form of a trapezoid. The
parallel sides 6 Dm. and 32 m., and their distance apart 124
dm. His neighbor has a square garden of equal area. Find
the side.
9^ The diagonal of a trapezium is 82 ra. Find the area of
the trapezium if the altitudes of the two triangles made by the
diagonal are 1.8 Dm. and 200 dm.
10. The sides of a rectangle are 28 m. and 63 m. Find the
side of a square equivalent, to the rectangle.
11. A square and a rectangle have the same perimeter, 90 m.
The length of the rectangle is twice its width. Compare the
area of the rectangle with the area of the square.
12. Compare the area of a rectangular field whose length is
four times its width, and perimeter 1,000 m., with the area of a
square field whose perimeter is 800 m.
METRIC SYSTEM-- CUBIC MEASURE. 237
1. Learn .- 1,000 cubic millimeters = 1 cubic centimeter.
1,000 cubic centimeters = 1 cubic decimeter.
1,000 cubic decimeters = 1 cubic meter.
2. Write the following metric quantities as cubic meters
and decimals :
4 cu. m. 7 cu. dm. 8 cu. cm. 43 cu. m. 19 cu. dm. 8 cu. cm.
64,532 cu. cm. 34 cu. mm. 48,676 cu. dm. 46 cu. cm.
6,537 cu. m. 7 cu. cm.
3. Reduce 13.46 cu. m. to cubic decimeters ; 42,300 cu. dm.
to cubic meters; 12 cu. m. to cubic centimeters; 412 cu. cm.
to cubic decimeters ; to cubic meters,
4. How many cubic meters in a box 1.40 m. long, 1.10 m.
wide, and 0,6 m. deep?
6. How many cubic decimeters in a wall 45 m. long, 26 dm.
high, and 246 mm, thick?
6. How many bricks 20 cm. x 10 cm. x 5 cm. will it take
to build a solid wall 60 m. long, 2.5 m. high, and 65 cm. thick?
7. A hollow cannon-ball measures 6 dm. in diameter. If
the diameter of the hollow part is 4 dm., find the volume of
iron in the ball.
8. Compare the surface and volume of a cube whose edge
is 3 m, with the surface and volume of a cube whose edge is
Im.
9. A box measures 4 dm. by 3 dm. by 2 dm. Compare its
volume with the volume of a box if one dimension is doubled.
Compare it with another if two dimensions are doubled. Com-
pare it with a third if all three dimensions are doubled.
Note. —Is it necessary, to find the volume, to perform the work in Ex. 9?
10. How many sters of wood in a cubical pile, one edge of
which is 8 m. ?
Note. — In measuring wood, a cubic meter is called a ster (st.).
11. What is the cost of digging a cellar 12.5 m. long, 6.4 m.
wide, and 262 cm. deep, at the rate of 76/ a cubic meter?
238 METRIC SYSTEM -^CAPACITY.
(For table, see pa^e 305.)
1. A cubic decimeter of water is a liter (1.).
2. The prefixes are the same as in the table for determining
length. Write the table of liquid measure.
3. What name will you give to the tenth of a liter? The
hundi'edth of a liter ? The thousandth of a liter? What will
you call ten liters ? A hundred liters ? A thousand liters ?
4. How many cubic centimeters are there in a liter? In a
deciliter?
6. How many cubic millimeters are there in a milliliter?
6. Write 65 1. as dekaliters ; as deciliters ; as centiliters ;
at hektoliters ?
7. Write 2345 cl. as deciliters ; as liters ; as dekaliters.
8. Find the price of a liter at $5 a hektoliter.
9. Find the price of a liter at 3/ a centiliter.
10. How many hektoliters in a bin 4 m. long, 3 m. wide, and
2 m. deep?
11. A bin is 12 m. long, and 8 m. wide. How deep must it
be to hold 1,440 hektoliters of grain ?
12. How many liters in a tank 5.6 m. long, 3.26 m. wide, and
1.4 m. deep ?
13. A rectangular tank 4.8 dm. long, and 25 cm. wide, con-
tains 66 liters of water. What is the depth of the water?
14. 416 hektoliters of potatoes are put into a bin, 16 m. long,
and 6.2 m. wide. What is the height of the bin ?
15. Find the volume of a rectangular prism when the length
is 45 dm., breadth, 7 m., and height, 340 cm.
16. How many cubes, each with an edge of 1 dm., can be
put into a box 16 dm. by .8 m. by 6 dm., inside measurement?
17. Find the volume of a square pyramid, if the height is
4.5 cm. and a side of the base .8 dm.
18. The total surface of a cube is 2,400 square meters. Find
its volume.
METRIC WEIGHTS, 239
(For table, see page 306.)
1. The metric unit of weight is called a gram. It is the
weight of a cubic centimeter of pure ice-water.
2. The prefixes used with the terms meter and liter are
used with the gram. Give the name of the tenth of a gram ;
of ten grams; of the hundredth of a gram; of a hundred
grams ; of the thousandth of a gram ; of a thousand grams.
3. Write the table of metric weights.
4. Examine carefully the set of weights. Of what are
these weights made?
5. Weights representing the fractions of a gram are usually
made of thin sheets of aluminum or platinum. They are
graduated in the same way as the larger weights.
6. What is the weight of a liter of ice-water ? Of a cubic
millimeter of ice- water?
7. Change 6.4872 mg. to centigrams.
8. Change 2345 Dg. to decigrams.
9. Change 45789 eg. to kilograms.
10. How many centiliters of water will weigh 146 dg. ?
11. How many dekaliters of water will weigh 14.64 Kg.?
12. Add 44 dg.; 4.638 Dg.; and 2.45189 Hg.
13. From 16.4895 Dg. take 244.68 dg.
14. Multiply 2.48 dg. by 2.42, and express the result in
dekagrams.
15. Divide 148680 g. by 6.3, and express the result in kilo-
grams.
16. Divide 63.258 Dg. by 39 mg.
17. What will 474.6 Hg. of beef cost at 28/ a kilogram?
NoTB. — A kilogram is 2 J lb.
18. At $6.50 a ton, what will the coal cost to keep a fire a
week if 30 kilos (kilograms) are burned each day ?
19. Find the weight of water that may be contained in a
cistern 4 m. deep, 1.5 m. long, and 1.2 m. wida
240 ORAL.
1. A room is 10 ft. by 20 ft., and 10 ft. high. How many
square feet on the four walls ?
2. A room is 9 ft. square and 10 ft. high. What will it
cost to plaster it at 2b ^ a square yard ?
3. Jennie has a piece of ribbon 15 inches long which con-
tains 45 square inches. How wide is it?
4. Four square feet are what part of 4 feet square ?
5. A square contains 144 square feet. How long and how
wide is it?
6. How many feet long is a wall 11 ft. high, which contains
264 square feet, not including the top ?
7. 360 square feet ai-e .6 of the square feet in the floor of a
room 30 ft. long. How wide is the room ?
8. How many acres in a lot 20 rd. by 16 rd. ?
9. How many acres in a lot 28 rd. by 82 rd. ?
10. How many board feet in 8 boards, each 5 in. wide and
12 ft. long?
11. How many board feet in 40 boards, each 6 in. wide and
10 ft. long?
12. A 2-inch cube is made of inch cubes. One of the inch
cubes is what per cent of one row of the 2-inch cube ? What
per cent of one layer? What per cent of the whole cube?
13. Two inch cubes are what per cent of one row ? Of one
layer? Of the whole 2-inch cube?
14. Three inch cubes are what per cent of the 2-inch cube ?
Four inch cubes are what per cent? Six? Seven? Eight?
15. How many J-inch cubes will build a 2-inch cube ?
16. How many cords of wood in a pile 16 ft. long, 4 ft. wide,
and 8 ft. high? In a pile 24 ft. long, 8 ft. wide, and 4 ft. high?
In a pile 40 ft. long, 4 ft. wide, and 16 ft. high ? In a pile 32
ft. long, 4 ft. wide, and 8 ft. high ?
Note. — Picture the cord as a unit, and see how many units there are. Do
not change to cubic feet.
ALGEBRAIC ADDITION. 241
In Algebra we have two kinds of quantities, positive and
negative. The sign + is prefixed to positive quantities and the
sign — to negative quantities.
1. If we call gain iti business positive, what shall we call
the loss ?
2. If 15° below zero is negative, what should 16° above
zero be called ?
3. If direction to the east is called positive, what should
direction to the West be called ?
Note. — In Algebra negative quantities are as real as positive quantities,
and compared with positive quantities mean opposite in direction or effect.
4. Change the following expressions, using positive terms
for negative : (a) — 4 south latitude ; (b) I traveled — 5 miles
to the east; ((?) I lost — $2; {d) Subtract — 6; (e) Add— 4.
■ +
6. -7,-6,-6,-4, -3, -2,-1, 0,+l,+2,+3,+4,+5,+6,+7,+8.
1. +4 + (+2)t= + 6. 2. -4 + (-2) =-6.
3. +4+ (-2) = + 2. 4. -4 + (q.2) = -2.
(a) To illustrate (1) : place your pencil on + 4, and move in the positive
direction two points. Where are you ? What is the algebraic sum of two pos-
itive quantities ?
(6) To illustrate (2) : place your pencil on — 4, and move in the negative
direction two points* Where are you ? What is the algebraic sum of two
negative quantities ?
(c) To illustrate (3) : place your pencil on +4, and move in the negative
direction two points. Where are you ? What is the algebraic sum of a posi-
tive and a negative quantity, when the positive quantity is the larger ?
(d) To illustrate (4) : place your pencil on — 4, and move in the positive
direction two points. Where are you ? What is the algebraic sum of a posi-
tive and negative quantity, when the negative quantity is the larger ? •
6. Add — a, — 4a, — 7a, and — 2a.
7. What is the sum of 8 J and — 4 J ?
Unite the terms in each of the following algebraic expressions:
8. ix + ^x—8x. 9. 4:hc + She + 2bc.
10. 4 a: -f 4 a; — 5 a;. 11. 2xy — 5 xy + 4:xy.
12. 3 a — 2 a + 6 a. 13. 8 6 + 2 J — 6 J.
242 ADDITION.
1. Add 14 a, — 8 a, 8 a, — 2 a, — 4 a, 7 a, — a, 18 a.
Note. — Combine the positive quantities, and then the negative, before
finding the algebraic sum.
2. Find the sum of — 16 aJ, aJ, 7 aJ, — aJ, 11 aJ, — 2 ab.
8. Find the sum of 15 a\ - 6 a^ 5a\ - 8 a^.
Add the following examples :
4. 2a^ + 4a:y 5. 3a-2J + 6c
bac — 2xtf 2a + Sb — 5c
Zac -\-2xy 4a— b + 2e
— 2ac + Sxy — 5a — 46 — 3(?
6. 46 + 3 (?, -3 6, — 5(?-2 6, — 5^, -66 + <?.
7. 6a-5(? + 6, -56 — 4a: + 4tf-3.
8. 5a6 - 8 a26 + a^^y + a:y2, 4 a2j _ 7 ^62 - 4a?y + 5 arj^,
3 a62 + 4 a26 _ 3 a^Jy + ixf.
9. a8 + a2 + a, 2 a3 + 3 a2 - 2a, 8 a8 - 4 a2 + a.
10. — 3a: + 2y + «, rr- 3y + 22, 2a: + 3y — 2.
11. 5 a6 -f 6 6tf — 7 ac, 3 a6 — 9 6tf + 4 a^, 3 6(? + 6 a^.
13. 68 + 4 62(-- 3 6(?2, _4 62(?-5 6c2-c8, 3 62^ + 4 6c*.
13. 2:8 + ^ + ^ + 5^ 3a;8_4^aJ8_6aa; + 5, 3a:8-3ca:2
-7aa;-ll.
14. 2a - 36 + c, 15a- 216 -8(?,'3a + 24 6 + 7(?.
15. 5a — 3(? + d, 6 — 2a + 3d, 4(? — 2a — 3d.
16. 4a - 2a6 + 8a6 + 156, - 2a + 4a6 - 2a6 — 12 6,
J _ 2 a6 + 4 a6 — a, 2 a6 — 2a — 4 6 — a6.
17. 2a-56 + 2tf, 26-5tf + 2a, 2tf-5a + 2 6.
18. 6a + 36 + 7, -5a-56-ll, -7a-126 + 6,3a
+ 46-9.
19. 4 a6 + 6tf - 3 X, 3 a6 + 7 6(? - 3 a;, 3 6(? + 7 a; + 4, 4 a;
- 2a6 + 7, a6 + 2a; + 6tf — 3.
20. 7a: + 3y + 82 - 4, 525 - 7 + 3a? — 8y, 3y — 6a? + 6,
- 2as, 2 -4a? + 3y- 22.
21.
6a-56 + 2d, 4a-26 + 3d, 5a-26-4<i.
ALGEBRAIC SUBTRACTION, 243
Be sure that the pupils understand positive and negative quantities, page 241.
1. Subtraction is the process of finding the difference be-
tween two quantities.
--H wf^^ — ^«^ ►— +
-7,-6, -5, -4, -3,-2,-1,0, +1, +2, + 3, + 4, + 5, +6, +7.
Subtract :
12345678
Min. 4-4 4-4 2-2 2-2
Sub. 2-2-2 2 4-4-4 4
2-2 6-6-2 2 6-6
To Illustrate (1), place your pencil on the subtrahend, + 2, and move it to
the minuend, + 4. In what direction did you move ? Over how many spaces ?
What is the algebraic difference ?
To illustrate (3), place your pencil on the subtrahend, — 2, and move it to
the minuend, + 4. In what direction did you move ? and over how many
spaces ? What is the algebraic difference ?
To illustrate (6), place your pencil on the subtrahend, + 4, and move it to
the minuend, + 2. In what direction did you move, and over how many
spaces ? What is the algebraic difference ?
Illustrate 2, 4, 6, 7, and 8 in the same way.
2. In (1) which is the greater, the minuend or subtrahend?
Is the difference positive or negative ? In (2) which is the
greater, the minuend or subtrahend ? Is the difference positive
or negative ? Test the others in the same way.
3. When you subtract a larger number from a smaller, the
difference is always . Which of the eight examples above
illustrate this ?
4. When you subtract a smaller number from a larger, the
difference is always . Which of the eight examples above
illustrate this?
5. From a study of these eight examples, when is the alge-
braic difference found by adding the minuend and subtrahend ?
when by subtracting them ?
244 ALQEBRAW 8UBTMACTI0N.
Note. — The pupils at first should be required to tell in each ease whether
they are subtracting a less from a greater or a greater from a less, This will
be- less confusing than to give them the usual rule for subtraction.
1. 7 a 2. 8x
4a — 5x
3.
-9y
-4y
4. - 7 6 6. 12 ab
2b Sab
6. -11^ 7. 2x
- 7z 11a:
8.
_3y
9y
9.-7 abc 10. - Qxy
8 abc ~ 9 xy
1. 14 a 12. ~26
-7a 86
13.
2c
9c
14. 2d 16. —lixy
-7d - 2xy
6. 1 a — ib - 6c
3a + 26-4tf
17.
ix -2y - 5z
6* + Bz-' a
4a-66-2(? -2a:-2y-10« + a
18. From 6a — 26 + 5<? take a - i + 2 tf ,
10. From 6a: — 2y — 3^ take Sx-4y + 7z.
20. From 4:ab ^2ae + 3bc + 2bd take 4ab -}- Zao -^ 2bc
+ 4Jd.
21. From 8 63 _- 6 a6(? - 5 c3 take 6b^ + 5c^ -11 abc.
22. From
26* - 8aW + 6a62 _, 2a% take a% ^Zab^ + 4a^b» - 36*.
23. From 9 a — 6 6 + (? take 2a + 66-^2<?-fll.
24. From 7 a — 5 a: + 4 take 8 a — 9 a: + y^,
25. From 4a6c - 7aj + 3y — 24 takelabo + 8aj— 4y + 38.
26. From a + 6 take a — 6.
27. From x — y take » + y.
28. From 7a'-f5y— 3a take a; — 7 y + 5 a.
29. From \x — c take — 3 a6 + (?,
Reduce to its simplest form :
30. 6a-|-2a;-[2a + 6a!-(4a-3 a;)].
31. 7 a - [3 a + 7 « - (3 a - 3 a; - 5 a --4aj)].
82. 8aj- {6y ^ [4«- (4aj + 2«)]}.
83. (a + 6 — tf) - (a — 6 + (?) + (6 — a ^ o) — (a — a -- 6),
ALOBBBAtC MULTIPLICATION. 245
1. Multiply (^hy t^.
(fi = ax ax a. a^t:^axa. .'. a^ X a^ =^ (a X a X a) x {a X a)
^aXaXaXaX€t:^a\
a. a* X a^ =^ a^ c* X c^ = c?.
Note. — In each of these examples you notice that the exponent of the
product is found by adding together the exponents of the factors. This is
a general law.
Multiply s
8. a^bya*.
4.
cfic by ac^.
6.
l^hjhx.
6. a* by a^
7.
ah: by oa^.
8.
chc by c£b2.
9. a26bya^.
10.
aj*y by o^i/\
11.
SCy22 by jriygS.
12. a^c3 by a2c.
Id.
3/8z by xyz.
14.
c8c?2 by al^d.
Multiply :
16. 2 a2 by Qa\
16.
6 ft by 4 **.
17.
2aftby 3a2ft2.
18. 6a;yby5a^y2,
10.
3 ft2c by 3 6c2.
20.
3a2a^by4a»ar*,
21. If a man is $2 in debt, how many dollars has he ?
We say hd has -^ $2. If another man has 2 times as much, would he be
out of debt, or would he be deeper in debt ?
22. -2x2 =
-4.
A negaiive quanwiy muiLipii
gives a negative product.
eu uy a p
23. Multiply:
X X
X
XX X
X
3 2
1
0-1-2
- 3
3a; 22;
X
_a; -2a;
-Sx
In these examples the multiplicand remains the same, and the multiplier is
one less each time. Notice that the products are also one x less each time.
One X less than x is 0. What is one 3t less than ? What is one x less than
— X? Here we see that a positive quantity multiplied by a negative gives a
negative result.
24. 4 a; 25. 2abx 26. — Ic^d^ 27. — 8 ah?
-8 --6 3 4
25. State three laws of multiplication learned from this page.
246 ALGEBRAIC MULTIPLICATION.
1. On page 245, Ex. 23, make the multiplicand — x. The
products will be — 3 a:, — 2 a:, — a:, 0, a;, 2 a:, 3 a?. The multi-
plier is one less in each case. Do the products grow smaller or
larger? Two negative quantities multiplied together give a
positive product.
2. - 4 a26a s. - 2a?^y3 4. - 3 «%? 5. - 12 cd^
^7 ^7 ^7 - 7
6.-5 alf^c 7.-8 xyz 8.-9 a^d? O. — 7 a^
-The' -2axz - 3a»d - 4a^y*
10. Multiply 2^ + 2 xy -\' 2 yz + y^ by ^ xy.
1 1. Multiply a^ - 3 a^J 4- 4 a2j2 + J4 by 2 a%\
12. Multiply a^ - ^a^y + i3^y^ - 2a;y + / by Sa:^^.
13. Multiply a8 - 2 a2^ - 3 ac2 + 2 08^3 by 5 a^.
14. Multiply a^ + l^+ c^ — ab — be — ae hy abc.
15. Multiply 3 a^^y" + 2 aryS - 5 a:8y - 2 aryS by 5 a:2y2.
16. Multiply 4 a* - 2 a^J - 3 a%^ - 2 a2 by 4 a^b^.
17. Multiply 2 a^J - 3 a J2 _ 4 a?62 4. 2 aJ by 3 aJa.
18. Multiply a3 - 6 ay + 2 y2 _ 2 a^ by 6 aV-
19. Multiply a* + 6 a262 + 6* - 4 a^j _ 4 oA^ by 2 afts.
20. a^ -^ ay + y^ 21. 2 a; — 4 y
g - y a? - 2y
o^ + a^y + ay^ 2x^ — \xy
— g^y ^ ayi ^ f - 4 a;y + 8 yg
08 -y3 2a?«-8a3^ + 8y2
22. Multiply a:2 + 4a; + 5bya: + 3.
28. Multiply a* - 2 a^J + 3 a2J2 - 5 aJ^ + 7 J* by a 4- 4 6.
24. Multiply ar* — a:3_|_2:2_2,_|_ibya: + l.
25. Multiply l+3a;'-7a;2byl_6aj + 4a:2,
26. Multiply ar» + a:-2 by a^^ + a:- 4.
27. Multiply a8 - 3 a2 + 3 a - 1 by a2 + 3 a + 1.
28. Multiply a;9 — 2ajy + y^ by a; — y.
ALGEBRAIC DIVISION, 247
Note. — Since division is the inverse of multiplication, it follows that the
sign of the quotient must be + when the divisor and dividend have like signs
and — when the divisor and dividend have unlike signs.
If a» X a« = a» + 2 = a*, it follows that a^^a^= a^-^ =z a*.
1. Divide 15 a%^cf^ by 3 atA
15a2j3^ 96jc8y423
3a6c2 -12tc2y22 ^
8. a*JV ^ ^2j^ ^ ? 4. 25 aWd^ ^ -.5ab^ = ?
5. - 18 a%^c -5- 2 ac = ? 6. 8 a^JyV ^ a;y222 ^ ?
7. 21 oayS -j- - 3 ay = ? 8. - 28 a^Je? ^ _ 7 oA = ?
9. - 18a^^ - 6aa; = ? 10. 15 aa;y2 ^ -3 ay = ?
11. 3a2| 12a^- 9a»6 + 6a2g.
4 a3 - 3 aJ + 2 (?
12. - 8 ya j3 gyg + 12 bxyz — 9 y^g
— X — ^bx + 3 y
13. — 12 x^yz + 9 a:y2 — 6 xt/^z -i 3 a;y = ?
14. 25 a^bc - 15 a^c + 5 a36<?a;2 _^ _ 5 ^2^ = ?
15. 6 aV - 8 aV + 12 a2y4 ^ 2 ay = ?
16. - 86 a^a; + 54 a2x- 18 aa: ^ 18 rr == ^
17. 3 ic* - 12 a:8 + 15 a^ -J- - 3 a:2 = ?
18. 15 a%c - 10 a3J*(?5y2 ^ 5^258^2 ^ ^ 5 aJ^ = ?
10. 4 a? + 36 aar^ - 16 2; ^ - 4 a; = ?
20. 3 a8 - 9 a26 - 6aJ2 ^ _ 3 a = ?
21. Divide 6 a2 J -- 8 a% -\- 6 a2J2 _ 9 aJ by 3 oi.
22. Divide 8 xy® + 4 a:2y2 _ 4 ^.^ -f 12 2?y^ by 4ajy2.
23. Divide 2 (fie - 4: a^c^ ^ 6 a^(^ -^ 8 a^c by 2 a2e.
24. Divide 4 a^ar^ - 2 aV ^. 4 ^2^ _ 6 aV by 2 a2ar2.
25. Divide 6 a;Sy3 -h 12 a:*y^ - 18 xY - 24 a^^y® by 6 3^1/^.
26. Divide 18 a:"y2 + 27 aV - 45 a»y8 by - 9 ar^y.
27. Divide 14 cfib^ + 28 a^b^ - T aSjs _ 2I a^je by 7 a^P.
28. Divide 15 a«y^ - 12 ay + 18 aY + 21 aV by - 3 aV
29. Divide 8 J^a^ + 24 J^a;^ + 16 6«a6 - 40 ft^a* by 8 bhfi.
30. Divide 21 a^J^^ + 14a362c2 _ 7a26c3 by lobe.
248 ALGEBRAIC DIVISION.
1.
a:8_8a^y + 8a^ — y8 [ ^a _ 2 xy + y* Divide the ftrrt term of
J ^ ^ o ^ ^ term of the divisor. What
— Qchf + 2xy^ -^ y^ is the first term of the quo-
— x^y -\- 2xy^ — }^ tient? Multiply the whole
divisor hy this term. What
is the product? Subtract it from the dividend. What is the remainder?
Consider the remainder as a new dividend, and proceed as i^t first.
2. 3.
X — y a8 - 3 a6c + 6» + c» |a + 6-hc
ofiy — y^ —a%'-'a*e'^Babc + 6» + c«
x^ — gV - a»6 ^ ofeg -^ 06c
«*y^ — y* - a*c + aft'— 2 a6c + 6» + c«
ggaya _ gys — g^c — abc — ac'
{Cy«— y4
gy' — y*
X* — y*
X — x'j
a6»
— abo -f
ac» + 68 + c«
— abc +
- aftc
ac2 _ 62c + c8
- 62c - 6c2
4-
^ a!* + ir2y» + y* ^ ?
15 a* ^ a^a ^ 4 oaj -
4. a^ — y8^a — y=:? 5. a^
6. a^ - 9 aa? -]- 12 ahP + Sb a^x
7. 6aJ* + 21iB3y + 31a:2y2 + 27iFy8 - 5y*^3a;« + 6»y -
0. aJ^-2a;y + y2-«a^a;-y-2:^?
10. 2:8 + y8 4- 2® - 3 2:^25 -^ a: + y + 2 = ?
11. a6 + /-^a; + y=?
12. Divide Sa^ + Sa^J + 4a62 ^ ja by 2a + h.
13. Divide ar^ + Ha; + 30 by a: + 5.
14. Divide 9a:2 _ 3^ _ 2 by 3a: ~ 2.
15. Divide 9^ - 18a:2 ^ 26a: - 24 by 8;r -. 4.
16. Divide a* —6* by a — 6.
17. Divide o^ - Sa^J + 3a6a _ ja by a^ ^ 2ai + R
GENERAL SUMMARY, 249
1. Mathematics is the science that treats of measuring of
quantities to ascertain their properties and relations.
2. This measuring demands a unit of measurement, as 1 ft,
1 oz., 1 two-dollar bill, 1 tenth, 1 doz., etc.
3. Measuring quantity by a unit of measurement demands
number.
4. Number answers ihe question, How many ?, or shows
the ratio of the quantity measured to the unit of measure-
ment.
5. Arithmetic is the science of numbers and the art of using
them.
6. Algebra is that branch of mathematics which reasons
about quantity by the use of letters.
7. Qeometry is that branch of mathematics which treats of
space and its relations, and the measurement of lines, angles,
surfaces, and solids.
NOTATION AND NUMERATION.
8. Notation is the art of expressing numbers by symbols or
characters.
9. Numeration is the art of reading numbers that are ex-
pressed by figures.
10. There are two systems of notation, the Arabic and the
Roman.
11. The Arabic is the system in general use, and is so called
because it is supposed that it was introduced into Europe by
the Arab3.
12. The Arabic system employs ten characters, called fig-
ures^ to represent numbers, thus, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
13. The first of these characters is called naught, cipher,
zero. The other characters are called digits. By means of
these characters any number can be written.
250 NOTATION AND NUMERATION.
1. The position occupied by any figure gives it its value.
A figure standing alone or immediately at the left of the deci-
mal point represents single units, or units of the first order.
2. Ten units of any order make one unit of the next higher
order, and the orders increase in value from right to left.
3. A figure at the left of the single-units figure represents
tens of the unit, or units of the second order. A figure at the
right of the single-units figure represents tenths of the unit, or
units of the first decimal order,
4. For convenience three orders form a period.
6. From right to left the names of the periods are, uniU^
thatiaandSj millions^ billionSj trilUonSy quadrillions^ quintillionSy
sextillionSj septillions^ octillions^ nonillionSj decillionSj undecillionSj
duodedllionSy tredecillions^ qiLattuordecillions^ quindecillions^ sex-
decillionSy septendecillions, octodeeillions, novemdecillions, vigirv-
tillions,
6. At the right of the decimal point, and reading from left
to right, the names of the decimal orders are tenths, hundredthsy
thousandths, ten-thousandths, hundred-thousandths, millionths, ten-
miUionths, hundred-millionths, billionths, ten-billionths, etc,
7. Use the word and in reading numbers in place of the
decimal point only.
Write in figures :
8. Forty thousand, six and five tenths.
9. Twenty thousand, four hundred and four hundredths.
10. Three hundred and three hundred-thousandths.
11. Three himdred three hundred-thousandths.
12. Twenty-eight million, two thousand eighty-five and six-
teen millionths.
13. Six quadrillions, two hundred trillions, three hundred
six millions, forty-four, and five hundred-billionths.
14. One hundred sixteen trillion, two thousand and one mil-
lion, four hundred six himdred-millionths.
NOTATION AND NUMERATION, 251
Note. —See Part II, pages 162, 163.
Read the following numbers :
1. 406.00608. 40,040,040,040. 4,060,001,604.00004.
2. 1.0106487. 111,010,001,070. 28,014,643,005,004.72.
3. 100001.01. 200,200,002,002. 58,076,104,074.01016.
4. 36,071,026,108,000,010,042. 5000.0000004648.
6. 1,010,100,000,600,010,002,074. 16004.0001017484.
6. 100.002. .102. 600.16. .616. 300.025. .326.
7. 216.00112. 100.008008. 106.000000106.
8. 1000.001. 711,468.143464. 1,111,111,111.000111.
9. 216.000216. 114,106.00048. 2,002,002,020.0202.
10. 400,000,010.01. 7,000,000,007.7. 4,400,040,004.004.
Express the following numbers in figures :
11. Five million, three, and five millionths.
12. Four billion, two million, sixty-four thousand, nin,e.
13. Six million, six thousand, six hundred six and six hun-
dredths.
14. Sixteen billion, fourteen million, twelve thousand, twenty-
two, and twelve hundred-thousandths.
16. Foriy-seven thousand twenty-four ten-billionths.
16. One thousand six and twelve thousand four hundred-
millionths.
17. Eighty-one million, five hundred eight thousand, one
hundred seventy-eight and six thousand thirty-two ten-thou-
sandths.
18. Two hundred million, seventy-two thousand, seven hun-
dred fifty-four and two ten-billionths.
19. One billion, twenty million, three hundred four thousand,
fifty and one" hundred five thousandths.
20. Six billion, eight hundred seventy million, twenty-eight
thousand two hundred six and one hundredth.
21. One hundred twenty-five thousand ten and sixty-seven
thousandths.
262 ROMAN NOTATION,
1. In this system seven letters are used to express numbers :
Letters I V X L C D M
Value 1 5 10 50 100 500 1000
2. All other numbers are expressed by writing two or more
of these letters side by side.
8. The following four principles must be learned:
(a) When a letter is followed by another of equal or less
value the number expressed is equal to the mm of the simple
values, thus, 111 = 3, XV = 15, CC = 200, LXX = 70.
(6) When a letter is followed by one of greater value than
itself, the number denoted by the expression is the difference of
their values, thus, XC=90, IX = 9, CD = 400.
(c) When a letter is placed between two letters, each of
greater value, its value is subtracted from the sum of the other
letters to find the value of the expression, thus, XIV = 14,
DXL = 640, XIX = 19.
(d) A dash or bar placed over a letter multiplies its value by
1000, thus, V = 5000, XTV = 14,000, CLX = 100,060.
Express in Arabic notation :
4. XLV, CCCV, MMM, DXC, MMD.
6. LXV, XLVI, XCIV, XCIX, LXXXIX.
6. CIX, DLIV, MDCI, MXCV, MDCCCXIX.
7. XCI, CMVl, MCDX, CCXC, CDCCXLIX.
8. DCXC, DCCX, CLXXV, XXIX, CCCCXLVIII.
Express in Roman notation :
9. 16, 24, 88, 52, 65, 78, 89, 91.
10. 156, 204, 660, 478, 892, 999, 312, 463.
11. 1186, 1776, 1890, 1896, 1900, 1902, 1492, 1886. '
12. 1209, 1680, 1756, 1876, 1879, 1905, 1910, 5648.
ADDITION. 253
1. Addition is the process of uniting two or more numbers
into one number. Each of the quantities added must have the
same measuring unit ; that is, they must be of the same kind.
2. The numbers to be united are called addends.
3. The sum is the number obtained by adding.
4. The sign of addition is +? and is read plus.
5. The sign of equality is =, and is read equaln. Thus
5 + 4 = 9 is read, 5 plus 4 equals 9.
6. There are two principles to be memorized :
(a) Only like numbers can be added.
(6) The sum is the same denomination as the addends.
7. Drill on the following forty-five combinations till accuracy
and rapidity are secured.
,.1. o2..23 .34 .345. 7456. .4567.
^'1' "^'1' ^'2'l' '2'1' ^'3'2'1' ''3'2'l' '4 3'2'l'
q5678 .^56789 ..6789 .o6789.
9, 4' 3I 2' 1; 10. 5' 4' 3' 2' 1' ^1' 5' 4* 3' 2' ^^' 6' 5' 4' 3'
10 T 8 9 1.789 ..8 9. .^ 8 9. .» 9. .0 9
13, g, g,^; 14, ^, g, g; 15, ^, g , 16, g, ^, 17, g, 18, g.
8. All problems in addition are simply repetitions of these
problems, though the numbers may not always occur in the
7 8
same order. Thus the numbers may be 5. or «» the sum re-
maining the same.
9. Enlarge each of these combinations thus : —
(a) Place a tens figure before one of the addends, thus,
12 22 32 ,
J _2 2 "^"-
(J) Place a tens figure before each of the addends, thus,
26 35 65 ^.„
24 44 84 ®^-
254
ADDITION.
1.
2.
3.
4.
346
7966
23756
868963
465
9664
12762
527878
664
6643
66431
869667
642
6432
96246
787238
426
3219
53669
984794
268
2190
86725
689468
683
9083
92368
948973
832
6886
68649
494747
327
4776
26735
636836
796
7792
76482
966775
6.
6.
7.
8.
7.32
81.078
427.36
4.84
28.397
16.004
61.037
132.468
11.016
8.74
44.074
230.067
248.318
164.8
8.74
57.8
.428
82.063
236.7
9.732
1.08
8.166
85.798
162.875
9. Add six hundred twenty miHion, two hundred six thou-
sand, four hundred eight ; nine million, three thousand, four ;
twenty-three million, fifteen thousand, five hundred four ; seven
million, thirty-two thousand, seventy-five; three million, four
thousand, forty-eight
10. Add thirty-six and twenty-eight thousandths ; twenty
and eight hundred five ten-thousandths; forty-one thousand
two hundred eight hundred-thousandths ; two million, three
thousand, one hundred ten and three thousand sixty-two ten-
thousandths ; five tenths ; twelve hundredths.
11. Add 5.4307; 48.6512; 7564.02; 314.065; 864.23;
1026.0087; 2346.002004; 86.24 ;. 1209.00643 ; 3109.02041;
3287.0074; 6.007; 704.0049; 1010.00101; 4.07; 16.00109.
SUBTRACTION. 256
1. Subtraction is the process of taking one number from
another, or it is the process of finding what part of a gfiyen
number remains when a part has been taken away.
2. Mimiend is the number from which another number is
taken, or it is the given number from which the part is taken.
3. Svbtrahend is the number taken away from another
number, or it is the given part which is taken from the minuend.
4. The result of an example in subtraction is called
difference or remainder.
5. The sign of subtraction is — . It is read minus. Thus
9 — 5 is read 9 minus 5, and indicates that 5 is to be subtracted
from 9.
6. The eighty-one primary facts of subtraction should have
been learned while learning the facts of addition. If not, each
pupil should be made perfectly familiar with them now. See
Page 253.
7. Enlarge each combination as suggested in addition on
Page 253.
8. From 763 take 486.
763 We cannot take 6 units from 3 units, so we take 1 ten from the
486 ^ tens, leaving 5 tens. We add this ten which equals 10 units to the
27^ 3 units, making 13 units. 6 units from 13 units leaves 7 units, which
we write in units' column. 8 tens from 5 tens we cannot take, so we take
1 hundred, leaving 6 hundreds. We add this one hundred, or 10 tens, to 6 tens,
making 15 tens. 8 tens from 15 tens leaves 7 tens, which we write in tens'
column. 4 hundreds from 6 hundreds leaves 2 hundreds, which we write in
hundreds' column. The remainder is 2 hundreds, 7 tens, and 7 units, or 277.
9.
10.
11.
12.
13.
14.
342
604
738
820
701
812
163
467
419
644
418
619
16.
16.
17.
18.
19.
20.
8052
5409
8025
6317
3020
4004
5148
3645
7184
3189
2185
3074
>6
BUBTBACTION.
Subtract:
1.
2.
3.
2030406060
7988862
4421618
1468194782
4726845
2567487
4.
6.
6.
20040060070
75103040
800206080
16417381246
67068172
434184165
7.
8.
9.
60407030809
40000000
84000605045
41626818714
23456789
42613417632
10. To what number must 472687 be added to make
604032?
11. How much less than 3002020004 is 1436817486?
12. What number must be subtracted from 2001004000
that the remainder may be 981607106?
18. What number must be added to 647683 to make
1047102?
14. From two million, two thousand four, take sixty-eight
thousand seventy-one,
15. How much greater is one billion than two himdred
thirty-two million, forty-five thousand, seven hundred thirty-two ?
16. Find the difference between two hundred and two
thousandths, and two hundred two thousandths.
17. From thirteen thousandths take forty-one millionths.
18. 479.0128 -(114.016 + 96.0074)=?
19. If the whole quantity is 45602, and one of the parts is
26715, find the other part.
20. The subtrahend is 1054608 ; the minuend is 4610072.
What is the difference ?
MULTIPLICATION, 267
1. Multiplication is the process of repeating a quantity a
certain number of times, or it is the process of finding the sum
of several equal numbers.
2. The multiplicand is the number repeated.
3. The multiplier shows how many times the quantity is to
be repeated.
4. The product is the result of the process of multiplication.
5. The sign of multiplication is x , and is read times or
multiplied hy ; thus, 4 X $3 is read 4 times $3 and $3 x 4 is
read f 8 multiplied by 4. In both cases 4 is the multiplier, and
shows the ratio of the product to the multiplicand.
6. Learn the following principles of multiplication : —
(a) The multiplier denotes ratio, and is always an abstract
number.
(6) The product is always of the same denomination as the
multiplicand.
7. The sixty-four facts of multiplication, as found in mul-
tiplication tables, should be thoroughly memorized.
8. The multiplicand and multiplier are sometimes called
factors,
9. Multiply 684 by 476. 10. Multiply 5600 by 130.
684 5600
476 130
Omit ciphers in mul-
tiplying, and annex to
4104 = 684 X 6 units 168 the product as many
47880 as 684 x 7 tens, or 70 units 66 ciphers as have been
2736)J0 = 684 X 4 hundreds, or 400 728000 omitted.
325584 units
11. To prove an example in multiplication : —
(a) Use the multiplicand as the multiplier.
(6) Divide the product by either factor. The quotient will
be the other factor.
258
MUL TIPLIGA TION.
Multiply !
1.593
566
6. 2876
186
11. 92646
675
16. 695736
3928
2.672
748
7. 6542
347
12. 37595
793
3.878
896
8. 9734
296
4. 839
456
9. 6542
347
5. 789
658
10. 9678
978
13. 45674 14. 82457 15. 83729
962 406 596
17. 843934 18. 836719 19. 547386 20. 840673
7926 5849 9657 6489
21. 874007 22. 900608 23. 960087 24. 930807 26. 670509
9047 4705 7008 9060 7060
26. What will 943 barrels of flour cost at $6 a barrel ?
(a) $ 6 (b)
9 6
943
18
24
54
$5668
943
$ 6
$5658
Method (a) is correct and needs no explanation.
Pupils should be allowed to use the other form for
brevity. It is important, however, that they should
keep clearly in mind that (6 is the number to be
repeated in each case.
27. How many bushels of potatoes can be raised on 678
acres at the rate of 87 bu. an acre ?
28. Multiply seven thousand sixty-four by nine thousand
six hundred five.
29. Multiply seven thousand ninety-six by five thousand
seventy.
30. Multiply seven hundred sixty-five thousand, six hun-
dred seventy-seven by eight thousand sixty-eight.
31. "What is the value of 8679 acres of land at $467 an
acre?
32. Multiply the sum of 96142 and 87310 by three times
their difference.
DIVISION. 259
1. Division is the process of finding how many times one
number is contained in another, or it is the process of sepa-
rating a number into equal pai-ts.
Note. — At 3^ each how many peaches can be bought for 16 cents?
15^ ji. 3^ z= 5. This illustrates the tirst definition, or division. I paid 15^ for
6 peaches. What did each cost ? 15^ -r 5 = 3^. This illustrates the second
definition, or as it is sometimes called, Partition.
2. The dimdend is the number that is to be divided or
separated into parts.
3. The divisor is the number by which we divide.
4. The quotient is the result obtained by division.
5. The remainder is the part of the dividend remaining,
when the divisor is not contained an exact number of times.
6. The sign of division is -s-, and is read divided hy. It
indicates that the number before the sign is to be divided by
the number after it.
7. Division is expressed in four ways : —
(a) Fractional method, thus V = 3.
(6) Using sign, -i-, thus, 12-5-4 = 3.
(c) Using sign, : , thus, 12 : 4 = 3. Ratio method.
(d) Working method, thus 4 )12 .
8. Division is the converse of multiplication. Multiplica-
tion is thus a proof of division.
9. Divide 6048 by 36.
168 36 is not contained in 6 thousands. 36 is contained in 60
36)6048 hundreds, 1 hundred times. Write the 1 over hundreds' figure
3600 o^ *^^® dividend. 36 times 1 hundred are 36 hundred. 36 hun-
2440 dred from 60 hundred leaves 24 hundred, or 240 tens. 240 tens
2160 ^^^ ^ ^^^^ ^^^ ^^^ ^®°®* ^^ ^"^ contained in 244 tens, 6 tens times.
— — 36 times 6 tens are 216 tens. 216 tens from 244 tens leaves 28
^^^ tens, or 280 units. . 280 units and 8 units arc 288 units. 36 is
?2? contained in 288 units, 8 units times. 36 times 8 units are 288
units. There is no remainder.
Note. — Teachers should explain that ciphers marked in the example are
not usually written.
260 DIVISION.
Divide : —
1. 2862 by 27. 2. 6680 by 84.
3. 11997 by 43. 4. 84894 by 58.
6. 83927 by 89. 6. 109604 by 94.
7. 79692 by 916. 8. 61074 by 783.
9. 13,746,232 by 386. lo. 56,882,034 by 694.
11. 441,239,442 by 4863. 12. 489,877,092 by 8721.
18. 26,446,182,662 by 62598. 14. 26,015,867,442 by 75407.
15. 4,680,423,662 by 27418. 16. 1,047,298,812 by 86414.
17. The dividend is 531288 and the divisor 628. What is
the quotient?
18. How many times can 826 be subtracted from 1,066,666 ?
19. By what must 4006 be multiplied to make 366,684?
20. The dividend is 17,590,800 and the quotient 5136.
What is the divisor?
21. The product of two factors is 403326, and one of the
factors is 126. What is the other factor?
22. The multiplier is 246 and the product 562620. What
is the multiplicand?
23. What number will divide 866,775 exactly 196 timefi?
24. What must be taken from 232,696 that the remainder
may be exactly divisible by 612 ?
25. What is the nearest number to 682186 that will con-
tain 321 without a remainder?
26. The sum of 366 equal numbers is 8, 113, 815. What
is each number? ^
27. The product is 10,365,051 and the multiplicand is 3021.
What is the multiplier?
28. If 13 horses and 15 cows cost $2820, and the average
price of a horse is $130, what is the average price of a
cow?
29. A rectangle contains 4824 sq. ft. If it is 72 ft. long,
how wide is it?
PROPERTIES OF NUMBERS. 261
1. Numbers are integral^ fractional^ or mixed.
2. Integral numbers are divided into two classes, even and
odd.
3. An even number is a number that is exactly divisible by
2, as, 4, 8, 32, etc.
4. An odd number is a number that is not exactly divisible
by 2, as, 5, 9, 13, 41, etc.
6. Integral numbers are also classified as prime or com-
posite.
6. Factors of a number are the numbers that multiplied
together will produce the number, as 4 and 3 are the factors of
12; 2, 3, and 5 are the factors of 80.
7. A prime number is a number that has no integral fac-
tors except itself and 1, as 5, 11, 29, etc.
8. A composite number is a number that has integral fac-
tors, as, 9, 25, 36, etc.
9. Factoring is the process of finding the factors of a
number.
10. Prime factors of a number are the prime numbers which
exactly divide the given number.
11. Learn : To resolve a number into its prime factors,
divide the number by any prime factor, and do the same with
each quotient until the quotient becomes a prime number.
The several divisors and the last quotient are the prime factors
required.
Find the prime factors of : —
12. 105 13. 429 14. 8735 15. 3224
16. A divisor of a number is a number that exactly divides it.
17. Name all the divisors of 45, 51, 96, 81, 32.
18. A common divisor of two or more numbers is a divisor of
each of them.
19. What divisors are common to 9 and 12 ? 24 and 36 ?
45 and 60 ? 6, 9, and 12 ? 14, 21, 28, and 35 ?
262 GREATEST COMMON DIVISOR.
1. The Oreatest Common Divisor of two or more numbers
is the largest number that will exactly divide each of them.
2. Find the greatest com- 3. Find the greatest com-
mon divisor of 46, 60, 75. mon divisor of 92 and 115.
115
92
23
1 Divide the greater
4 number by the less,
the divisor by the re-
ihainder, and thus con-*
3 )45, 60,75 Divide by any com- 92
5M5 20 25 ™^^ prime factors of 92
^ — T^ — -r — z all the numbers. Do —
Q 4. K
^> ^> ^ the same with the quo-
tients, till the quotients have no com- tinue until there is no remainder,
mon factor. The product of all the The last divisor will be the greatest
divisors will be the greatest common common divisor,
divisor.
Find the greatest common divisor of :
4. 18, 27, and 45 5. 42, 56, and 84
6. 909 and 1414 7. 1917 and 2556
8. 350 and 475 9. 759 and 1155
LEAST COMMON MULTIPLE.
10. The Least Common Multiple of two or more numbers is
the least number that is exactly divisible by each of them. It
contains all the prime factors found in each of the numbers,
and no other factors.
11. Find the least common multiple of 8, 12, and 20.
Divide the numbers by any prime factor of
2 )8—12-20 two or more of them ; write the quotients and
2^4— 6—10 undivided numbers beneath, divide as at first,
^-^ — ^ — -z and so continue until the numbers in the last
~ line are prime to each other. The product of
2x2x2x3x5 = 120 the divisors and the numbers in the last line
will be the least common multiple.
Find the least common multiple of :
12. 28, 42, 63, 108 13. 171, 592, 703
14. 65, 78, 104, 130 15. 115, 161
Note. — When the numbers are not readily factored, find the greatest
common divisor by the second method. Divide one number by the greatest
common divisor and multiply the quotient by the other number.
SUMMARY OF DEFINITIONS AND PROCESSES 263
OF COMMON FRACTIONS,
1. A fractional unit is one of the equal parts of a unit.
2. A fractional number is a collection of fractional units.
3. A fraction is one or more of the equal parts of a unit.
4. The denominator of the fraction shows the number of
equal parts into which the integral unit has been divided. It
therefore gives the name to the fraction.
5. The numerator of the fraction shows the number of the
parts taken to form the fraction.
6. The numerator and denominator are the terms of the
fraction.
7. Fractions are classified with respect to their value into
proper and improper fractions.
8. A proper fraction is one whose numerator is less than
its denominator, i.e., its value is less than a unit, thus, §, 5, etc.
9. An improper fraction is one whose numerator is equal to
or less than its denominator, i.e., its value is equal to or greater
than a unit, thus, f , f .
10. Fractions are classified with respect to their form into
simple and complex fractions.
11. A simple fraction is one whose terms are integers, as §.
12. A complex fraction is one which contains a fraction in
2i
one or both of its terms, as -j- .
4
13. A mixed number is a number consisting of an integer and
a fraction, as 4J.
14. Write two proper and two improper fractions.
15. Name the numerator and denominator of each.
16. Write a complex fraction having a simple fraction in
both terms. Write a complex fraction having a mixed number
in both terms.
17. Reduction of fractions is the process of changing their
form without changing their value.
264 ItEDUCTION OF FRACTIONS.
To change a mixed number to an improper fraction: —
1. Change 9f to an improper fraction.
9| ae J^ In one unit there are 6 sixths. In 9 nnits there are 9 times 6
sixths or 64 sixths. 54 sixths and 6 sixths are 69 sixths.
Learn : Change the integer to an equivalent fraction having
the denominator of the fraction, and then unite the two frac-
tional parts.
2. Change the following to improper fractions : -—
9S; 15|; 46H; 38i| ; 16|; 96t; 41^^; 29?; 87^; 16^.
To change an improper fraction to a whole or mixed number: —
8. Change V to a mixed number.
%^- = 5f Since there are 5 fifths in 1 unit, in ^ there must be M many
units as five is contained times in 27 ^ or 6f .
Learn : Since a fraction is an expression of division, the
rule follows : Divide the numerator by the denominator.
4. Change to whole or mixed numbers : — ^^^ ; V » \i\ \\\
M^; ¥7'-; ¥/; W; W; ¥/s W; W-
To change a fraction to its lowest terms : —
5. Change §g to its lowest terms.
5 I M = f Since dividing both terms of the fraction by the same number
does not change its value, we divide both terms by the common
factor 6 and our result is J.
Learn: Divide both terms by a common factor till no
common factor can be found. When no common factor can be
found by inspection, find the greatest common divisor of the
terms, and divide both terms by it.
6. Change to lowest terms: — H; If; tW; if*; HI; 4ti;
/irV^; §M; im; HH; H5; JS»; HI; t^iPA-
REDUCTION OF FRACTIONS^ ADDITION, 265
To change fractions to equivalent fractions having the same
denominator : —
1. Change i and f to 12ths.
1. X I = i®w f X J = -^w Since multiplying both numerator and denomi-
nator by the same number does not change the
value of the fraction, we multiply both terms by such a number bs will change
the denominator to the required denominator.^
2. Change to the same denominator.
I, §, I to 12ths. i, i, i, tV to 48ths.
§, ^ to 15th8. i, hh \ ^ 36th8.
J, f , 5 to 24th8. I, §, i, A to 60ths.
Fractions having the same denominator are called like
fractions or similar fractions.
ADDITION OF FRACTIONS.
3. Add f and f .
Since only like or similar fractions can be added,
4 ^^ 30* 5 ^^ 3u find the least common multiple of the denominators
hi ^ /u ~ §8 ~ -^^Tf ^or the new denominator and make the fractions
similar.
Add:
—
4.
h h i, ^^'
6.
h h H> a-
8.
h A> T^TTJ i-
10.
H, ih T^2, «.
6. h
T*5> lV> f •
7. h
5. \h i-
9- «,
h ihH-
11. /„
T^> ^> tV'
In the following examples add the integers and fractions,
separately.
12. 61| + 112J+78| + 176|. 13. 216| + 14l4^f +86^1+415^.
14. 412iJ+371^ + 211^+911i. 15. 86J+ 116^5 + 28TV+196f.
16. 74f+83| + 136j+672|. 17. 69/5 + 64^^+75/^+118^-^.
18. 96Tar+71/5+461§ + 164/3f. 19. 64^ + 961+84^+178/,.
20. 46^{+66t!V+436H+126|. 21. 45^+8lH + 66iJ+148/j.
22. 18^+94f+397/5+326ii. 23. 65/T.+42^f +18i|+211H.
266 SUBTUACTWN AND MULTIPLICATION OF FliACTIONS.
1. From J take J.
I = II . I = /f Change both fractions to similar fractions having the
jj __ 1 1 ^®*^^ common denominator, then find the difference of the
3* ~ iTf — 2* numerators, and write it over the common denominator.
2. From Hi take 7j.
Hi = 11^, = lOf 5
"7 714— "^14 '^^^ process is identical with that of subtraction
' F = < if - J tl of integers.
3. 45^ - 18| = ? 4. 76^ - 34f = ?
5. 12V^ - 96| = ? 6. HIt^j -_ 74| = ?
7. 127ii - 48H = ? 8. 79i - 46A = ?
9. 219i - lllf = ? 10. llBil - 87ii = ?
11. 18^ + 7J - 3i + 44 = ? 12. 48i - 6i - 2i - 15^ = ?
MULTIPLICATION OF FRACTIONS.
To multiply a fraction hy an integer^ or an integer by a
fraction : —
1. (a) Multiply J by 5.
j X 5 = J^5^ = 3| Since the numerator expresses the number of parts,
the fraction is multiplied by multiplying its numerator.
(6) Multiply /^ by 6.
-7 X 5 = J = 34 Dividing the denominator of the fraction by the in-
teger multiplies the fraction, since it increases the size
of the parte without increasing their number.
Multiply the following : —
2.
«by8.
3.
T-V by 5.
4.
T^T by 8.
6.
H by 9.
6.
i\ by 29.
7.
U by 26.
8.
«i by 84.
9.
11 by 17.
10.
75 by H-
11.
625 by %l
12.
407 by M-
13.
27 by f .
14.
32 by A.
16.
22 by ^5.
16.
36 by ^,.
17.
24 by IJ.
18.
li by 33.
19.
« by 927.
20.
54 by ii.
21.
H by 576,
MULTIPLICATION OF FRACTIONS.
267
To multiply a fraction by a fraction: —
1. Multiply I by f.
^ of I — ^ This means find | of {. First find ^ of { or f J is
__ i_Q_i 2 times ^ or } = J. This worlc may be much shortened
»~*^ff~8~3 by using canceliation, hence cancel all factors common
or to the numerators and denominators, and multiply the
a q 1 remaining factors of the numerator for a new numer-
2. y^ Z := - ator and the remaining factors of the denominators for
3 ^ ^ 3 a new denominator.
Multiply :
2. ! by |.
6. «byH.
10. ii by IS.
3. «by#.
7. MbyH.
11. Hby!?.
4. llbyH.
8. ebyf?.
12. H by U-
6. fibytf.
9. MbyH.
13. I? by a.
14. Multiply 15§ by 12*.
or
15J
_12i_
8
180
191ft
Multiply :
16. 85| by 24i.
18. 15J by 95.
21. 25| by 32i.
24. 69 J by 72«.
27. i|xifxi«xH=?
29. 5ixftxftxH=?
81. 8fxHx6jxe=?
33. 2ixixJx3|x5J=?
36. Hxi5xftxHxi«5=?
Change mixed numbers to im-
proper fractions, and multiply
as in the first illustration, or
as in second illustration with-
out changing. J x f = ^^^ .
i X 15 = 3. 12 X f = 8. 12 X
15 = 180. Use this method when
the numbers are small.
16. 25i by 2|.
19. 24i by 12i.
22. 27i by I64.
25. 76i by 6J.
17. 16§by6|.
20. 27T^n by 205.
23. 23| by 49f .
26. 86f by 27?.
28. 12§x8jx4tVx^^=?
30. i?x2ix3§x2ixi«=?
32. 3JXt75X2jX«=?
34. 6x7ixHx3T\=?
36. 12jxl8ix2VV=?
268 DIVISION OF FBACTI0N8.
To divide a fraction or a mixed number by an integer: —
1. (a) Divide f by 3.
1 of f =s ^ "^^^^ means find J of J. This is the same as multipUca-
Q tion, page 267, or divide the numerator of the fraction by
or ^-^6 = ^ ^^^ integer.
(J) Divide * by 3.
^ X f = ^. ^^ ^bi^ case we cannot divide the numerator, so we multiply
the denominator, since multiplying the denominator divides
the fraction.
((?) Divide 25§ by 6.
) 2~~i-. 1) Divide as in whole numbers, 6 is contained in 26, four
4j^ times and 1 remainder. 1 = |. |4- f = J. | -^ 6 = /j, or,
2^ 25% = ^ 7 ^) change the mixed number to an improper fraction.
S/xjAi ''' = '^- y-« = H = *A-
Divide :
2. If by 12. 3. HI by 26. 4. t', by 3. 5. ?} by 5.
6. §1 by 7. 7. 25f by 5. 8. 32} by 4. 9. 40| by 12.
10. 37i by 15. 11. 62i by 25. 12. if by 12. 13. Hi by 11.
To divide an integer by a fraction : —
1. Divide 6 by §.
Change
the int^^r to the same de-
6 = -yi. ¥-^§ =
18 + ;
2 _ 9 nomination as the fraction, and divide
the numerators.
or
Or
6-!-J = 6x
1 = 9
Multiply the integer by the fraction
inverted,
because multiplying by the
reciprocal of a number
is the same as dividing by
that number.
Divide :
2. 12 by f.
3.
8byf.
4. ISby/^.
6. 45byi«.
6.
16 by |.
7. 52by/j.
8. 87byH.
9.
231 by M.
10. 330 by 4f .
11. 74 by 7f.
12.
308 by 8^.
13. 264 by 9 .
14. 166 by lOg.
15.
288 by 19 J.
16. 176by6j.
17. 60 by 1^
18.
54 by h
19. 126 by i.
DIVISION OF FRACTIONS.
269
To divide ajraction by a fraction: —
1. (a) Divide ^ by f.
10
9
6
3'
10
2
3
24 = IjV
9 +3
(J) Divide f by f.
H + §J = 25
(c) Divide | by %.
I H- f = I X I = V- = 3|
cal of the number, we obtain the rule
Divide :
Divide the numerators for a new numerator
and the denominators for a new denominator.
Change the fractions to equivalent fractions
having their least common denominator, then
divide the numerators.
On the principle that dividing by any num-
ber is the same as multiplying by the recipro-
Multiply by the inverted divisor.
2. i by §.
6. 4J by 2i.
10. 2i by IJ.
14. 69i by 41.
3. I by ?.
7. S^bylJ.
U. 1\ by li
15. 42i by 2f.
4. H by H.
8. ^ by 2^
12. 2§ by t.
16. 12i by If.
5. I by 2i
9. 5« by IJ.
13. I by i.
17. 6| by J.
COMPLEX FRACTIONS.
Reduce to simple fractions:
L. ^.
6?
11.
14.
2.
Hi
5f
3 2i
6.
X j X
I
+ i
V- X /t X IJ
9.
2i-li
4iV
12.
- 2| + 6| - If
I X § X i X H
1§
2J + li
17. -r':
21.
5V
^ 4i - Z\
3i
18. '^^'^
^
i
402H
22. l'«
27|
7|
15.
X A X T^y
- X ^ X 71
19.
23,
12A
81'
m
7.
3ii
\ X f
10.
i xi?
4i + 3g -
13.
2^
-3| + 2
X95
16
20.
24.
I Xl3i
t X7i
I off
i of if ■
270 PROBLEMS INVOLVING FRACTIONS,
1. In a certain school ^q of the pupils are under 8 ; J
between 8 and 12; i between 12 and 14; -^ between 14 and
16 ; and 27 are over 16. What is the whole number of pupils ?
2. If two men can do a piece of work in 6 J days, what part
of it can one man do in 3 J days ?
3. Three-sevenths of a certain number exceeds i of the
same number by 25. What is the number ?
4. If a man can build |f of a wall in 6 J days, how long will
it take him to finish it?
5. If tS: acres of land cost $18/y, how much must be paid
for 784 A.?
6. If I5 A. of land cost $76, what will IfV of an acre cost?
7. If a man owns /y of a mill, and sells f of his share for
110,000, what is the value of the entire mill?
8. Find the cost of 57i yd. of cloth | yd. wide, if 39]^ yd.
of the same cloth % wide cost f 118.50.
9. If a man owns f of a mill, and sells f of his interest for
$4,000, what is the value of the mill ?
10. A farmer raises 4i tons of hay on 2| acres of land. How
many tons can he raise on 12§ acres?
11. How many square inches of tin will be required to make
a box 9 in. long, 3| in. wide, and 3i in. deep?
12. If J of A's money is increased by J of i of his money,
the sum will equal $198. How much money has he ?
13. A boy's money diminished by i and i of itself, equals
$1.32. How much has he ?
14. A owned J of a factory, and sold f of his share to B,
who sold f of his share to C, who sold i of what he bought to
D. What part of the factory did each then own ?
15. A owns 79^% acres of land, B 9^^^ acres less, while C
owns 26i| acres less than A and B together. How many acres
have B and C ?
16. If 5^ bu. of wheat cost $6.60, how much will $121^ buy ?
SUMMARY OF DECIMAL FRACTIONS, 271
1. The word decimal comes from the Latin word decemy
which means ten.
2. A decimal fraetiony usually called a decimal, is a fraction
therefore whose denominator is some power of ten.
3. The denominator is not usually written, but is shown
by the position of the decimal point. •
4. In reading decimals, read as if the decimal were an
integral number and add the name of the lowest decimal place.
5. Read the following: 1.5; 2.06; 3.007; 4.0016;
6.00025; 9.000164; 200.02; 20.002; 300.003; .303.
6. Write the following decimally: three tenths; eleven
hundredths; one hundred twelve hundred-thousandths; six
hundred four thousandths ; six hundred and four thousandths ;
eighteen and fifteen hundred-thousandths; two and one half
tenths ; forty-five hundred and forty-five hundredths.
7. Notice the similarity of sound and difference of value
in the following : — 101000 ; 100.001 ; .101.
To change a decimal to a common fraction.
8. Change .25 to a common fraction.
.25 = Jyi = i Express the decimal as a common fraction, and reduce it
to its lowest terms.
Change the following decimals to common fractions : —
9. .75 10. .64 11. .032 12. .12^ 13. .87i
14. .3i 15. .00125 16. .0005 17. .024 is. .1625
19. 1.6§ 20. .161 21. .008i 22. .081J 23. .OOj
24. .0726 25. 8.661 26. 22.0i 27. 2.03t 28. 75.25
To change a common fraction to a decimal,
1. Change | to a decimal.
8 )3.000 f means 3 -r 8. Perform the indicated operation by annexing
.375 ciphers to the numerator and dividing by the denominator.
Change to decimals :
2. ^ 3. J. 4. tV 5. ^V 6. tVV- 7. t%. 8. 16^V
9. 6^ 10. 7^^. 11. ^y\,. 12. iJ. 13. i^. 14. iM. 15. if.
272 ADDITION AND SUBTRACTION OF DECIMALS,
1. Add 2.514; 6.7; 18.1206; 8.24.
2.514
6.7
18 1206 Write the units of the same order in the same column, and add
QOA ^ ^ whole or integral numbers.
35.67,45
2. Add 4.616; 7.1;* .4689; 7.46819; 8.01.
3. Add .612; 6.0; 9.2178; 6.00623; .2002.
4. Add 10.017; 266; 8.001; 3.1206; 0.46; 3.07.
5. Add eighty-two and three hundred sixteen thousandths ;
one and two hundredths; four and one hundred two ten-thou-
sandths ; six thousandths ; fourteen.
6. Add four tenths; twenty-four hundredths; eighteen
thousandths; one hundred nine hundred-thousandths; one
thousand two hundred one millionths ; seventeen hundredths ;
seventy-five ten-millionths.
7. Add 2i tenths; 16i thousandths; 19^ hundredths; 2i
thousandths ; 46^ thousandths.
8. From 2.79 take 1.07.
2 79
1 07 Write units of the same order in the same column, and subtract as in
— — inte&ral numbers.
1.72
9. From 6.6 take .49. 10. From 2.106 take .0004.
11. From 4.01 take 2.004. 12. From 3.04 take 1.906.
13. From 3.05 take 1.076. 14. From 6.4 take 4.806.
15. From 76.01 take 61.964. 16. From 0.716 take 0.6418.
17. From .025 take .000487. 18. From 400 take .004.
19. From 1.046 take ,00687. 20. From 100 take .0001.
21. From &\ hundredths take 4^ thousandths.
22. From two take two hundredths.
23. From two hundred take two hundredths.
24. From twelve hundredths take twelve millionths.
26. From three hundred and three thousandths take three
hundred three thousandths.
MULTIPLICATION OF DECIMALS, 273
1. Multiply 3.5 by .5.
{n\ '^ jK (}i\ % ^ (^) "^^ means find ^i^ of 3.6. ^ of 3.5 is .36, then
A Vk 5 A are 5 times .36 or 1.75.
/^ z^—z (p) 35 multiplied by 6 are 176, and tenths multiplied
I . 75 ^-i^ -^y tenths gives hundredths, hence the result is 176 hun-
dredths, or 1.75.
Multiply as in whole numbers, and make the decimal places in the product
equal the sum of those in the multiplicand and multiplier.
Multiply : —
2. .7 by .465 3. .5 by .064 4. .6 by .049
5. .15 by .628 6. .07 by .085 7. .24 by .184
8. .09 by .007 9. .08 by .009 10. .06 by .007
II. .462 by .005 12. .074 by .641 13. 075 by .028
14. 1.007 by 2.005 15. .0046 by .00098
16. 2078 by .0047 17. 40.079 by .046
18. 62.174 by 2.16 19. 7.216 by .463
20. 5.002 by 5.06 21. 10.005 by .105
22. 16 by 1.0705 23. 148 by .00148
24. 5.028J by .064 26. 48.072^ by .081^
26. 99.9i by .666J 27. 1000 by .055*
28. Multiply 48062 thousandths by 4078 hundredths.
29. Multiply 608 millionths by 32 ten-thousandths.
30. Multiply 15 hundred-thousandths by 76 ten-millionths.
31. Multiply 25 hundreds by 25 hundredths.
32. Multiply 16 thousands by 16 thousandths.
33. What is the product of one tenth and one tenth?
34. What is the product of one tenth and one thousandth?
35. What is the product of one thousandth and one thou-
sandth ?
36. What is the product of one thousandth and one mil-
lionth?
37. What is the product of one thousand and one mil-
lionth?
38. What is the product of one million and one thousandth ?
274
DIVISION OF DECIMALS.
1. Divide 12.4 by 4.
3 j^ Notice that in dividing a decimal by an integer the decimal point
4*^12 4 ™^^^ ^ placed in the quotient when the point in the dividend is
^ — '-^ reached. Also notice that in short division it is better to place the
quotient above the dividend.
2. a. Divide 125 by .6 b. Divide .125 by .06.
25 Q 2 5 Since multiplying the divisor and
Ca^\6U25\0 CM\05^\12 5 dividend by the same number does not
\ / \ I — \ A V >' \ ^ \ ^ change the quotient, we multiply both
terms by that number which will make the divisor an integer, then the rule for
the decimal point is the same as in the first illustration.
3. Learn: Move the decimal point in both divisor and
dividend to the right as many places as is necessary to make
the divisor a whole number, then divide as in whole numbers
and place the decimal point in the quotient when the point in
the dividend is reached.
Dii
ride : —
4.
91.512 by 3.72
21.
1.46475 by 3.15
6.
177.66 by 31.5
22.
1501 by 31.6
6.
151.411 by 6.13
23.
1.3792 by 8.62
7.
44.691 by 73.1
24.
46.224 by 96.3
8.
683.76 by 8.4
26.
3.7284 by 4.78
9.
20.88 by 8.7
26.
.78387 by .087
10.
62.26 by 7.8
27.
.82848 by .096
11.
61.41 by 6.9
28.
199.525 by 57.5
12.
6.175 by .9
29.
3604.68 by 52.7
13.
471.42 by 9.7
30.
299.052 by 639
14.
872.64 by 86.4
81.
7.20252 by 7.41
15.
197.316 by 20.3
32.
229059 by .279
16.
24.412 by 71.8
33.
.0721512 by .91:
17.
.20976 by .46
34.
303107 by 8.17
18.
32.7 by 65.4
35.
36.926 by 781
19.
469.56 by 8.6
36.
29.632 by 64
20.
1.1439 by .31
37.
67.643 by 1.73
PROBLEMS INVOLVING DECIMALS, 275
1. Find the cost of 4J cords of wood at $3.76 a cord ; 6^
tons of hay at $12.50 a ton and 47f bu. of potatoes at 56/ a
bushel.
2. How many times will a wheel 4 ft in diameter revolve
in going 2i miles ?
3. If a road rises 3.75 ft. in every 50 ft., how much does
it rise in a mile ?
4. A man sold a mill for f 14,600, which was ,04 more than
he paid for it How much did he pay for it ?
5. A bushel even measure contains 2160.42 cubic inches.
If this is .783 of a heaped Jbushel, how many cubic inches are
there in a heaped bushel ?
6. My gas meter registered Nov. 1, 59,600 feet, and on
Oct. 1, 56,400 feet. What is the amount of my gas bill for the
month of October at $1.10 a thousand feet?
7. A meter is 39.375 inches. How many yards are there
in 18 meters?
8. If a man travel 29.6 miles a day, in how many days will
he travel 1016.168 miles?
9. How many rods of fence will inclose a rectangular field
that is 76.08 rd. long and 46.48 rd. wide ?
10. How much pure iron in 64,148 lb. of iron ore, if .75 of it
is pure iron ?
11. If 40.5 yd. of cloth are bought for $263i, what will
18,76 yaixls cost?
12. Twenty-five hundredths of a farm cost $1200. What
will seven-tenths of it cost?
13. For the roof of a building 9000 tiles are to be used.
What will they cost at $7.62^ a thousand?
14. Two men bought 2160 acres of western land, and divided
it so that one man received .37^ of it, and the other man
received the remainder. How many acres did each man
receive ?
276 SUMMARY OF MENSURATION.
1. Mensuration is the process of computing the lengths of
lines, the areas of surfaces, and the volumes of solids.
2. A solid is a portion of space bounded by a surface or
surfaces. It has three dimensions, length, breadth, and thick-
ness.
8. A surface is the boundary or limit of a solid. It has two
dimensions, length and breadth.
4. A line is the boundary or limit of a surface. It has only
one dimension, length.
6. A straight line is a line that does not change its
direction.
6. A curved line is a line that changes its direction at every
point
7. Parallel lines are lines which have the same direction.
8. An angle is the difference in direction of two lines.
9. When one straight line meets another, two angles are
formed. If the angles are equal, they are right angles and the
lines are perpendicular to each other.
10. An acute angle is smaller than a right angle.
11. An obtuse angle is larger than a right angle.
12. A polygon is a plane figure bounded by straight lines.
13. Polygons are named from the number or relations of
their sides or from their angles.
14. A quadrilateral is a polygon of four sides.
15. A parallelogram is a quadrilateral whose opposite sides
are parallel.
16. A rectangle is a right-angled parallelograjn.
17. A square is an equilateral rectangle.
18. A rhomboid is an oblique-angled parallelogram.
19. A rhombus is an equilateral rhomboid.
20. A trapezoid is a quadrilateral only two of whose sides
are parallel.
21. A trapezium is a quadrilateral having no parallel sides.
SUMMARY OF MEASUREMENTS,
277
To find the area of parallelograms.
s
b
SoLuaxe.
Bectanffle.
Bhomboid.
General formulas : S =z ab.
a =
Let S equal the
surface or area, b the
base, and a the alti-
tude. Any two of
these being given, the
other may he found.
S
6 =
S
b a
Write these formulas as rules, thus : The area of a rectangle
is found by multiplying the base by the altitude. Divide the
area of a rectangle by the base to find the altitude, etc.
Using formula, find the missing term : —
1. Base 20 ft., altitude 12 ft., area x.
Altitude 9 ft., area 720 sq. ft., base x.
Base 17 ft., area 85 sq. ft., altitude x.
Base 15 ft., altitude 24 ft., area x.
Base 12 ft., area 96 sq. ft., altitude x.
Altitude 15 ft., area 180 sq ft,, base x.
5/ Let S = area, a, altitude, b and 6',
the bases.
Greneral formulas:
2.
3.
4.
5.
6.
Trapozold.
a = 2-
,S'
b =
ia
(6 + 6')
Write these formulas as rules.
Using formula, find the missing term : —
7. Bases 20 ft. and 30 ft, altitude 10 ft., area x.
Bases 15 ft. and 35 ft., area 500 sq. ft., altitude x,
Base (b) 110 ft., altitude 50 ft., area 4625 sq. ft.
8.
9.
Find
b\
10. There is a house-lot with four straight sides, two of
which are parallel, 80 ft. apart, and measuring 120 and 132 ft.
What is the value of the lot at 25/ a square foot
278 SUMMARY OF MEASUREMENTS— TRIANGLES,
1. A triangle is a tbree-sided polygon.
2. A right triangle is a triangle having a right angle.
3. An equilateral triangle is a triangle having three equal
9ides.
4. An isosceles triangle is a triangle, two of whose sides are
equal.
6. A scalene triangle is a triangle no two of whose sides are
equal.
6. Any side upon which a triangle rests is the base.
7. The angle opposite the base is the vertex,
8. The altitude of a triangle is the perpendicular drawn
from the vertex to the base or the base extended.
General formulas:
5 ^ h ab. a = 2f.
o
6 = 2-. Write these
b a
formulas as rules.
Using the formula, find the missing term : —
9. Base 40 ft, altitude 25 ft, area x.
l(J. Base 28 ft, altitude 20 ft., area x.
11. Altitude 45 ft, area 900 sq. ft, base x.
12. Base 30 ft, area 540 sq. ft., altitude x.
13. What is the area of a piece of ground in the form of a
triangle, when the length of one side is 80 rd. and the per-
pendicular distance to the vertex of the opposite angle is
60 rd.?
14. A triangular piece of land contains 2^A. If the altitude
is 36 rd., what is the base ?
15. At $85 an acre find the value of a triangular piece of
land whose base is 24 rods and altitude 21 rods.
16. The area of a triangle is 1080 sq. ft. and the altitude is
64 ft Whatis the base?
SUMMARY OF MEASUREMENTS. 279
1. A circle is a plane figure bounded by
a curved line, every point of which is equally
distant from the center, or a circle is a regu-
lar polygon of an infinite number of sides.
2. The circumference of the circle is its
boundary line.
3. The diametery AB^ is a straight line passing through the
center of the circle and touching the circumference at both ends.
4. The radius is a straight line joining the center and the
circumference of a circle. It is one-half of the diameter.
5. The ratio of the circumference of a circle to its diameter
is 3.1416.
To find the circumference or diameter of a circle : —
Let c = circumference, i2, the radius, 2 i2, the diameter, and
TT z= 3.1416. General formulas ; c = 2 irB, 2 E z= - ,
Using the formula, find the missing term and state the rule :—
6. Diameter 5 ft., circumference x.
7. Diameter 9 ft., circumference x,
8. Circumference 21.9912 ft., diameter x.
9. Circumference 26.1328 ft., diameter x,
10. What is the distance one-half round a circular piece of
ground, which measures 180 ft. across the middle?
11. How far has a carriage gone when one of its wheels,
measuring 3^ ft. in diameter, has made 1200 revolutions?
12. There are two circles drawn from the same center. The
circumferences measure 196 ft. and 264 ft. respectively. Find
the width of the ring formed by these two circles.
13. What is the circumference of a circle whose i-adius is
28 ft. ? Of one whose diameter is 14 rd. ?
280 SUMMARY OF MEASUREMENTS.
To fold the area of circles : —
General formulas: ^
^ IT ^ TT
Write these formulas as rules.
1. Find the area of a circle whose diameter is 30 ft.
2. Find the area of a circle whose circumference is
814.16 ft
3. Find the diameter of a circle whose area is 392.7 sq. rd.
4. Find the radius of a circle whose area is 28.2744 sq. ft.
5. Find the circumference of a circle whose area is 78.64
sq. rd.
6. Find the area of a cLrcle whose radius is 12 ft.
7. Find the area of a circle whose radius is 20 ft.
8. What is the circumference of a circular fountain whose
area is 872| sq. yd. ?
9. Find the diameter of a circle whose area is 144 sq. ft
Of another circle whose area is 36 sq. ft How do these two
diameters compare ?
10. A horse is tied to a stake by a rope 20 ft. long. Over
how many square feet can he graze ?
11. The perimeter of a square and the circumference of a
circle are each 15.708 ft Find tiie difference in area.
12. What is the area of a semicircle, if the diameter of the
circle is 72 ft ?
13. How many yards are there in the radius of a circle
whose area is 706.86 sq. yds. ?
14. If the area of a circle is 476.24 sq. ft, what is the diame-
ter of the circle ?
16. The areas of two circles are to each other as 4 to 16.
Find the diameter of the smaller when the diameter of the
greater is 60 ft.
SUMMARY OF MEASUREMENTS — SPHERES. 281
To find the surface of a sphere : —
1. A sphere is a solid bounded by a curved
surface every point of which is equally distant
from its center.
Let S == surface, R = radius, 2 jB = diameter,
(7 = cii*cumf erence, and 7r = 3»1416.
General formulas :
S=2RC. S=i7rtP. 2B=^\f^.
The surface of a sphere is equal to four times the square of
the radius multiplied by 3.1416, or it is the product of the
square of the diameter and tt.
2. Find the surface of a sphere whose tadius is 6 ft.
3. Find the surface of a sphere whose diameter is 10 ft.
4. Assuming the earth to be a sphere 7960 miles in diame-
ter, how many square miles are therein its surface ?
5. Find the diameter of a sphere whose surface contains
1000 square inches.
6. Find the radius of a sphere whose surface is 314.16
sq. ft.
7. Find the circumference of a sphere whose surface is
804.2496 sq. ft.
Volume of a sphere .* —
A sphere may be regarded as composed of pyramids whose
bases taken together form the Surface of the sphere, whose tops
are at the center, and whose height is the radius. Hence the
rule, multiply the surface by one-third of the radius.
Formula : v = iirlP x - = 1^.
1. Find the volume of a sphere whose radius is 2 ft.
2. Find the volume of a sphere whose diameter is 8 ft.
3. Find the volume of a Sphere whose circumference is
31.4156 ft.
282 SUMMARY OF MEASUREMENTS,
Surface of a cylinder : —
General formulas : Let I = length, or altitude, then 2 irS?
= area of bases, and 2 irRl == convex surface.
Find the entire surface of the following cylinders : —
1. Radius of base 5 in., length 20 in.
2. Diameter of base 12 in., length 24 in.
3. Diameter of base 6 ft., length 40 ft.
Surface of a cone : —
General formulas: Let h = slant height, then hwR = convex
surface, and hirll + ttS? = the entire surface.
1. Find the surface of a cone, when the radius of the base
is 4 in. and slant height 8 in.
2. A tent in the form of a cone has a slant height of 20 ft.
and a diameter of 30 ft. How many square yards of cloth are
required to make it?
3. The circumference of the base of a cone is 75.3984 ft,
and the slant height is 50 ft. Find its entire surface.
Surface of a right pyramid : —
The convex surface of a right pyramid is equal to the perim-
eter of the base multiplied by half the slant height. To this
must be added the area of the base to find the entire surface.
1. Find the entire surface of a square pyramid whose slant
height is 45 ft. and each side of the base 18 ft.
2. Find the convex surface of a pyramid whose slant height
is 36 ft. and the base a hexagon whose sides are each 15 ft.
Surface of prisms : —
Multiply the perimeter by the height to find the surface of
the sides. To this add the area of the top and bottom to find
the entire surface.
Find the entire surface of the following rectangular prisms: —
1. Length 16 J ft., width 12 ft., height 11 ft.
2. Length 20 J ft, width 18j ft, height 14 ft
SUMMARY OF MEASUREMENTS. 283
Volume of a cylinder or prism : —
To find the volume of a cylinder or prism multiply the area
of the base by the height. Formula for the volume of a cylin-
der is V = lirR*. I = length.
1. Find the volume of a cylinder when the radius of the
base is 8 in. and length 21 in.
2. Find the volume of a cylindrical iron vat 4J ft. in
diameter and 10 ft, deep.
3. Find the volume of a rectangular prism whose altitude
is 40 ft. and the sides of the base 5 ft and 9 ft. respectively.
4. The volume of a cylinder is 144 cu. ft. The diameter
is 4 ft. Find the entire surface.
6. Find the capacity in cubic feet of a cylindrical cistern
6 ft. in diameter and 9 ft. deep.
Volume of a pyramid or cone : —
To find the volume of a pyramid or cone multiply the
area of the base by one third of the altitude. Formula for the
volume of a cone,
V =
a = altitude.
Find the volume of the following : —
1. A pyramid whose base contains 9 J sq. ft. and whose
height is 12 ft.
2. A cone the radius of whose base is 9 in. and its altitude
20 in.
3. A cone the diameter of whose base is 16 in. and its
altitude 70 in.
4. A pyramid whose base is 6 in. by 9 in. and altitude
15 in.
6. Find the volume of a cone the circumference of whose
base is 36 in., and whose altitude is 5 ft
284 AREAS OF SIMILAR FIGURES.
1. Similar plane figures are those having the same shape,
but not necesBarily the same size.
2. The area9 of similar plane figures are proportional to the
squares of their corresponding or like lines ; i.e., the area of a
2-in. square : the area of a a^in. square ==4:9.
3. The diameters of 2 circles are 8 ft, and 4 ft What is
the ratio of their areas ?
4. If the area of a hexagon is 20 sq, ft., what is the area of
a similar hexagon each of whose sides is three times as long ?
5. A square whose side is 3 ft. is what part of a square
whose side is 6 ft. ?
6. A circle whose diameter is 2 ft. is wh^t part of a circle
whose radius is 2 ft ?
7. A circle whose diameter is 4 ft is how many times a
circle whose diameter is 6 in. ?
8. A horse is tied to a stake so that he can gra^e over 250
square feet of land. Another horse is tied by a rope 3 times
as long. Over how much land can the second hor^e gra^e ?
VOLUMES OF SIMILAR SOLIDS.
The volumes of similar solids are proportional to the cubes
of their corresponding line.
1. How many 2-in. cubes are equal in volume to an 8-in.
cube ?
2. A sphere whose diameter is 3 in. is what pai*t of a sphere
whose diameter is 6 in. ?
«
3. If a cannon ball weighs 42 lb., what will one weigh whose
diameter is 3 times as great?
4. If a cubical block of wood one foot long weighs 4 lb.,
find the weight of a cubical block 6 ft. long.
6. The weight of a cube of metal each edge of which meas-
ures 4 in. is 18^ lb. What is the weight of a cube of copper
each edge of which measures 5 in. ?
SUMMARY OF PERCENTAGE. 285
1. Percentage is a system of calculations by hundredths.
2. Per cent means by hundredths*
3 Any per cent may be expressed in three ways : a, as a
common fraction, j^^, 5^77 ; i, as a decimal fraction, .05, .005 ;
e, with the symbol, 5%, i%.
4. Express the following as decimals and common frac-
tions: 6%, 23i%, 215%, 1%, 1%, 1%, i%, 212i%, 3ri%.
5. Express as decimals and with the symbol: i, J, ^, i, f,
}y h §) J) l> 4j tj» S«
6. The Jase is a whole of which a part is taken as a per-
centage.
7. The rate per cent is the number of hundredths taken of
the base.
8. The peroenta^^ Is the number obtained by taking a num-
ber of hundredths of the base*
9. These three terms are closely related, and any two being
given the other may be found.
10. $20 Base Here the three terms arp used as an example
06 Per Cdtlt ^ multiplication, The base is the multiplicand,
ffl,^ ^^ ,^ , the per cent is the multiplier, and the percentdge
$1.00 Percentage j^ the product.
11. From our knowledge of the principles of multiplication
three rules may be derived corresponding to the three cases of
percentage.
12. Letting p stand for percentage, r for rate per cent, and
b for base, we may express these three rules by three formulas :
p =± bfi r =t p •+■ b; b i=sp -h r.
13. State these formulas as rulea
To find any per cent of a mmber.
14. Find 9% of 346.
Qg p = br. The percentage is found by multiplying the base by the
oa\a ^^^ expressed either as a decimal or a common fraction.
Find : 16. 20 % of $600. 16. 87^ % of $240.
17. 16§ cf^ of $437. 18. 98 % of $846.
286 SUMMARY OF PERCENTAGE.
To find what per cent one number is of another : —
1. What per cent of 36 is 12 ?
12 is Jf or J of 36 ^ _.P . i^i^i^je ^^^ percentage by the base to find the
i = ^^i% rate. This division may be expressed as a common
fraction, and then changed to hundredths, or divide as in division ol decimals.
2. 378 is what per cent of 1800 ?
.21 3. What per cent of 61i is 12i ?
^^^^^llm^ 4. What per cent of *210 is $42 ?
??^ 6. What per cent of $1842 is $73.68 ?
1800 ^' ^^a* P^r c®^* ^^ ^^^ ^s ^^^
7. 150 is what per cent of 1876?
8. $261 is what per cent of $348 ?
To find a number when some per cent of it is given : —
1. 50 is 10 % of what number?
(a) 10% = 50
100^^ = itf X^ r 500 ^}^^ fi^^ ^% and then 100% ;
or * ^^ *
^^ Divide the percentage by the rate.
(6)/10 )50/00;
2. $16.50 is 6% of how many dollars?
3. 644 yd. are 35% of how many yards?
4. $83.10 is 66§% of what sum?
5. J lb. is 1% of how many pounds?
6. $2.35 is 16§% of what sum?
7. tV is 33i% of what fractional number?
8. 375 is 125% of what?
9. Find the number of which 12 is 81%.
10. Find the number of which 216 is 44j%.
11. Find the number of which 175 is 31i%.
12. 120 is 88|% of what number?
13. 960 is 12i% of what number?
14. $200 is i% of what number?
SUMMARY OF PERCENTAGE -- PROFIT AND LOSS. 287
1. Cost is the sum paid for an article.
2. Selling Price is the amount paid by the buyer to the
seller. It is always equal to the cost plus the gain, or to the
cost minus the loss.
3. Profit is the excess of the selling price over the cost
4. Loss is the excess of the cost over the selling price.
5. All formulas and rules in Percentage apply to Profit
and Loss. (See pages 285, 286.)
6. A farm which cost $4750 was sold so as to gain 18%.
What was received for it ?
7. A lot of goods was bought for $6124.50, and sold at a
loss of 4%. What was the selling price ?
8. A farmer sold a cow for $32.20 and thereby lost 8%.
Find the cost.
9. A house was sold for $3203.20 at a gain of 12%.
What was the cost?
10. A dealer sold a piano for $270.90, and lost 10%. At
what price should he have sold the piano to gain 10% ?
11. A farm was sold at a loss of 20%. If the loss was
$516, what was the cost of the farm ?
12. A merchant, who sells goods at a gain of 15%, clears
$315. Find the cost.
13. A farm was sold for $6000, which was at a gain of 12%.
What would have been the loss per cent had it been sold for
$4000?
14. If I buy goods at 20% below cost, and sell at 20%
above cost, what per cent do I gain ?
15. A man lost 20% of his money, and then gained 10% of
what he had left. If he then had $352, how much had he
at first?
16. A man drew out 33 J % of his bank deposit, and paid 25%
of it for a house worth $1800. What was his original bank
deposit ?
288 SUMMARY OF PER0EN1AQE ^ COMMISSION.
1. Commission is the sum paid to an agent for the trans-
action of business.
2. The one for whom the business is transacted is called
the principal or employer.
8. The one performing the service is called agents factor^
broker^ collector^ or eommission merchant.
4. Agents receive money for their employers by collecting
debts or selling property, aad their commission is some per
cent of the money received.
6. Agents also expend money for their employers, and
their commission is some per cent of the money paid out.
6. The Net Proceeds is the amount returned to the em-
ployer after deducting commission and other charges.
7. All rules and formulas in Percentage apply to Commis-
sion. See pages 285 and 286.
8. An agent's commission at 3% is $414. What amount
of goods did he sell ?
9. I sent my agent f 6150 to invest in flour after deducting
his commission at 2^%. What amount did he invest ?
10. An agent bought for me 800 bbl. of flour at $4.75 a
barrel. If his commission was 2^ and other charges $31.50,
what was the entire cost to me ?
11. An agent received $60 for selling potatoes at 50 cents a
bushel on a commission of 4%. How many bushels did he
sell ?
12. An agent sold goods for $4500, and remitted to his
employer $4349.50. What was the rate of his commission ? •
13. After selling goods an agent deducted $112 commission,
and sent his employer $5488. What rate of commission did
he receive ?
14. An agent returned to his employer, as net proceeds of a
sale, $6165.75 after deducting $93.75 for expenses and a com-
mission of 2j%. What was the amount of the sale?
SUMMARY OF PEROENTAOE — INSURANCE, 289
1. Insurance is a contract by which the insurer promises to
pay the insured for any loss resulting from certain events, like
fire, flood, storm, accident, or death.
%, From these different causes arise the different kinds of
insurance, as fire insurance, accident insurance, life insurance,
etc,
3. The policy is the written agreement between the insur-
anoe company and th^ person insured.
4. The premium is tiie sum paid for insurance. It is
usually a certain per cent of the amount insured,
h. All rules and formulas in Percentage apply to Insur-
Jince. See pages 285 and 286.
6. Mr. Clark injured his house for $4200 at |%. What
was the premium ?
7. Mr. Ingham paid $42 for insuring his house for $2800.
What was the rate of insurance ?
8. Mr. Brown paid $57.30 to insure his house at li%.
What sum was named in the policy ?
e. A building is insured for f of its value at 1^. What is
the value of the building if the premium is $72.36?
10. C's house worth $12,000 is insured for § of its value.
What is the rate of premium, if he pays $96 for the insurance ?
11. A building which cost $40,000 is insured for | of its
value at 2^%. If it should be totally consumed by fire, what
would be the loss to the owner? To the insurance company?
13. Find the value of the property when the premium at
i% is $30.00.
13. What is the rate of insurance when a policy for $130,000
costs $2487.70 premium?
14. A man 26 years old takes out a $5000 life-insurance
policy payable in 20 years. If he pays an annual premium of
$46.60 a $1000, how much will his insurance cost him if he
lives till the policy falls due ?
290 SUMMARY OF PERCENTAGE -- TAXES.
(Review pages 144, 145.)
1. To pay the expenses of a town, city, county, or state
government money is collected from the citizens.
2. A tcix is the money levied upon persons or property for
public purposes.
3. A, poll tax is a tax levied upon each voter, without regard
to the amount of property that he owns.
4. The property tax is a tax on property, and is usually a
certain per cent of the assessed valuation of the property.
6. Property may be either Personal or Real,
6. Assessors are persons chosen to estimate the value of
each piece of real estate.
7. From the whole tax subtract the poll-tax, if any, the
result will be the property tax. Divide the property tax by the
assessed value of the property to find the tax rate. Multiply
each man's property by the tax rate to find his tax.
8. The whole tax of a town is $16,020 and the taxable
property is $784,750. The number of polls is 260, each assessed
11.25. If A's property is assessed at $10,500, what is his tax?
9. A tax of $29,692 is to be assessed in a certain town, the
property of which is valued at $530,000. There are 4246 polls
at $2 each. What is Mr. C's tax, whose property is assessed at
$8,400?
10. A tax of $485.25 was paid when the rate of taxation
was $.00 J. Find the value of the property.
11. A tax of $673.50 was paid on property valued at $44,900.
What was the rate ?
12. My property, which cost me $15,600, is taxed at § of its
value. What is the rate of taxation, if my tax is $31.20?
13. In a certain town the valuation of the property amounts
to $1,720,000. The town raises $28,780 by taxation. There
are 840 persons upon whom a poll-tax of $1.50 each is assessed.
Find the rate of taxation on $1,000.
SUMMARY OF PERCENTAGE — TRADE DISCOUNT. 291
1. Manufacturers and wholesale dealers issue price-lists of
their goods.
2. A discount from this list-price is usually made. This
discount varies as the condition of the market varies.
3. Frequently in trade several discounts are made, as 20%
and 10% off. This means that first a discount of 20% is made,
and then a discount of 10% from the remainder is made.
4. What is due on a bill of $500, subject to a discount of
20% and 10%?
(a) 100% - 20% = 80%. (i) 20% of $500 = $100.
10% of 80% = 8%. $500 - $100 = $400.
80% - 8% = 72%. 10% of $400 = $40.
72% of $500 = $360. $400 - $40 = $360.
5. If the price of an organ is $108 after discounts of 20%
and 10%, what is the list-price ?
6. I bought some goods which were listed at $800, at 20%
below the list-price, and sold them at 10% below the list-price.
How much did I gain ?
7. What per cent did I gain in Ex. 6 ?
8. On a bill of goods amounting to $1200, which is better
for the purchaser, and how n?uch better, 55% discount or two
successive discounts of 50% and 5% ?
9. The net price of an invoice of goods was $4074, the pur-
chaser having been allowed 30% and 3% off. What was the
list-price ?
10. Find the net amount of a bill of $1250, discounts being
25% and 4%. Find a single discount equivalent to these two
discounts.
11. An invoice of goods was listed at $12,000. A merchant
bought the goods at discounts of 20%, 10%, and 5%, and sold
them at 35% above net cost prices. At what price did he sell?
At what per cent below list-price did he sell ?
292 SUMMARY OF PERCENTAGE ^ STOCKS AND BONDS.
1. For definitions see pages 196 and 197.
2. The cost, the selling price, the dividend, the brokerdgd,
each is some % of the par value.
3. Let c = cost ; « = selling price \ d ±t itite of dividetid ;
b = brokerage ; n = number of shares 5 p =^ market Value of
one share; i= income; i;=par value | d'c±rate of inveStlneilt.
tf = n (^ + 6). « = n (p — 6). n = — —— . i = n vet,
i J i 8 J, t)d
n = —^. d = — . n = . d =
vd nv p — p
Write these formulas as rules.
4. What is the cost of 75 shares of stock at 92, brokerage
5. What are the proceeds from the sale of llO shares of
state bonds at 104, brokerage J % ?
6. How many shares of mining Stock at 126| can bd bought
for S31,750, brokerage J ^ ?
7. If I own 76 shares of 5 % stock, what will be my annual
income ?
8. How much must be invested in U.S. 4's at 121i, broker-
age i%, to realize an income of $3600 ?
9. Find the rate per cent of dividend wheh 81 fchares of
stock yield an annual income of f 155.
10. C sold stock at 97^, brokerage \ %, receiving 112,464.
How many shares did he sell ?
11. If I buy 6% stock at 80, what per cent shall I make on
my investment?
12. What must be the price of stock when $4200 worth of
stock is bought for $3570?
13. If I invest $8976 in U.S. 4's at 102, what is my annual
income ?
14. How many shares of R.R. stock at 91, and brokerage J,
can be bought for $16949.26 ?
SUMMARY OF PERCENTAGE -^ INTEREST. 298
1, Interest Is the price paid for the use of money.
2. The principal i& the sum of money for the use of which
interest is paid.
8. The amount ig the sum of the principal and interest.
4. The rate of inter eit is the interest on one dollar for one
year, It is always a specified number of hundredths of the
principal.
5. Interest is the percentage; the principal is the base;
and the rate is the rate of interest,
6. Interest is the most common of the applications of per-
centage. Because of the element of time that is involved it is
the most difficult of the applications.
7. Find the interest on $720 for 1 yr. 3 mo. 26 d. at 6%.
Point off two places or
$ 7.20 = int. for 2 mo. or 60 d. move the decimal point two
43.4^0 =3 int. for 1 yp. places to the left to find the
3,60 =?? iRt. for 1 ipo. interest for 2 mo. or 60 d.
2.40 = int» for 20 4 "^^^^ ®^^^ multiples and
.72 «= int. fpr 6 d. parts of this sum as will
^mgrr ^n ' u n ^ o o/jj j./?^ gi^c the interest for the re-
^^lil ^ l^'l' ^^ • 1^ ^'' Q '"''' oa ^* ^t ^f q^ir^d time at 6%. A^d to
$47.60 = int. for 1 yr. 3 mo. 26 d. at 5%. such parts as will give the
interest at the required rate.
8. Find the interest on |820 for 3 yr. 2 mo. 13 d. at 7%.
9. Find the interest on 1474.60 for 2 yr. 8 mo. 6 d. at 6%.
10. Find the interest on $4128 for 3 yr. 7 mo. 20 d. at 7^.
11. Find the interest on $274.80 for 1 yr. 5 mo. 18 d. at 6%.
13. Find the interest on $378.20 for 2 yr. 3 mo. 25 d. at 4^.
13. Find the interest on $26745 for 4 yr. 8 mo. 21 d. at 5^.
14. Find the interest on $304.86 for 1 yr. 7 mo. 9 d. at 6%.
16. Find the interest on $65392 for 2 yr. 3 mo. 10 d. at 6i%.
la Find the interest on $960.70 for 1 yr. 6 mo. 20 d. at 5%.
17. Find the interest on $78805 for 2 yr. 8 mo. 22 d. at 4i%.
18. Find the interest on $8615.60 for 3 yr. 10 mo. 16 d. at 6 %.
294
SUMMARY OF PERCENTAGE ^ INTEREST.
1. Find the interest on $240 from June 6, 1900, to Aug. 12,
1902.
From June 6, 1900, to June 6, 1902 :
From June 6, 1902, to Aug. 6, 1902 -.
From Aug. 6, 1902, to Aug. 12, 1902 :
% 2.40 = int. for 2 mo.
14.40 = int. for 1 yr.
14.40 = int. for 1 yr.
M = int. for 12 d.
f 31.68 = int. for 2 yr. 2 mo. 12 d.
; 2 yr. This method ol sob-
: 2 mo. tractiiig dates is the one
. 12 d. ^'^^ gcnendly used. It
is not necessary to have
the work written out as
in the example. Let
the pnpils do the work
mentaUy, and write
only the resnlt.
Find the amount of : —
2. $246.75 from Aug. 10, 1898, to June 8, 1902, at 6^%.
3. $408.90 from June 18, 1899, to Oct. 20, 1901, at 6%.
4. $540.50 from Jan. 8, 1899, to Feb. 2, 1903, at 6i%.
6. $124.84 from Nov. 10, 1898, to Nov. 16, 1902, at 6%.
6. $264.60 from Feb. 16, 1899, to Aug. 2, 1903, at 5^%.
7. $647.28 from Dec. 15, 1898, to Nov. 5, 1902, at 6%.
8. $124.40 from June 8, 1899, to Sept. 14, 1901, at 4j%.
9. $762.40 from Dec. 3, 1900, to Feb. 9, 1903, at 6%.
10. $345.60 from March 25, 1899, to July 11, 1902, at 5%.
11. $465.70 from Jan. 9, 1900, to Sept. 29, 1902, at 6%.
12. $567.80 from Oct. 1, 1902, to March 26, 1904, at 4%.
13. $678.90 from Nov. 17, 1901, to April 27, 1903, at 6%.
14. $789.10 from Dec. 6, 1901, to May 18, 1903, at 5j%.
15. $891.20 from July 5, 1900, to Feb. 26, 1902, at 6%.
16. $912.30 from Aug. 11, 1902, to March 2, 1904, at 6%.
17. $123.40 from Sept. 1, 1902, to April 21, 1903, at 6%.
18. $234.50 from Oct. 7, 1901, to May 2, 1903, at 4i%.
19. $345.60 from Nov. 15, 1902, to June 25, 1903, at 6%.
20. $636.20 from Dec. 7, 1901, to Sept. 19, 1903, at 4%.
21. $75126 from Nov. 2, 1901, to Aug. 17, 1903, at 6%.
22. $46785 from May 2, 1903, to Nov. 17, 1905.
23. $76107 from Aug. 19, 1902, to April 6, 1904.
SUMMARY OF PERCENTAGE — PARTIAL PAYMENTS. 295
(Review pages 171-175.)
cwb cU/vnxx/vui/ UM/t^v i>W>tA/tXi>t C3ut b % .
1. A promissory note is a written promise of one person to
pay another person or any one to whom he may order it paid a
certain sum of money.
2. The payee is the person to whom the money is to be
paid.
3. The maker is the person who promises to pay the money.
4. Partial payments are payments in part on notes.
6. An indorsement is a record of a partial payment, with
the date of payment, made upon the back of tlie note.
6. A note of 1400 was dated Apr. 21, 1901. The indorse-
ments were: June 27, 1902, $125; Dec. 9, 1902, |200.
What was due Oct. 9, 1903?
$400 = Principal
28.40 = Int. to June 27, 1902
$428.40 = Amt. to June 27, 1902
125.00 = 1st payment
$303.40 = New principal
8.19 = Int. to Dec. 9, 1902
$311.59 = Amt. to Dec. 9, 1902
200.00 = Payment
$111.59 = New principal
15.58 = Int. to Oct. 9, 1903
$127.17 = Amt. due Oct. 9, 1903
Find the amount of the principal
to the time of the first payment. If
the payment equals or exceeds the
interest, subtract the payment from
the amount and regard the re-
mainder as a new principal, and
proceed in the same manner with
the remaining payments.
If the payment is less than the
interest, find the amount of the
principal to a time when the sum of
the payments equals or exceeds the
interest due.
296 SUMMARY OF PERCENTAGE -^ BANK DISCOUNT.
(For note see 297, Ex. 2. Reriew pages 185-188.)
1. Bank diseount is the interest kept by a bank for advan-
cing money on a promissory note, draft, or bill of exchange
before it becomes due.
2. Tlie proceeds, avails^ or cash value of a note is the face
of the note less the discount.
3. If a note is written so as to be payable a certain num-
ber of months after date, calendar months are to be understood.
When the time specified is a certain number of days, use that
exact number of days in finding the date of maturity.
4. The time from the day of discount to the date of
maturity is called the term of discount
6. Days of grace have been abolished by statute in toliny
of the States. Use them or not according to the custom of the
place in which you live.
6. Find the proceeds of a note for $500, dated May 9,
1902, due in 60 days and discounted June 3, 1902.
May 9 + 60 d. = July 8. $5.00 = 60 d. $600
May 22 June 27 2.60 = 30 d. 2>92
June 30 July ^ .416 = 5 d. $497.08
July _8 35 = term of $2,916 = Bank Proceeds
60 discount Discount
7. Add the time of the note to the date of the note to find the divte of
maturity. Find the number of days from the day of discount to the date
of maturity to find the term of discount. Compute the interest on the face of
the note for the term of discount at the given I'ate, to find the discounti Sub-
tract the discount from the face of the note to find the proceeds. If the hote
is an interest-bearing note, first find the amount of the note at maturity, and
use this amount as the basis for discount.
Find the proceeds of notes as follows : -^
Face.
Date.
Time. Day oft* DliOOuifT
8.
$260
March 1
2 mo. March 22
9.
$3G4
April 4
60 days May 1
10.
$586
June 17
90 days July 15
11.
$697
July 20
70 days Aug. 11
Rate.
/*'• ItttEHtEBT
%• ]^ATE OF.
CHECKS — NOTES — RECEIPTS. 29 7
Check.
-jQoXLo/^A/.
Note — Individual — Time — Negotiable.
_cuf/teyu cLcuto ci! joAoTn/Ux/e^ to ^fixxA^ fccv
oV O'bcUA/,
jQoXto/'Ui^
rLo. jQu>e^
3. Note — Joint and Several — Interest Bearing.
cMru^f viAx>^VY\,\AiA^ to jpxi/Uy to
uM^tfiy t/TvteA/tA t o/t b % .
298 NOTES — CHECKS — RECEIPTS,
1 . Note — Demand — Joint — Non-negotiable .
$-
"TLua^ J^CcLOKy^n/,-
1^-
CItu cLeywuxTuL u>-t/ |aAx>TmXu^ to ^pxxA^ to-
T/oXtu^ \ajcaa/\xA^
-JOoXLcxAA'.
Receipt in Full.
Bo-CiXooOy,-
R/ex!/e>tAM>cL of/-
i/w tuXt op O/tt cLe/wbou'vulxi^ to cLcutc
-jQoiXaAA/,
Receipt for Part Payment.
MXAA^' cW>aAMyyi/_
^RajcaaamA^ op_
CWU OyOOOttTLt.
1^—
. jQo^tLoA/Ci^
CUBE ROOT* 299
(Review Involution.)
1. On page 211, we learned that (25)^= (20 + 5)2= (20)2
+ 2 (20 X 5) + 62. To cube a number we must multiply the
square of the number by the number itself.
(20)2 ^. 2 (20 X 6) + 62
20+6
(20)8 + 2 (202 X 5) .[. (^20 X 62)
(202 X 6) + 2 (20 X 52 ) + 68
(20)8 + 8 (202 X 6) + 8 (20 X 62) + 58.
Substftuting t for 20 and u for 5, we have the formula t^ + 3 thi -{-S tu^ -j-
u' ; i.e., every perfect cube consists of four parts, viz., the tens figure cubed,
plus three times the tens figure squared times the unit figure, plus three times
the tens figure times the units figure squared, plus the units figure cubed.
2. Using this formula, write the cube of :
36 74 48 63 85 72 28
17 58 62 84 95 38 49
8, To extract the cube root of a number is to find one of
three equal factors.
4. Extract the cube root of 15,626.
^ ^ First point off the number into periods of three
f + 3 in« + 1 _ 15,625 figures each, to find how many figures we are to have
8 ttt* + w \j _ jj in our root. What is the greatest number whose
"__ cube is not more than 15 ? Place the 2 above the
q"^ - AA *®"® period. Cube the tens figure, and subtract it
■52 from the tens period. What is the remainder ? Place
^^2 beside it the first figure of the next period. The next
Ztu =150 p^j^ Qf ^.jjg formula IsSthi; of this only 3 «« is known.
125 rpj^ig jg called the trial divisor. How many times is
w* = 125 12 contained in 76 ? Notice that 12 is only a trial
divisor, and allowance must be made for the rest of the formula. Place
the 5 units over units period. Find the value of 3 t^u, and subtract. What is
the remainder? Place the next figure of the power beside the remainder.
Find the value of 8 t w^, and subtract. What is the remainder ? Place the
next figure of the power beside it. Find the value of w^, and subtract. Is
there any remainder ? What is the cube root of 15,625 ?
* The remaining pages of this book may (without loss) be entirely omitted uniesa
required by the course of study.
300 CUBE ROOT.
1. Find the cube root of 15,626.
Fig. I.
Fig. 2.
Ffg. 3. Fig. 4.
The entire cube In Fig. 1 represents 15,625.
The part marlced A represents the largest tens figure cubed. Fig. 2 repre-
sents the part left after the tens cube has been removed. B, C, and D are three
solids, each as long and wide as the tens cube. 3 x (20)^ x 5 = contents of
B, C, and D. Fig. 3 represents what is left after B, C, and D have been taken
away from Fig. 2. How long, wide, and thick are E, F, and G ? What, then,
are their cubic contents ? Fig. 4 represents the part of Fig. 3 that is left
after E, F, and G are removed. This is a little cube. What are its dimensions ?
What is its cubic contents ? What part of the formula represents each of the
eight parts of the cut-up cube ?
54,872 804,367 167,464
438,976.
32,768 941,192 912,673
274,626.
29,791 110,592 753,671
614,126.
42,875 147,649 406,224
185,193.
39,304 3,048,626 6,639,203
7,880,599.
Note. — When there are more than two figures
in the root, let t of the
formula represent all of the root known, and begin again with 3 «*tt, and
repeat.
CUBE ROOT, 301
1. Find the cube root of 84,328,125.
34 • sifi • 125 ^* ^"^^ ^® depth of a cubi-
cyr _ ^ cal box whose volume is 175,616
^4 _ Q #2, 8. A cubical block of stone
Yq5~" contains 857,875 cubic inches.
2 What is the area of one side ?
7" 4, A hall in the form of a
cube contains 357,911 cubic feet.
3072 = J ^~ At f .90 a square yard, how much
(3 X 322) {) 15601 ,^1 it Q^^ ^ carpet the floor ?
15360_= 8 ihi 5. How long, wide, and high
2412 is a cubical pile of wood contain-
2400 = 3 tu^ ing 32 cords?
125 e. A rectangular solid is 348 ft.
125 = w3 j^j^g^ 216 ft. wide, and 729 ft.
high. Find the edge of a cube containing an equal number of
cubical units.
7. A cube measures 5 in. on an edge. A second cube has
8 times the volume of the first. By how much does the length
of an edge of the second cube exceed that of an edge of the
first cube ?
8. A cubical block of stone contains 50,653 cubic feet.
What is its surface area ?
9. What is the edge of a cube which contains as much as a
solid 7 ft. long, 3 ft. 6 in. wide, and 1| ft. high?
10. What is the number of square inches in one face of a
cubical block whose contents are 74,088 cubic inches?
11. Find the cube root of:
43,614,208 130,323,843 354,894,912.
41,068,625 303,464,448 751,089,429.
14,348,907 258,474,853 27,081,081,027.
96,071,91 2 807,546,875 1,871,880,681.
302 EXCHANGE,
If William Andrews of Boston owes John Blackmer of Chi-
cago a sum of money, he can pay the debt in several ways :
1. He can buy a postroffice order at the Boston post-office
payable to Mr. Blackmer at the post-office at Chicago.
2. He can buy an express order at the office of an express
company payable to Mr. Blackmer at any express-office in Chi-
cago of the same company.
8. If he has money deposited at any bank, he can write a
check, and send it to Mr. Blackmer. (See Lesson 31.)
4. He can buy a draft at a bank payable to Mr. Blackmer
in Chicago.
Copy the following draft, and explain each item :
$Z,600. Boston, Jan. 10, 1899.
Ten days after date, pay to
the order of William Andrews
Twenty-jive Hundred Dollars.
Value received and charge the same to the account of
Mercjiants National Bank.
William Jones, Cashier.
To the First National Bank,
Chicago.
Mr. Andrews writes on the back:
Pay to the order of John Blackmer.
William Andrews.
He then sends the draft to Mr. Blackmer in Chicago, who
takes it to the First National Bank for acceptance, which is
done by the cashier writing the word "Accepted," and his name
underneath, across the face.
Sometimes the words " At sight " are written before " Pay to.*'
These are called sight drafts, and are payable on presentation.
Drafts may be used for collecting debts as well as for paying
them. Exchange is thus seen to be a method of making pay-
ments in distant places by means of drafts.
EXCHANGE. 303
1. The following are the rates charged 'for express orders
to any part of the United States or Canada :
15.00, 6/. $20.00, 10/. $40.00, 18/, $75.00, 26/.
10.00, 8/. 30.00, 15/. 60.00, 20/. 100.00, 80/.
2. The following are the rates for post-office money orders :
$5.00, 5/. $20.00, 10/. $40.00, 15/. $75.00, 25/.
10.00, 8/. 30.00, 12/. 50.00, 18/. 100.00, 30/.
8. The cost of a draft varies. In the draft in Lesson 129,
if the Boston banks have but little money on deposit in Chicago,
they will charge Mr. Andrews a certain per cent for the draft.
On the other hand, if they have large sums of money there that
they want at home, they will gladly sell Mr. A. the draft at a
discount.
4. Calling the rate of premium i%, find the cost for send-
ing the following sums of money by Post-office Order, by Ex-
press Order, and by Draft: $25, $50, $65, $80, $100.
5. Find the cost of a draft on New York for $800, when
exchange is i% premium.
The premium = i% of $800 = $1.00.
.-. the cost = $800 + $1.00 = $801.00. An9.
6. How large a sight draft on Chicago can be purchased
for $4,010, when the exchange is J% premium?
The cost of $1.00 = $1.0025.
$4,010 -5- $1.0025 = $4,000. Am.
7. What will be the cost of a 3 mo. time draft for $3,000
at i % premium ?
The premium = J% of $3,000 = $15.00.
The discount of $3,000 for 3 mo. = $45.00.
.-. the cost = $3,000 + $15.00 - $45.00 = $2,970. Am.
Find the cost of the following drafts :
8. $700, premium ^%, payable at sight.
9. $1,200, discount J%, payable in 90 days at 6%.
10. $2000, premium i%, payable in 30 days at 6%.
304 MISCELLANEOUS FACTS FOR REFERENCE.
A square, used in shinglingy etc., is 100 sq. ft
A hand is 4 in., used in measuring horses.
A size is ^ in., used by shoemakers.
A span is 9 in., a fathom 6 ft. used by sailors.
A pace is 3 ft., used in estimating distances.
A league is 3 miles, used in measuring distances at sea.
A load is one cubic yard of earth.
A perch is 24J cubic feet, used in measuring stone and masonry.
A long ton is 2240 lb., used in buying coal at the mines, and by
custom-house officers in collecting duties.
A barrel of flour weighs 196 lb. ; a barrel of beef or pork, 200
lb. ; a quintal of fish, 1 00 lb. ; a keg of nails, 100 lb.
A bushel of oats weighs 32 lb. ; barley, 48 lb. ; rye or corn, 56
lb.. ; wheat or potatoes, 60 lb. ; a firkin of butter, b& lb.
A gallon is 231 cu. in., or 7\ gaL* fill a cubic foot
One bushel, even measure, contains 2,150.42 cu. in. or 1^ cu. ft.*
One bushel, heaped measure, contains 2,688 cu. in. or 1^ cu. ft*
A chain is 66 ft., used by surveyors.
A bundle of paper contains 2 reams ; 5 bundles, a bale.
A folio is paper folded in 2 leaves for a book ; a quarto or 4to, 4
leaves; an octavo Or 8vo, 8 leaves; a duodecimo or 12mo, 12
leaves.
Shingles are packed in bunches. 4 bunches make 1,000. The
price is always given by the thousand.
1,000 shingles, laid 4 in. to the weather, will cover a square, or
100 sq. ft. ; 900 shingles when laid 4^ in.
A lath is 4 ft long, and 1^ in. wide. 50 or 100 laths make a
bunch. 1 bunch of 50 will cover 3 sq. yd., allowing for waste.
A section of land is one mile square, or 320 rd. x 320 rd.
A brick is 8 in. long, 4 in, wide, and 2 in. thick. 22 bricks make
1 cu. ft. of wall.
Wall-paper is 18 in. wide, and 24 ft. long, a single rolL
* Approximately.
METRIC SYSTEM.
305
OTANDARD UWriH
Meter (m.).
Square meter (sq. m.)<
Cubic meter (cu. m.).
Liter (1.).
Gram (g.).
LENGTH.
10 mm. = 1 cm.
10 cm. = 1 dm.
10 dm. = 1 m.
10 m. =1 Dm.
10 Dm. = 1 Hm.
10 Hm. = 1 Km.
VOLUME.
1,000 cu. mm. = 1 cu. cm.
1,000 cu. cm. = 1 cu. dm.
1,000 cu. dm. = 1 cu. m.
1 cu. m. = 1 ster (at.) of wood.
SURFACE.
100 sq. mm. = 1 sq. cm.
100 sq. cm. = 1 sq. dm.
100 sq. dm. = 1 sq. m or centar (ca.).
100 sq. m. =1 sq. Dm. or ar (a.).
100 sq. Dm. = 1 sq. Hm. or hektar (Ha.).
100 sq. Hm. = 1 sq. Km.
CAPACITY.
10 ml.
= lcl.
10 cl.
= ldi.
10 dl.
= 11.
101.
= 1D1.
10 Dl.
= 1HL
10 HI.
= 1K1.
PREFIX.
MiUi-
Centi-
Deci-
Deka-
Hekto-
Kilo-
M3rria-
10mg<
10 eg.
10 dg.
10 g.
10 Dg.
10 Hg.
WEIGHT.
= leg.
= ldg.
= lg.
= lDg.
= lHg.
= lKg.
1,000 Kg. = 1 ton (T.).
ABBRE-
VIATION.
(m.)
(c.)
(d.)
(D.)
(H.)
(K.)
CM.)
RATIO.
.001.
.01
.1
L
10.
= 100.
= 1,000.
= 10,000.
306
TABLES OF WEIGHTS AND MEASURES.
LINEAR MEASURE.
12 inches (in.) =r 1 f oot (ft.). 6i yards, or 16i feet = 1 rod (rd.).
3 feet = 1 yard (yd.). 320 rods, or 6280 feet = 1 mile (m.).
SQUARE MEASURE.
144 square inches (sq. in.) = 1 square foot (sq. ft.),
9 square feet
304 square yards, or ^
272i square feet J
100 square rods
640 acres
= 1 square yard (^q. yd.).
= 1 square rod (sq. rd.).
= 1 acre (a.).
= 1 square mile (sq. ul).
SOLID OR OUBIO MEASURE.
1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.).
27 cubic feet = 1 cubic yard (cu. yd.).
WOOD MEASURE.
16 cubic feet = 1 cord foot (cd. ft.).
8 cord feet, or 1^^^^^ (cd.).
28 cubic feet / ^ ^
128 cubic feet
LIQUID MEASURK
4 gills (gi.) = l pint (pt.).
2 pints = 1 quart (qt.).
4 quarts = 1 gallon (gal.).
1 gal. = 231 cubic inches.
AVOIRDUPOIS WEIGHT.
16 ounces (oz. ) = 1 pound (lb. ).
2000 pounds = 1 ton (t.).
2240 pounds = 1 long ton
MISCELLANEOUS TABLE.
12 units = 1 dozen.
12 dozen = 1 gross.
12 gross = 1 great gross.
20 units = 1 score.
24 sheets = 1 quire.
20 quires = 1 ream.
DRY MEASURE.
2 pints (pt.) = 1 quart (qt.).
8 quarts = 1 peck (pk.).
4 pecks = 1 bushel (bush.).
1 bushel =3 2160.42 cubic inches.
dROULAR MEASURE
60 seconds (") = 1 minute (').
60 minutes = 1 degree (°).
380 degrees = 1 circumference (clrc.).
TIME MEASURE.
60 seconds (sec.
.) = 1 minute (m.).
60 minutes
= lhour (h.).
24 hours
= lday (d.).
7 days
= lweek (wk.).
366 days
= 1 common year (c. yr.)
366 days
= 1 leap year (1-yr.)
100 years
=rlcentiU7 (C.).
5?^
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