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University of California • Berkeley

The Theodore P. Hill Collection

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Early American Mathematics Books

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WENTWORTH SMITH MATHEMATICAL SERIES

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JUNIOR HIGH SCHOOL
MATHEMATICS

BOOK I

BY
GEORGE WENTWORTH
DAVID EUGENE SMITH

AND

JOSEPH CLIFTON BROWN

GINN AND COMPANY

BOSTON • NEW YORK • CHICAGO • LONDON
ATLANTA • DALLAS • COLUMBUS • SAN FRANCISCO

DAVID EUGENE SMITH, AND JOSEPH CLIFTON BROWN

ENTERED AT STATIONERS' HALL

617.7

Wbe gtftcnaam l$tti€

GINN AND COMPANY- PRO-
PRIETORS • BOSTON • U.S.A.

PEEFACE

A proper curriculum for junior high schools and six-year
high schools demands, in the opinion of many teachers, a course
in mathematics which introduces concrete, intuitional geometry
and the simple uses of algebra in the lower classes. This book
is intended to meet such a demand for the lowest class.

Arithmetic furnishes the material for the first half of the
book. The second half of the book is devoted to intuitional
and constructive geometry, a subject which is more concrete
than algebra and which adhiits- of more simple illustration.
Both arithmetic and geometry are arranged with respect to
large topics, the intention being to avoid the lack of system
which so often deprives the student of that feeling of mastery
which is his right and his privilege. These large topics are
set forth clearly in the table of contents.

In this book there is gradually introduced the algebraic
formula, so that the student is aware of the value of algebra
as a working tool before he proceeds to the study of Book II.
The text is so arranged that with entire convenience to the
teacher and student the work in arithmetic may be taken
parallel with the work in geometry, or -the work in geometry
may precede the work in arithmetic.

Any remodeling of the elementary curriculum that sacrifices
thorough training in arithmetic would be a transitory thing,
and any anaemic course in mathematics that leaves a student
too languid intellectually to pursue the subject further with
success is foredoomed to failure. This book gives to arithmetic
the place due to its fundamental importance, and it adheres
to a sane and usable topical plan throughout the development

iii

iv PREFACE

of the various subjects treated. Because of this the authors
believe that they have here produced a textbook suited to the
needs of a raj)idly growing class of schools and that they have
not failed in any respect to adhere to the best standards of
mathematics and pedagogy. As to material for daily drill,
teachers should consult page 105.

Book II is devoted to algebra and arithmetic, each making
use of the important facts presented in Book I and each
including those large and important topics which are valuable
in the elementary education of every boy and girl. The two
books thus work together to a common purpose, the first being
the more concrete and preparing by careful steps for the
second, and the second blending with the first in presenting
to the student a well-organized foundation for the more formal
treatment of the mathematics which naturally follows.

The authors take great pleasure in being able to include in
this work a number of decorative illustrations of early mathe-
matical instruments and their uses, by Mr. T. M. Cleland.
They feel sure that teachers and students will welcome this
innovation in the preparation of textbooks in mathematics,
and will appreciate such a combination of the work of the
artist with that of the mathematician.

CONTENTS

PART I. ARITHMETIC

CHAPTER PAGE

I. Arithmetic of the Home 1

II. Arithmetic of the Store 33

III. Arithmetic of the Farm 55

IV. Arithmetic of Industry 67

V. Arithmetic of the Bank 77

Material for Daily Drill '. 105

PART 11. GEOMETRY

I. Geometry of Eorm Ill

11. Geometry of Size 155

III. Geometry of Position 215

IV. Supplementary Work 237

Tables for Reference 247

Index 249

JUNIOR HIGH SCHOOL
MATHEMATICS

BOOK I

PART I. ARITHMETIC
I. ARITHMETIC OF THE HOME

Nature of the Work. You have already completed the
arithmetic which treats of ordinary computation, and have
probably learned how to use the common tables of meas-
ures and how to find a given per cent of a number. You are
therefore now ready to consider the most important appli-
cations of arithmetic — those which relate to the home, the
store, the farm, the most common industries, and the bank.

In taking up these applications of arithmetic we shall
review the operations with numbers and shall pay par-
ticular attention to those short cuts in computation that
the business man needs to know and that are useful in
various kinds of work. We shall also take up again the
important subject of percentage, which enters into every
kind of business, and shall treat of it from the beginning.

In this book notes in this type are chiefly for the teacher's use.

The teacher is advised to spend a little time in discussing this
page, in order to take stock of what has already been studied and
to set forth the general nature of the work which is ahead of the
students. The reading of a few interesting problems which the
students will meet adds a motive to the work.

ARITHMETIC OF THE HOME

Reckipts

Paymknts

f(f20

/c/20

fam^.

^xMyfv 0-71' ka,rvcL

ja/yv.

B'a//o.e.v

S^uviru ■nm^^t&

?]'1civ6-t&Qy

'^uytfu j^tk&v

fSaak

Swa&vcloy

^a.Lm&&

fSalcin<s.&

80
00

/^.^.

Cash Account. It is important that every boy and girl
should early in life form the habit of keeping a personal
cash account. It is customary to write the receipts on the
left side of the account (called the debit side) and to write
the payments on the right side (called the credit side), thus :

Here a shows cash on hand when this page of the
account is begun ; 5, c^ and d are receipts ; e is the sum
of these items on the day the account is halanced. On the
right side g^ A, and i are amounts paid. To find the amount
on hand we subtract 80 (f, the sum of 5(f., 25^, and 50^,
from $3.80, and find that the balance is |3.00, which we
write at j and again at /. To check the work, //, A, i, and ./
are added, and the sums, at e and A;, must agree. In
keeping a cash account the symbol | is usually omitted.

Check. To check the work in addition always add a
column from the top down after having added it from
the bottom up. Every computation should be checked.

One of the first things one must learn in business is the neces-
sity of checking every computation. Teachers should impress this
important business rule upon the students. When the other opera-
tions are met in this work, the teacher should explain the proper
checks if necessary, but these checks should be known to the students
from the work of previous years.

CASH ACCOUNTS 3

Exercise 1. Cash Accounts

G-iven the followitKj items^ make out the cash accounts and
balance them :

1. Receipts: Oct. 25, cash on hand, 45 (/^.; Oct. 26, earned
by errands, 26 (^; Oct. 27, earned by errands, 35(/'. Payments:
Oct. 26, ball, 20(f,; Oct. 28, bat, 25(f. Balanced Oct. 29.

2. Receipts: Sept. 7, cash on hand, |3.20 ; Sept. 8,
earned by errands, 40 (j^; Sept. 9, gift from father, 50 (^;
Sept. 10, earned by cleaning automobile, SO(f^. Payments:
Sept. 9, football pants, |1.75 ; Sept. 10, share in football,
50(^; Sept. 11, book, 30(^. Balanced Sept. 12.

3. Receipts: Nov. 5, cash on hand, 75(/!; Nov. 5, earned
by caring for furnace, 20 (^; Nov. 6, earned by cleaning
automobile, 25(/5 ; Nov. 7, earned by caring for furnace, 16 (j-.
Payments: Nov. 7, cap, 75 (^. Balanced Nov. 8.

4. Receipts: May 1, cash on hand, $38.75 ; May 3, J. C.
Williams, |20 ; May 5, R. S. James, |36.85. Payments:
May 4, groceries, |8.75 ; meat, |2.80. Balanced May 6.

5. Receipts: May 9, cash on hand, $275.25; May 10,
R. J. Benjamin, $73.75; May 12, S. K. Henry, $250.75.
Payments: May 9, rent, $85; May 12, clerks, $75; May 15,
coal, $22.50; May 17, account book, $1.50; May 22, tele-
phone, $2.75; May 23, gas, $5.60. Balanced May 23.

6. Receipts: Aug. 1, cash on hand, $178.50; Aug. 2,
P. M. Myers, $87.60; Aug. 3, A. B. Noyes, $49.75; Aug. 5,
R. L. Dow, $78.80. Payments : Aug. 2, rent, $50 ; Aug. 3,
electric light, $9.75 ; Aug. 4, gas, $4.80 ; Aug. 5, J. P. Sin-
clair, wages, $16.75; Aug. 5, telephone, $2.50; Aug. 6,
M. R. Roe, wages, $15. Balanced Aug. 7.

Problems of this kind should be made up by the students, who
should be urged to keep their personal accounts.

4 ARITHMETIC OF THE HOME

Household Account. The following will be found a con-
venient form for keeping an account of household expenses :

1^20

Rkceipts

Payments

d'e^t.

^^k o-ru koyrvcL
Re/nt

^/S

f8
f

20
¥-0

mitk, / w^L

J(Sy&

f
2

^0
26
36
36
30
¥0

(^kwv&k
^koAyUyif

^tav-&Qy
Helji., 2 da..

f
3

76
60

(^a-Qyk v&<^' d
!Scitam/Sy&

^.oqA an kayyvd

Cfi

The balance should agree with the cash on hand. The
balance is found by adding the items of payments and sub-
tracting this sum from the sum of the receipts. In this
case we have |85.92 — |39.91. The sum of the receipts
will then agree with the sum of the payments and the
balance. Bookkeepers usually write the balance in red.

HOUSEHOLD ACCOUNTS 5

Exercise 2. Household Accounts

1. Extend the 'account on page 4 from Sept. 16 to
Sept. 24 by including the following receipts and payments :
Receipts : Sept. 18, Mrs. Adams, loan repaid, |3.50 ; house-
hold allowance, |20. Payments: Sept. 18, telephone, |1.75;
vegetables, 65 (^; Sept. 19, trolley tickets, $1; shoes, |4 ;
ribbon, 75(f ; Sept. 21, umbrella, $2.50; belt, 75(f ; Sept. 23,
laundry, 90(^; charity, $1.50; groceries, $5.15; ice, 45 (^.

Make out and balance the following accounts :

2. Receipts: Oct. 9, cash on hand, $15.42; Oct. 10,
allowance, $15 ; Oct. 12, loan repaid by Mr. Green, $5.80.
Payments: Oct. 10, insurance, $2.50; help, $1.50; lecture,
50(f ; Oct. 11, dress goods, $5.80; trimming, $1.10; Oct. 12,
raincoat, $6.50; shoes repaired, $1.25; Oct. 13, groceries,
$3.90; milk, $2.80; Oct. 15, carfare, 30^; flowers, 40(^;
dressmaker, 3 da. @ $1.80. Balanced Oct. 16.

3. Receipts: Nov. 18, cash on hand, $12.67; Nov. 19,
allowance, $22.50. Payments : Nov. 19, cleaning gloves,
25 (f ; dress goods, $9.60 ; Nov. 20, telephone, $1.90 ; remov-
ing garbage, 75(^ ; Nov. 21, gas, $2.30 ; electric light, $1.80 ;
magazines, 30(f; Nov. 22, books, $2.60; groceries, $7.30;
meat, $1.20; charity, $1.25. Balanced Nov. 23.

4. Write an account setting forth the reasonable expenses
of a week for a family of two adults and three children, to
include groceries, meat, milk, gas, electric light, telephone,
and such other items as would be found in an account of
such a family in your vicinity. Take the balance on hand
as $4.80 and the weekly allowance for household expenses,
exclusive of rent and clothes, as $16.

For Ex. 4 the students should be asked to make inquiry at home
as to prices and reasonable purchases.

6 ARITHMETIC OF THE HOME

Ifeed for knowing about Per Cents. John Adams is
manager of the baseball team of a junior high school.
He finds that he can buy 4 bats at 50^ each. The next
day he sees that they have been marked down 10 per cent.
Hence he should know what per cent means.

It is probable that everyone in the class knows. In that case some
student may tell the meaning ; otherwise the subject may be dis-
cussed after page 7 has been read. This page is merely a preparatory
reading lesson for those who may not already have studied percent-
age, which subject will now be considered.

John's mother wishes to buy a dress for his sister. The
price was |12, but the dress has been marked down 15 per
cent. If she wishes to buy the dress, she should know
what this means. Do you know what it means ?

The teacher has to make a report of the number of
students tardy or absent last week. She says in the report
that 4 per cent of the students were tardy and 3 per cent
were absent. Do you know what this means?

There were 10 questions on an examination in arithmetic
and a boy answered 90 per cent of them correctly. Do you
know how many questions he answered correctly ? Do
you know how many he did not answer correctly ?

A man wishes to buy an automobile. He can buy a
new one of the kind he likes for $1200, but one of the
same make that is nearly new is offered for 30 per cent
less than this price. Do you know how to find what the
man would have to pay for the second-hand car?

Do you know the meaning of the symbols 10^^, 15%,
4%, 3%, 90%, and 30%?

It is not necessary that any student should be able to answer these
questions. What is of importance is that each member of the class
should see that per cents are frequently encountered and that every-
one must know what they mean.

PEE CENTS 7

Per Cent. Another name for '' Imndredtlis " is per cent.
For example, instead of saying " ten hundredths " we may
say " ten per cent." The two expressions mean the same.

If John Adams finds that some bats which are marked
$2 can be bought for 10 per cent less than the marked
price, this means that they can be bought for -^^1 oi' tV'
less than $2 ; that is, they can be bought for |2 less j^ of
12, or |2 - 10.20, or *1.80.

Symbol for Per Cent. There is a special symbol for per
cent, %. Thus we write 20% for 20 per cent, or 0.20.

Hundredths written as Per Cents. Because hundredths
and per cents are the same, any common fraction with 100
for its denominator may easily be written as per cent. Thus

iter = 5% 1^ = 1% ^<ts =

^75% 1||:

= 135%

Exercise 3. Reading Per Cents

All 'Work or a

Read the followini/ as per cents :

^' 100- ^' 100- '^' 100-

10.3^.

1Q 8 5

o 6 K 12 Q 35
'*• 100- ^- 100- ^- 100-

11 "^0
^^' 100-

1A 06

1^- TOO-

3 8 fi 15 Q 5 0 19 8 0 i c 1 0 0

*• 10 0- '^•10 0- ^* 10 0* ■•■•*• TOO* ^^' TO'O'

Bead the following as hundredths:

16. 6%. 18. 22%. 20. 43%. 22. 66%, 24. 100%.

17. 9%. 19. 37%. 21. 50%. 23. 72^%. 25. 300%.

26. Read |-^ as per cent, and 125% as hundredths.

G-i 161- 1 37i -3- 871-

27. Read as per cents: — ^, — ^, — ^, — ^, -^—, — ^.

^ 100 100 100 100 100 100

8 ARITHMETIC OF THE HOME

Important Per Cents. Mr. Fuller says that he sold a
used car for 50% of what it cost him. He sold the car
for how many hundredths of what it cost him? Express
the answer as a common fraction in lowest terms.

If a baseball team played 36 games and lost 25% of
them, how many hundredths of the games did it lose ?
Express the answer as a common fraction in lowest terms.

From these two examples, and by dividing 25% by 2,
we see that

50%=^ 25% =1 12J%=i

If this circle is called 100, how much is the shaded
part? What per cent of the circle is the shaded part?

Read 0.33|^, using the words "per cent";
using the word '' hundredths." How many thirds
are there in 1 ? How many times is 33^% con-
tained in 1? Then 33 J% is what part of 1 ?

From these two examples, and by taking 2 x 0.33^ and
^ of 0.33^, we see that

mfo=i 66f%=f 161% =i

Read 0.2 and 0.20, using the word '' hundredths " in
each case ; using the words " per cent." What relation
do you see between 0.2 and 0.20? between 0.2 and 20%?
between 1 and 20%?

What is the relation of -^ to -^q? to 40% ?

What is the relation of 0.60 to f ? of 0.80 to f ?

From answers to these questions we see that
20% =1 40% =1 60% =f 80% =f

In the same way it is easily seen that
75% =1 37i%=§ 62^%=! 87i%=^

Such per cent equivalents should be drilled upon so thoroughly
that the mention of one form automatically suggests the other.

IMPORTANT PER CENTS 9

Exercise 4. Finding Per Cents

All work oral

1. A woman- Avho had set aside |16 for household
expenses for a week finds that she has spent 50% of it.
How much money has she spent?

Find 50% of each of the following numbers :

2. 18. 3. 20. 4. 36. 5. 64. 6. 400.

7. If this square represents a box cover which contains
100 sq. in., how many square inches are shaded ?
How many fourths of the cover are shaded ? How
many hundredths of the covej: are shaded?
What per cent of the cover is shaded ?

8. A man with an income of $4 a day spends 25% of
it for food. How much money does he spend for food ?

Find 25% of each of the following numbers :

9. 16. 10. 36. 11. 48. 12. 5. 13. 240.

Find the values of the following :

14. 20% of 75. 17. 80% of 45. 20. 75% of 800.

15. 40% of 25. 18. 331% of 75. 21. 87^% of 1600.

16. 60% of 35. 19. 16f % of m. 22. 66|% of 6000.

23. If you answer correctly 66|^% of the questions on
an examination paper and there are 12 questions in all,
how many questions do you answer correctly ?

24. If a boy is at bat 10 times and makes base hits
20 % of the times, how many base hits does he make ?

25. If a merchant gains 33^% on $1500, what fractional
part of $1500 does he gain ? How much does he gain ?

10 ARITHMETIC OF THE HOME

Per Cents and Common Fractions. We have learned that

87^ 175 7
*^''~100~200~8\

To express per cent as a common fraction, write the number
indicating the per cent for the numerator and 100 for the
denominator, and then reduce to lowest terms.

It is sometimes convenient to use one form, and some-
times another. Thus, if we are multiplying by 26.9%, it is
easier to think of the multiplier as 0.269 ; but if we are
multiplying by QQ^%, it is easier to think of it as §.

We knoy that |- may be reduced to hundredths by
multiplying each term by 33^. We then have

To express a common fraction as per cent, reduce it to
a common fraction with 100 for the denominator, and then
write the numerator followed hy the symbol for per cent.

We may also reduce ^ to hundredths by dividing 2 by 3, thus :

I = 2 H- 3 = 2.00 -^ 3 = 0.66§ = 66§%.

Exercise 5. Per Cents and Common Fractions
Express as common fractions in lowest terms:

1. 12%. ■ 3. 36%. 5. 64%. 7. 29%. 9. 17^%.

2. 24%. 4. 31%. 6. 32%. 8. 45%. 10. 66.6|%.

Express the following as per cents :

11. |. 13. |. 15. f. 17. |. 19. ^. 21. Jj.

12. |. U. f. 16. |. 18. I. 20. 3^. 22. ^.

PER CENTS AS DECIMALS 11

Per Cents as Decimals. Since 26.5% and 0.255 have the
same value, we see the truth of the following:

To express as a decimal a number written with the per cent
sign, omit the sign and move the decimal point two places to
the left, prefixing zeros if necessary.

When we omit the per cent sign we must indicate the hundredths
in some other way, as by moving the point two places to the left.
Thus 21% = 0.02i or 0.0225 ; 325% = 3.25 ; 0.7% = 0.007.

Exercise 6. Per Cents as Decimals

Examples 1 to 13, oral

1.. If 3% of the students of this school are absent to-day
how many are absent out of every 100 ? out of every 200 ?

2. How much is -^ of $400? 1% of $400?

3. How much is 0.05 of |100 ? 5% of $100? 5% of
$200? 5% of $2000? 5% of $4000?

Read as decimals, whole numbers, or mixed decimals:

4. 35%. 6. 27%. 8. 325%. 10. 365%. 12. 400%.

5. 72%. 7. 42%. 9. 225%. 11. 425%. 13. 200%.

14. Express £% as a decimal ; as a common fraction.

15. Express |^% as a decimal; as a common fraction.

16. Express 666|^% as a decimal ; as an improper fraction.

17. How much is 0.35 of $300 ? 35% of $300 ?

18. How much is 2| x $650 ? 2.75 x $650 ? 275% of
$650? 31 X $750? 3.25 X $750? 325% of $750?

Express as decimals, whole numbers, or mixed decimals:

19. 24%. 21. 331%. 23. 300%. 25. 3000%.

20. 36%. 22. 0.8%. 24. 250%. 26. 0.008%.

12 ARITHMETIC OF THE HOME

Decimals as Per Cents. Since per cent means hundredths,
to express a decimal as per cent we have to consider only
how many hundredths the decimal represents.

1. Express 0.5 as per cent.

Since 0.5 = 0.50, we see that 0.5 is the same as 50 hundredths,

or 50%.

2. Express 0.625 as per cent.

Since 0.625 = 62.5 hundredths, or 62j hundredths, we see that
0.625 is the same as 62.5%, or

3. Express 0.00375 as per cent.

Since 0.00375 = 0.00yVo% = ^-OOf, we see that 0.00375 = |%.

4. Express 4.2^ as per cent.

Since 4.2^ = ^^, we see that 4.2^ = 425%.

Therefore, to express a decimal as per cent, write the per
cent sign after the number of hundredths.

Exercise 7. Decimals as Per Cents

Examples 1 to 6, oral

1. A foot, being 0.33^ yd., is what per cent of a yard ?

2. A peck, being 0.25 bu., is what per cent of a bushel ?

3. A quart, being 0.125 pk., is what per cent of a peck?

4. Express 0.24 mi. as per cent of a mile.

5. Express 1 oz. as per cent of 10 oz.; of 100 oz.

6. Express 0.2 ft. as per cent of 1ft.; of 2 ft.

Express the following as per cents :

7. 0.7. 9. 0.42. 11. 0.625. 13. 6.66|.

8. 0.8. 10. 0.39. 12. 0.375. 14. 0.00875.

FINDING PER CENTS 13

Finding Per Cents. We have now found that certain
important fractions like |-, ^, ^, ^, ^, and -^ are easily
expressed as per cents ; that any fraction can be expressed
as a per cent by first reducing it to hundredths ; and that
per cents can easily be expressed as common fractions.

If we wish to find 50%, 25%, 121%, 33^%, 16f%,
or 20% of a number, it is easier to use the equivalent
common fraction, as in the first of these problems:

1. If Kate wishes to buy a suit that is marked $12,
and finds that it is to be marked down 12^%, how much
will the suit cost her after it is marked down ?

Since 12j% = -J, we need simply to take -J of |12, and this is the
amount the suit is to be marked down.

Then we have 1 of $12 = $1.50 ;

and $12 - $1.50 = $10.50, cost.

2. Robert wishes a Boy Scout suit. The marked price
of the suit he wants is $7.50, but he finds that to-morrow
at a bargain sale this suit is to be marked down 15%. How
much will it then cost him ?

Since 15% = 0.15 we should find
0.15 of $7.50. We then have

0.15 of $7.50 = $1,125,
and $7.50 - $1.13 = $6.37, cost.

The dealer will probably de-
duct only $1.12, neglecting the 5,
in order to make the computation
easier and to take advantage of

the half cent himself. There is no general custom as to using a frac-
tion of a cent. The student should consider ^^ or more as a whole
cent in each operation except in cases of discount.

To find a given per cent of a number^ express the per cent
as a common fraction or as a decimal and multiply the num-
ber hy this result.

$7.50

0.15

1.12

3750

$6.38, cost

750

$1.1250

14 AKITHMETIC OF THE HOME

Exercise 8. Finding Per Cents

Exmnples 1 to 20, oral

1. A man spends 20% of his income for rent. What
fraction of his income does he spend for rent?

2. A man saves 10% of his income. His income is
$90 a month. How much does he save in a month ?

3. A girl spends for clothes 25% of her allowance.
Express this per cent as a common fraction.

Express as common fractions or as whole or mixed numbers :

4. 7%. 7. 75%. 10. 1%. 13. 100%. 16. 125%.

5. 30%. 8. 60%. 11. 2%. 14. 200%. 17. 250%.

6. 40%. 9. 90%. 12. 4%. 15. 500%. 18. 375%.

19. How much is 10% of 420 lb.? of $250 ? of .30 yd.?

20. Milk yields in butter about 4% of its weight. How
much butter will 25 lb. of milk yield ?

21. If a merchant pays l^f. apiece for pencils and sells
them at a profit of 50% on the cost price, what is the
selling price per hundred ?

22. If a butcher buys a certain kind of meat at 15(^ a
pound and sells it at a profit of 20% on the cost price, at
what price does he sell it per pound ?

23. If a dealer pays 7^^. a quart for milk and sells it at
a profit of 20 % on the cost price, how much does he receive
for 200 qt. of milk ?

Find the values of the folloiviruj :

24. 30% of 2400. 27. 75% of |6280.

25. 16% of $3600. 28. 27% of 3700ft.

26. 35% of $825. 29. 3^%. of $4275.

HOME PROBLEMS 15

Exercise 9. Reading the Gas Meter

1. Here is a picture of the three dials on a gas meter.
The left-hand dial indicates ten thousands, the middle
dial indicates thousands, and the right-hand dial indicates

hundreds. The dial shows that 64,300 cu. ft. of gas has
passed through the meter. If this is the reading for May 1,
and the reading for April 1 was 62,300, how much was
the gas bill for April at 80 (^ per M (1000 cu. ft.)?

In the above case more than 64,300 cu. ft. has passed through the
meter, but we read only to the hundred last passed. Notice that the
middle dial is read in the opposite direction from the others.

2. Read this meter. If the reading was 52,700 a month
ago, how much is. the gas bill for the month at $1 per M ?

3. If the gas consumed by a family in January was
3200 cu. ft., and that consumed in June was 10% less
than m January, how much was the gas bill for each of
these, months at fl.25 per M?

4. A ce;:'tain gas company deducts 10% from bills paid
before the tenth of the month. If the readings are 72,400
and 74,200 on the first days of two consecutive months,
and the rate for gas is |1.30 per M, how much will be
saved by paying the bill before tlie tenth of the month ?

ARITHMETIC OF THE HOME

Exercise 10. Reading the Electric Meter

1. Mr. Jacobs lights his store by electricity. The elec-
tricity is measured in kilowatt hours (K.W.H.), and the
meter shows thousands, hundreds, tens, and units. He
reads the meter in a way similar to that in which he
reads the gas meter. Notice that the second and fourth
dials are read clockwise and that the first and third are
read counterclockwise. Read the meter shown:

The technical meaning of the K.W.H. in the science of electricity
need not be considered in the mathematics of the junior high school.

2. If Mr. Jacobs in June uses 40 K.W.H. at 15 (f and
5 K.W.H. at 8(f, how much is his bill for the month?

Companies charge large users differently according to the number
of lights or machines. For example, in most places 10 lights would
cost more in proportion than 200 lights.

3. If Mr. Jacobs in the month of March uses in his house
18 K.W.H. at 14 (^ and has a reduction of 4% on the bill if
paid before the 15th of April, how much is his bill if he
takes advantage of the reduction ?

Find the amount of each- of the following hills :

4. 12 K.W.H. at 15(^, 3 K.W.H. at 7f., less ^(f per
K.W.H. on account of prompt payment.

5. 28 K.W.H. at 16 (f, 6 K.W.H. at 10 (f, less ^ per
K.W.H.

The student should actually read the electric meter in the school
or at his home if this can be done conveniently.

PROBLEMS OF PEECENTAGE 17

Three Important Problems of Percentage. There are three
important problems of percentage. The first, which is by
far the most important one, is that of finding a given per
cent of a number. This has already been considered. The
second problem of importance is to find what per cent one
number is of another, and this will be considered on page 18.
The third problem of importance is to find the number of
which a given number is a given per cent, and this will
be considered on page 21.

The problems on pages 18 and 21 depend upon the
following principle :

Given the product of two factors and one of the factors, the
other factor may he found hy dividing the product hy the
given factor.

That is, if we have given 10, the product of 2 and 5, we
can find the factor 2 by dividing the given product by 5
and the factor 5 by dividing the product by 2.

In this review of percentage we shall confine the applied problems
largely to those relating to home interests, but shall occasionally
introduce other problems for the purpose of variety.

Exercise 11. Product and One Factor

All work oral

The first number in each of the following examples being
the product of two factors, and the second number being one
of these factors, find the other factor :

1. 72, 8. 4. 36, 9. 7. 90, 9. 10. 300, 10.

2. 72, 9. 5. 80, 10. 8. 60, 6. 11. 9, 3.

3. 36, 4. 6. 80, 8. 9. 300, 30. 12. jV ^•

13. I am thinking of the number which, multiplied by 9,
gives the product 63. What is the number ?

18 ARITHMETIC OF THE HOME

Finding what Per Cent One Number is of Another. The

second problem of importance in percentage mentioned on
page 17 is to find what per cent one number is of another.
For example, if in a certain test Anna solved correctly
8 problems out of 12, what per cent of the problems did
she solve correctly?

Here we have a certain per cent x 12 = 8 ; that is, we have the
product (8) and one factor (12), to find the other factor (a certain
per cent). Therefore

8 -4- 12 = 0.66^ = G6g%, the per cent solved correctly.

Exercise 12. Product and One Factor

All ivork oral

1. If there are 80 students enrolled in a class and 3 of
them are absent to-day, what per cent are absent ?

2. On an automobile trip of 60 mi., what per cent of the
distance has a man made when he has traveled 30 mi. ?

3. During a series of games Fred was at bat 36 times
and made 9 base hits. What was his batting average ?

4. If a baseball team wins 14 games out of 20 games
played, what per cent of the games does it win ?

Find ivhat 'per cent the second number is of the first :

5. 4, 2. 9. 6, 2. 13. 45, 15. 17. 64, 8.

6. 4, 3. 10. 40, 30. 14. 75, 15. 18. 64, 16.
•7. 4, 4. 11. 30, 15. 15. 68, 34. 19. 36, 18.

8. 5, 2^. 12. 90, 45. 16. 72, 36. 20. 15, 7^

21. If a book has 240 pages, what per cent of the num-
ber of pages have you read when you have read through
page 24 ? when you have read through page 48 ? when
you have read through page 72? through page 120?

PROBLEMS OF PERCENTAGE 19

Application to Written Exercises. You now understand
the second important problem in percentage. We shall
consider once more the first problem, and then the appli-
cation of the second to written exercises.

1. A man bought an automobile for |800 and sold it
at a profit of 25% on the cost. How much did he gain?

Here we have to find 25% of $800, or 0.25 of |800 ; that is, we
have two factors giveu, 0.25 and $800, to find the x^roduct. Therefore

^*^ ^'^^® 0.25 X $800 = $200.

That is, the man gained $200.

2. A man bought an automobile for $800 and sold it at
a profit of $200. What per cent of the cost did he gain ?

Here we have $200 equal to some per cent of $800 ; that is, we
have given the product ($200) of two factors and one of the factors
($800), to find the other factor. Therefore we have

$200 -^ $800 = 0.25 = 25%.

That is, the man gained 25% of the cost.

Exercise 13. Product and One Factor

1. $260 is what per cent of $5200? of $7800?

2. $22.50 is what per cent of $450? of $67.50?

3. $20.24 is what per cent of $506 ? of $404.80 ?

4. $58.20 is what per cent of $931.20? of $349.20?

5. A foot is what per cent of a yard ? of 2 yd. ? of 8 yd. ?

6. A quart is what per cent of a gallon ? of 16 gal.?

7. £ is what per cent of |- ? |^ is what per cent of |; ?

8. 35%iswhatper cent of 70%? of 140%? of 210%?

9. 33^% is what percent of 66f % ? of 1 ? of 1331%?
10. 66|% is what per cent of 33^%? of 66f%? of 2 ?

20 AEITHMETIC OF THE HOME

11. If an employer reduces the working day for his men
from 9 hr. to 8 hr., what is the per cent of reduction ?

12. If a class had 20 examples to solve on Monday and
28 on Tuesday, what was the per cent of increase ?

13. If you have read 78 pages in a book of 300 pages,-
what per cent of the pages have you read?

14. A man's income is $3300 a year and he spends
$1914. What per cent of his income does he save ?

15. In a certain city 1152 out of the 2400 pupils in
school are boys. What per cent are boys ?

16. In a certain school 360 out of 750 pupils are girls.
What per cent are girls ?

17. If a class devotes 42 min. a day to arithmetic one
year, and 45 min. a day the next year, what is the per cent
of increase per day ?

18. If the last chapter of a book is numbered XXXV
and you have finished reading Chapter VII, what per cent
of the chapters have you read ?

19. The purity of gold is measured in carats, or twenty-
fourths, 18 carats, or 18 carats fine, meaning ^ pure gold.
What is the per cent of pure gold in a 14-carat ring ?

20. What is the per cent of pure gold in a watch case
that is 18 carats fine ? in a chain that is 10 carats fine ?
in an ingot of pure gold ?

21. If 3 members of a class of 48 have not been either
tardy or absent during the year, and 6 members have not
been absent, what per cent have not been either tardy or
absent ? What per cent have not been absent ?

22. Stockings are marked 35(f a pair or 3 pairs for $1.
What per cent on the higher price does a customer save
who buys 3 pairs for fl instead of paying 35(f a pair?

PROBLEMS OF PEECENTAGE 21

Finding the Number of which a Given Number is a Given
Per Cent. The third important problem in percentage
mentioned on page 17 is to find the number of which a
given number is a given per cent.

For example, if a baseball team has lost 9 games, which
is 45% of the number of games played, how many games
has the team played?

Here we have given the fact that 45% of the number of games
is 9. In other words, we have given the product (9) of two factors
and one of the factors (0.45), to find the other factor.

Therefore we have 9 ^ 0.45 = 20.

That is, the team has played 20 games.

Check. 45% of 20 = 9.

Exercise 14. Product and One Factor

Examples 1 to 13, oral

1. If 20% of the inhabitants of a certain city are school
children and there are 20,000 school children in the city,
what is the total population of the city ?

2. If 23% of the inhabitants of a certain city voted at a
certain election and there were 2300 ballots cast, what was
the total population of the city ?

3. A player reached first base 20 times, or 33^% of the
times he was at bat. How many times was he at bat ?

4. An automobile was sold second-hand for $480, which
was 40% of the amount paid for it originally. How much
was paid for it originally?

5. If you pay 30% of the expenses of a camping trip
and pay |12, what are the expenses of the trip ?

6. In a certain school there are 170 boys, which is 85%
of the number of girls. How many girls are there ?

22 ARITHMETIC OF THE HOME

Find the numbers of 7vhich the folloiving are the given
per cents :

7. 72, S%. 14. 222, 3%. 21. 175, 871%.

8. 81, 9%. 15. 222, 2%. 22. 62.8, 66|%.

9. 72, 12%. 16. 87|, 121%. 23. 32.4, 16f %.

10. 75, 15%. 17. 99, 31%. 24. 680, 34%.

11. 88, 8%. 18. 66f, 331%. 25. 220, 110%.

12. 63, 9%. 19. 312, 37|%. 26. 2120, 106%.

13. 63, 7%. 20. 250, 62^%. 27. 34.2, 3.8%.

28. $6.30 is 15% of what sum of money?-

29. 112.48 is 16% of what sum of money?

30. 23 ft. 10 in. is 22% of what length?

31. If a woman pays |15.20 for some groceries which is
80% of the list price, what is the list price?

32. A woman set some eggs, and found that 15% of
them failed to hatch. If the number failing to hatch was
66, how many eggs did she set ?

33. A man saves |1350.40 a year, which is 32% of his
income. How much is his income ?

34. A certain school has 20% of its pupils in the sixth
grade, which numbers 58. How many pupils are there
in the whole school?

35. A man sells some vegetables for $90, thereby gain-
ing 20% on the cost of raising them. What per cent of
the cost of raising the vegetables is the selling price ?
How much is the cost of raisinof them ?

From tlie above problems it will be seen that tlie third importaut
problem of percentage is not nearly so practical as are the first and
second problems previously studied.

HOME ACCOUNTS

Exercise 15. Expense Accounts

In the following family expense account for 4 mo., the
income being $125 a month, find:

1. The amount saved each month.

2. The total of each item for 4 mo., including savings.

3. The per cent which each total is of the grand total.

Expense Account

Jan.

Feb.

Mar.

Apr.

Total

Food

24.10

23.30

24.80

22.15

Household

Rent

18.00

Fuel (average) . .

4.20

Gas (cooking) .

2.20

2.35

2.15

2.25

Electricity . . .

1.60

1.80

1.30

1.45

Help

1.80

2.25

1.30

1.00

Furnishings . . .

4.20

.65

2.30

9.60

Personal

Clothing ....

7.00

2.00

24.00

Car fare . . . .

.80

1.25

.90

1.20

Insurance

Health and accident

1.10

Life (average) .

4.30

Benevolences

Church ....

3.70

4.20

2.60

2.80

. Charity ....

1.30

2.60

.80

3.20

Education, Recreation

1.90

2.30

4.70

3.60

Incidentals ....

7.30

1.30

2.60

7.38

Savings

Total ....

These per cents should be carried to the nearest tenth.

24 ARITHMETIC OF THE HOME

Problem Data. The following price list may be used in
solving the problems on page 25 and similar problems.

The data may also be secured by the students through inquiry
at home or at some grocery. This list may be made the basis of prac-
tical problems in simple domestic bookkeeping. The object is, of
course, to make arithmetic as real as possible, and when this purpose
has been served, the student should proceed to other topics.

Allspice, 10 (^ per can; $1 per dozen cans.
Asparagus, 35 <^ per can ; $4 per dozen cans.
Bacon, American, 28 ^ per pound.

Sliced, in jars, 30^ per pound; $3.25 per dozen jars.
Breakfast cereal, 14^ per package ; $1.60 per dozen packages.
Cinnamon, 10 ^ per can ; $1 per dozen cans.
Cloves, 30 i per pound ; 50 <^ per 2 pound box.
Cocoa, half-pound cans, 25^; $2.75 per dozen cans.
Coffee, Maracaibo, 20 ^ per pound ; 5 lb. for 85 ^.

Java and Mocha, 35^ per pound ; 5 lb. for $1.60.

Old Government Java, green, 27<^ per pound ; 5 lb. for $1.30.
Crackers, Saltines, 25 ^ per tin ; $2.75 per dozen tins.

Ginger snaps, 8^ per carton; 90^ per dozen cartons.
Domino sugar, 5 lb. for 60 ^.
Flour, buckwheat, 6(^ per pound; a bag of 24j lb., $1.30.

Self-raising, 3 lb. for 19 (^ ; 6 lb. for 35^.

Wheat, 5^ per pound ; $6 per barrel of 196 lb. ; 90^ per sack
of 24 J lb.
Granulated sugar, 8 <^ per pound.
Herring, 15 ^ per can ; $1.75 per dozen cans.
Honey, 8-ounce bottles, 30^; $3.25 per dozen bottles.
Loaf sugar, 11 ^ per pound.

Macaroni, 12^ per package ; 25 packages for $2.75.
Maple sirup, pints, 25 <^ ; gallon cans, $1.45 ; $16.50 per dozen cans.
Olive oil, 40 <^ per pint.

Olives, 32 <^ per bottle; $3.75 per dozen bottles.
Soups, half -pint cans, 10 <^; pint cans, 16^; quart cans, 28^;

$3.25 per dozen quart cans.
Sugar sirup, half -gallon cans, 50 <^; 5-gallon cans, $4.
Tea, Black India, 50 <^ per pound.

English Breakfast, 48 <^ per pound.

HOUSEHOLD ECONOMICS 25

Exercise 16. Household Economics

1. If a family wishes a dozen cans of cocoa, what per
cent is saved in buying at the dozen rate ?

In such cases reckon the per cent on the higher price.

2. If a family wishes 5 gal. of sugar sirup, how much
is saved in buying a 5-gallon can instead of 10 half -gallon
cans ? What per cent is saved ?

3. How much does a hotel manager save in buying
120 gaL of maple sirup by the dozen gallon cans instead
of by the single can ? What per cent does he save ?

4. In which is the per cent of saving greater, in buying
honey by the dozen bottles instead of by the bottle, or in
buying maple sirup by the dozen cans instead of by the can?

5. If a woman wishes 24|- lb. of buckwheat flour, how
much does she save in buying it by the bag?

6. What per cent is saved in buying seK-raising flour by
the 6-pound package instead of by the 3-pound package ?

7. By inquiry at home, make out a grocery list for a
week, from page 24. Make two pages of a home account,
the left-hand page showing the amount received, and the
right-hand page showing the amounts spent for groceries.

8. Make out a bill for six items of groceries, making
the proper extensions and footing. Receipt the bill.

Unless the students recall this from their preceding work in arith-
metic, the teacher should take it up at the blackboard.

9. If a man uses 2320 cu. ft. of gas in April, how much
is his gas bill for that month at 80 (f per 1000 cu. ft. ?

10. At the beginning of the month a gas meter registers
14,260, and at the end of the month 17,140. How much
is the gas bill for the month, at |1 per 1000 cu. ft.?

26 ARITHMETIC OF THE HOME

Exercise 17. Heating the House

1. A man put a hot- water heater in his liouse at a cost
of 1540, and found that he used 12 T. of coal last season,
the coal costing |7.60 per ton. How much did he spend
for the heater and fuel ?

2. If the house in Ex. 1 was heated for 204 da., what
was the average cost of the fuel per day ?

3. If the house in Exs. 1 and 2 had 9 rooms, what
was the average cost of the fuel per room per day ?

4. A man has a steam-heating plant in his house. Last
winter it consumed 22|^T. of coal costing $7.25 per ton.
How much did the coal cost ?

5. If 15% of the coal in Ex. 4 was lost in ashes, how
many pounds of coal were lost in ashes?

6. If the house in Ex. 4 has 14 rooms and is heated
for 190 da. in a year, what was the average cost of the
fuel per day and the average cost per room per day?

7. If 85% of the weight of coal is used in producing
heat in a furnace, how many tons of coal are transformed
into heat by a furnace that burns 17 T. in a season?

8. A man used 14 T. of coal in his furnace in a season,
but on buying a new furnace he used 8^% less coal. At
|6.75 a ton, how much did he save on the coal?

9. If a heating plant costing |525 averages 12 T. of
coal per year at $7.25 a ton and furnishes the same amount
of heat as one costing $375 and averaging 14 T. of coal
per year at the same price, which costs the more money
in 4 yr., money being worth 5% and the depreciation on
each plant per year being 10% of the original cost? How
much more ?

HOUSEHOLD ECONOMICS 27

Exercise 18. The Family Budget

1. Last year Mr. Stone received an income of $3000.
He set aside certain per cents of his income as follows:
rent, 15%; heat, 3%; light, 1|% ; food, 28%; wages,
5|-% ; incidentals, 7% ; other personal expenses, 15% ;
books, music, church, and pleasure, 8%. How much
money did Mr. Stone allow for each of these purposes?

2. Mr. Stone in Ex. 1 really paid for rent, |320; for
heat, 152.75; for light, |28.50; for food, $608.75; for
wages, |135; for incidentals, $175.50; for other personal
expenses, $302.80 ; and for books, music, church, and
pleasure, $167.75. How much did each item of expendi-
ture differ from the estimate and how much did Mr. Stone
save during the year?

3. In Ex. 2 what per cent of the amount spent for rent
and food was spent for rent and what per cent for food ?

4. Mr. Sinclair has an income of $175 a month. He
pays during the year for rent, $480; for heat and light,
$85.75; for food, $675.80; for clothing, $168.40; for in-
surance, $54.75 ; and for other expenses, $250. What per
cent of his income does he save ?

5. In Ex. 4 what per cent of his income does Mr. Sinclair
pay for rent ? for food ? for heat and Ught ?

6. If a man's income is $225 a month and he spends
$600 a year for rent, what per cent of his income does he
spend for rent and what per cent is left for other purposes ?

7. If a family with an income of $2200 a year spends
16% of its income for rent and 26% for food, what amount
does it spend for each of these items ?

Students should be encouraged to prepare family budgets at
home, with the help of their parents.

28 AEITHMETIC OF THE HOME

Exercise 19. Household Economics

1. A grocer sells coffee in half-pound packages at 19 (^
a package and in 4-pound cans at |1.40 a can. If a woman
wishes 4 lb., what per cent does she save in purchasing
by the can ?

2. If you can buy Dutch cocoa in ^-pound boxes at 24 (^
a box or in 4-pound cans at $2.65 a can, and you wish 4 lb.,
what per cent do you save in purchasing by the can ?

3. If you can buy maple sirup at 48 (f a quart or in gallon
cans at $1.50 a can, and you wish 1 gal. of sirup, what per
cent do you save in purchasing by the can ?

4. If a woman can buy corn at 15 (f a can or |1.50 per
dozen cans, what per cent does she save on 4 doz. cans in
buying by the dozen?

5. If a woman can buy soup at 20 (^ a can or |2.10 per
dozen cans, what per cent does she save on 2 doz. cans in
buying by the dozen ?

6. A woman can buy a bushel of potatoes for 80^ or a
peck for 25^. If she needs a bushel of potatoes, what per
cent does she save if she buys by the bushel?

7. A woman can buy ^ doz. cans of tomatoes for 750
or 1 can for 15 (f. If she needs ^ doz. cans, what per cent
does she save if she buys by the half dozen?

8. If flour costs $7.40 a barrel (196 lb.) or 5^ a pound,
what per cent does a family save in purchasing flour by
the barrel if it requires 196 lb.?

9. If a family's ice bill averages $1.75 a month, and ice
costs 35 (^ per 100 lb., how many pounds does the family
use? If by having a better ice box there is a saving of
10% in the amount of ice used, how many pounds are
used ? How much is now the average ice bill per month?

HOUSEHOLD ECONOMICS 29

10. If you can buy some chairs for $24 cash or |3 down
and |3 a month for 8 mo., what per cent do you save if
you pay cash?

In Exs. 10-12 interest is not to be considered at this time. It
should be mentioned incidentally as a subject to be studied later.

11. If you can buy a sewing machine for |40 cash or
by paying $4 a month for a year, what per cent do you
save if you pay cash ?

12. If a reduction of 10% is allowed on all electric-
light bills paid before the tenth of each month, what amount
would be saved in 4 mo. if advantage is taken of this rule
in the account on page 23 ?

13. After the holidays the price of toys in a certain store
was reduced 40%. How much would you save by waiting
until after the holidays to buy a mechanical toy that was
marked |3.50 before the reduction ?

In the following problems use the current market price as
found hy inquiry at home or at the store :

14. Find the per cent which you can save in purchasing
each of the following in 5-pound packages instead of by
the pound: sugar, starch, prunes, raisins.

The teacher may omit Exs. 14-16 if desired. A few such problems,
in which the students supply the data, serve, however, to make the
subject more real.

15. Find the per cent saved in purchasing each of the
following by the dozen cans : tomatoes, corn, peaches.

As a matter of economy it should be noticed that it is not always
good policy to purchase in large amounts because the material may
deteriorate or be wasted.

16. Find the per cent saved in purchasing potatoes by
the bushel instead of by the peck.

ARITHMETIC OF THE HOME

Exercise 20. Miscellaneous Problems

1. Mr. Anderson earns $28 a week. He spends 20% of
his income for rent, 26% for food for the family, 6% for
fuel and lights, 18% for clothing for the family, 10% for
church and charity, and 2% for incidentals. How much is
left each year for other expenses and for savings ?

Although 1 yr. = 52^ wk., or 52|- wk. in leap years, 52 wk. is
always to be taken as a year in problems of this type.

2. The girls in a class in millinery need 20 yd. of a
certain quality of ribbon. They can buy this ribbon at
22 (^ a yard, or 5 yd. for |1. What per cent will be saved
if they take the latter plan ?

3. The goods for a certain dress cost |7.80 and the
buttons and trimmings cost |2.20. The cost of making
the dress is 60% of the cost of the materials. If a dress
of like quality and style can be bought for |15, what per
cent is saved by buying the dress ready made ?

4. Make out a blank like the one shown below, but
extended to include your entire school program, and com-
pute the per cent of time devoted to each subject :

Time of
Recitation

Subject

Minutes or
Recitation

Per Cent
of Total

Minutes
OF Study

Per Cent
OF Tota l

9-9 35
940-1015

Arithmetic
English

35
35

5. Some girls made 30 pieces of candy from the follow-
ing recipe: 3 cups granulated sugar, 15(^; 1^ cups milk, 3^;
^ cake chocolate, 2^(^; an inch cube of butter, 2(^. The
fuel cost them 2(f, and they sold the candy at the rate of
3 pieces for 5(^. What per cent was gained on the cost?

REVIEW 31

Exercise 21. Review Drill
Add, and also subtract, the following :

1. 2.

1750.68 ■ 1680.01
298.98 297.56

1630.27
429.68

5 ft. 4 in.
2 ft. 6 in.

Multiply the following :
5. 6.

$298.63 1342.80
27 92

1674.39
129

ttvo decimal places :

2 ft. 7 in.

Divide as indicated, to

9. $426.34-7. 10.

3469.1-16. 11.

4271 -^ 0.6.

12. $275 is what per cent more than $200 ?

13. $200 is what per cent less than $275? what per
cent less than $300 ?

14. If 17 cars cost $13,600, how much will 9 cars cost?

15. How much is 72% of 3501b.? of $3500? of 3^?

16. How much is 175% of $2500? 12% of $625? ^% of
$2000? 1% of $1200? 11% of $2400?

Write the answers to the following :

17. 39,987 + 46,296. 22. CDXL = (?).

18. 73,203-59,827. 23. 32^ ft. = (?) yd.

19. 341 X $42,346. 24. 18|- gal. - (?) qt.

20. $29.75-25. 25. 4|bu.= (?)pk.

21. I of 25 ft. 4 in. 26. 16 mi. = (?) yd.

In all such drill work the teacher should keep a record of the time
required by the students to solve the problems. Each student should
strive to improve his record when reviewing the page later in the year.

82 ARITHMETIC OF THE HOME

Exercise 22. Problems without Numbers

1. If you have an account with several items of income
and several items of expenses, how do you proceed to
balance the account ?

2. How would you proceed to make out a household
account for a week ?

3. How do you find what per cent of the week's income
is spent for household expenses ?

4. If you know the income of a household and know
what per cent of the income is allowed for food, how do
you find the amount allowed for food?

5. If you know what fraction of his income a man
spends for rent, how do you find what per cent he spends
for rent ?

6. If you wish to know before you receive the gas bill
the amount of gas consumed at your home next month,
how will you proceed to read the meter?

7. If you know the cost of tomatoes per dozen cans
and the cost per can, how will you find the per cent of
saving of a person who purchases a dozen cans at the
dozen rate instead of by the can ?

8. If you know the recipe for making cake for a certain
number of persons, how will you change the recipe if you
are making enough for a certain other number of persons?

9. If a man wishes a set of dining-room furniture and
finds that, by waiting a week, he can buy it at a mark-
down sale at a certain rate per cent off the regular price,
how will you find the amount he will save by waiting?

10. If you know how much a man paid for rent last
year and how much more he pays this year, how will you
find the per cent of increase ?

ARITHMETIC OF THE STOEE 33

II. ARITHMETIC OF THE STORE

Nature of the Work. Fred Dodge applied for a position
in a store. The manager asked him if he could add a
column of figures quickly and correctly, and if he could
compute quickly in his head. Fred thought he could, but
when the manager tested him it was found that Fred was
lacking in two things: he had not been taught to check
his work, and he did not know the common short cuts
in figuring that are used in all stores.

Fred found that the arithmetic work which he needed
most was addition, making change, and multiplication.
We shall briefly review these subjects.

In this review special attention will be given only to such topics
as are not generally treated in the elementary arithmetic which
precedes this course.

Oral Addition. In adding two numbers like 48 and 26
mentally, it is better to begin at the left. Simply think
of 68 (which is 48 + 20) and 6, the sum being 74. This
is the way the clerk in the store adds 48 (^ and 26 (f.

Exercise 23. Addition

All work oral

Add the following, beginning at the left and stating only
the results:

1. 2. 3. 4. 5. 6.

29 68 75 38 45 68
15 23 21 24 25 27

7. 8. 9. 10. 11. 12.

75 76 88 95 75 80
25 26 75 25 30 75

34 ARITHMETIC OF THE STORE

Exercise 24. Addition

See how long it takes to copy and add these numbers^ clteck-
ing the additions and writing the total time :

1. 2. 3. 4.

,1287.75 1349.08 |476.82 . $495.53

425.90 346.58 345.46 228.69

381.92 238.46 38.69 642.95

360.58 190.84 248.60 ^ 72.38

1458.65

$394.38

$298.05

$482.60

97.86

26.95

342.60

234.65

230.95

700.00

20.07

381.90

48.38

83.56

78.68

32.83

621.04

75.08

380.95

300.00

10.

11.

12.

$2080.60

$3064.45

$4148.20

$3275.25

3679.70

817.66

876.42

842.35

909.36

4239.58

3192.68

3065.06

517.38

86.38

2124.45

2095.05

2310.25

3098.07

629.00

812.40

3096.65

2901.94

5082.00

3028.60

13.

14.

15.

16.

$6240.45

$4063.45

$3083.95

$2438.65

239.76

398.43

498.76

480.00

3865.42

2200.75

298.80

3557.76

396.37

4346.68

3763.^

463.48

900.48

328.93

2989.95

3086.75

1637.00

4043.68

4263.49

2945.50

ADDITION AND SUBTRACTION 35

Oral Subtraction. In subtracting mentally it is better to
begin at the left except in making change. In the case
of 52-28 think simply of 32 (which is 52-20) and take
8 from it, leaving 24.

T'his subtraction may be treated by the process of making change,
next described. Students should be familiar with both processes.

Making Change. If you owe 64 (f. to a merchant and
give him ^1, he says, " 64 and 1 are 65, and 10 are 75,
and 25 are $1," or, briefly, " 64, 65, 75, |1," at the same
time laying down 1(^, 10 (^, and 25 (^.

The merchant will first lay down the coin or coins that will bring
the amount up to a multiple of 5 ; then the largest coin or coins that
will bring it up to a multiple of 2.5 ; and so on.

Exercise 25. Subtraction

All work oral
Subtract the following numbers :

1. 2. 3. 4. 5.

47 47 47 56 73

30 33 39 28 34 _ __

Imagine yourself selling goods at a store and receiving in
each case the first amount given, the goods costing the second
amount. State the amount of change due in each case, and
state what coins and bills you would give in change :

8. tl; 84f 11. 13; |2.20. 14. $5; $2.65.

9. $2; 11.25. 12. 14; |3.56. 15. $10; $7.75.

10. $2; $1.78. 13. $5; $2.28. 16. $5; $2.35.

The teacher should ask the students to find how a cash drawer is
arranged, and should describe the cash register. A little work in
making change with real or toy money may profitably be given.

36 ARITHMETIC OF THE STORE

Exercise 26. Subtraction

See how long it takes to copy these numbers, to subtract, and
to check by adding each difference to its subtrahend ; write
the total time with the last result :

74,856

24,965

44,430

34,008

36,278

18,986

36,898

30,975

75,500

38,990

78,006

60,900

34,965

29,009

38,869

36,969

10.

11.

12.

$275.68

$220.85

$308.06

$600.04

46.99

165.90

88.79

189.86

13.

14.

15.

16.

$278.00

$470.41

$309.20

$202.70

149.96

82.64

67.64

32.96

17.

18.

19.

20.

$402.64

$300.00

$408.72

$472.92

89.85

183.75

45.86

88.96

21.

22.

23.

24.

$309.92

$482.60

$300.00

$425.30

43.48

193.84

285.68

226.98

25.

26.

27.

28.

$329.80

$408.73

$496.05

$506.00

49.96

229.84

309.78

329.80

MULTIPLICATIOK 37

Oral Multiplication. When Fred went to work in the
store he found that he often needed to multiply rapidly.
For example, if he sold 7 yd. of cloth at 45^ a yard, he
needed to find the total selUng price at once, without using
a pencil. He found that it was usually easier to begin
at the left to multiply. In the case of 7 x 45(^ he simply
thought of 7 X 40 (j^, or |2.80, and 35(^, making |3.15 in all.

Exercise 27. Multiplication

Examples 1 to 12, oral
Multiply the following, beginning at the left:
1. 2. 3. 4. 5. 6.

45 38 32 56 56 65

_6 ^ _7 _8 _9 _7

7. 8. 9. 10. 11. 12.

72 77 56 45 55 35

_8 _3 _^ _8 j; _8

Multiply the following :

13. 43 X 473. 21. 355 x 926. 29. 35 x 6464.

14. 38 X 308. 22. 280 x 628. 30. 42 x 8480.

15. 29 X 247. 23. 84 x 6088. 31. 68 x 9078.

16. 6Q X 385. 24. 29 x 4756. 32. 39 x 4030. -

17. 425 X 736. 25. 63 x 2798. 33. 203 x 3405.

18. 520 X 826. 26. 42 x 4802. 34. 330 x 4143.

19. 332 X 509. 27. 34 x 3006. 35. 223 x 6062.

20. 477 X 805. 28. 23 x 3989. 36. 447 X 3095.

Teachers who care to give the check of casting out nines may do
so at this time. Algebra is required, however, for its explanation.

38 ARITHMETIC OF THE STORE

Short Cuts in Multiplication. You have already learned
hi arithmetic; that there are certain short cuts in multipli-
cation. These short cuts can be used extensively in the
store. The most important ones are as follows :

To multiply hy 10, move the decimal point one place to the
right ; annex a zero if necessary.

To multiply hy 100 or 1000^ inove the decimal point to the
right tivo or three places respectively ; annex zeros if necessary.

To 77iultiply hy 5^ multiply hy 10 and divide hy 2.

To multiply hy 25^ multiply hy 100 and divide hy 4.

To multiply hy 125^ multiply hy 1000 and divide hy 8.

To multiply hy 33 j, multiply hy 100 and divide hy 3.

To multiply hy 9, multiply hy 10 and subtract the iiiulti-
plicarid.

To multiply hy 11^ midtiply hy 10 and add the multiplicand.

Exercise 28. Short Cuts

Find the residts mentally tvhenever you can
Multiply^ in turn^ hy 10, hy 100^ hy 5^ hy 25^ and hy 125 :

1. 6456 9248 25,192 23,848 22,200

2. 8168 9.376 19,920.8 25.088 58.752

3. 5776 24.432 ,56,246.4 23.048 46.832

Multiply^ in turn., hy 33 j, hy 0, and hy 11 :

4. 46,977 67,053 15,240 17,604 13,806

5. 441.54 466.74 1639.2 457.05 96.816

6. 483.66 190.56 1804.5 20.166 69.306

Multiply, in turn, hy 5, hy 25, and hy 50 :

7. 15,384 56,812 87,824 756.52 73.728

8. 86,988 47,752 93,104 527.24 43.332

SHORT CUTS IN MULTIPLICATION 39

Multiply the following :

9. 10 X 10.35. 20. 331 x 45. 31. 1000 x |7.62.

10. 10 X 1225. 21. 331 X 288. 32. 50 x |4220.

11. 10 X 17.75. 22. 331 X 585. 33. 125 x |3200.

12. 50 X 1652. 23. 100 X $45. 34. 331 x |345.

13. 50 X 1345. 24. 100 X *33. 35. 16| x 186.

14. 25 X *544. 25. 1000 x |65. 36. 16f x |696.

15. 25 X 1280. 26. 25 x $85.35. 37. 5 x 40,364.

16. 25 X $428. 27. 121 x $4400. 38. 5 x $15,680.

17. 675 X $35. 28. 121 X $4088. 39. 125 x $408.

18. 25 X $5.20. 29. 125 x $560. 40. 125 x $4.08.

19. 10 X $4.80. 30. 125 x $5600. 41. 16| x 7200.

42. How much will 25 books cost at 80 (^ each?

43. How much will 25 yd. of cloth cost at 16 (f. a yard ?

44. How much will 50 cans of corn cost at 14 (^. each?

45. How much will 4 books cost at 750 each?

46. How much will 25 yd. of cloth cost at 24 (f. a yard ?

47. How much will 12^ yd. of cloth cost at 48 (f a yard ?

48. How much will 80 doz. pencils cost at 56<f a dozen ?

49. How much will 75 books cost at 60(/! each?

50. How much will 75 coats cost at $5 each ?

51. How much will a man earn in 48 wk. at $25 a week ?

52. How will 3|- doz. cans of tomatoes cost at 12 ^ a can ?

53. At $7.50 each, how much will 11 tables cost?

54. At $8.25 each, how much will 9 desks cost?

55. At $9.60 each, how much will 25 chairs cost?

56. At $42.50 each, how much will 11 typewriters cost?

40 ARITHMETIC OF THE STORE

Product of an Integer and a Fraction. In the store we
frequently have to find the product of an integer and a
fraction. For example, we may need to find the cost of
^ yd. of velvet at $2 a yard. As we have learned,

To find the product of a fraction and an integer^ multiply
the numerator of the fraction hy the integer and write the
product over the denominator.

Before actually multiplying^ indicate the multiplication and
cancel common factors if possible.

Reduce the result to an integer^ a mixed number, or a
common fraction in lowest terms.

For example, to multiply ^ by 18. Since we have ^,
if we have 18 times as much we shall have

;^xll 22 ^1

While we use ^^ as an illustration, we seldom find in a store any
need for a common fraction with a denominator larger than 8.

Exercise 29. Multiplication

All work oral
Multiply the following, using cancellation when possible :

1. f of 6. 6. 48 X |. 11. -j^ of 48. 16. I of 960.

2. I of 8. 7. 50 X iV 12. 24 x |. 17. | of 272.

3. 48 X |. 8. 40 X 3^. 13. 128 x |. 18. 864 x |.

4. 80 X |. 9. I of 35. 14. 132 x ^V l^- 484 x |.

5. 50 X f . 10. I of 88. 15. I of 36. 20. 192 x |.

Multiplication of a fraction by a fraction is not so frequently
needed in the store as the work given above.

MULTIPLICATION 41

Multiplication involving Mixed Numbers. In the store
we frequently need to find such a product as 12^ X 16, as
in reckoning the cost of 12|-yd. of cloth at 16^ a yard.
In this particular example we should simply think of -1-1-^
of 16, or 200, and state the result at once as |2. But in
general, as we have already learned.

To multiply a mixed number hy an integer^ multiply
separately the integral and fractional parts of the mixed
number by the integer and add the products.

For example, 7 X 2| = 14-^ = 191

To multiply a fraction by a fraction^ multiply the numer-
ators together for the numerator of the product and the de-
nominators together for the denominator of the product.

This case, familiar to the student, is mentioned here for the sake
of completeness.

To multiply a mixed number by a mixed number, reduce
each to an improper fraction and multiply the results, using
cancellation whenever possible.

For example, ^x^ = ^x^ = ^^ = 14|.

Exercise 30. Multiplication

Multiply the following :

15 X ^.

9| X 320.

15.

31 X 251

18 X 2|.

9|x688.

16.

12| X 15f .

36 X 5|.

10.

35f X 18.

17.

25| X 34f.

80 X 9|.

11.

17| X 65.

18.

14|xl6|.

48 X 7-I-.

12.

261 X 84.

19.

18| X 231

36| X 48.

13.

251 X 320.

20.

181 X 27|.

321 X 45.

14.

31x51

21.

15| X 28|.

42 ARITHMETIC OF THE STORE

Use of Aliquot Parts in Multiplication. As you have al-
ready learned and would naturally infer from page 38 and
from a few examples already met, it is easier to multiply
|1 ^, and ^ than it is to multiply 12|(f, 16|(^, and 331(^.
Such parts of a dollar or of any other unit are often called,
as we have learned, aliquot parts.

At 33-J^^ each, 15 books cost 15 times $^, or $5.

At 16-|<f each, 24 rulers cost 24 times |^, or |4.

At 121(^ each, 16 notebooks cost 16 times ||^, or $2.

While goods are seldom marked 16^<f. or $^, they are
often marked 6 for |1, which is the same thing.

Exercise 31. Aliquot Parts

All tvork oral

1. At 12^ (^ a yard, how much will 32 yd. of cloth cost ?

2. At 16|^(^ a yard, how much w411 36 yd. of cloth cost ?

3. At 33^ (^ a yard, how much will 39 yd. of cloth cost ?

4. At 8 notebooks for |1, how much will 24 notebooks
cost? How much will 32 notebooks cost?

5. At 16|^(^. each, find the cost of 42 glass vases.

6. At the rate of 6 pairs for |1, how much will 48
pairs of scissors cost?

7. How many children's coats can be cut from 7^ yd.
of cloth, allowing 2^ yd. to a coat ?

State the products of the following :

8. 9 X S^^. 12. 24 xl6|(^. 16. 16 xl2^(jfe.

9. 18 X SSy. 13. 66 X 16f (f. 17. 48 x 12|(^.

10. 27x33^(^. 14. 48xl6|(f. 18. 72xl2i(f.

11. 150x33^(^. 15. 72xl6f(^. 19. 96xl2^<f.

MULTIPLICATION

Cash Checks. In many of the large stores the clerks
are reqmred to fill out cash checks like the following:

Sold by No. 29 Amount rec'd, |^^, 00

^ ycL

/6j ^^

1.50
.80

• .^6

3
//
16

7^
60
68

cfS
07

Total

Chang

e due

Exercise 32. Cash Checks

Make out cash checks for the following sales :

1. 31 yd. cotton @ 18(^., 24yd. velveteen @ ^1^., 161yd.
dimity @ 30(/!. Amount received, $30. ^

2. 81yd. gingham @ 30 (f, ^Ijd. madras @ 38 (^, 9f yd.
silk @ $1.25. Amount received, $20.

3. 241yd. linen @ 38 (^, 221yd. linen suiting @ 85 (^,
4| yd. dimity @ 28 <^. Amount received, $30.

4. 61 yd. India linen @ 42(f., 81 yd. cheviot @ $1.35,
18 yd. cotton @ 12|^(f. Amount received, $20.

5. 14 yd. muslin @ 25 (^., |^yd. velvet @ $3, 6|yd. India
linen @ 45(f, 12|-yd. lining @ 11 (f.. Amount received, $10.

6. 24| yd. muslin @ 24(f, 3^ yd. velvet @ $2.40, 26| yd.
lining @ 12(f., 6f yd. silk @ $1.60, 5| yd.* suiting @ S0(^,
6^ yd. ribbon @ 30 (f. Amount received, $50.

The teacher should ask the students to make problems similar to
those given above, using the local prices of common materials.

44 ARITHMETIC OF THE STORE

Discount. When goods are sold at less than the marked
price, the reduction in price is called discount.

Local examples should be mentioned and the students should be
asked for illustrations of discount within their experience.

List Price. The price of goods as given in a printed
catalogue or list issued by the manufacturer or by the
wholesale house is called the list price. In stores where
the goods are marked, this is called the marked price.

Discount is usually reckoned as a certain per cent or
as a certain common fraction of the list price or marked
price, thus: 20% off, 33-|% off, ^ off, and so on.

Net Price. The price of goods after the discount has
been taken off is called the net price.

Cash Discount. A discount allowed because the pur-
chaser pays at once is called a cash discount.

For example, a Boy Scout suit may be marked |6, but
owing to the desire of the dealer to clear out his stock at
the end of the season he may mark it to sell for 10% off
for cash. The suit will then be marked $6 less 10% of $6,
or $6-10.60, or $5.40.

Trade Discount. When merchants, jobbers, or manufac-
turers sell to dealers they often deduct a certain amount
from the list price. This reduction is called a trade discount.

Such terms as retail dealer, wholesale dealer or jobber, and manufac-
turer should be explained by the teacher if necessary.

A special form of trade discount is allowed for very large orders.
This is called a quantity discount.

The terms of discount are often stated thus : 2/10, 1/30,
iV/60, these symbols meaning 2% discount if the bill is paid
within 10 da., 1% if paid within 30 da., net (no discount)
thereafter, and the bill to be paid within 60 da.

DISCOUNTS 45

Exercise 33. Discounts

Examples 1 to 15 , oral

1. If some goods are marked $20, and 10% discount
is allowed, what is the selling price ?

2. If a book marked 800 is sold at a discount of 25%,
this is how many cents below the marked price ?

3. If a merchant buys |800 worth of goods and is
allowed 10% discount in case he pays for. them at once,
how much does he save by prompt payment?

Find the discounts on the following at the rates specified :

4. tSO, 10%. 8. $40, 25%. 12. |120, 10%.

5. 125, 10%. 9. $88, 25%. 13. |150, 20%.

6. $50, 20%. 10. $60, 50%. 14. $160, 25%.

7. $25, 20%. 11. $90, 50%. 15. $250, 50%.

16. If goods marked $475 are sold to a dealer at a
discount of 20%, how much do they cost him?

17. If a merchant marks a lot of suits at $24.75 each,
and sells them at -J- off, what is the net price of each ?

Students should be asked to watch advertisements in the news-
papers to see what discounts are offered and should state the prob-
able reasons for these discounts.

Given the marked prices and rates of discount as follows,
find the net prices :

18. $17.50, 10%. 23. $21.60, 121%. 28. $48.60, 25%.

19. $16.50, 20%. 24. $72.30, 33^%. 29. $27.70, 50%.

20. $27.75, 20%. 25. $38.70, 33^%. 30. $35.00, 10%.

21. $64.40, 25%. 26. $43.50, 16|%. 31. $34.75, 20%.

22. $86.60, 25%. 27. $86.40, 16f %. 32. $65.25, 20%.

46 AKITHMETIC OF THE STORE

Several Discounts, lii some kinds of business two or
more discounts are frequently allowed. For example, a
dealer may buy hardware listed at $200 with discounts
of 20%, 10% (20% and 10%, often called and written
simply 20, 10). This means that 20% is first deducted
from the list price, and then 10% from the remainder.

The list price is |200.

The list price less 20% is |160.

Then |160 less 10% is $144, the net price.

In reckoning discounts the student should discard every fraction
of a cent in the several discounts.

Exercise 34. Several Discounts

Examjdes 1 to 5, oral

1. From tlOO take 10%, and 5% from the remainder.

2. From |800 take 25%, and 1% from the remainder.

3. From |600 take 20%, and 20% from the remainder.

4. A dealer bought some goods at a list price of f 100,
with discounts of 10%, 10%. How much did he pay?

Eind the net prices of goods marked and discounted as
follows :

5. 1400, 20%, 30%. 9. $1300, 15%, 10%.

6. $1400, 35%, 5%. 10. $800, 20%, 10%.

7. $800, 20%, 15%. 11. $600, 15%, 10%.

8. $1200, 12%, 4%. 12. $550, 10%, 10%.

13. What is the difference between a discount of 50%
on $1000, and the two discounts of 25%, 25% ?

14. Is there any difference between a discount of 5%,
4%, and one of 4%, 5%, on $900 ? How is it on $600 ?

SEVERAL DISCOUNTS 47

Sample Price List. The following is a price list of
certain school supplies, with the discounts allowed to
schools and dealers when the prices are not net:

Composition books, |4.80 per gross, less 5%

Drawing compasses, 1.75 per doz., less 10^, 5%

Drawing paper, 9 xl2, 1.30 per package, less 10%

Drawing pencils, 4.80 per gross, less 20%

Penholders, 3.40 per gross, less 12%, 4%

Pens, 0.65 per gross, less 25%, 10%

Rulers, 0.40 per doz., net

Thumb tacks, 0.45 per 100, less 30%

Tubes of paste, 4.15 per gross, less 10%, 6%

Exercise 35. Purchases for the School

1. A school board wishes to buy 8 packages of drawing
paper and 200 tliumb tacks. How much will they cost?

In this exercise use the above price list.

2. How much will 12 gross of pens and 2 gross of com-
position books cost ? 4 gross of pens and 3 doz. rulers ?

3. There are 18 students in a class, and eacli student
needs compasses and a ruler. How much will all these
drawing instruments cost the school ?

4. If a dealer sells pens at a cent apiece, how much
does he gain per gross ? If he sells penholders at 30 each,
how much does he gain per gross ?

5. A dealer buys 3 gross of rulers and 2 gross of
drawing pencils. He sells the rulers and pencils at 5(f
each. How much does he gain in all?

6. If a dealer buys 2 gross of tubes of paste for mount-
ing pictures and sells the tubes at 5(/', each, how much
does he gain on the purchase ? how much per gross ?

ARITHMETIC OF THE STOEE

Bill with Several Discounts. The following is a common
form of a jobber's bill of goods with several discounts :

Burhngton, la., TTlcmf /3, 1920
Mv. ^ea. c/. mUteA., ^Uoa. RcujolcU, Jcv.

mougU of R03ERTS ^ STONE, Jewelers

, ^ 1072 Passaic Avenue
Terms: 20%, /0%

mOAf

8 cicKi. a^o-cyyiQy @ f/^.J^S
Le^ 20%, /0%

//8
2Cf

00
¥-0

f^7

^0
^7

f06

Exercise 36. Bills

Make out hills for the following :

1. 4 doz. sweaters at |34. Discounts 10%, 5%.

2. 16 doz. files at |7.30. Discounts 25%, 20%.

3. 625 yd. taffeta at |1.35; 240 yd. velvet at *1.80.
Discounts 15%, 10%.

4. 6 doz. pairs hinges at |4.50 ; 12 doz. table knives at
19.20. Discounts 20%, 10%.

5. 16 doz. locks at $4.50 ; 4 doz. mortise locks at $4.85.
Discounts 20%, 8%.

6. 840 yd. taffeta at $1.10 ; 12 gross pompons at $150 ;
4 doz. pieces braid at $21.60. Discounts 10 %, 5 %.

7. 960 yd. silk at $1.75; 640 yd. lawn at 27(^ ; 860 yd.
taffeta at $1.05. Discounts 10%, 5%, 5%.

DISCOUNTS

Receipted Bill. The following is a receipted bill for
some goods purchased by a retail merchant from a jobber :

Ckicago, III, ^e.^. //, 1920
M^. a. m. cAcnow.^, fatiet, ML

bought of g^^^^ ^ TIFFANY, Jewelers

8378 Burlington Ave.
Terms: i'%

^£^.

70
f

¥-0

In this case Mr. Nourse is the debtor (Dr.), since he is in
debt for the amount; the firm of Starr & Tiffany is the
creditor (Cr.), since it trusts Mr. Nourse, or gives him
credit. On this bill only a single discount was allowed.

A receipt may also be written separately instead of appearing on a
bill. Such a receipt should be dated, and should be in substantially

this form: "Received from the sum of dollars for ."

The receipt should be signed by the creditor.

The subject of commercial discount is of great value because of its
extensive use not only in wholesale transactions but even in bargain
sales. Students should understand that some of the reasons for allow-
ing discounts are buying in large quantities and paying cash down
or within a specified time, and they should become familiar with the
ordinary deductions from list prices. It is well, for obvious reasons,
to consider bills and receipts, preferably of a local character, in con-
nection with the study of this topic.

ARITHMETIC OF THE STOEE

Invoice. A bill stating in detail a list of items and prices
of goods sold is called an invoice. A sample invoice follows :

St. Paul Minn., M^.. /^, 1920

1^1. ^i^ni&Qy Ir'. ^ii/}v{>U'V, 3a^xaa, cA.^.

'Bough of RANDALL BROTHERS

IMPORTERS OF DRY GOODS AND FANCY GOODS

Terms: /O clcL.

2S6

^-0\ ^0, ^/, ^-2,

f /, ^0 676 331^

/q2

if^Y

^0, 39, ^/\ ^0\

3f, ^f, ^2\ //^ 32^^ //-

^-06

6^7

In the above invoice 41^ means 41 J^; 41^ means 41|,
or 41^; and 41^ means 41|^. This is the customary way
of indicating the number of yards in pieces of goods. The
numbers 236 and 427 refer to the price list in which the
goods are described. The numbers 14 and 8 indicate the
number of pieces (pc.) bought. The number of yards of
the first is 576 and of the second 324.

The expression '' Terms : 10 da." means that Mr. Dunbar
has 10 da. in which to pay for the goods. Such an invoice
may or may not mention the discount allowed.

There is no essential arithmetic difference between a
bill of o-oods and an invoice of a wholesale dealer.

INVOICES 51

Exercise 37. Invoices

Make out invoices for the folloiving :

1. 24doz. caps @ $15.50, 21doz.'ties @ |8.40.

2. 32 pc. ribbon @ |1.25, 14 pc. @ $1.30, 48 pc. @

$1.75, 24 pc. @ $1,121 32 pc. @ $1,371 48 pc. @ $1.2:5,
36 pc. @ $1.40, 64 pc. @ $1.60.

3. 3 carloads coal, 21,700 lb., 24,200 lb., 25,100 lb., @
$5.20 per short ton; 3 carloads coal, 22,700 lb., 21,900 lb.,
20,400 1b., @ $5.75 per short ton. Terms: 4% for cash.

4. 1 gro. cans sardines @ $3.40 per doz., 9 doz. cans
shrimps @ $1.75, 8 doz. tins herrings @ $2.50, 6^ doz.
cans lobster @ $2.88, 32 cans mackerel @ 16§(/', 4|- doz.
cans kippered herrings @ $2.65. Terms: 4% for cash.

5. 8 doz. packages codfish @ $1.80, 9 doz. cans salmon @
$2.40, 15 doz. cans caviar @ $3.50, 18 doz. cans Yarmouth
bloaters @ $2.40, 24 doz. cans tongue @ $8.75, 16 doz.
cans baby mackerel @ $1.80. Terms: 3|^% for cash.

6. 12 doz. jars meat extract @ $3.40, 24 doz. cans
chicken @ $3.15, 24 doz. cans beef @ $2.40, 18 doz. cans
soup @ $3.40, 9 doz. cans clam chowder @ $3.75, 16 doz.
cans clam juice @ $1.10. Terms: 3%, 2%.

7. 8 armchairs @ $6.75, 24 kitchen chairs @ $1.25,
12 kitchen tables @ $2.25, 6 bedroom sets @ $42.50, 24
rockers @ $8.25, 16 dining tables @ $12.50, 9 sideboards
@ $16.66|. Terms: 6%, 4%.

8. 20 pc. linen containing 40\ 40^, 38^, 42, 41i, 402, 40,
40, 401, 412, 43^ 393^ 401, 421, 402, 40i, 41^, 42, 39i, 40 yd.,
@ 75(f.; 16 pc. silk containing 41^, 42i, 40, 392, 42, 40^,
38, 411, 43^ 443, 39^ 402, 401, 411, 40, 392 yd., @ $1.40.
Terms: 6%, 3%.

52 ARITHMETIC OF THE STORE

Exercise 38. Review

1. Make out an invoice for the following goods purchased
May 7 and paid for May 10, terms 2/10, 1/20, N/SO:
10 bolts dress linen, 10, 12, 11^ 122, iqs^ 113^ 12, lOi, 112,
11^ yd., @ 48 (^ a yard; 24 bolts French nainsook, 12 yd. each,
@ 18(f a yard; 30 bolts mercerized lingerie batiste, 24 yd.
each, @ 22(^ a yard; 32 bolts imported lawn, 10 yd. each,
@ 460 a yard. Insert names and find the net cost.

2. In Ex. 1 find the net cost if the payment is made
on May 20.

3. Make out an invoice for the following goods purchased
Sept. 20 and paid for Oct. 5, terms 2/20, iV/90 : 5 No. 264
plows @ 142.50 less 20%; 3 No. 178 self-dump hayrakes
@ 118.60 less 15%; 9 No. 325 hay stackers @ |46.50 less
15%. Insert names and find the net cost.

4. A jobber offers his customers discounts of 15%, 10%,
but the invoice clerk made a mistake on a bill of $85 and
gave a single discount of 25%. How much did the error
cost the clerk or the jobber ?

5. A manufacturer lists a desk at |52 less 25%, and a
rival manufacturer offers a similar desk for |57 less -|. If
the first dealer increases his discount to 25%, 3%, which
will be the lower net price and how much lower?

6. A retail dealer being allowed a discount of 20%, 2%,
find the net price of the following goods purchased from a
wholesale dealer at the list prices stated: 1 dining table,
$34; 8 chairs @ $2.85; 1 buffet, $32.50; 1 rug, $26.

7. A dressmaker bought the following bill of goods,
receiving a trade discount of 8% and a cash discount of
6%: 8^ yd. broadcloth @ $3; 13|yd. silk lining @ 80(f;
3 yd. trimming @ $2.40. What was the net price ?

EEVIEW 53

Exercise 39. Review Drill

1. Multiply 488 by 1; by 0.5; by 50% ; by 50.

2. Multiply 24 by 331% ; by 25% ; by 12|%.

3. Multiply f by I ; | by f ; f by |.

4. Dividelbyi;ibyl;f byl;lbyf.

5. Multiply 8 by ^; 8 by 0.125; 8 by 125.

6. Divide 4800 by 100 ; by 10 ; by 20 ; by 40.

7. Add ^ and J; |- and J; ^ and ^.

8. Find the values of 1-f ; l-|-f ; 5| - 3|.

9. Express as decimals and also as common fractions:
371%; 16|%; 621%; 66f % ; 871%; 831%; 331%.

10. Express 12 ft. as inches; 96 in. as feet; 69 ft. as
yards ; 51 yd. as feet ; 16 gal. as quarts ; 16 qt. as gallons ;
3 lb. 4 oz. as ounces ; 32 oz. as pounds.

11. 1500 is what per cent of |400 ?

12. 1500 is what per cent more than |400 ? than $250 ?

13. 1400 is what per cent less than $500 ? than $800 ?

14. If 8 typewriters cost $500, how much will 15 cost?

15. If 9 desks cost $63, how much will 36 desks cost ?

16. If the rent of an apartment is $720 for f yr., how
much is the rent for a year ?

Perform the following operations :

17. 41f in. + l^in. 22. 25% of $1.60.

18. 5fin.-2|in. 23. $2.75^-11

19. 2| X 271 in. 24. f of 34^ ft.

20. 6| ft. ^ 3^;^ ft. 25. I of 9 ft. 4 in.

21. 3.75 --2|. 26. 75% of 6 1b. 4 oz.

64 ARITHMETIC OF THE STORE

Exercise 40. Problems without Numbers

1. If you buy a fishing rod and sell it at a certain rate
per cent above cost, how do you find the selling price ?

2. If a man buys, a car and sells it at a certain per
cent below cost, and you know how much he paid to keep
the car and how much he saved by using it, how do you
find whether he gained or lost, and the amount?

3. If you know the number of school days in the year
and the number of times you were absent from school dur-
ing the year, how do you find the percentage of absences ?

4. If you know the number of times and the percentage
of times a train arrives on time during a certain month,
how do you find the number of runs the train makes ?

5. If you sell a person a certain bill of goods, and he
hands you more than the required amount of money, how
do you proceed to make change ?

6. If you know a man's income and the amount which
he pays for rent, how do you find what per cent of his
income he pays for rent?

7. If you know the rate and amount of an agent's
commission, how do you fiiid the selling price ?

8. How do you express a given per cent as a decimal ?
as a common fraction ?

9. If you know what a certain per cent of a number
is, how do you find the number?

10. How do you find what per cent one given number
is of another given number?

11. If you sell a man a number of articles at a certain
price each, and this price is an aliquot part of a dollar,
what is the shortest method of finding the amount due ?

ARITHMETIC OF THE FARM 55

III. ARITHMETIC OF THE FARM

Nature of the Work. The farming industry is one of the
largest in our country. There are between six and seven
milhon farms in the United States and their total area is
nearly 900,000,000 acres. These farms are worth, with
their . buildings and machinery, over $40,000,000,000 and
they produce about 110,000,000,000 annually. From these
immense sums it will be seen how important is the farm-
ing industry and how necessary it is to know some of the
problems relating to it.

Every boy and every girl who lives on a farm ought to
know how to measure a field and find its area, how to keep
farm accounts, and how to make the necessary computa-
tions relating to the dairy, the crops, and the soil. Even
boys and girls who live in villages and cities should,
for their general information, know something of these
matters, just as those who live on the farm should know
something about the problems of the city.

The work of measuring land and computing such volumes as the
farmer uses is taken up in the geometry in this book, but the famil-
iar case of the area of a rectangle is assumed to be understood.

Teachers in village and city schools will find in the following
pages a sufficient number of problems for their purposes, but the
entire topic may be omitted if necessary. In rural schools, however,
additional problems should be drawn from the locality in which each
school is placed. Agricultural products, the soils, the customs, the
wages, and the prices all vary greatly in different sections of our
country, and the teacher should encourage the students to bring
to school problems which relate to local interests. Problems about
irrigation are important in some states, while in others they are
quite unheard of ; the alfalfa crop is very important in certain parts
of the country, while in others it is not ; land is laid out in sections
in some states, while it is never so laid out in others. .Tlie teacher
should be guided by a knowledge of these various customs.

56 AEITHMETIC OF THE FARM

Exercise 41. Cost of Wastefulness

1. If a farm wagon that cost |60 is left out in the yard
instead of being kept in the shed, it will last about 6 yr.,
but if kept under cover, it will last about twice as long.
What per cent of the cost, not considering the interest on
the money, does a farmer pay for his carelessness per year
if he leaves the wagon out of doors?

2. A farmer after threshing his wheat had 16 T. of
straw left. A ton of this straw contains 10 lb. of nitrogen
worth 15 (^ a pound, 18 lb. of potassium Avorth 6(^ a pound,
and 2 lb. of phosphorus worth 12(f a pound. If the farmer
wastes the straw instead of using it on the soil as fertilizer,
how much money does he waste ?

3. It is computed that a certain kind of farm machinery
depreciates in value as follows, if reasonably good care is
taken of it: 10% of the original value the first year, 8% of
the original value the second year, 6 % of the original value
the third year, 3% of the original value the fourth year
and each year thereafter. A machine of this kind cost a
farmer |240. He did not take proper care of it, and at
the end of 4 yr. it was worth only |115. The per cent of
the original value thus lost in the 4 yr. was how much
more than the per cent that would have been lost had
proper care been taken of the machine ?

4. The strip of waste land along each side of a woven-
wire fence is 2 ft. 6 in. wide, the strip along a barbed-wire
fence is 3 ft. wide, and the strip along a rail fence is 4 ft.
6 in. wide. How many rods of each kind of fence cause
a waste of 1 acre of land on one side ? At |90 an acre,
compare the losses occasioned by the various tjrpes of
fences for . the inside of a rectangular field 40 rd. wide
and 60 rd. long fenced in on the four sides.

FARM ACCOUNTS

Farm Accounts. Many careful farmers keep systematic
accounts of the receipts and expenditures for each of their
fields, as well as for the farm as a whole. In the problems
of the following exercise an itemized statement is given of
expenditures for a 20-acre field of corn.

*" Exercise 42. Farm Accounts

1. In the following account supply the missing amounts:

Expenses

Apr.

Plowing, 5| da.

@ $4.80

Harrowing, 2i da.

@ $4.50

2 bu. seed corn

@ $1.00

Planting, 2 da.

@ $4.80

May

Replanting, 1 da.

@ $1.25

Harrowing, 2 da.

@ $4.50

Plowing, 4 da.

@ $4.75

June

Plowing, 3 da.

@ $4.75

Plowing, 3 da.

@ $4.75

July

Plowing, 3 da.

@ $4.75

Sept.

Cutting, 6 da.

@ $4.00

Nov.

Husking, 718 bu.

@ ^

Rent of land @ |5.50
Total

per acre

2. If the receipts in the above case came from the sale
of 718 bu. of corn at 62 (f, with $40 worth kept on hand,
the expenditures are what per cent of these receipts ?

3. In the field of Exs. 1 and 2 find the net profit.

In rural schools it is desirable to secure or to have the students
secure local accounts of this kind. This is the best way to make the
subject real to those who are studying it.

AEITHMETIC OF THE FARM

Exercise 43. Farm Records

1. Ralph's father explamed to him what was meant by
grading the cows, keeping their records for milk, and sell-
ing the poor cows. He showed him this farm record:

With Systematic

Grading

Without Systematic Grading

Annual cost

Annual

Annual cost

Annual

Herd

of food

profit

Herd

of food

profit

per cow

134.28

$34.02

$31.05

$13.34

33.37

30.19

40.58

8.13

47.11

25.83

38.30

22.22

30.72

38.27

37.40

22.33

('.

31.19

48.80

33.78

38.93

31.57

42.81

32.88

14.91

Ralph and his father then figured out the average annual
cost of food and profit per cow, in each class of herds.
They did this by dividing by 0 the total of each of the
four columns. What were the results ?

2. At a certain experiment station the five most profit-
able and the five least profitable cows compared as follows:

Grade of Cows

Cost of

Feed
AND Care
Annually

Pounds of

Butter

Fat

Annually

Average

Cost of

1 LB. of

Butter Fat

Five most profitable cows
Five least profitable cows

$50.54
52.02

304

189

Compute the average cost of 1 lb. of butter fat for the
most profitable and for the least profitable cows.

PROBLEMS OF THE DAIRY ' 59

Exercise 44. Problems of the Dairy

1. A farmer sells 26,250 lb. of milk to a creamery in a
certain month. The milk averages 4.2% by weight of
butter fat. With how many pounds of butter fat does the
creamery credit the farmer in that month ?

2. If 6 lb. of butter fat are needed in making 7 lb. of
butter, what is the value of the butter produced from
1236 lb. of butter fat, the butter being worth 340 a pound ?

3. A certain dairy sells to a creamery milk averaging
3.75% of butter fat. The butter fat weighs 630 lb. How
many pounds of milk does the dairy sell?

4. A farmer has two cows, one supplying 986 lb. of milk
testing 3.1% butter fat in a certain month, and the other
8121b. testing 4.2%. If the creamery allows the farmer
32 (^ a pound for butter fat, which cow pays him the more
and how much more, the feed and care costing the same ?

5. A creamery uses 7500 lb. of milk in a week. The
skim milk amounts to 80% of the whole milk and con-
tains 0.6% butter fat. How many pounds of butter fat
are lost in the skim milk ?

6. A farmer owns a herd of 18 cows that average 25 lb.
of milk per head daily. This milk tests 3.5% butter fat,
and the butter fat is worth 26.5 0 per pound. How much
does the farmer receive in 30 da. for the butter fat?

7. A herd of 24 cows averages 22 lb. of milk per cow
daily, and another herd of 18 cows averages 28 lb. per cow.
The milk of the first herd averages 5% butter fat and that
of the second herd 3.5%. How many more pounds of
butter fat are produced by the first herd per week?

In rural communities there should be special computations on
such subjects as rations, cost of labor, and the income from cows.

JMl

ARITHMETIC OF THE FARM

8. Given the following table showing the number of
pounds of nitrogen, phosphorus, and potassium in 1000 lb.
of each of five kinds of feed, find the per cent of each of
these three ingredients in each of the five kinds:

Feed

Nitrogen

Phosphorus

Potassium

Wheat straw . .
Timothy hay . :
Clover hay .

Corn

Wheat ....

5.0
12.0
20.0
17.1

23.7

0.8
1.5
2.5
3.0
4.0

9.0

11.8

15.0

3.4

4.3

9. The following table shows the amount of protein
and carbohydrates in certain kinds of feed:

Feed

Weight of a

Bushel in

Pounds

Pounds per Bushel

Protein

Carbohydrates

Rye

Barley ....

Corn

Oats

56
48
56
32

5.0
4.0
3.5
3.0

39
32
40
19

A dairy cow of average size requires daily about 2 lb. of
protein and 12 lb. of carbohydrates. When corn is 62^ and
oats 41(^ per bushel, which is the cheaper food, considering
the protein alone ? considering carbohydrates alone ?

10. In Ex. 9 the weight of the protein in a bushel of rye
is what per cent of the weight of the rye ? Answer the
same question for barley ; for com ; for oats. Answer
the same questions for the carbohydrates.

The weight of a bushel varies in the different states.

FEEDING CORN 61

Exercise 45. Feeding Corn

1. When corn was selling at 55^ a bushel, a farmer de-
cided to feed his corn to his cattle. He estimated that the
increase in the value of the cattle, from the corn alone, was
60 (^ for each bushel used for feed. What was the per cent
of increase in the value of the corn by using it as feed ?

2. A record of the result of feeding corn to hogs was
kept on several farms. On one farm, when corn was sell-
ing at 55 (^ a bushel, it was found that the increase in the
value of the hogs was equivalent to 82^ per bushel of
corn fed to hogs. What was the per cent of increase in
the value of the corn by using it as feed ?

3. On another farm the figures of Ex. 2 were 52^ a
bushel for corn when sold and 75 <^ a bushel when used
as feed. What was the per cent of increase in the value
of the com by using it as feed ?

4. A bushel of corn contains about |- lb. of nitrogen, ^ lb.
of phosphorus, and ^ lb. of potassium. How many pounds
of each of these, substances are contained in the crop from
a 20-acre field yielding 62 bu. of corn to the acre ?

5. If it costs tll.80 per acre to grow a crop of corn
and haul it to the market, where it is sold at 60 <f per
bushel, what is the net profit from a 25-acre field if the
rent on the land is |9 per acre and the land yields an
average of 58 bu. of corn per acre ?

6. A cow is fed daily 6.51b. of corn worth 60^ per
bushel of 56 lb. What will be the cost of the corn fed
to the cow in 2 mo. of 30 da. each ?

7. If a bushel of corn, whea fed to hogs, will produce
9.5 lb. of pork, how much will it cost to produce 1 lb. of
pork when com is 620 a bushel?

AEITHMETIC OF THE FAEM

Exercise 46. Farm Income

1. A farmer owns two farms of the same size and value.
One he runs himself and the other he lets on shares. The
following table shows the itemized income of each farm :

Dairy products .
Wool ....
Eggs and poultry
Domestic animals
Crops ....

Find the total receipts on each farm.

2. In Ex. 1 find the per cent of increase of each item
of income on the home farm over the corresponding item
on the rented farm.

3. The following table shows the itemized expenses of
the two farms mentioned in Ex. 1 :

Home Farm

Rented Farm

$348.60

$103.75

43.75

28.80

316.80

123.50

637.50

321.60

1072.80

785.30

Home Farm

Rented Farm

Labor

. $212.60

$140.30

Fertilizers

. 92.30

12.00

Feed

. 63.50

58.60

Maintaining buildings . .

. 31.75

63.40

Maintaining equipment .

. 15.50

42.50

Taxes and miscellaneous .

. 82.50

80.75

Find the total expenses of each farm, the difference in
each pair of items, the difference in the total expenses,
and the net gain of each farm.

4. In Ex^ 3 find what per cent more was paid for labor
on the home farm than on the rented farm.

5. In Ex. 3 find what per cent more was paid for
maintaining buildings on the rented farm than on the
home farm.

SOILS AND CROPS 63

Exercise 47. Soils and Crops

1. The soil t)f an acre of rich land in the Corn Belt,
plowed to the depth of 6|^ in., is estimated to weigh
2,000,000 lb. and to contain 8000 lb. of nitrogen, 2000 lb.
of phosphorus, 35,000 lb. of potassium, and 15 T. of cal-
cium carbonate (Hmestone). Express each of these weights
as per cent of the total weight.

The figures given in the problems on this page are, as usual in
such cases, only approximations, because soil and crops vary greatly
in different places. The figures are, however, always based upon
scientific results as obtained in agricultural experiment stations.

2. The plant food liberated from the soil during an
average season is 2 % of the nitrogen, 1 % of the phos-
phorus, and ;| % of the potassium contained in the surface
stratum of 6|^ in. mentioned in Ex. 1. Find the number
of pounds of each of these elements liberated at these
rates from a 100-acre field in 1 yr.

3. The grain in a 100-busliel crop of corn takes from
the soil 100 lb. of nitrogen, 17 lb. of phosphorus, and 19 lb.
of potassium, and the stalks take 48 lb., 6 lb., and 52 lb.
respectively. Express each of the first three as per cent of
the total weight of the corn, allowing 56 lb. to the bushel.
Allowing 60 lb. of stalks to produce 1 bu. of shelled corn,
express each of the last three as per cent of the total
weight of the stalks.

4. Given that 1 T. of clover hay contains 40 lb. of nitro-
gen, 5 lb. of phosphorus, and 30 lb. of potassium, express
each as per cent of the total weight of the hay.

5. If 50 bu. of wheat, weighing 60 lb. per bushel, con-
tains 12 lb. of phosphorus, 13 lb. of potassium, 4 lb. of
magnesium, 1 lb. of calcium, and 0.1 lb. of sulphur, each
is what per cent of the weight of the wheat ?

64 ARITHMETIC OF THE FARM

Exercise 48. Good Roads

1. A teamster had to haul 7^T. of barbed wire a dis-
tance of 13 mi. over poor roads from the railway. He found
that he could haul only 1500 lb. to a load and that it took
him a full day to make the round trip. How long did it
take him to haul the 7^ tons of wire, and how much did
it cost at 1 5 per day for man and team ?

2. In Ex. 1 what was the cost of hauling 1 ton 1 mi.,
or the cost of 1 ton-mile, as it is ordinarily called ?

3. In Ex. 1, after a new state road had been constructed,
the teamster found that with the same team he could haul
2^ T. to the load and make the round trip in 1 da. How
much did it then cost to haul a ton of wire ? What was
the cost per ton-mile ?

4. Comparing the results in Exs. 2 and 3, what per
cent less was the cost per ton-mile in Ex. 3, owing to
good roads ? What was the per cent of time saved in
hauling 7^ T.?

5. A farmer lives 10 mi. from the railway. The road
was formerly so bad that with a two-horse team he could
haul only 30 bu. of wheat to the load, and it took 1 da. to
make the round trip. At $5 a day for man and team, how
much did it cost per bushel to haul the wheat ?

6. The roads were recently improved. The farmer can
now haul 75 bu. to the load. Allowing ^ da. for the trip,
find the cost per bushel of hauling the wheat now.

7. Comparing Exs. 5 and 6, how much more money due
to improved roads does the farmer get per bushel ?

8. Taking 60 lb. as the weight of a bushel of wheat
and comparing Exs. 5 and 6, what has been the per cent
of reduction in cost of cartage per ton-mile ?

REVIEW 65

Exercise 49. Review Drill

1. Express 0.4% as a decimal fraction.

2. Express 2.8 as per cent.

3. Express 2^ qt. as per cent of 1 gal. ; of 5 gal.

4. Express an inch as per cent of 1 yd. ; of 3 yd.

5. Express 8J%, 16|%, 33^%, 62^%, 66|%, 83^%,
and 87-|% as common fractions in lowest terms.

Find the value of each of the following :

6. 50% of 860. 11. 0.7% of 275 lb.

7. 25% of 621. 12. 0.08% of 56,000.

8. 37^% of $9600. * 13. 225% of 4800.

9. 133J% of 119.56. 14. 2.25% of 4800.
10. 187^% of 119.28. 15. 300% of 156.7.

Multiply as indicated:

16. 275 X 3468. 17. 39.6 x 31.78. 18. 0.432 x 687.2.

Divide to two decimal places :

19. 1^3.7. 20. 0.27-^0.5. 21. 68.01 h- 0.7.

22. $30 is what per cent of 1360? of $3600? of $36,000?

23. 8in. is what percent of 12 in.? of 12 ft.? of 12 yd.?

24. A pint is what per cent of 1 qt. ? of 1 gal. ? of 6 gal. ?

25. 56 ft. is 8% of what distance ? 4% of what distance?

26. 7ft. 6 in. is 10% of what distance? 8% of what
distance ? 33J% of what distance ?

27. 751b. is 25% less than what weight?

28. A bill of goods amounting to $725 is allowed a
discount of 15%. Find the net amount.

66 ARITHMETIC OF THE FARM

Exercise 50. Problems without Numbers

1. If you know the expenditures and the receipts for
a year on a farm, how do you find the net profit or loss ?

2. In Ex. 1 how do you find the average net profit
or loss per acre ?

3. If you know the dimensions of a rectangular field
in rods, how do you find the area in square rods? in acres?

4. If you know the total annual cost of food for the
cows on a farm and the number of cows, how do you find
the average cost of food per cow ?

5. If you know the average profit per cow on a farm
and the number of cows, how do you find the total profit?

6. If you know the average per cent of butter fat in
the milk from a certain herd of cows and the number of
pounds of milk delivered at a creamery, how do you find
the number of pounds of butter fat in this milk ?

7. If a farmer has two cows, and knows the amount
of milk furnished by each in a year and the per cent of
butter fat in the milk of each cow, how does he find which
cow produces the more butter fat?

8. If you know the weight of nitrogen in a ton of
clover hay, how do you find the per cent of the nitrogen ?

9. If you know the per cent of nitrogen in a ton of
clover hay, how do you find the weight of the nitrogen?

10. If you know the per cent of nitrogen in clover hay,
how do you find the amount of clover hay necessary to
produce a given amount of nitrogen ?

11. If you know the length of a rectangular field, how
do you find the width that must be fenced off so as to
inclose just an acre of land ?

ARITHMETIC OF INDUSTRY 67

IV. ARITHMETIC OF INDUSTRY

Nature of the Work. We have thus far considered three
important topics, the home, the store, and the farm, and
have seen that each has its special kinds of problems. We
shall now consider the problems of more general industries,
such as manufacturing establishments of various kinds.

Teachers should draw problem material from local industries
whenever possible. No textbook can do more than give a general
survey of the subject, and it must always endeavor to present prob-
lems which are not too technical to be generally understood. In
certain localities, however, where some single industry is prominent,
more technical problems can safely be given because the students
will naturally be familiar with the terms used.

In order to solve the problems which arise in the shop,
it is necessary to review the operations with numbers. We
shall therefore briefly review the operations with fractions.

Exercise 51. Fractions

1. Some plaster ^ in. thick is coated with a finer plaster
^in. thick. How thick is the plaster then?

2. A plate of brass -^ in. thick is laid on a plate of iron
^ in. thick. What is the total thickness ?

3. An iron rod of diameter |^ in. is covered with a brass
plating ^ in. thick. What is now the diameter of the rod ?

4. A table 4 ft. 4| in. long and 3 ft. 2^ in. wide has a
molding ^ in. thick put around it. What is then the
perimeter of the table ?

5. A boy is making a dog kenneL One of the pieces
of wood is 4 ft. 3 in. long, and from this he saws a piece
2 ft. 4^ in. long. How long is the piece which remains ?

68 AEITHMETIC OF INDUSTRY

6. A girl has a piece of ribbon 2^ yd. long. She uses
14^ in. for a hat. How much ribbon has she left?

7. From a board 14 ft. long a man saws off a piece
2 ft. 3|- in. long and another piece 2 ft. 7| in. long. How
long is the remaining part ?

8. To a piece of cloth 4|- yd. long another piece 8 J in.
long is sewed, and then 18;^ in. is cut off and used for
making a bag. How many yards of cloth are left?

9. A plate of glass 18|^ in. by 23|- in. was set in a
picture frame that covered it ^ in. from each edge. What
are the dimensions of the glass not covered by the frame?

Perform the following additions :

10.

i+l+l-

14.

i + l + 1^-

18.

1I + I + 3A-

11.

l+l+l-

15.

i + i + A-

19.

2| + i + lA-

12.

i+i+l-

16.

i+l+A-

20.

3| + 2|+xV

13.

l+l+l-

17.

l + f+tV-

21.

H + H+a-

22. In making a dress ruffle 4-^ in. wide when finished
enough cloth must be allowed to turn in |- in. on one side
and -^ in. on the other. Find the width of cloth needed.

23. A gas fitter, in running a pipe into a schoolroom,
has four pieces of pipe respectively 8 ft. 9^ in., 6 ft. 2| in.,
8 ft. 3;| in., and 9 ft. 4 in. long, and finds he has 3 ft. 7 in.
more than he needs. What is the length required?

Perform the following subtractions :

24. 61 -2|. 28. 71 -5|. 32. 8 in. - 2| in.

25. 8J-3|. 29. 41 -2|. 33. 9 1 in. - 6-^ in.

26. 61 -2f. 30. 6-4^. 34. 8 in. - 1 3^ in.

27. 7|-4f. 31. 5-2^^. 35. 9in. -3^in.

EEVIEW OF FRACTIONS 69

Division by a Fraction. We know that there are 3 thirds
in 1, or that 1 h- ^ = 3. From this we see that 6 -f- ^ = 6 x 3.

That is, 6h-| = 3x6,

and Q^^ = half as much = — - — = 9.

That is, 6 -V- f = I of 6.

Therefore, to divide hy a common fraction, multiply hy the
reciprocal of that fraction.

That is, 15 -^ - = 5 X ;^ = 25.

5 ^

Similarly, -._ = |x_ = -.

We shall now review both multiplication and division of fractions.

Exercise 53. Fractions

1. A tin cup is found to hold ^ pt. When it is ^ full
the cup contains what part of a pint ?

2. If a jar has a capacity of ^ qt., what part of a quart
will it contain when it is ^ full? when it is ^ full?
What part full must it be to hold 1 pt.?

Perform the following operations:

ixf.

4x|.

15.

ixi.

21.

ixA-

*-!•

10.

i-l-

16.

i-l-

22.

i-*-

l-i-

11.

l-i.

17.

l-i-

23.

A-i-

ix|.

12.

ixT6-

18.

|xf.

24.

2xA.

I-I-

13.

i-A-

19.

i-l-

25.

2-^A-

l-l-

14.

A-4.

20.

i-i-

26.

A-2.

70 ARITHMETIC OF INDUSTRY

Exercise 53. The Pay Roll

1. The following is a week's pay roll of a manufacturer :

PAY ROLL

For the week ending Feb. 7, 1920

No.

Name

No. OF Hours per Day

Total
Time

Wages

PER

Hour

Total
Wages

:f.

M. S. Rowe

S21

T. D. Bell

42J

S. M. Lee

■^i

-r}

Louis Shea

C. P. Grove

L. S. Cram

27i

Totals

Fill each space marked with an asterisk (*).

Make out pay rolls for a week and insert names, when the
men's numbers, the hours per day^ and ifhe wages per hour
are as follows :

2. No. 1: 7^, 7|, 7^, 8, 8, 3^, 60(^; No. 2: 8, 8, 8, 8,
8, 4, 55(^; No. 3: 8, 8, 7, 7, 8, 4, 62(f.

3. No. 1: 7, 8, 8, 8, 7|, 4, 62^(f ; No. 2: 8, 7, 8, 7, 6,
4, 60(f; No. 3: 8, 7^, 8, 8, 8, 4, 48^(f ; No. 4: 8, 8, 8, 8,
7^, 4, 65(^; No. 5: 8, 7, 8, 8, 8, 4, 72(^.

4. No. 1: 8, 7, 6, 8, 8, 4, 72J(f ; No. 2: 8, 7, 6, 8|, 8^,
4, 64(^; No. 3 : 8, 6, 6, 8, 8, 4, 62(^; No. 4: 7, 8, 7|,
6t^, 8, 4, 57(f ; No. 5: 8, 6, 7, 7|-, 6^, 4, 48.^.

5. No. 1: 7, 7, 8, 8, 8, 4, 62|(^; No. 2: 8, 8, 7|, 7^, 7|,
4, 65^; No 3: 8, 8, 8, 7f, 7|, 4, 58^; No. 4: 7^, 7^, 7^,
8, 7|, 3|, 62(^; No. 5: 8, 8, 8, 8, 8, 4, 63|(^.

THE PAY ROLL

6. Fill each space marked with an asterisk in the fol-
lowing pay roll, allowing double pay for all overtime:

PAY ROLL For the week ending Jan . 10 , 1920

No.

Name

No. OF Hours per Day

Total
Time

WAGEiS
PER

Hour

Total
Wages

1
2
3

R. S. Jones
M. L. King
J. M. Mead

'1/
1/

'is/

52J

40
50
62J

*
*
*

*
*

Totals

Before assigning Ex. 6 the teacher should explain that from one
and a half to two times the regular hourly wage is usually paid for
overtime, and that the check (y^) in the above pay roll means full
time for the day. In this pay roll the full time is 8 hr. except on
Saturday, when it is 4 hr. The symbol {/ means 8 hr. + 2 hr.
overtime. A dash ( — ) indicates absence. Part time, like 6^ hr.,
is indicated as above on Friday for King. Since the allowance
for overtime is double that for regular work, Jones's time is
8 + 8 + 8 + 8 + 8 + 4 regular time and 2 x (2 + ij + 1^) overtime,
or 54 hr. in all. Explain the significance of the parentheses.

Make out 'pay rolls (inserting names) when the men's
numbers, the hours per day, and the wages per hour are as
follows, 8 hr. constituting a day''s work except on Saturday,
when it is 4 hr., and double pay being given for overtime :

7. No. 1: 8, 9, 8, 9, 8, 5, 67^(f ; No. 2: ^, 9, 9^, 8, 8,
4, 65<^; No. 3: 8, 8, 8, 10, 8, 6, 62(f.; No. 4: 8, 9, 9, 9, 8,
4, 60(^; No. 5: 8^, ^, 9, 8, 8, 5, 60 (^,.

8. No. 1: 8, lO; 8, 10, 8, 6, 60(^; No. 2: 9^, 8, 8, 9, 8,
4, 62(^; No. 3: 10, 10, 10, 10, 8, 5, 62^(^; No. 4: 8, 8, 8,
8, 10, 9, 62-|(^; No. 5: 8, 8, 8, 9, 8^, q\, 65(j^.

72 ARITHMETIC OF INDUSTRY

Exercise 54. The Iron Industry

1. What is the weight of a steel girder that is 18^ 10''
long and weighs 46^ lb. to the running foot ?

2. What is the cost of 16' 4" of iron rod, 4|- lb. to the
foot, at l^<f, a pound?

3. The wooden pattern from which an iron casting is
made weighs 6|% as much as the iron. The pattern
weighs 67^ lb. How much does the casting weigh?

4. If steel rails weighing 180 lb. to the yard are used
between New York and Chicago, a distance of 980 mi.,
how many tons of rails will be required for a double-
track road between these cities?

5. An iron tire expands 1^^% on being heated for
shrinking on a wheel. A certain wooden wheel needs a
tire 16' 8" in circumference. How much longer will the
tire be when thus heated?

6. If 3.5% of metal is lost in casting, how much metal
must be melted to make a casting to weigh 77.2 lb.?

Since 100% - 3.5% = 96.5%, 77.2 is 96.5% of the weight.

7. In a certain blast furnace the casting machine turns
out 40 pigs of iron per minute, averaging in weight 110 lb.
each. If this machine runs for 312 da., 16 hr. a day, how
many long tons (2240 lb.) of pig iron will it turn out ?

8. Some years ago the average daily wages paid to em-
ployees in a certain mill was |1.90, and the men worked
11 hr. a day, 6 da. in the week. At present the average
daily wage is $2.60 and the men work 8 hr. a day,
5 da. in the week and 5 hr. on Saturday. Considering the
wages per hour, what has been the per cent of increase ?
Considering the hours per dollar of wages, what has been
the per cent of decrease?

MISCELLANEOUS PROBLEMS 73

Exercise 55. Miscellaneous Problems

1. Sea Island cotton is usually shipped in bags of 150 lb.,
while Alabama cotton is shipped in bales of 500 lb. How
many bags of Sea Island cotton at 28 (^ a pound will equal
in value 42 bales of Alabama cotton at 11<^ a pound?

2. The average number of wage earners engaged in the
manufacture of cotton goods during a recent year was
379,366. The value of the materials was |431,602,540
and the value of the finished products was |6 76,569,335.
What per cent of value was added by manufacture ?

3. The United States produced 10,102,102 bales of
cotton in the year 1900 and 11,068,173 bales in the year
1915. What was the per cent of increase ?

4. The total value of the cotton raised in the United
States in a recent year was $627,861,000, and the number
of bales was 11,191,820. Find the average value of a bale.

5. During a recent year the United States produced
11,000,000 bales of cotton and used only 7,000,000 bales.
The amount used in this countrv was what per cent of
the total amount produced ?

6. During a recent year 86,840 sq. mi. of cotton terri-
tory was invaded by the boll weevil. The total area
infected at the end of the year was 409,014 sq. mi.
What was the per cent of increase for the year?

7. In the days when cotton cloth was woven by hand
an experienced weaver could turn out 45 yd. of cloth per
week. At present a workman operating six power looms
in a cotton mill will produce 1500 yd. per week. How
long would it have taken the worker to do this with the
hand loom ? What is the per cent of increase in output
per man with the power looms ?

74 ARITHMETIC OF INDUSTRY

8. Before the invention of the cotton gin a laborer could
separate in a day only 1^ lb. of lint from the seed. At
the present some gins turn out 10 bales of 500 lb. each
per day. Such a machine does the work of how many men ?

9. In making a silk lamp shade the following materials
were used : 1^ yd. silk @ 11.10, 2^ yd. silk fringe @ |1.84,
£yd. silk net @ $2.20, 1 frame costing 60 (^. The labor and
overhead charges amounted to $3.25. The shade was
marked $14.50 but was sold at a discount of 10%. Find
the gain per cent over the total cost.

Overhead charges, also called overhead or burden, means the general
expense of doing business.

10. By repairing an automobile engine a mechanic in-
creased its horse power 7|-% and reduced the amount of
gasoline necessary to run it 3%. Before the repairs were
made the engine developed 40 H.P. and used 2 gaL of
gasoline on a 20-mile trip. How much gasoline per horse
power did it use on a 50-mile trip after it was repaired ?

The letters H.P. are commonly used for horse power.

11. It is desired to construct an engine that will generate
102.5 H.P. net, that is, actually available for use. It is
found that 18 % of the horse power generated is lost. This
being the case, what horse power must be generated ?

12. How many fleeces of wool averaging 6£ lb. each
must be used to make a bale of wool weighing 250 lb.,
and how many pounds will be left over?

13. If a wool sorter can sort 80 lb. of wool in a day,
how many days will it take him to sort a shipment of
24 bales of 250 lb. each ?

14. After scouring (cleaning) a shipment of 12,000 lb.
of wool it weighed only 5240 lb. What per cent of the
original weight was lost by scouring?

REVIEW 75

Exercise 56. Review Drill

1. Add 147.832, 29.68, 575, 0.387.

2. From 1000 subtract the sum of 148.9 and 9.368.

3. Multiply 78.4 by 9.86.

4. Divide 0.8 by 0.13 to three decimal places.

5. How much is a profit of 14|^% on a sale of cotton
goods which cost $1275.50 ?

6. Find the commission at ^% on goods sold for $15,000.

7. Goods listed at $1450 are sold at a discount of 6%,
10 %. Find the selling price.

8. How much is the profit, at 12|-% on cost plus over-
head charges, on the sale of goods which cost $1645.75,
the overhead charges being $268.50 ?

9. How much is the loss on a house which cost $4500,
including all charges, and which was sold at a loss of 6 % ?

10. Some goods which cost $750, including all charges,
were sold for $675. What was the per cent of loss ?

Write the results of the following :

11. 84in. = (?)ft. 16. 2sq.ft. = (?)sq. in.

12. 84oz.= (?)lb. 17. 2cu.ft.= (?)cu. in.

13. 84 ft. = (?) yd. 18. 2 sq. yd. = (?) sq. ft.

14. 84pt. = (?)qt. 19. 72in. = (?)ft.

15. 84qt.= (?)pt. 20. 72in. = (?)yd.

21. Make out an imaginary personal account of six items
on each side, and balance the account.

22. Write a bill of $45 for rent due ; a check to pay
half this rent ; a receipt for this part of the rent ; a note
for the rest of the rent.

JMl

76 ARITHMETIC OF INDUSTRY

Exercise 57. Problems without Numbers

1. If you know the wages per hour and the number of
hours worked each day by each man, without overtime,
how do you find the total wages due all the men in a
shop in a week ?

2. If you know the regular wages due a man per hour,
the wages for overtime, and the number of hours he works
each day in a week, some of these being overtime, how do
you find his total wages for a week ?

3. If you know the weight of a steel girder per running
foot and the length of the girder, how do you find the
total weight?

4. If you know the weight of a girder and its length,
how do you find its weight per running foot ?

5. If you know the weight of each of several bales of
cotton and the price paid per pound, how do you find
the gain or loss to a firm that buys this cotton on a basis
of 500 lb. to the bale ?

6. If you know the shipping rate per hundredweight
from where you live to Liverpool, and know the weight of
a shipment, how do you find the total charge for freight ?

7. If a farmer has wheat to ship to Chicago and knows
the freight rate per bushel or carload and the number of
bushels or carloads, how does he find the freight charges ?

8. If a farmer knows the charges for shipping a certain
number of bushels of corn to Chicago, how does he find
the freight rate per bushel ?

9. If you know the average weight of a fleece of wool
and the number of fleeces a sheep grower has, how do
you find the number of bales of wool weighing 250 lb.
each and the amount, if any, that will be left over?

ARITHMETIC OF THE BANK 77

V. ARITHMETIC OF THE BANK

Saving. A boy who puts l(f each day into a toy bank
will have enough in six months to buy a catcher's mitt or
three baseball bats. A girl who saves 10(f a day will have
enough in a month or two to buy a good pair of shoes.
A man who saves |1 a day will have enough in a few
years to buy a building lot in some place where he may
care to live. We often get the things that we need for
comfortable living or the things that give us legitimate
pleasure, by saving a little at a time.

The following suggestion to teachers will be found helpful :
Begin with a brief discussion of the need and value of saving
money. There are many of us who never learn how to save money
wisely. Many of us prefer to gratify our immediate desires rather
than to provide for the future. Why is this a bad plan? On the
other hand, there are others of us who, in order to save for the future,
deny ourselves the things which it would be real economy to buy.
There is always the temptation to live extravagantly. Extravagance
includes not only living beyond our means but also spending money
foolishly. Every boy and every girl should begin early in life to form
the habit of saving, no matter if it be but a few dollars a year.

Exercise 58. Saving

1. A man wishes to buy an automobile that costs $780.'
If he saves $2 every week day, in how many weeks will
he save enough to buy the car?

2. A boy wishes to buy a camera that costs $6.60. If
he saves 10 (f every week day, in how many weeks will he
save enough to buy the camera?

3. A girl wishes to buy a purse that costs 60 (^. If she
saves 5 0 every week day, in how many weeks will she save
enough to buy the purse ?

78 ARITHMETIC OF THE BANK

4. A man who has been smoking six cigars a day,
which he buys at the rate of three for a quarter, decides
to give up smoking and save the money. How much will
this saving amount to in 5 yr. ?

In all such problems the year is to be considered as 365 da. (or
313 da., excluding Sundays), although in 5 yr. there will probably be
one leap year and there may be two leap years, and although a year
need not have exactly 52 Sundays.

5. A boy earns some money by selling papers. He finds
that he can easily save 150 a day, excluding Sundays. If
he does this for 5 yr., how much will he save in all ?

6. A woman in a city has a telephone for which she is
charged 5 ^ for each call. She finds that she can economize
by reducing the number of her telephone calls on an
average four a day, including Sundays. If she does this
for 3 yr., how much will she save ?

Make out accounts, inserting dates, items of receipts and
payments, and the balances, given the following :

7. On hand, |4.20. Receipts: 20(^, 40(^, 25<f, fl, 30(^,
65f Payments: 25(^, 30(^, 10(^, 75(^.

"V 8. On hand, $4.30. Receipts: 10(f, 15^, 60(^, 55(^.
/Tayments: 300, 420, 750, 50, 300, 600, 200, 50.

9. On hand, $4.63. Receipts: 100, 120, 320, 120, |1.10.
Payments: 250, 350, 50, 240, $1.50, 250, 500.

10. On hand, $5.10. Receipts: $1.25, 350, 750, 420,
680, 700, $1.22. Payments: $1.50, 700, 50, 100.

11. On hand, $1.30. Receipts: $1, 400, 50, 150, 100,
'"250, 120, 160, 100. Payments: 100, 200, 50, 50, 220, 30.

12. On hand, $3.20. Receipts: 700, 600, 100, 30, 150,
100, 250, 50, 50, 300, 520, 30 0. Payments: $1.50, 500,
900, 700, 180, 120, 300, 450, 250.

SAVINGS BANKS 79

Bank Account Essential. One thing that is essential at
some time to everyone who hopes to succeed is a bank
account. A reliable person may "open an account," as it
is called, as soon as he begins to save even small amounts.

People who are saving money usually keep it in a bank
until they have enough for investing permanently. Certain
kinds of banks, such as savings banks and trust companies,
not only guarantee to take care of all money left with them
by depositors but also pay a certain per cent of interest.
National banks also generally allow interest on what are
called inactive accounts ; that is, deposits that remain undis-
turbed for some time.

Many schools have found it interesting and profitable to organize
school banks, electing the officers and carrying on a regular banking
business, either with small amounts of real money placed on deposit
by students and transferred by the teacher to some bank or trust
company, or with imitation money. Such exercises should not, how-
ever, interfere with the work in computing.

Savings Bank. To deposit money a person goes to a
bank, says that he wishes to open an account, and leaves
his money with the officer in charge. The officer gives
him a book in which is written the amount deposited,
and the depositor writes his name in a book or on a card,
for identification. When he wishes to draw out money,
he takes his book to the bank, signs a receipt or a check
for the amount he desires, and receives the money, the
amount being entered in his book.

Students should be told of the advantages of opening even small
accounts at a savings bank. A boy who deposits |1 a week for 10 yr.
in a bank paying 2 % every 6 mo. and adding it to the account, will
have $631.54 in 10 yr., and a man who puts in $10 a week will have
about ten times as much, or $6317.16.

The class should be told about trust companies, which take charge
of funds, manage estates, and pay interest on deposits.

80 ARITHMETIC OF THE BANK

Exercise 59. Saving

1. How much will 25^ saved each working day, 310 such
days to the year, amount to in 10 yr.?

2. If a boy, beginning at the age of 14 yr., saves 25^
a day for 310 da. a year and deposits it in a bank, how
much has he when he is 21 yr. old, not counting interest ?

Interest and withdrawals are not to be considered in such cases.

3. If a man saves $3.25 a week out of his wages and
continues to do this 52 wk. in a year for 12 yr., how much
money will he save ?

4. If a father gives his daughter on each birthday until
and including the day she is 25 yr. old as many dollars as
she is years old, depositing it for her in a savings bank,
how much has she when she is 25 yr. old ?

5. A merchant saves $750 the first year he is in business.
The second year he saves one third more than in the first
year. The third year his savings are only 85% as much as
the second year. The fourth year they increase 30% over
the third year. How much does he save in the four years?

6. A man works on a salary of |18 a week for 52 wk.
in a year. His expenses are |3.75 a week for house rent,
60% as much for clothing, 300% as much for food as for,
clothing, and 20% as much for other necessary expenses as
for food. How much of his salary can he deposit each year
in the savings bank ?

7. A clerk had a salary of $12 a week two years ago and
a commission of 2 % on goods he sold. That year he worked
50 wk. and sold $4800 worth of goods. Last year his salary
was increased 25%, his rate of commission remaining the
same. He worked 48 wk. and sold goods to the amount
of $56^0. How much was his income increased ?

INTEREST 81

Interest. If Mr. James has a house and lot worth
15000, and rents it to Mr. Jacobs at $42 a month, his
income from the rent is 12 x |42, or |504 a year. This
is a little more than 10% of the value of the property,
but out of it Mr. James has to pay for various expenses,
such as insurance, repairs, and taxes.

If Mr. James has $5000 and lends it to Mr. Jacobs at
the rate of 6 % a year, his income from this transaction is
6% of $5000, or $300 a year.

Money paid for the use of money is called interest. In the
above illustration about the lending of money $300 is the
interest for 1 yr., 6% is the rate of interest, and $5000 is
the principal.

Schools do not require, as formerly, the learning of many defini-
tions. What is necessary is that the student should use intelligently
such terms as interest, rate, and principal.

Men often have to borrow money to carry on their busi-
ness. For example, a merchant may wish to buy a lot of
holiday goods, feeling sure that he can sell them at a
profit. In this case it is good business for him to borrow
the money, say in November, taking advantage of all cash
discounts allowed, and then to repay the money in January
after the goods are sold. If, for example, he needs $1000
for 2 mo. and can borrow it from a bank at the rate of
6% a year, he will have to pay 3^ of 6% of $1000, or 1%
of $1000, or $10, a sum which he can easily afford to pay
for the use of the money.

To find the interest on any sum of money for part of a
year first find the interest for 1 yr. and then find it for the
given part of a year.

Formal rules for such work need not be memorized. An example
or two may profitably be worked on the blackboard before studying
the next page. ,

82 ARITHMETIC OF THE BANK

Exercise 60. Interest

Examples 1 to 27, oral

Find the interest on the following amounts for 1 yr. at the
given rates:

1. $1000, 5%. 6. $150, 4%. 11. $400, 6%.

2. $1000, 6%. 7. $250, \<^o. 12. $400, 5^%.

3. $1000,4^%. 8. $500, 5%. 13. $1000, 3^%.

4. $2000, 5%. 9. $600, 6%. 14. $1000, ^\%

5. $3000, 6%. 10. $800, 3^%. 15. $1000, 4f %.

^^/^(i ^Ae interest on the following amounts for 6 mo. at
the given rates :

16. $100, 6%. 18. $1000, 5%. 20. $3000, 4%.

17. $300, 6%. 19. $2000, 5%. 21. $1000, ^%

Find the interest on the following amounts for 1 yr. 6 mo.
at the given rates :

22. $1000, 4%. 24. $2000, 6%. 26. $5000, 4%.

23. $1000, 5%. 25. $5000, 5%. 27. $2500, 4%.

28. A man having $17,250 invested in business has
found that his net profits average 16% a year on the
investment. He is offered $25,000 for the business, and
he could invest the money at 4|^%. If he sells out and
retires, what is his annual loss in income ?

29. In April a coal dealer borrowed $66,420 at 5%.
With this he purchased his summer's supply of coal at
$5.40 a ton, his overhead charges being 30 (^ a ton. He
sold the coal at $6.68 a ton, the buyers paying for the
unloading and delivery, and he paid his debt in October
after keeping the money 6 mo. How much did he gain ?

INTEEEST FOR MONTHS AND DAYS 83

Interest for Months and Days. Suppose that a man bor-
rows from a bank $400 on Sept. 10, 1919, at 6%. What
will the interest amount to Aug. 7, 1920 ?

yr. mo. da.

1920 8 7 = second date

1919 9 10 = first date

10 27 = difference in time

Taking, as is usual, 30 da. to the month, the difference
in time is 327 da. We therefore have

109 i

— of 6% of $400 - _^^^_^^^ - ^-y- - ^2i-^^-

Hence the interest due August 7, 1920, is |21.80.

Banks usually lend money for a definite number of days or else
require payment to be made on demand. In either case they com-
pute the interest for the days that the borrower has the money and
not for months and days. To enable them to compute the interest
easily they have interest tables. Private individuals, however, occa-
sionally have to compute interest for months and days, and in that
case they may proceed as in the above problem.

It is a waste of time for the student to find the interest on very
small or very large sums of money, for very short or very long
periods, or at more than legal rates. A few such examples may be
given, however, for practice in computation. In general, interest is
now reckoned on such a sum as $750 rather than $749.75, and for
periods not exceeding 90 da. rather than one involving years, months,
and days. Teachers should advise the students that if the interest
is for more than 1 yr. they should first find it for the given number
of years, and then, by the above method, for parts of a year. Such
cases are, however, rapidly becoming obsolete. Banking facilities
make it rare to find interest periods for years, months, and days.

84 AEITHMETIC OF THE BANK

Interest for 30, 60, and 90 Days. In borrowing money
at a bank the time for which the money is borrowed is
usually 30 da., 60 da., or 90 da., except when repayment
is to be made on demand. Since 6 % is the most common
rate, it is convenient to be able to work mentally the
common types of interest examples.

How much interest must you pay if you borrow $500
from a bank at 6% for 60 da.? for 30 da.? for 90 da.?

Since 60 da. = ^\% yr. = ^ jr., the interest on |500 for 60 da. is
^ of 6% of $500, or 1% of $500, or $5.

For 30 da. the interest is ^ of 1% of $500, or $2.50.
For 90 da. the interest is f of 1% of $500, or $7.50.

From this work state a simple rule for finding the interest
at 6% for 60 da.? for 30 da.? for 90 da.?

Exercise 61. Interest

JEJxaTnples 1 to 15, oral
Find the interest at 6^o on the following amounts :

1. $400, for 60 da. 6. $840, for 60 da.

2. $650, for 30 da. 7. $350, for 90 da.

3. $725, for 60 da. 8. $450, for 30 da.

4. $875, for 60 da. 9. $950, for 30 da.

5. $900, for 90 da. 10. $860, for 30 da.

11. Find the interest on $600 for 60 da. at 5%.
The interest at 6% is $6, and so at 5% it is f of $6.

12. Find the interest on $3000 for 30 da. at b%.

13. Find the mterest on $240 for 90 da. at 5%.

14. Find the interest on $600 for 60 da. at 4%.

15. Find the interest on $1200 for 30 da. at 3%.

INTEREST 85

16. Find the interest on $400 for 2 yr. 10 mo. 27 da.

at 6%.

The interest for 2 yr. is 2 x 6% of $400, or $48, and for 10 mo. 27 da.
is $21.80, as found on page 83. Hence the total interest is $69.80.

As already stated, such examples are becoming more rare. A few
are given on this page, chiefly as exercises in computation.

Find the interest on the following :

17. $1250 for 2 mo. 17 da. at 5%.

18. $1500 for 7 mo. 23 da. at 6% ; at 5|-%.

19. $2400 for 8 mo. 11 da. at 5% ; at 6% ; at ^%.

20. $575 for 2 yr. 9 mo. 15 da. at 5% ; at 5|%.

21. $850 for 3 yr. 10 mo. 6 da. at 51% ; at 6%.

22. $925 for 4 yr. 10 mo. 6 da. at 6% ; at 5%.

23. A dealer bought 24 sets of furniture on Nov. 1, at
$50 a set, promising to pay for them later, with interest at
6%. He paid the bill on the following Jan. 16. What
was the amount of principal and interest?

24. A man borrowed $750 on Mar. 10, at 6%, and $1600
on Apr. 10, at 5%. He paid the entire debt on July 10 of
the same year. How much did he pay in all ?

25. A man borrowed $750 on May 1, at 5%, and $1800
on July 5, at 4J%. He paid both debts with interest on
Dec. 16 of the same year. How much did he pay in all ?

26. What is the total amount of principal and interest
on $950 borrowed Mar. 10, at 6%, and $1600 borrowed
May 15, at 5%, the payment in both cases being made
on Oct. 20 of the same year?

27. A man borrowed $750 on May 9, at 5%, and $625
on June 15, at 6%, each loan to run for 60 da. When
was each due, and how much was the total interest?

ARITHMETIC OF THE BANK

12000.= first principal
.02

$40.= int. first 6 mo.
2000.

Interest at Savings Banks. Savings banks usually pay
interest every six months or every three months. This
interest is added to the principal, and the total amount
then draws interest.

Compound Interest. When interest as it becomes due is
added to the principal and the total amount then draws
interest, the investor is said to receive compound interest
on his money.

Compound interest
is not commonly used,
but if one collects in-
terest when due and at
once reinvests it, he
practically has the ad-
vantage of compound
interest. The method
of finding compound
interest is substantially
the same as that used
in simple interest.

For example, how
much is the amount of
$2000 in 2 yr., deposi-
ted in a savings bank
that pays 4% annu-
ally, the interest being
compounded semiannu-
ally? How much is the
compound interest ?

12040.= amt. after 6 mo.
.02

= int. second 6 mo.

$40.80
2040.

$2080.80

$41.62

2080.80

$2122.42

$42.45
2122.42
$2164.87
2000.
$164.87 = int. for 2 yr.

amt. after 1 yr.

int. third 6 mo.

amt. after li yr.

int. fourth 6 mo.

amt. after 2 yr.

Simple interest for the
same time is $160, or $4.87 less than the compound interest.

Here the compound interest has been found exactly, but savings
banks pay interest only on the dollars and not on the cents.

SAVINGS BANKS

Savings Bank Account. The following is a specimen
account at a savings bank which pays interest at the rate
of 4^^ a year, the interest being payable semiannually,
on January 1 and July 1, " on the smallest balance on
deposit at any time during the previous interest period:

Date

Deposits

Interest

Payments

Balance

1922

July

600

July

675

Sept.

120

555

Dec.

615

Dec.

550

1918

Jan.

561

May

200

761

July

772

The smallest balance durrug the first interest period is
$550.50. Interest is computed on the dollars only, the
cents being neglected. At 4% per year the interest for
6 mo. on |550 is 2% of |550, or $11. In the second period
the smallest balance is $561.50, and therefore the interest
is 2% of $561, or $11.22.

Some banks allow interest from the first of each month;
others from the first of each quarter; others, as above,
from the first of each half year. The interest is computed
on the smallest balance on hand between this day and the
next interest day, and is usually added every half year,
although it is sometimes added every quarter.

Students should ascertain the local custom as to savings banks.

88 ARITHMETIC OF THE BANK

Exercise 62. Compound Interest

Find the amount of principal and interest at simple interest^
and also at interest compounded in a savings hank annually :

1. $3000, 2yr., 5%. 6. $2750, 4 yr., ^%.

2. $3000, 4yr., 6%. 7. $825.50, 5 yr., 3^%.

3. $2000, 4yr., 4%. 8. $2000, 6 yr., 4%.

4. $3250, 4yr., 3%. 9. $625.50, 4 yr., 4|%.

5. $3750, 4yr., 3%. 10. $875.50, 3 yr., 4^%.

Find the amount of principal and interest^ the interest being
compounded in a savings hank semiannually:

11. $400, 3yr., 4%. 16. $600, 2 yr., 4%.

12. $600, 2yr., 4%. 17. $2000, 2 yr., 4|%.

13. $850, 2yr., 6%. 18. $2000, 3 yr., 4%.

14. $900, 3yr., 3%. 19. $3000, 2 yr., 3^%.

15. $900, 3yr., 4%. 20. $3000, 4 yr., 4%.

21. If you deposited $140 in a savings bank on July 17,
1919, and $35 on Feb. 9, 1920, and if you have made no
withdrawals, to how much interest are you entitled July 1,
1920? In this bank on July 1 and Jan. 1 interest on each
deposit at 4% per year is credited from the day of deposit
if on the first day of a month, and otherwise from the
first day of the following month.

22. If a man deposits $1500 in a savings bank on Jan. 1,
$215 on Feb. 1, $140 on May 7, $270 on Sept. 11, and $243
on Dec. 3, and makes no withdrawals, how much will he
have to his credit on the following Jan. 1 ? In this bank on
July 1 and Jan. 1 interest on each deposit at 4% per year is
credited from the day of deposit if on the first day of a month,
and otherwise from the first day of the following month.

POSTAL SAVINGS BANKS 89

Postal Savings Bank. The United States government
conducts a savings bank in connection with the post office.
Although all savings banks are carefully regulated and in-
spected by the state governments, there are many persons
who are willing to take the smaller rate of income which
the postal savings bank pays, because of the fact that our
government guarantees the payment of their money.

Any person of the age of 10 yr. or over may deposit
money in amounts of not less than $1, but no fractions
of a dollar are accepted for deposit. No one can deposit
more than |100 in any one calendar month or have a
balance at any time of more than $500, exclusive of
accumulated interest. Deposits may be made at the larger
post offices, and a depositor receives a postal savings certifi-
cate for the amount of each deposit. Interest is paid by the
government at the rate of 2% for each full year that the
money remains on deposit, beginning on the first day of
the month next following the one in which the deposit is
made. Interest is not paid for any fraction of a year.
A person may exchange his deposits in sums of $20 or
multiples of $20 for bonds bearing interest at 2J%.

Exercise 63. Postal Savings Bank

All work oral
Find the interest for 1 yr. on the following deposits :
1. $30. 2. $40. 3. $75. 4. $300. 5. $500.

Find the interest for 2 yr. on the following deports :
6. $50. 7. $60. 8. $100. 9. $200. 10. $500.

Find the interest for 1 yr. on a ^|% hond of:

11. $80. 12. $240. 13. $360. 14. $480. 15. $500.

ARITHMETIC OF THE BANK

Bank of Deposit. When a man has money enough ahead
to pay his bills by checks, he will find it convenient to
have an account with a bank such as merchants commonly
use,, sometimes called a hanh of deposit.

Such banks do not pay interest on small accounts, the
deposit being a matter of convenience and safety. If a
man wishes to open an account he sometimes has to give
references, for banks do not wish to do business with
unreliable persons. A man's credit in business is always
a valuable asset.

In some sections of the country banks receive deposits
under two classes of accounts, savings accounts and check-
ing accounts. In the former case they act as savings
banks ; in the latter, as banks of deposit. For the purposes
of the school it
is not necessary
to consider this
difference further.
Students should,
however, investi-
gate the local cus-
tom in the matter.

DEPOSITED FOR CREDIT OF

IN THE

SECOND NAIIONAL BANK

OF THE CITY OF NEW YORK
191

RTTT.Q

DOLLARS

CENTS

rniN

CHECK ON B'K

Deposit Slip. A

man, when he de-
posits" money or
checks in a bank,
fills out a deposit
slip similar to the
one here shown.

Sometimes the depositor enters the name of the bank on which
each check is drawn ; sometimes the receiving teller at the bank does
this by writing the bank's number; and sometimes it is not entered
at all. These are technicalities that do not concern the school.

BANKS OF DEPOSIT 91

Exercise 64. Deposit Slips

Write or fill out deposit slips for the following deposits,
inserting the name of the depositor and of the bank :

1. Bills, $375; silver, $60; check on Garfield Bank,
$87.50; check on Miners Bank, $627.75.

2. Bills, $423; gold, $175; silver, $235.75; check on
Corn Exchange Bank, $736.90.

3. Bills, $135; check on Second National Bank of
New York, $425 ; check on Chase National Bank, $75.40.

4. Bills, $1726; gold, $100; silver, $200; check on
Merchants Bank, $245.50 ; check on Union Bank, $275.40.

5. Bills, $1275 ; checks on Harriman National -Bank,
$146.50, $200 ; checks on Jefferson Bank, $325, $86.50.

6. Gold, $100 ; checks on First National Bank, $175,

$240, $32.80 ; checks on Sherman Bank, $37.42, $61.85.

7. Bills, $2475; silver, $275.50; check on Case Bank,
$43.50 ; check on Miners National Bank, $250.

8. Bills, $345; silver, $350.75; gold, $480; check on
Merchants National Bank, $455 ; check on Farmers Trust
Co., $262.50 ; check on City Bank, $1000.

9. A man deposited $475.75 in cash to-day, a check for
50% of a debt of $675 due him, and a check in payment
for 45 yd. of velvet at $2.25 a yard less 33-^% discount.
Make out a deposit shp.

10. A merchant received cash for 8 doz. forks @ $14.75,
5|- doz. teaspoons @ $13, a watch costing $40.50, and
4 clocks @ $7.75. He also received a check on the Lincoln
Trust Co. for 3 doz. dessert spoons @ $17.75 and 4^ doz.
nutcrackers @ $9. He deposited all tliis in a bank. Make
out a deposit slip.

ARITHMETIC OF THE BANK

Check. A check book containing checks and stubs, substan-
tially as follows, although often varying in certain details,
is given the depositor when he opens an account.

No. c?^^

New York, 7;ra.y i'/, IQi'^

^econb J^ational pank

'Pay to the order of

I jlcyyv&§y.

..^.^...^^^.^.^...-^^.........^Dollars

jla/yyv&O' fC. Roa&v^

Check

The person to whom a check is
payable is called the payee. In the
above example Myron P. Jones is
the payee. A check may be made
payable to " Self," in which case
the drawer alone can collect it; or
to the order of the payee, as in the
above check, in which case the payee
must indorse it, that is, he must
write his name across the back ; or
to the payee or " bearer," or to

'' Cash," in which cases anyone can collect it.

The indorsement made by Mr. Jones would appear on the

back in the form here shown :

No.

8^6

D^iemOAf^/./^^O 1

m. S>. fcyyv^

For

^'hO-&£AA.&Qy

Amt. fS^.'-^ \

Stub

The teacher should explain to the mtfuyyv i. fcyyie^

class the nature of checks, the different
ways of filling them out, and the end
on which they should be indorsed. The teacher should explain the
advantages of the various methods of making the checks payable
and the students should write or fill out various styles of checks.

CHECKS 93

Exercise 65. Bank Deposits

1. If your deposits in a bank have been |58.65, $43,
*25, 180, 195, 125.75, |12.50, and $9.50, and you have
drawn checks for |8.25, $16.30, $15.75, $16.48, and $25,
what is then your balance at the bank ?

2. A man earning $22.50 a week deposits $15 every
Saturday, and each Monday gives a check for $4.50 for
his board. What will be his balance in 13 wk.?

3. If your deposits in a bank have been $68.45, $92.30,

$47.60, $38.50, $78.75, and $96.70, and you have drawn
checks for $8.55, $23.65, $8.58, $48.75, and $34.60, what
is your balance ?

4. A merchant having $980.75 in the bank deposits
during the next week $185.50, $97.85, $135.50, $86.85,
and $236.80. He gives checks for $89.65, $37.20, $93.60,
$15.20, $248.70, and $39.80. What is now his balance?

5. A man having $825.60 in the bank gives a check
for $128.75. He then deposits checks for $75.80, $126.75,
$234.80, and $42.80. During this time he gives checks
for $125.80 and $24.75. What is now his balance?

6. A merchant having $828.50 in the bank deposits
$567.80, $245.50, $89.65, $482.86, $429.50, and $376.50,
and draws checks for $427.50, $38.95, $67.82, $568.70, and
$122.58. He also pays by check a bill for $125.40 less
10%, another bill for $86 less 4%, and another for $48.75
less 6%. What is now his balance?

7. A merchant having $1026.92 in the bank deposits

$488.75, $928.75, $386.48, $442.80, $196.85, $327.75,
and draws checks for $96.75, $286.75, $342.80, $438.50.
He also gives a check for $230 plus interest for 4 mo. at
5%. What is now his balance ?

94 ARITHMETIC OF THE BANK

Promissory Note. A paper signed by a borrower, agree-
ing to repay a specified sum of money on demand or at a
specified time, is called a promi8sory note, or simply a note.

The sum borrowed is called the principal, or, if a note
is given, the face of the note.

The sum of the principal (or face) and the interest is
called the amount of the note.

A note should state the date, face, rate, person to whom
payable, and time to run (time before it is due to be paid),
and that it has been given for value received by the maker.
The. following is a common form for a time note :

//^.- New York, S'e.Uyiia.vxf 7, 19^/

cfi/^ 'nvo-ntA^ after date, for value received, I promise

to pay to jla/i^ faknoxyyv or order,

^eA)-emt%f-^iA>-& ^ « ^..,_..^-,...v.^...v.v^Dollars,

with interest at 6%

S^xckm^k/ <S^'bajyi(S^[yQ/

The following is a common form for a demand note:

fso!^ New York, Tnaif 2, \92/

On demand, for value received, I promise to pay to

..Ra6-£At R. fan&a^ or order,

with interest at 6%.

PKOMISSORY NOTES 95

Parties to a Note. The person named in a note as the
one to whom it is payable is called the "payee. The person
who signs a note is called the maker.

Indorsing a Note. If the payee sells the note, he must,
when it is payable to himself or order, indorse it.

A note is indorsed by the payee by writing his name
across the back. The indorser must pay the note if the
maker does not.

A note payable to John Johnson or hearer may be sold
without indorsement. Such notes are not common.

If the payee wishes to sell the note without being respon-
sible for the payment in case the maker should fail to pay
it, he may write the words " without recourse " across the
back, and write his name underneath. This means that he
relinquishes all title to it and that the buyer cannot come
back (have recourse) on him. The following are the forms :

Indorsement in Blank Indorsement in Full Limited Indorsement

^afi'yi fak

'noxyyv

raA4

to tk& o-vd&v

jo-k'w jaA'Tuuyyv

Teachers should explain fully the meaning of these several indorse-
ments, and should have the students indorse notes properly.

Rate of Interest. The United States borrows money at
rates of about 3% to 3|-%. Savings banks pay depositors
about 3% or 4%. In cities, on good security, borrowers
usually pay from 4% to 6%.

When a note bears interest, but the rate is not specified,
it bears interest at a certain rate fixed by the law of the
state. In many states this rate is 6%.

In most states, if a note falls due on a Sunday or a legal
holiday, it is payable on the next business day.

96 ARITHMETIC OF THE BANK

Exercise 66. Promissory Notes

1. Compute the amount of the first note on page 94.

2. Write a promissory note, signed by A and payable
to B, for 1 75, due in 1 yr., at 6%. Find the amount.

3. F. H. Ryder borrows |750, at 6%, for 1 yr., from
M. P. Read. He gives a note payable to Mr. Read or
order. Mr. Read sells the note to F. N. Cole. Make out
the note, indorse it, in full, and find the amount.

4. Make out a note like the one referred to in Ex. 3,
but for 1 725. Indorse it in blank and find the amount
due at the end of the year. Write a check for this amount.

5. Make out a note like the one referred to in Ex. 3,
but for $1250. Indorse it without recourse and find the
amount due at the end of the year. Write a check for
this amount.

6. Write a note for |275, bearing interest at 6% and
payable in 6 mo. Insert names and dates, and indorse it
payable to the order of John Ball, with a second indorse-
ment by which Mr. Ball transfers it to James Clay.

Make out and indorse, payable to the order of the buyer,
the following notes, and find the amount due on each :

Maker Payee Buyer Face Rate Time

7. A.N.Cole P. R. Carr J.R.Hall |775 6"% 4 mo.

8. A. R. Doe E.F.Dun M.L.King *650 5% 8 mo.

9. D.E.Bell E.L.Cree A. K. lies $550 6% 3 yr.

10. O.N. Olds B.R.Hall B.S.Hill |350 5% 5 mo.

11. S.M,Roe C.N.King O.M.Coe $225 6% 2yr.

12. A. J. Burr G.F.Ray A. R. Carr $850 5J% 6 mo.

13. A. B. Bain G.F.Dow F.E.True $950 4^% 9 mo.

BANK DISCOUNT 97

Bank Discount. When a man borrows from a bank on a
time note he pays the interest in advance. Interest is not
mentioned in the note, because it has already been paid.

Interest paid in advance on a note is called discount.

Teachers should call the attention of the students to the fact that
the same word is used for bank and commercial (trade) discount,
explaining that the mathematical process is the same in both cases ;
that is, finding some per cent of a number.

Unless otherwise directed, always call 30 da. a month.

Proceeds. The face of a note less the discount is called
the proceeds.

What are the discount and proceeds of a note for |225
for 6 mo. at 5% ?

The discoudt (interest) for 1 yr. is 5% of $225, or $11.25.
The discount for 6 mo. is \ of $11.25, or $5.63.
The proceeds are $225 - $5.63, or $219.37.

Exercise 67. Bank Discount
Find the discounts and the proceeds on the following ■:

1. $300, Imo., 6%. 11. $300, 60 da., 5%.

2. $500, 30 da., 6%. 12. $575, 2 mo., 4J%.

3. $750, 2 mo., 5%. 13. $400, 4 mo., 5^%.

4. $475, 3 mo., 6%. 14. $800, 2 mo., 3|%.

5. $825, 2 mo., 5%. 15. $5000, 63 da., 5%.

6. $500, 90 da., 6%. 16. $3350, 93 da., 6%.

7. $475, 6 mo., 6%. 17. $1250, 10 da., 6%.
8.- $800, 90 da., 6%. 18. $2500, 15 da., 6%.
9. $150, 45 da., 6%. 19. $1500, 20 da., 6%.

10. $600, Imo., 5J.%. 20. $1250, 45 da., 5%.

98 ARITHMETIC OF THE BANK

Find the discounts and the proceeds on the following :

21. $675, 30 da., 6%. 27. $3000, 90 da., 5%.

22. $750, 90 da., 5%. 28. $4500, 90 da., ^%.

23. $850, 30 da., 6%. 29. $3750, 30 da., 4^%.

24. $3500, 60 da., 5%. 30. $136.75, 30 da., 6%.

25. $4250, 60 da., 6%. 31. $275.50, 60 da., 5%.

26. $4500, 90 da., 5%. 32. $42,000, 30 da., 5%.

33. Make out a 60-day note for $450, dated to-day, pay-
able to R. D. Cole's order at some bank. Discount it at 6%.

34. Make out a 30-day note for $350, dated to-day, pay-
able to Frank Lee's order at some bank. Discount it at 5%.

35. Make out a 60-day note for $960, dated to-day, pay-
able to Ray Lang's order at some bank. Discount it at 5^^.

36. Make out a 90-day note for $3000, dated to-day,
payable to L. D. Baldwin's order at some bank of which
you know. Discount it at 6%.

37. A man's bank account shows deposits of $175.50,

$68.50, $50, $300, $40, $75, $100, $125, and $500 ; checks
drawn, $43.75, $125.50, $62, $5, and $125.35. He needs
$4500 to start him in business and wishes to keep about
$500 in the bank. How much money, to the nearest $100,
should he borrow ?

38. If the man in Ex. 37 makes out a note for this amount
for 90 da. at 6%, how much discount must he pay ? What
are the proceeds ? What are the proceeds for 60 da.?

39. A. D. Redmond has to pay a debt of $2000 less 10%.
He has in the bank $587.60, and has $327.50 in cash in
his safe. He wishes to leave about $500 in the bank and
about $100 in his safe. How much, to the nearest $50,
must he borrow? Discount the note for 30 da. at 6%.

"DISCOUNTING NOTES 99

Commercial Paper. If a dealer buys some goods for the
fall trade, but does not wish to pay for them until after
the holidays, he may buy them on credit, giving his note.
The manufacturer may need the money at once, in which
case he will indorse the note and sell it to a bank or to a
note broker for the face less the discount. Such notes are
commonly called commercial paper.

For example, if you give a manufacturer your note for
1500, dated Sept. 1 and due Jan. 1, with interest at 5%,
and he, needing the money, discounts the note at a bank
Sept. 1 at 6%, what are the proceeds?

Face of the note ....... |500.

Interest for 4 mo. at 5% 8.33

Amount due at maturity . . . . . 1508.33

Discount for 4 mo. at 6% . . . . 10.17

Proceeds $498.16

The manufacturer may not need the money Sept. 1,
and so he may put the note away in his safe and let it
lie there drawing interest. But if he needs the money
Sept. 16 he may then decide to discount the note at a
bank. We shall then have

Face of the note ....... $500.

Interest for 4 mo. at 5% . . . . . 8.33

Amount due at maturity $508.33

Discount for 107 da. at 6% . . . . 9.07

Proceeds $499.26

Banks usually compute the discount period in days,
and the discount by tables based on 360 da. to the year.

If the banks themselves need more money, they may rediscount
this paper at the Federal Reserve Bank. The details of the Federal
Reserve Bank need not be considered in the schools.

100 ARITHMETIC OF THE BANK

Six Per Cent Method. The following short method, com-
monly known as the Six Per Cent Method, has been referred
to already (page 84), and is convenient not only in com-^
puting interest but also in discounting notes.

Find the interest on $420 for 5 mo. 10 da. at 6%.

Since 2 mo. = ^ yr., the rate for 2 mo. is -^ of 6%, or 1%.

The interest for 2 mo. is 1% of $420 = $4.20
The interest for 2 mo. more = 4.20

The interest for 1 mo. more is | of $4.20 = 2.10

The interest for 10 da. is ^ of $2.10 =• JO

The interest for 5 mo. 10 da. = $11.20

Therefore the interest at 6% for 60 days is 0.01 of the
principal, for 6 days is 0.001 of the principal, and for other
periods the interest can be found from this interest.

The rule is conveniently stated as follows:

For 30 da. take § of 1% ; for 60 da., 1% ; for 90 da., ij %.

Since bank notes usually run for 30 da., 60 da., or 90 da., since
6% is the most common rate, and since we can tell the discount for
60 da. by simply glancing at the face of the note, we can often find
mentally the discount on bank notes for the usual periods.

Exercise 68. Six Per Cent Method

1. Find the interest on $4250 for 60 da. at 6%, first by
the Six Per Cent Method, then by cancellation, and finally
by the ordinary method of finding the interest for 1 yr. and
then for the fractional part of a year. Write a statement
telling the advantage of the Six Per Cent Method.

2. Using the three methods mentioned in Ex. 1, find
the interest at 6% on $875 for 90 da. ; on $2500 for 30 da.

3. A note for $1275 is discounted for 60 da. at 6%. Find
the discount and the proceeds.

SIX PER CENT METHOD 101

Find the discounts at 6^0 on notes for the following amounts:

4. $3000, for 90 da. 10. |250, for 3 mo. 8 da.

5. 12550, for 90 da. 11. $800, for 3 mo. 15 da.

6. $4575, for 30 da. 12. $750, for 1 mo. 18 da.

7. $3575, for 90 da. 13. $950, for 3 mo. 20 da.

8. $4625, for 90 da. 14. $2175, for 3 mo. 15 da.

9. $8250, for 30 da. 15. $6500, for 1 mo. 18 da.

16. A note for $1500 is discounted for 30 da. at 6%.
Find the discount and the proceeds.

17. A note for $3750 is discounted for 90 da. at 6%.
Find the discount and the proceeds.

18. A note for $1250 is discounted for 90 da. at 5%.
Find the discount and the proceeds.

Find the discount at 6% and deduct ^ of this.

19. A man wishes to borrow about $7500 for 60 da.
The bank offers to lend it to him at 5%. If he makes out
a note for $7600 and discounts this, how much more than
$7500 will he receive from the bank ?

20. A speculator buys some property for $30,000. He
pays $9600 down and borrows the balance for 90 da. at 5%.
How much discount must he pay on the note ?

21. How much greater, if any, is the discount on a note
for $2500 discounted for 60 da. at 6% than on one for
$5000 discounted for 30 da. at 6%?

22. A man needs $9750 to pay for some goods. If he
gives a note for $9800 for 30 da. at 6%, will the proceeds
be more or will they be less than the amount he needs?
How much more or how much less ?

23. A firm gives its note for $12,500, discounting it for
90 da. at 5|^%. How much is the discount?

102

AKITHMETIC OF THE BANK

Exercise 69. Miscellaneous Problems

1. Make out a 60-day note for |950, dated to-day,
payable to M. W. Gross or order at some bank in your
vicinity, sign it X. Y. Z., and discount it at 6%.

2. Fill out the blanks in a table like the following and
compute the discount at 6 % on all the notes mentioned :

Time in
Days

To BE
FOUND

Face of Note

|280

$575

|850

$1200

Discount
Proceeds

3. George Lang sold his farm of 120 A. to Fred Ray at
$95 an acre. Ray paid |8000 in cash and gave a 90-day
note without interest for the balance. If Lang discounted
the note at 6% the day it was made, how much did Lang
actually receive for the farm ?

4. A man deposits $420 in a savings bank on July 1,
$48.50 on July 19, $41.30 on Aug. 9, and $72.90 on Dec. 7.
His withdrawals are $20.50 on July 29, and $51 on Dec. 22.
The next year he deposits $39.80 on Feb. 4 and $126.40
on Apr. 14, withdrawing $38.50 on Feb. 23. The savings
bank pays 1 % every three months, on Jan. 1, Apr. 1, July 1,
and Oct. 1, on the smallest balance in even dollars during
the preceding quarter. Find the man's balance on Oct. 1
following his last deposit.

REVIEW 103

Exercise 70. Review Drill

Write in common numerals :
1. Seven thirty-seconds. 2. Ninety-six thousandths.

Add, and also subtract:

3. 4.

4346.8 9185.48

4878.46

4008.06

3946.8 7369.72

2398.59

869.58

Multiply, and also divide :
7. 259.2 by 2.88. 8. 946.96 by 6.23. 9. 95.19 by 5.01.

Find the sum, difference, product, and both quotients of:
10. I, f. 11. I, |. 12. I, f. 13. ^, 1 14. i, if.

By both quotients of | and § is meant f -^ f and f -^ f •

Find the interest on the following :

15. $275, 2yr., 6%. 16. $275, 2 yr. 8 mo., 6%.

Find the discounts on the following bills:

17. $475, 6%. . 18. $8734.75, 6%, 3%.

19. Find the discount on $725 for 60 da. at 4%.

20. Find the interest on $12,500 for 5 mo. at 4|-%.

21. Find the interest on $675 for 1 yr. 8 da. at 6%.

22. A workman has $1250 in the savings bank Jan. 1,
on which he receives 3^% interest. At the end of 6 mo.
he takes out this money and puts the $1250 with accumu-
lated interest in another bank where he receives 4 % interest.
How much has he to his credit after the money has been
in the second bank for 6 mo. ?

104 ARITHMETIC OF THE BANK

Exercise 71. Problems without Numbers

1. Given the face, rate, and time, how do you ascertain
the interest due on a note ?

2. Which pays the better interest, if the money is left
undisturbed for a given number of years, a promissory note
or a savings-bank deposit at the same rate per year? Why?

3. How do you fill out a deposit slip ? After entering
the items, what operation do you perform ? How do you
make sure that the result is correct?

4. If you know a man's balance in a bank a week ago
and his deposits and checks since, how do you find his
balance now?

5. How do you find the discount on a promissory note ?
How do you find the proceeds ?

6. If a note drawing a certain rate of interest is dis-
counted on the day it is made, at the same rate, are the
proceeds greater than the face, or equal to it, or less ?
Why is this ?

7. How can a manufacturer discount a claim against
a purchaser, the claim not being yet due ? How- is the
discount found?

8. If you know the proceeds and the discount, how do
you find the face of a note ?

9. If you know the face of a note and the proceeds,
how do you find the discount?

10. If you know the face of a note, the proceeds, and
the time, how do you find the rate of discount ?

11. If you have money in a postal savings bank, how
much higher rate of interest will you receive if you ex-
change it for government bonds ?

MATERIAL FOR DAILY DRILL

EXERCISE 1

Taking (a) 8.32, (b) 1M.8, (c) 16£4, (d) O.i
(e) 2.496, or (f) 0.2912, as the teacher directs:

L Add it to 0.732 + 9 + 7.29 + 68.4 +L726 4- 0.85. -a i[^* ^

2. Subtract it from 25.865 + 2L854 + 78.146. - /3^»;^^ ]^/g

3. Multiply it by 125, using a short method.

4. Divide it by 0.13.

5. Find ^ of it; 87|% of it.

EXERCISE 2

TaUng (a) $8.64, (b) $12.96, (c) $17.28, (d) $21.60,
(e) $25.92, or (f) $38.88, as the teacher directs:

1. Add it to $15 + 10.76 + 12.88 + |9.36 + $2.75.

2. Subtract it from |2.63 + |8.13 + 120.75 + $16.87.

3. Multiply it by 66|^, using a short method.

4. Divide it by |1.08.

5. Find 12|% of it; 37|% of it; 62^% of it.

This Material for Daily Drill is so arranged as to give daily
practice in the fundamental operations. By first going through all
the exercises with the number denoted by (a), and then with the
one denoted by (b), and so on, more than a hundred different
exercises will result, or more than one exercise for each school day
of the half year, giving enough for a selection.

,106

106 MATEEIAL FOR DAILY DRILL

EXERCISE 3

Taking (a) $8,96, (b) $13A4, (c) $17-92, (d) $22.40,
(e) $26.88, or (f ) $31.36, as the teacher directs :

1. Add it to 119 + $287.30 + $2.75 + $48.60 + $42.86.

2. Subtract it from $4.63 + $10.14 + $27.82 + $9.86.

3. Multiply it by 750, using a short method.

4. Divide it by $1.12.

5. Find 25% of it; ^% of it; 250% of it.

EXERCISE 4

TaUng (a) $9.28, (b) ^$13.92, (c) $18.56, (d) $23.20,
(e) $27.84, or (f) $32.48, as the teacher directs :

1. Add it to $37.62 + $0.27 + $150 + $3.98 + $48.60.

2. Subtract it from $25.37 + $17.26 + $14.96 + $5.04.

3. Multiply it by 125, using a short method.

4. Divide it by $1.16.

5. Findfof it; 75% of it; f of it; 37^% of it; 3.75% of it.

EXERCISE 5

TaUng (a) $9.92, (b) $14.88, (c) $19.84, (d) $24.80,
(e) $29.76, or (f) $34.72, as the teacher directs:

1. Add it to $3.09 + $17+ $0.75 + $27.68 + $9.32.

2. Subtract it from $2.80 + $15.06 + $19.87+ $10.13.

3. Multiply it by 37^, using a short method.

4. Divide it by $1.24.

5. Divide it by | ; by 2|.

MATERIAL FOR DAILY DRILL 107

EXERCISE 6

TaUng (a) $10.56, (b) $1584, (c) $21.12, (d) $26.40,
(e) $31.68, or (f) $36.96, as the teacher directs :

1. Add it to $0.29 + $28.70 + |15 + |3.28 + |4.96.

2. Subtract it from |140 + 172.36 + 127.64.

3. Multiply it by 33^, using a short method.

4. Divide it by 8 ; by 33 ; by |2.64.

5. Divide it by 16^.

EXERCISE 7

Taking (a) $10.88, (b) $16.32, (c) $21.76, (d) $27.20,
(e) $32.64, or (f ) $38.08, as the teacher directs :

1. Add it to 17.33 + $26 + $0.48 + $7.88 + $2.94.

2. Subtract it from $75 + $37.42 + $12.58.

3. Multiply it by 6.25.

4. Divide it by 8; by 17; by 34; by $2.72; by $1.36.

5. Divide it by 6| ; by 111

EXERCISE 8

Taking (a) $11.20, (b) $16.80, (c) $22.40, (d) $33.60,
(e) $28, or (f) $39.20, as the teacher directs :

1. Add it to 9 times itself, using a short method.

2. Subtract it from 11 times itself.

3. Multiply it by 37^.

4. Divide it by 8 ; by 7 ; by 5 ; by 35 ; by $1.40.

5. Divide it by 8f ; by 4| ; by 2^^.

108 MATERIAL FOR DAILY DRILL

EXERCISE 9

Taking (a) $11.52, (b) $17.28, (c) $23.04, (d) $28.80,
(e) $34.66, or (f ) $40.32, as the teacher directs :

1. Add it to 11.20 + $0.92 + |17 + $3.75 + |28.67.

2. Subtract it from $32.75 + $19.82 + $10.18.

3. Multiply it by 62.5.

4. Divide it by 2 ; by 4 ; by 8 ; by 16 ; by 32 ; by 36.

5. Multiply it by f. Divide it by 2|.

EXERCISE 10

Taking (a) $11.84, (b) $17.76, (c) $23.68, (d) $29.60,
(e) $35.52, or (f) $41.44, as the teacher directs :

1. Add it to $12 + $16.75 4- $0.82 + $2.98 + $48.20.

2. Subtract it from $75 + $37.80 + $42.60 + $17.90.

3. Multiply it by 37.85.

4. Divide it by 2 ; by 4 ; by 8 ; by $0.37.

5. Multiply it by 0.12^; by ■^; by 12^%.

EXERCISE 11

Taking (a) 1216, (b) 182.4, (c) 24.32, (d) 0.4256,
(e) 3.648, or (f) 0.304, as the teacher directs:

1. Add it to 9 + 15.75 + 2| + 5f.

2. Subtract it from 1300. %

3. Multiply it by 0.365.

4. Divide it by 3.04 ; by 0.7 ; by 40 ; by 400 ; by 4000.

5. Divide it by 12f ; by 6^; by ^.

MATERIAL FOR DAILY DRILL 109

EXERCISE 12

Taking (a) 124.8, (b) 18.72, (c) 24.96, (d^) 0.312,
(e) 3.744, or (f) 0.4368, as the teacher directs:

1. Add it to 3.848+148.276 +175 + 48.76 + 9.009.

2. Subtract it from 4.6273+74.896 + 56.215.

3. Multiply it by 12.5, using a short method.

4. Divide it by 2 ; by 4 ; by 8 ; by 13 ; by 3.12 ; by 6|.

5. Find 12J% of it, using a short method.

EXERCISE 13

TaUng (a) 13A4, (b) 20.16, (c) 2.688, (d) 0.4704,
(e) 4.032, or (f) 0.336, as the teacher directs:

1. Add it to 72.8796+182.08+7.087+72.6 + 0.983.

2. Subtract it from 2.786 + 46.93 + 53.17.

3. Multiply it by 342.87.

4. Divide it by 3 ; by 7 ; by 8 ; by 0.042; by 3.36.

5. Of what number is it 75% ? |? 7|%? |% ?

EXERCISE 14

TaUng (a) 1.408, (b) 21.12, (c) 2.816, (d) 0.352,
(e) 4.224, or (f) 0.4928, as the teacher directs:

1. Add it to 482.76894 + 9 + 0.987 + 0.7236 + 483.

2. Subtract it from 0.7+276.93+14.963 + 5.037.

3. Multiply it by f of \.

4. Divide it by 0.8 ; by 4.4 ; by 0.352 ; by 2.2 ; by f .

5. Of what number is it 80% ? 120% ? 1^ ? | ?

110 MATERIAL EOR DAILY DRILL

EXERCISE 15

Taking (a) 15.36, (b) 23,04, (c) 3.072, (d) 0.384,
(e) 4.608, or (f) 0.5376, as the teacher directs:

1. Add it to 0.2702 + 298.742 + 0.7298 + 7017+ 2983.

2. Subtract it from 36.7 -f 92L006 +78.239 + 21.761.

3. Multiply it by 1221

4. Divide it by 9f .

5. Of what number is it 125%? IJ? |? 12^%?

EXERCISE 16

Taking (a) 15.68, (b) 2.352, (c) 313.6, (d) 0.392,
(e) 470.4, , or (f ) 5488, as the teacher directs :

1. Add it to 0.1271+ 2789.762 + 2936 + 7064 + 0.8729.

2. Subtract it from 48.789 + 968.32 + 3429 + 6571.

3. Multiply it by itself.

4. Divide it by 0.0784.

5. What per cent is it of 5 times itself ? of half itself ?

EXERCISE 17

Taking (a) 2.88, (b) 26.4, (c) 124.8, (d) 17.04,
(e) 34.56, or (f) 69.12, as the teacher directs:

1. Add it to 125% of itself.

2. Subtract it from 200% of itself.

3. Multiply it by 0.5% of itself.

4. Divide it by 0.15; by 0.075; by 1.5.

5. Of what number is it 12%? 1.2%? 120%?

PART II. GEOMETRY
I. GEOMETRY OF FORM

First Steps in Geometry. Thousands of years ago, when
people began to study about forms, they were interested in
pictures showing the shapes of objects ; these they used in
decorating their walls, and later in showing the plans of
their houses and their temples and in representing animals
and human beings. As land became valuable they showed
an interest in measuring objects, fields, and building ma-
terial. When they wished to locate places on the earth's
surface and when they began to study the stars, it was
necessary that they should consider position. From very
early times, therefore, the ideas of form, size, and position
have interested humanity.

There are three things which we naturally ask about
an object: What is its shape? How large is it? Where
is it ? It is these three questions that form the bases of
the kind of geometry which we are now about to study.
There are also other questions which we might ask about
the object, such as these : How much is it worth ? What
is its color ? Of what is it made ? None of these questions,
however, has to do with geometry.

The teacher will recognize that demonstrative geometry is not
touched upon directly by the three questions above set forth.
Another question might be asked relating to all three, namely,
How do you know that your statement is true ? It is this question
which leads to the proof of propositions. For the present we are
concerned almost exclusively with intuitional and observational
geometry as related to the questions of shape, size, and position.

Ill

112 GEOMETRY OF FORM

Geometric Figures. You are already familiar with such
common forms as the square, triangle, circle, arc, and cube.
Such forms are generally known as geometric figures.

Angle. Two straight lines drawn from a point form an
angle. The two straight lines are called the sides of the
angle, and the point where they meet is called the vertex.

The three most important angles are the right jangle.,
the acute angle^ which is less than a right angle, and the
obtuse angle., which is greater than a right angle.

Right Angle Acute Angle Obtuse Angle

If necessary, the teacher should explain what we mean when
we say that an angle is greater than or less than another angle.
This is easily done by slowly opening a pair of compasses.

Acute angles and obtuse angles are called oblique angles.

Triangle. A figure bounded by three straight lines is
called a triangle.

Equilateral Isosceles Right Acute Obtuse

The five most important kinds of triangles are the equi-
lateral triangle, having all three sides equal; the isosceles
triangle, having two sides equal ; the right triangle, having
one right angle ; the acute triangle, having three acute
angles; and the obtuse triangle, having one obtuse angle.

The side opposite the right angle in a right triangle is
called the hypotenuse of the right triangle.

The sum of the sides of a triangle is called the perimeter.

ANGLES AND TEIANGLES

113

Exercise 1. Angles and Triangles

Examples 1 to 6, oral

1. Point to three right angles in the room.

2. Point, if possible, to two straight lines on the wall
or on a desk which form an acute angle.

3. Point, if possible, to two straight lines in the school-
room which form an obtuse angle.

4. Which is the greater, an acute angle or an obtuse
angle ?

5. How many right angles all lying flat on the top of
a table will completely fill the space around a point on
the table?

6. If one side of an equilateral triangle is 6 in., what
is the perimeter of the triangle ?

7. Draw a right angle as accurately as you can by the
aid of a ruler.

8. Draw an acute angle and an obtuse angle, writing
the name under each.

9. In this figure name
by capital letters the tri-
angles which seem to you
to be right triangles.

10. In the same figure name by a small letter each of
the acute angles.

11. In the same figure name by a capital letter each of
the obtuse triangles.

12. What kinds of angles are represented in the figure
by the letters o, ^, r, s ? Write the name after each letter.

13. If two straight lines intersect, what can you say as
to any equal angles ?

114 GEOMETRY OF FORM

Quadrilateral. A figure bounded by four straight lines
is called a quadrilateral.

The rectangle, square, parallelogram, and trapezoid, the
four most important kinds of quadrilaterals, are shown below.

OJ E2\

Rectangle Square Parallelogram Trapezoid

A quadrilateral which has all its angles right angles is
called a rectangle.

A rectangle which has its sides all equal is called a
square.

A quadrilateral which has its opposite sides parallel is
called a parallelogram.

A quadrilateral which has one pair of opposite sides
parallel is called a trapezoid.

It is not necessary at this time to give a formal definition of
parallel lines. The students are familiar with the term.

We shall hereafter use the word line to mean straight line unless
we wish to use the word straight with line for purposes of emphasis.

Polygon. A figure bounded by straight lines is called a
polygon. The quadrilaterals shown above are all special
kinds of polygons, and a triangle is also a polygon.

Polygons may have three, four, five, six, or any other
number of sides greater than two.

The side on which a polygon appears to rest is called
the base of the polygon.

The sum of all the sides of a polygon is called the
perimeter of the polygon.

The points in which each pair of adjacent sides intersect
are called the vertices of the polygon.

In the case of a triangle the vertex of the angle opposite
the base is usually called the vertex of the triangle.

DRAWING

116

Congruent Figures. If two figures have exactly the same
shape and size, they are called congruent figures.

Drawing Instruments. The instruments commonly used

in drawing the figures in geometry are the compasses,

the ruler, the protractor, and the right

triangle. The compasses are used

for drawing circles as here

shown and also for laying

off distances on paper.

A protractor of the general type
here shown is convenient for
use by students, and with
its aid angles of any
number of de-
grees can be
drawn.

For work out of doors a surveyor measures angles and
finds levels by means of a transit such as is here shown.

Each student should have a ruler,
a pair of compasses, and a protractor,
since the constructions studied in this
book can be made only by their use.

If necessary such familiar terms
as circle, radius, diameter, arc, and
circumference should be explained
informally. They are more for-
mally stated later.

On pages 117 and 119 and later
in the work some interesting illus-
trations of ancient instruments are
given. Students often make similar
instruments for use in geometry.

116 GEOMETRY OF FORM

Constructing Triangles. We often have to construct
triangles of various shapes and sizes. We shall first con-
sider the following case:

Construct a triangle having its sides respectively equal to
three given lines.

Let Z, m, n be the given lines.

It is required to construct a
triangle with Z, w, n as sides.

Draw a line with the ruler and

on it mark off with the compasses

a line AB equal to L ^

It is more nearly accurate to do this

with the compasses than with a ruler.

With A as center and m as radius draw a circle ; with
B as center and n as radius draw another circle cutting
the first at C. Draw AC and BC.

Then because AB=l, AC=m, and BC=n it follows
that ABC is the required triangle.

Show why it is not necessary to draw the whole circle in either case.
Teachers should informally explain to the students the methods
commonly used in lettering a line, an angle, and a triangle.

Exercise 2. Triangles

1. Construct a triangle with sides 2 in., 3 in., 4 in.

Construct triangles with sides as follows :

2. 3 in., 4 in., 5 in. 5. |- in., £ in., 1 in.

3. 1^ in., 2 in., 21 in. 6. 2^ in., 2| in., 2| in.

4. 11 in., 2|in., 3 in. 7. 3 in., 3 J in., 3^ in.

In schools in which the metric system is taught it is desirable to
use the system in this work. The necessary metric measures often
will be found on protractors such as the one shown on page 115.

ANCIENT INSTRUMENTS

117

118

GEOMETRY OF FOEM

-.C,,

Isosceles Triangle. In the case studied on page 116 we
see that the three sides need not all be equal. If two
sides are equal we have to construct an isosceles triangle.

Construct a triangle having two sides each equal to a given
line and the base equal to another given line.

The base of an isosceles triangle is always taken as the side which
it not equal to one of the other sides.

Let AB be the given base and let I
be the given line.

Then with center A and radius I
draw a circle, and with center B and
radius I draw another circle, or pref-
erably only an arc in each case.

Let the two arcs or the two circles
intersect at the point C.

Then ABC is the triangle required.

Equilateral Triangle. From the preceding case we see
that if the base is equal to each of the other sides, we
shall have an equilateral triangle.

I-

Exercise 3. Isosceles and Equilateral Triangles

1. In making a pattern for the tiles used in the floor
shown below it is necessary to draw an equilateral triangle
of sid© 1 in. Draw such a triangle.

rTTTTTTl

Construct isosceles triangles with bases
1 in. and equal sides as follows :

2; fin. 3. I in. 4. 1^ in. 5. 2 in. 6. 2^ in

Construct equilateral triangles with sides as follows :
7. fin. 8. I in. 9. 1^ in. 10. 1| in.

11. 21 in.

ANCIENT INSTRUMENTS

119

120 GEOMETEY OF FORM

12. Cut three isosceles triangles of different shapes from
paper and fold each through the middle so that one of the
equal sides lies exactly on the other. What inference can
you make as to the equality or inequality of the angles
which are opposite the equal sides? Write the statement
as follows : In an isosceles triangle the angles opposite the
equal sides are equal.

13. Draw three equilateral triangles of different sizes.
With a protractor measure each angle in each of the
triangles. What inference can you make as to the number
of degrees in each angle ? Write the statement, beginning
as follows: The number of degrees in each angle of, etc.

14. From Ex. 13 what inference can you make as to
the number of degrees in the sum of the three angles of
an equilateral triangle ? This is the same as the number
of degrees in how many right angles ?

15. Draw three triangles of various shapes and investi-
gate for each the conclusion drawn in Ex. 14. This is
most easily done by cutting
them from paper and then

cutting off the three angles /\ /\ /^\g/

in each case and fitting them ^
together. Write the statement, beginning as follows: In
any triangle the sum of the three angles is equal to, etc.

16. From the truth discovered in Ex. 15, find the third
angle of a triangle in which two angles are 75° and 45°.

17. In a certain right triangle one acute angle is 30°.
How many degrees are there in the other acute angle ?

In this work the student is led to discover by experiment
various important propositions to be proved later in his work in
geometry. Teachers may occasionally find it advantageous to
develop simple proofs in connection with this intuitional treatment.

PEEPENDICULAES

121

Perpendicular. A line which makes a right angle with
another line is said to be perpendicular to that line.

One of the best practical methods of constructing a
line perpendicular to a given line and passing through a
given point is shown ^
in this illustration.

Place a right tri-
angle ABC so that^C
lies along the given
line. Lay a straight-
edge or ruler along
AC, as in the left-hand figure. Since you wish the per-
pendicular line, or perpendicular, to pass through the point
P, slide the triangle along MN until AB passes through the
point P, as shown in the right-hand figure. Then draw
a line along AB, and it will be perpendicular to the line
XY and will pass through the point P.

Exercise 4. Perpendiculars

1. Draw a line XY and mark a point P about |-in.
below it. Through P construct a line perpendicular to XY,
by the- above method.

2. Through a point P on the line XY construct a line
perpendicular to XY, by the above method.

3. Construct a right triangle in which the two shorter
sides shall be 1^ in. and 2 in.

4. Construct a square having its side 2 in.

5. Draw a picture showing how two carpenter's squares
can be tested by standing them on any flat surface with
two edges coinciding and two other edges extending in
opposite directions.

122

GEOMETRY OF FORM

Other Methods of Constructing Perpendiculars. There are
other convenient methods of constructing perpendiculars.

From a given point on a given straight line construct a
perpendicular to the line.

Let AB be the given line and P be
the given point.

With P as center and with any con-
venient radius draw arcs intersecting _
AB at X and Y.

With X as center and XY as radius draw a circle, and
with Y as center and the same radius draw another circle,
and call one intersection of the circles C. p

With a ruler draw a line from P to C,

From a given point outside a given
straight line construct a perpendicular to XlT
the line.

Let AB be the given line and P be
the given point. How are the points • ' c

X and Y fixed ? Then how is the point C fixed ? Draw
the perpendicular PC.

Exercise 5. Perpendiculars

1. In making a pattern for a tiled floor hke the one
here shown it becomes necessary to draw a
square 1 in. on a side. Construct such a
square, using the first of the above methods.

2. Construct a rectangle as in Ex. 1, using
the second, of the above methods.

3. Given two points on a given line, construct perpen-
diculars to the line from each of them.

ANCIENT INSTKUMENTS

123

124

(^eo:metry of form

-^-

Bisecting a Line. To divide a line into two equal parts
is to bisect it. In constructing the common figures we often
have to bisect a line. We can bisect a line
roughly by measuring it with a ruler,
but for accurate work we have a much
better method.

Bisect a given line.

Let AB be the given line. What is
now required ? With A and B as centers
and with radius greater than ^AB draw ^^*

arcs. The most convenient radius is usually AB itself.

Call the points of intersection X and Y. Draw the
straight line XJ, and call the point where it cuts the
given line M.

Then XY bisects AB at M.

This is much more nearly accurate than it is to measure the line
with a ruler and then take half the length.

Bisecting an Angle. To draw a line from the vertex of
an angle dividing it into two equal angles is to bisect it.

Bisect a given angle.

Let AOB be the given angle.

What is now required ?

With 0 as center and with any con-
venient radius draw an arc cutting OA
at X and OB at Y.

With X and Y respectively as centers and with a radius
greater than half the distance from X to Y draw arcs and
call their point of intersection P. Draw OP.

Then OP is the required bisector.

This is much more nearly accurate than it is to measure the angle
with a protractor and then take half the number of degrees.

SIMPLE CONSTRUCTIONS 125

Exercise 6. Simple Constructions

1. Draw a line 4.5 in. long. Bisect this line with ruler
and compasses. Check the construction by folding the paper
at the point of bisection, making a fine pinhole through one
end of the line to see if it strikes the other end.

2. Construct a triangle having two of its sides 3 in., the
third side being less than 6 in.

3. Construct a triangle having its sides respectively
2 in., 2.5 in., and 3 in.

4. Draw a line 4 in. long, and at a point 1 in. from
either end construct a perpendicular to the line.

5. Is it possible to construct a triangle having its sides
respectively 3 in., 2 in., and 1 in. ? If not, what is there
in the general nature of these lengths which makes such
a triangle impossible ?

6. With a protractor draw an angle of 35°. Bisect
this angle and check the work with the protractor.

To draw the angle of 35° draw a line, mark a point O upon it,
lay the hypotenuse of a triangular protractor on it, sliding it down
slightly so that the center of the circle rests on 0. Lay a ruler on
the protractor from 0 along the line of 35° and mark a point on the
paper. Remove the protractor and draw a line from 0 to the point.

Construct the triangles whose sides are as follows and bisect
all three angles of each triangle :

7. 4 in., 3 in., 41 in. 10. 3J in., 4^ in., 5-| in.

8. 5 in., 7 in., 8 in. 11. 1^ in., 4^ in., 5 in.

9. 6 in., 3 in., 5 in. 12. 3|- in., 3^ in., 3|- in.

13. In Exs. 7-12 what do you observe as to the way
in which the three bisectors meet? Write a statement of
your conclusion, beginning as follows: The bisectors of
the three angles of a triangle, etc.

126 GEOMETKY OF FOKM

Constructing an Angle equal to a Given Angle. In copy-
ing figures we often have to construct an angle equal to
a given angle. This leads to the following construction :

From a given point on a given line construct a line which
shall make with the giveyi line an angle equal to a given angle.

d/^

D \C

.'if

Let F be the given point on the given line PQ and let
angle AOB be the given angle.

What is now required ?

With 0 as center and with any radius draw an arc cut-
ting OA at C and OB at D.

With P as center and with OC as radius draw an arc
cutting PQ at M.

With M as center and with the straight line joining C
and Z> as radius draw an arc cutting the arc just drawn at
iV, and draw PN.

Then the angle MPN is the required angle.

Exercise 7. Simple Constructions

Construct triangles with sides as follows and bisect all
three of the sides of each triangle :

1. 5 in., 6 in., 7 in. 4. 3 in., 3|- in., 4|^ in.

2. 4 in., 4 in., 7 in. 5. 2^ in., 4 in., 4 in.

3. 3| in., 4^ in., 1^ in. 6. 3^ in., 3^ in., 4 in.

Interesting figures may be formed by connecting the points of
bisection and shading in various ways the parts thus formed.

SIMPLE CONSTRUCTIONS 127

7. Construct a triangle ^i^Cwith AB =1 in., AC=1^ in.,
angle A = 30°, and then construct another triangle XYZ
with Xr=lin., XZ^l^in., angle X= 30°. Are the
triangles ABC and XYZ congruent?

8. From Ex. 7 write a complete statement of the
truth inferred, beginning as follows: Two triangles are
congruent if two sides and the included angle of one are
respectively equal to, etc.

9. Construct a triangle ABC in which angle ^ = 30°,
angle J5 = 60°, AB = 1^ in., and then construct another
triangle XYZ in which angle X= 30°, angle r= 60°,
XF=l^in. Are these triangles congruent? What is
the reason ? Write a complete statement of the truth
inferred, as in Ex. 8.

10. Construct a triangle ABC in which AB = 1 in.,
BC=l^m., CA=l^m., and then construct another triangle
XYZ in which Xr=lin., YZ^l^in,, ZX=l^m. Are
these triangles congruent ? Write a complete statement
of the truth inferred, as in Ex. 8.

11. Construct a triangle with angles 30°, 60°, and 90°,
and another triangle with sides twice as long but with
angles the same. Are these triangles congruent ? Are
triangles in general congruent if the angles of one are
respectively equal to the angles of the other ?

12. As in Ex. 11, construct two triangles with angles 45°,
45°, and 90°, one with sides three times as long as the other.

13. Try to construct a triangle with angles 45°, 60°,
and 90°. If you have any difficulty in making the con-
struction, write a statement of the cause.

14. Try to construct a triangle with angles 45°, 45°, and
100°. If you have any difficulty in making the construction,
write a statement of the cause.

128

GEOMETRY OF EORM

Parallel Lines. One of the most common constructions
in making architectural and mechanical drawings is to draw
one line parallel to another line. For
practical purposes one of the best plans
is to place a wooden or celluloid tri-
angle ABC with one side ^C on the
given line, lay a ruler along another
side AB, and then slide the triangle along the ruler to the
position A'B'C (read ^-prime, i?-prime, (7-prime). Then
B'C is parallel to ^a

A triangular protractor like the one shown on page 115 of this
book may be used for the above purpose.

Draftsmen in offices of architects or in machine shops
often use a T-square as here shown. As the part MN slides
along the edge CD of a draw-
ing board, the part OP moves
parallel to its original position.
Drawing EF and sliding the
T-square along, we can easily
draw lines parallel to EF. A
second T-square may slide along
BD if the board is rectangular,
and thus lines can be drawn perpendicular to the line EF
or to any lines parallel to it.

When the lines are very long, this is the best method.

Draftsmen also use a parallel ruler like the one here
shown. They also use a cylindric ruler, rolling it along

the paper as a guide for r-^ q

parallel lines. In gen- T) (J

eral, however, the plan , // //

of sliding a triangle along

a ruler is one of the simplest and at the same time is

accurate. It should be used in the exercises which follow.

ANCIENT INSTRUMENTS

129

130

GEOMETRY OF FORM

Dividing a Line. We often need to divide a line into a
given number of equal parts; that is, to solve this problem:

Divide a given line into any given yiumher of equal parts.

Let AB be the gfiven line, and ^ ,, ,r ^

let it be required to divide AB ^ ^"--V^ \ \ \ \
into five equal parts. "q""A^ \ \

Draw any line from A, as AX. ^s^-X

Mark off on AX with the com-
passes any five equal lengths AP, FQ, QE^ BS, and ST.

Draw TB, and then, by sliding a triangle along a ruler,
draw SO, BJST, QM, and FL parallel to TB.

Then AB is divided into five equal parts, AL, LM, MN,
NO, and OB.

The material for another very simple method may be
easily prepared by the student. Let him rule a large
sheet of paper with
several parallel lines
at equal intervals,
and number these
lines as shown on
the edge. If it is
desired to divide the
line AB into five
equal parts, place the
paper on which AB
is drawn over the
ruled paper so that
the line 0 passes
through A and the
line 5 through B. Lay the ruler along each ruled line in
turn and mark each point of division. In this way the
four required points of division may be accurately found.

SIMPLE CONSTRUCTIONS 131

Exercise 8. Simple Constructions

1. Draw a line 5 in. long and divide it into nine equal
parts by using ruler, triangle, and compasses.

Construct triangles whose sides are as follows^ and con-
struct a ferpendicular to each side at its midpoint:

2. 4^ in., 4J in., 5 in. 4. b^ in., 5J in., 6^ in.

^ 3. 3|in., 3|in., 5| in. 5. 4 in., 51 in., 6^ in.

The teacher should ask for the inference as to the meeting of the
three perpendicular bisectors of the sides of a triangle.

6. With a protractor draw an angle of 45°. With ruler
and compasses bisect this angle. Check the construction
by folding the paper ; by using the protractor.

7. Draw any triangle, bisect the sides, and join the
points of bisection, thus forming another triangle. With
ruler and triangle test to see whether the sides of the
small triangle are parallel to those of the large triangle.

8. Repeat Ex. 7 for a triangle of different shape. What
general law do you infer from these two cases ?

9. Draw a line 4^ in. long and divide it into seven
equal parts.

10. Construct a square 3 in. on a side. If the figure is
correctly drawn, the two diagonals will be equal. Check
by measuring the diagonals with the compasses.

If such words as diagonal are not familiar they should be explained
by the teacher when they are met. It is desirable to avoid formal
definitions at this time, provided the students use the terms properly.

11. Construct two parallel lines and draw a slanting
line cutting these lines so that eight oblique angles are
formed. Name the various pairs of angles in the figure
that appear to be equal.

132

GEOMETRY OF FOEM

Geometric Patterns. By the aid of the constructions
described on pages 116-130 it is possible to construct a
large number of useful and interesting patterns, designs
for decorations, and plans for buildings or gardens.

To secure the best results in this work the pencil
should be sharpened to a fine point and should contain
rather hard lead, and the lines should be drawn very fine.

Exercise 9. Geometric Patterns

1. By the use of compasses and ruler construct the
following figures :

The lines made of short dashes show how to fix the points needed
in drawing a figure, and they should be erased after the figure is
completed unless the teacher directs that they be retained to show
how the construction was made.

2. By the use of compasses and ruler construct the
following figures :

It is apparent from the figures in Exs. 1 and 2 that the radius of
the circle may be used in drawing arcs which shall divide the circle
into six equal parts by simply stepping round it.

GEOMETRIC PATTERNS

133

3. By the use of compasses and ruler construct the
following figures, shading such parts as will make a
pleasing design in each case :

4. By the use of compasses and ruler construct the
following figures, shading such parts as will make a
pleasing design in each case:

5, By the use of compasses and ruler construct the
following figures:

In such figures artistic patterns may be made by coloring jiortions
of the drawings. In this way designs are made for stained-glass
windows, for oilcloths, for colored tiles, and for other decorations.

134

GEOMETRY OF FORM

6. By the use of compasses and ruler construct the
following figures, leaving the dotted construction lines:

As stated on page 133, artistic patterns may be made by coloring
various parts of these drawings. Interesting effects are also j)ro-
duced in black and white, as in the designs in Ex. 9 on page 135.

7. Draw a line 1^ in. long and divide it into eighths of an
inch, using the ruler. Then with the compasses construct
this figure.

It is easily shown, when we come
to the measurement of the circle, that
these two curve lines divide the space
inclosed by the circle into parts that
are exactly equal.

By continuing each semicircle to
make a complete circle another inter-
esting figure is formed. Other similar
designs are easily invented, and stu-
dents should be encouraged to make
such original designs.

8. In planning a Gothic window this drawing is needed.
The arc BC is drawn with A as center
and AB as radius. The small arches
are drawn with A, D, and B as centers
and AI) as radius. The center P is
found by using A and B as centers and
AE as radius. How may the points D,
E, and F be found ? Draw the figure.

GEOMETEIC PATTERNS

135

9. Copy each of the following designs, enlarging each
to twice the size shown on this page :

r^^

M i^^

This example and the following examples on this page may be
omitted by the class at the discretion of the teacher if there is not
enough time for such work in geometric drawing.

10. This figure shows a piece of inlaid work in an
Italian church. Construct a design of this general nature,
changing it to suit your taste.
Construct the figures as accu-
rately as you can.

11. Construct a design for par-
quetry flooring, using only com-
binations of squares.

12. Repeat Ex. 11, using com-
binations of squares and equilat-
eral triangles.

13. Repeat Ex. 11, using combinations of squares, rec-
tangles, and equilateral triangles.

14. Construct a design for a geometric pattern for lino-
leum, using only combinations of circles and squares.

15. Repeat Ex. 14, using only combinations of circles
and equilateral triangles.

16. Repeat Ex. 14, using only combinations of circles,
squares, and equilateral triangles.

136

GEOMETRY OF FORM

Drawing to Scale. The ability to understand drawings,
maps, and other graphic representations depends in part
upon knowing how to draw to scale.

Thus, if your schoolroom is 30 ft. long and 20 ft. wide,
and you make a 'floor plan 3 in. long and 2 in. wide, you
draw the plan to scale, 1 in. representing 10 ft. We indi-
cate this by writing: ''Scale, 1 in. = 10 ft." We may also
write this: "Scale, 1 in. = 120 in.," or "Scale 3^." We
often write 1' for 1 ft. and 1" for 1 in., so that the scale
may also be indicated as l'^=10'.

The following shows a line AB drawn to different scales :

The line ^5. drawn to the scale ^. 1 I

The line AB drawn to the scale ^. I 1

The figures shown below illustrate the drawing of a
rectangle to scale. In this case the lower rectangle is a
drawing of the upper
one to the scale ^, or
1 to 2, or 1" to 2".

Notice that the area of
the lower rectangle is only
^ that of the upper one.
When we draw to the scale ^
we mean that the length of every line is ^ the
length of the corresponding line in the original.
Whatever the shape of the figure, the area will
then be -^ the area of the original figure.

Maps are figures drawn to scale. The scale is usually
stated on the map, as you will see in any geography.
The scale used on a map is often expressed by means of
a line divided to represent miles, and sometimes by such
a statement as that 1 in. = 100 mi.

DRAWING TO SCALE

137

Exercise 10. Drawing to Scale

1. Measure the cover of this book. Draw the outline
to the scale ^.

This means that the four edges are to be drawn to form a rec-
tangle like the front cover, with no decorations.

2. Measure the top of your desk. Draw a plan to
the scale -|.

3. If a line 1 in. long in a drawing represents a dis-
tance of 8 ft., what distance is represented by a line 3J in.
long? by a line 4|^ in. long? by a line 1.5 in. long?

4. If the scale is 1 in. to 1 ft., what distance on a
drawing will represent 6 ft. 3 in. in the object drawn?

5. A drawing of a rectangular floor 20 ft. by 28 ft. is
6 in. by 7 in. What scale was used ?

6. A farmer plotted his farm as here shown, using the
scale of 1 in. to 40 rd. Find the dimensions of each plot.

J 1
1 1

i i

I Oats { Corn

Wheat 1 . |
1 1

Pasture

1 I O 1 Dwelling
1 Wood Lot 1 g ! and
j 1 g 1 Barns

7. A plan of a rectangular school garden is drawn to
the scale of 1 in. to 2 ft. 6 in. The plan is 18 in. long
and 12|-in. wide. What are the dimensions of the garden?

8. The infield of a baseball diamond is 90 ft. square.
Draw a plan to the scale of 1 in. to 20 ft.

138

GEOMETRY OF FORM

9. The field of play of a football field is 300 ft. long
and 160 ft. wide. Lines parallel to the ends of the field
are drawn at mtervals of 5 yd., and the goals, 18 ft. 6 in.
wide, are placed at the middle of the ends of the field.
Draw a plan to the scale of 1 in. to 60 ft. and indicate the
position of the goals ^nd of tli^ 5-yard lines.

10. A double tennis court is 78 ft. long and 36 ft. wide.
Lines are drawn parallel to the longer sides and 4 ft. 6 in.
from them, and the service lines are parallel to the ends
and 18 ft. from them. The net is halfway between the
ends. Draw a plan to any convenient scale.

11. The drawing here shown is the floor plan of a
certain type of barn. Determine the scale to which the

plan is drawn, find the width of the driveway in the barn,
the width of each horse stall, the width of each cattle stall,
and the dimensions of the box stall and the feed room.

DEAWING TO SCALE

139

12. A class in domestic science drew a plan for a model
kitchen in an apartment house, using the scale -^. If the
plan is 3 in. long and 2 in. wide, what are the actual
dimensions of the kitchen ?

13. A drawing was made of a lamp screen 20-| in. high.
The drawing being 2 j^ in. high, what scale was used ?

14. The drawing below is the plan for a concrete
bungalow. Find the scale used in drawing the plan.

VRRDROBE

BEDROOM

I O

TORCH
10-6 X 29 -(f

15. In Ex. 14 find the dimensions of the living room,
dining room, and smaller bedroom including wardrobe.

JMl

140 GEOMETRY OF FOEM

Accurate Proportions. Suppose that you measure a rec-
tangular room and find it to be 20 ft. long and 16 ft. wide,
and suppose that you measure a drawing of the room and
find it to be 10 in. long and 6 in. wide. You would con-
clude that the drawing is not a good one, because the
width should be, as in the room, -^ of the length.

An accurate drawing or picture must maintain the pro-
portions of the object

That is, if the width of the object is ^ of the length, the
width of the object shown in the drawing must be j of
the length in the drawing ; if the width of the object is ^
of the length in one case, it must be -§■ of the length in
the other case ; and so on for other proportions.

It is better at this time to explain informally the meaning of
proportion, as is done above. A more formal explanation of the
subject of proportion is given later in the book when it is needed.

Exercise 11. Accurate Proportions

1. A house is 36 ft. high and the garage is 20 ft. high.
If the house is represented in a drawing as 18 in. high,
how high should the drawing of the garage be ?

In all such cases the objects are supposed to be at approximately
the same distance from the eye, so that the element of perspective
does not enter.

2. A landscape gardener is drawing to scale a plan for a
rectangular flower garden 18 ft. long and 14 ft. wide. In
the drawing the length is represented by 6^ in. By what
should the width be represented ?

3. Draw a right triangle whose sides are 3 in., 4 in., and
5 in. respectively, and draw another right triangle of the
same shape but with the hypotenuse 1^ in. long.

SIMILAEITY OF SHAPE 141

Similarity of Shape. As we have already seen, it is
frequently necessary to draw a figure of the same shape
as another one, but not of the
same size. For example, an
architect or a map drawer may
reduce the original by using
a small scale, but if we are
making a drawing of a small
object seen through a micro-
scope we use a large scale. But whether ^^X '" V^"
the drawing reduces or enlarges the \ /

original, the shape remains the same. I I

Figures which have the same shape / V

are said to be similar. \ I

For example, here are two drawings
of a hand mirror. In outline each drawing is similar to
the mirror itself, and each is also similar to the other.

Figures which are similar to the same figure are similar
to each other.

Two maps of a state are not only similar in outline to the state
itself, but each is similar in outline to the other.

Exercise 12. Similarity of Shape

1. Construct three equilateral triangles whose sides are
respectively 2 in., 3^ in., and 5 in. Are they similar ?

2. Construct three rectangles, the first being 1^ in. by
21 in. ; the second, 3 hi. by 5 in. ; and the third, 2 in. by
2|- in. If they are not all of the same shape,
discuss the exception.

3. Construct a right triangle of the same
shape as this triangle but twice as high,
and another of the same shape but three times as high.

142 GEOMETRY OE FORM

Angles in Similar Figures. Here are two similar right
triangles, ABC and A'B'C, and in each triangle a perpen-
dicular (jo, p' respectively) is drawn from the vertex of
the right angle to the hjrpotenuse.

Are the figures still similar ? Are the sides proportional ?
What can be said as to the corresponding angles?

This brings us to another property of two similar figures,
namely, that the angles of one are equal respectively to the
angles of the other. That is, in similar figures.^ corresponding
lines are in proportion and corresponding angles are equal.

A close approximation to similar figures may be seen in the case
of moving pictures. The large picture shown on the screen is sub-
stantially similar to the small picture on the reel, although there is
some distortion, particularly around the edges.

Exercise 13. Similar Figures

All work oral

State which of the following pairs of figures are necessarily/
similar and state briefly the reasons in each case :

1. Two squares. 4. Two rectangles.

2. Two triangles. 5. Two isosceles triangles.

3. Two circles. 6. Two equilateral triangles.

7. State whether two parallelograms, each side of one
being 3 in. and each side of the other being 4 in., must
always be similar, and give the reason for your answer.

SIMILAR FIGURES

143

Similar Figures in Photographs. If you have ever used
a plate camera you have seen that there is a piece of
ground glass in
the back and that
an object in front
of the camera ap-
pears inverted on
this ground glass.
The reason is clear,
for the ray of light
from the point A of the flower passes through the lens of
the camera and strikes the plate at A. That is,

On a photographie plate the figure is similar in outline to
the original^ hut is inverted.

There is, of course, a slight distortion on account of the refrac-
tion of the rays of light in passing through the lens.

If the camera is 8 in. long and the object is 16 in. away
from the lens 0, an object 5 in. high will appear as 2i in.
high on the plate. That
is, since the length of the , ^^ ,— — -' Is in.

B' -21'

camera, B'O, is half the zim.f"^,^-"

distance of the object from

0, or half of OB^ we see that A'B\ the height of the object

on the plate, is half of AB, that is, half the real height.

Similarly, if the length of the camera is 10 in., and the
height of an object 18 ft. away is 5 ft., we can easily find
the height of the object on the plate as follows :
Reducing all the measurements to inches, we have
18 ft. = 18 X 12 in., and 5 ft. = 5 x 12 in.
10

Then

X 5 X 12 in. = 2| in.

18 X 12
The teacher is advised to solve this on the blackboard.

144 GEOMETRY OF FOEM

Exercise 14. Similar Figures in Photographs

1. A man 5 ft. 8 in. tall stands 16 ft. from a camera
which is 8 in. long. What will be the height of his
photograph? Explain by drawing to scale.

2. The photograph of a man who is 5 ft. 8 in. tall is
6 in. high, and the camera is 10 in. long. How far did the
man stand from the camera ?

3. If a boy's face is 8 ft. from a camera which is 10 in.
long, the height of the photograph of his face is what pro-
portion to the height of his face ? If he places his hand
2 ft. nearer the camera, the length of the photograph of
his hand is what proportion to the length of his hand ?

One of the first things a beginner has to learn in using a camera
is that objects appear distorted unless they are at about the same,
distance from the camera, especially if they are relatively near to it.

4. A tree photographed by a 4-inch camera at a distance
of 10 ft. appears on the photograph as 6 in. high. How
high is the tree ?

We see by this problem that heights and distances can often be
found by photography ; and, in fact, much difficult engineering work
is now done with the aid of photographs.

5. A camera is held directly in front of the middle of
a door and at a distance of 8 ft. from it. The door is
4 ft. by 7 ft. 6 in. and the length of the camera is 8 in.
Fmd the dimensions of the door in the photograph.

6. A 10-inch camera is placed at a certain distance
from a tree which is 50 ft. high, and a boy 5 ft. tall
stands between the tree and the camera. The height of
the boy in the photograph is 1|^ in., and the height of the
tree 8 in. Find the distance of both the boy and the tree
from tlie camera.

THE PA]^TOGHAPH 145

The Pantograph. Probably you have seen an instru-
ment which is extensively used by architects, draftsmen,
designers, and map makers in drawing plane figures simi-
lar to other plane figures. It usually consists of four bars
parallel in pairs, and is known as a pantograph.

In explaining the pantograph it becomes necessary to
speak of the ratio of two lines. By the ratio of 2 ft.
to 5 ft. is meant the quotient 2 ft. -^- 5 ft., or |-, and by
the ratio of ^ in. to 3 J in. is meant ^ in. -r- 3 J in., or ^.
Likewise, by the ratio of a line AB to a line CD is meant
the quotient found by dividing the length of the line AB
by the length of CD. This ratio is written AB/CD, or
ABiCD. If AB is half CD, then AB:CD = ^. This is
read "the ratio of AB to CD is equal to one half."

In the figure the bars are adjustable at B and JS. The
end A is fixed, that is, it remains in the same place while
the pantograph is
being used. A trac-
ing point is placed
at T and a pencil at
P, and BP and FU
are so adjusted as
to form a parallelogram PECB such that any required
ratio ABiAC is equal to CEiCT. Then as the tracer T
traces a given figure, the pencil P draws a similar
figure. If the given figure is to be enlarged instead of
reduced, the pencil and the tracing point are interchanged.

This discussion of the pantograph has little value unless the in-
strument is actually used by the students. A fairly good one can be
made of heavy cardboard or of strips of wood, and school-supply
houses will furnish a school with the instrument at a low cost.

A simple pantograph can be made by fastening a rubber elastic
at one end, sticking a pencil point through the other end, and placing
a pin for a tracer anywhere along the band.

146

GEOMETRY OF FOEM

BEDROOM
u'o"^n-9"

Exercise 15. The Pantograph

1. Draw a plan of your schoolroom to scale and then
enlarge it to twice the size with the aid of a pantograph.

This exercise should be omitted in case the school is not supplied
with a pantograph.

2. Find the map of your
state in a geography and re-
duce it to half the size by
using a pantograph.

3. By using a pantograph
reduce the size of this plan
of a cottage to two thirds its
present size.

This can be done by laying this
page flat on the drawing board
while someone holds the book. It is
better, however, to copy the plan on
paper and use the pantograph with
the drawing. It is desired that the
student should use the pantograph
a few times in connection with
various kinds of work in which it is really used in practical life.

4. By using a pantograph enlarge this sketch for a
child's coat to three- times the given size.

In addition to this, other similar drawings should be
made and then enlarged. Of late the pantograph has
come into extensive use by dressmakers for the purpose
of enlarging designs of this kind.

5. Draw a sketch of any object in the room and reduce
the sketch to one third its size by using a pantograph.

6. Draw a sketch of a tree near the school and enlarge
the sketch to five times its size by using a pantograph.

FIRST PLOORrLflN

SYMMETRY

147

Symmetry. If we place a drop of ink on a piece of
paper and at once fold the paper so as to spread the ink,
we shall often find curious and iuterest-
ing forms frequently resembling flowers,
leaves, or butterflies. These forms are
even more interesting if we use a drop
of black ink and a drop of red ink.
The interest in such figures comes from
the fact that they are symmetric^ that is,
that one side is exactly like the other.

In* this case we say that the figure is symmetric with
respect to an axis, this axis being the crease in the paper
or, more generally, the line
which divides the figure into
two parts that will fit each
other if folded over.

In architecture we often find
symmetry with respect to an
axis. For example, in this
picture of the interior of a
great cathedral we see that
much of the beauty and gran-
deur is due to symmetry.

This case is evidently one
of symmetry with respect to a
plane instead of with respect
to a line. We may also have symmetry with respect to a
center, that is, a figure may turn halfway round a point
and appear exactly as at first. This is seen in a circle, or,
among solids, in a sphere. It is also seen in the Gothic
window shown on page 148. Symmetry of all kinds plays
a very important part in art, not merely in architecture,
painting, and sculpture, but in all kinds of decoration.

148

GEOMETEY OF FORM

Exercise 16. Symmetry

•

1. Has this Gothic window an axis of symmetry? If so,
draw the circle and indicate the axis of symmetry. If it
has more than one axis of symmetry,

draw each axis of symmetry.

2. If the figure has a center of
symmetry, indicate this center in
your rough sketch by the letter 0.

3. Draw an equilateral triangle
and draw all its axes of symmetry.

4. Draw a square and draw all
its axes of symmetry.

5. Draw a plane figure with no axis of symmetry ; one
having only one axis of symmetry ; one having two axes
of symmetry; one having any number of axes of symmetry.

6. Draw the following designs in outline and indicate
by letters all the axes of symmetry in each design:

^^-"^

~-->^

/M^

>— <^

^<<ii^^

^2i--^^

7. Write a list of three windows in churches in your
locality which have axes of symmetry. If you know of
any window which has a center of symmetry, mention it.

The class should be asked to mention other illustrations of axes
of symmetry, as in doors and in linoleum patterns. There should
also be questions concerning planes of symmetry, as in a cube, a
sphere, a chair, animals, and vases. Objects in the schoolroom
offer a good field for inquiry.

CURVES

149

Plane Figures formed by Curves. We have already men-
tioned a number of figures formed by curve lines without
attempting to define them. We shall now mention these
again and shall discuss more fully a few of those which
occur most frequently in drawing, pattern
making, architecture, measuring, and the like.

This figure represents a circle with center
0, radius OA, and diameter BC.

The circle is sometimes thought of as the
space inclosed and sometimes as the curve
line inclosing the space. The length of this
curve is called the circumference^ and sometimes the curve
itself is called by this name.

It is not expected that the above statement will be considered as
a formal definition to be learned. All that is needed at this time is
that the terms shall be used properly. Teachers should recognize
that circle and circumference both have tv^o meanings, as stated above.

Another interesting figure, but one which is used not
nearly so often as the circle, is the ellipse. If we place two
thumb tacks at A and i?, say 3 in.
apart, and fasten to them the ends
of a stringy which is more than
3 in. long, draw the string taut
with a pencil point P, and then
draw the pencil round while keep-
ing the string taut, we shall trace
the ellipse.

It is evident that an ellipse has two axes of symmetry
and one center of symmetry.

The orbits of the planets about the sun are ellipses.

When facilities for drawing permit, the student should draw ellip-
ses of various sizes and shapes and should satisfy himself that two
ellipses are not in general similar.

150

GEOMETRY OF FORM

Solids bounded by Curved Surfaces. We have often
mentioned the sphere, and shall now speak of it and of
other solids bounded in whole or
in part by curved surfaces.

A sphere is a solid bounded by
a surface whose every point is
equidistant from a point within,
called the center.

We also speak of the radius
and diameter of a sphere, just as we speak of the radius
and diameter of a circle.

A cylinder is a solid bounded by two
equal circles and a curved surface as
shown in this figure.

The two circles which form the ends
are called the bases of the cylinder and
the radius and diameter of either base
are called respectively the radius and
diameter of the cylinder.

The line joining the centers of the two bases is called
the axis of the cylinder, and its length
is called the height or altitude of the
cylinder.

A cone is a solid like the one here
shown. It has a circular base and an
axis of symmetry from the center of the
ha^e to the vertex of the cone. The per-
pendicular distance from the vertex to
the base is called the height or altitude
of the cone. The words radius and
diameter are used as with the cylinder.

In this book we shall consider only cylinders and cones in
which the axes are perpendicular to the bases.

SOLIDS BOUNDED BY CURVES 151

Exercise 17. Solids bounded by Curves

1. If you cut off a portion of a sphere, say a wooden
ball, by sawing directly through it, but not necessarily
through the center, what is the shape of the flat section ?
Illustrate by a drawing.

2. In Ex. 1 is the section always a plane of symmetry ?
If not, is it ever a plane of symmetry, and if so, when ?

3. A cylinder is symmetric with respect to what line
or lines ? with respect to what plane or planes ?

4. Could a cylindric piece of wood, say a broom handle,
be so cut that the section would be a circle ? If so, how
should it be cut ? Could it be so cut that the section would
seem to be an ellipse ? Illustrate each answer.

The section last mentioned is really an ellipse, and this is proved
in higher mathematics.

5. Could a cylindric piece of wood be so cut that the
section would be a rectangle ? a trapezoid ? Illustrate.

6. Three cylinders of the same height, 4 hi., have as
diameters 3 in., 4 in., and 5 in. respectively. Can a section
in any one of them be a square ? Illustrate the answer.

7. Is a cone symmetric with respect ta any line ? to
any plane ? Illustrate each answer.

8. How could a cone be cut so as to have the section
a circle ? a triangle ? apparently an ellipse ? Illustrate.

The section last mentioned is really an ellipse, and this is proved
in higher mathematics. This is the reason why an ellipse is called
a conic section. Other conic sections are studied in higher mathe-
matics, and they are important in the study of astronomy, mechanics,
and other sciences.

9. How is the largest triangle obtained by cutting a
cone ? Illustrate the answer.

152

GEOMETRY OF FORM

Exercise 18. Review

1. If the rays of light from any object ABC pass
tlirough a small aperture 0 of an opaque screen and fall
upon another screen parallel to the
object, an inverted image A'B'O c
will be formed as here shown. If b
the object is 5 ft. long and 9 ft. a
from 0, how far from 0 must the
second screen be placed so that the image shall be 6 in.
long ? How far, so that the image shall be 8 in. long ?

2. With the aid of ruler and compasses, construct figures
similar to each of the following figures, but twice as large,
and indicate the axis, axes, or center of symmetry of each.

3. Draw the following figure about half as large again
and make it the basis for a pattern for lino-
leum, using other lines as necessary.

4. Draw a square and cut it into four tri-
angles by means of two diagonals. Describe
the triangles with respect to their being equi-
lateral, isosceles, or right.

5. Draw this figure about half as large
again and make it the basis for a pattern
for a church window, using other lines as
may be necessary for the purpose.

OUTDOOR PROBLEMS 153

Exercise 19. Optional Outdoor Work

1. Collect, if possible, several leaves of each of the
following kinds of tree : oak, elm, maple, pine, and poplar.
Are the leaves of each kind of tree approximately similar
to the other leaves of the same tree ? Has each of the
leaves an axis of symmetry ?

2. Do you know of any building lots or fields that are
triangular ? If so, make rough outline drawings of them.

3. Do any of the public buildings of your community
have cylindric columns ? If so, which buildings ?

4. Do you know of any church spires that are conic in
shape ? If so, which spires ?

5. What is the shape most frequently used m decorat-
ing the interiors of churches in your vicinity ?

6. Can you find an illustration of a Gothic window
in any of the churches in your vicinity ? If so, where ?

7. If there is a standpipe in yonr vicinity, what is its
shape ? What is the shape of most of the smokestacks of
the factories in your community ?

8. Notice the designs m the carpetmg, wall paper, and
linoleum exhibited by various stores. What general pattern
or design is most frequently used ?

9. If convenient, inspect a house that is being built
and compare the floor plan with the plans of the contractor
or architect. What scale was used in drawing^the plan?

10. Name illustrations of each of the following forms
that you have seen in your community : circle, rectangle,
cylinder, cone, sphere, trapezoid, and triangle.

As stated above, this work is purely optional. It is suggestive of
a valuable line of local questions.

154 GEOMETRY OF FOEM

Exercise 20. Problems without Figures

1. How do you construct a triangle, having given the
lengths of the three sides ?

2. How do you construct an isosceles triangle, having
given the base and one of the equal sides ?

3. How do you construct an equilateral triangle, having
given one of the sides ?

4. State two methods of drawing from a given point a
perpendicular to a given line.

5. If you have a line drawn on paper, what is the best ^
way you know to bisect it ?

6. How do you bisect an angle ?

7. How do you construct an angle exactly equal to a
given angle ?

8. How do you draw a line parallel to a given line ?

9. If you have a line drawn on paper and wish to
divide it into five equal parts, how do you proceed ?

10. How do you construct a six-sided figure in a circle,
the sides all being equal ?

11. How do you draw to a given scale the rectangular
outline of the printed part of this page ?

12. How do you draw a plan of the top of your desk to
the scale of a certain number of inches to a foot ?

13. When you know the scale which was used in draw-
ing a map, how do you find the actual distauce between
two cities which are shown on the map?

14. How can you enlarge a drawing by the aid of a
pantograph ?

15. How do you determine whether a figure is symmetric
with respect to an axis ?

GEOMETRY OF SIZE 155

II. GEOMETRY OF SIZE

Size. On page 111 we found that geometry is concerned
witli three questions about any object : What is its shape ?
How large is it ? Where is it ? Thus far we have con-!
sidered the shape of objects; we shall now consider size.

There are several ideas to be considered when we think
and speak of the size of objects, such as length, area, and
volume, all of which we may include in the single expres-
sion geometric measurement. That is, we shall not think
of size as including the measurement of weight, of value,
of hardness, and the like, but only as including the length
(width, height, depth, and distance in general), area
(surface), and volume (capacity) of figures.

Length. It seems very easy to measure accurately the
length of anything, but it is not so easy as it seems.
Linen tape lines stretch, steel tape lines contract in cold
weather, ordinary wooden rulers shrink a little when they
get very dry, and chains wear at the links and thus
become longer with age. But these matters are of less
moment than the carelessness of those who make the
measurements. If the members of your class, each by
himself, should measure the length of the walk in front
of your school, to the nearest sixteenth of an inch, and
not compare results until they had finished, it is likely
that each would have a different result. In fact, all
measurement is simply a close approximation.

One of the best ways of securing a close approximation
to the true result is to make the measurement in two
different ways. Never fail to check a measurement.

Just as we should always check an addition by adding in the
opposite direction, so we should always check a measurement of
length by measuring, if possible, in the opposite direction.

156 GEOMETRY OF SIZE

Outdoor Work. In connection with the study of the
size and position of common forms we shall first suggest
a certain amount of work to be done out of doors.

1. Measure the length of the school grounds.

To do this, drive two stakes at the appropriate corners,
puttmg a cross on top of each stake so as to get two points
between which to measure.

Measure from ^ to ^ by ^^

holding the tape taut ,and '^^^^^^^^^^^

level, drawing perpendic- -^^P^

ulars when necessary by means of a plumb line as shown
in the figure. Check the work by measuring from B back
to A m the same way.

2. Run a straight line along the sidewalk in front of
the school yard.

Of course for a short distance this is easily done by
stretching a string or a measuring tape, but for longer
distances another plan is
necessary. x p q % Y

If we wish to run a line
from X to F, say 300 ft., we drive stakes at these points
and mark a cross on the top of each so as to have exact
points from which to work. Now have one student stand
at X and another at F, each with a^lumb fine marking the
exact points. Then have a third student hold a plumb line
at some point P, the student at X motioning him to move
his plumb line to the right or to the left until it is exactly
in line with X and Y. A stake is then driven at P, and
the student at X moves on to the, point P. The point Q
is then located in the same way. In this manner we stake
out or " range " the Ime from X to F, checking the work
by ranging back from F to X.

OUTDOOE WORK 157

3. Measure the height of a tree by making on the ground
a right triangle congruent to a right triangle which has the
tree as one side.

To do this, sight along an upright piece of cardboard
so as to get the angle from the ground to the top of the
tree. Mark tne angle on the cardboard and then turn the
cardboard down flat so as to have an equal angle on
the ground. A right triangle can now easily be laid out
on the ground so as to be congruent to the one of which
the tree is one side. By measuring a certain side of this
right triangle, the height of the tree can be found.

4. Run a line through a point P parallel to a given
Ime AB for the purpose of laying out one of the two sides
of a tennis court.

Stretch a tape Ime
from P to any point
M on AB^ bisect the
line PM at 0, and
from any point iV on
AB draw NO. Pro-
long NO to (), making OQ equal to NO^ and draw PQ. Sup-
pose that ON is 20 ft. Then sight from N through 0,
and place a stake at Q just 20 ft. from O, Then P and
Q determine a line parallel to AB.

The proof of this fact, like the proofs of many other facts inferred
from certain of the exercises, is part of demonstrative geometry,
which the student will meet later in his course in the high school.

Outdoor work will be given at intervals and always by itself, so
that it can easily l)e omitted. The circumstances vary so much in dif-
ferent parts of the country as to climate, location of the school, and
other conditions, that a textbook can merely suggest work of this
kind which may or may not be done, as the teacher directs. A good
tape line, three plumb lines (lines with a piece of lead at one end),
and a pole about 10 ft. long will serve for an equipment for beginners.

158 GEOMETRY OF SIZE

Exercise 21. Practical Measurements of Length

1. Measure the length of this page to the nearest thirty-
second of an inch, checking the work.

If a ruler is used which is, as usual, divided only to eighths of
an inch, the student will have to use his judgment :.s to the nearest
thirty-second of an inch. The protractor illustrated on page 115 has
an edge on which lengths are given to sixteenths of an inch, and
such a scale may be used if laid along the edge of a ruler.

2. Measure the length of this page to the nearest twen-
tieth of an inch, checking the work.

The protractor illustrated on page 115 has an edge divided into
tenths of an inch. There is advantage in the student becoming-
familiar with the units of the metric system, even before he studies
the subject on page 205, since these units have come into use in
our foreign trade and in all our school laboratories. It is desirable
to know that 10 millimeters (mm.) = 1 centimeter (cm.) = 0.4 in.,
nearly; 10 cm.= 1 decimeter (dm.) ; 10 dm. = 1 meter (m.) = 39.37 in.

3. Measure the length of the longest line of print on
this page, to the nearest sixteenth of an inch, checking
the work.

If the student has a pair of dividers (compasses with sharp points),
this may be used to transfer the length to a ruler. This method
is usually more nearly accurate than to lay the ruler on the page.

4. Measure the length of jour schoolroom to the nearest
eighth of an inch, checking the work.

If the class works in groups of two, and each group checks its
result with care, there may still be some difference. In that case an
average may be taken. This is commonly done in surveying.

5. In the upper part of the opposite picture a man is
sighting across the stream in line with the front part of
his cap. He then turns and sights along the ground, as
shown by the other man standing near him. How does he
find the width of the stream by this method ?

LENGTH

159

160 GEOMETRY OF SIZE

6. Draw a liiie 6 in. long, and on it measure off
2.8 in. from one end and 2.3 in. from the other end. Check
the results by measuring the length of the intermediate
portion. What should that length be ? What do you find
it to be by actual measurement ?

7. Draw a line 3.9 in. long, and on it measure off
successive lengths of 1^-q in. and 1|- in. Check the results
by measuring the length of the remaining portion, as in
Ex. 6. What should that length be ? What do you find
it to be by actual measurement?

8. Draw a line 4^^^^ in. long, and on it measure off a
line 2^^ in. long. Check the results by bisecting the
original line as on page 124 and seeing if the point of
bisection falls at the end of the part measured off.

9. Draw a line 9|- in. long, and on it measure off
successive equal lengths of 31 in. Check the results by
dividing the original line into three equal parts by the
first method given on page 130.

10. Construct a square 1 in. on a side by the methods
already learned, and measure the length of each diagonal.
What do you find it to be by actual measurement ? If the
square is accurately constructed, what must be the length
of each diagonal to the nearest tenth of an inch; that is,
what is the square root of 2 ?

11. Construct a rectangle 3 in. high and 4 in. long.
Check the accuracy of the construction by measuring the
length of each diagonal. What do you find it to be' by
actual measurement ?

Such measurements give some idea of the accuracy required in a
machine shop. With the instruments which the students have, these
measurements are as nearly accurate as can be required, but for
practical purposes much closer approximations are often necessary.

LENGTH

161

162

GEOMETRY OF SIZE

Estimates of Area. There are several methods for esti-
mating areas, of which we shall now consider two. In
general it will be found that neither of these plans is
very practical, although both are used in certain difficult
cases of measurement. The most practical way of finding
areas is introduced on page 164.

1. If an area inclosed by a curve is drawn on squared
paper, the area may be estimated approximately by count-
ing the squares contained within the curve. The squares
on the boundary should be included or excluded according
as more than half their area is included or is not included
within the bounding line, and half the other squares that
are practically half within and half without should be
included. In certain cases the rule should be altered as
the peculiarity of the case requires.

For example, in this figure if each square represents 1 sq. in., the
area inclosed by this curve is approximately 33 sq. in., there being-
approximately 33 squares inclosed.

Paper such as that used in the illustration
is called squared paper or cross-section paper. It
can generally be bought at any stationer's.
When the paper is ruled into squares one tenth
of an inch on each side, there are, of course,
100 such squares in 1 sq. in.

In case it is not easy to purchase squared
paper, ruled in tenths of an inch, it is advis-
able for the student to rule some paper, drawing the lines with
the same care taken in making the other constructions of geometry.

2. The area inclosed by a curve drawn on thick paper or
cardboard may be estimated/oy cutting out the area to be
measured, weighing it, and comparing its weight with that
of a unit of area, such as a square inch cut from the paper.

Since this method considers weight, it is not geometric, and
furthermore it is not very practical in estimating small areas.

^ :^

i \_

A t

-I J-

1 ^

\ 1

^s. i

" — ■"

ESTIMATES OF AEEAS 163

Exercise 22. Estimates of Areas

1. Estimate the area of each of the following figures:

^^% ->5 ^^" ""^^

/ ^ ^ V ^ ^^

L ^ / X 1/ s^

J- V ^ \ -i \

/ t V X 1 i

^ ^ z ^

V U / V-

t /^ J^ i-A t

-/ ^ W -V

V t ^^- aS t

\^ J ^-- \-^ ^ '

^^-^^ ^^-^ \ ^=^ ^^

2. Draw the outline of a leaf on squared paper and

estimate the area.

Paper often comes ruled in millimeters, in which case the areas
can be found with greater accuracy, to square millimeters.

3. On a piece of squared paper draw a rectangle 2 in.
long and 1.7 in. wide, and divide it into two triangles by
drawing either diagonal. Estimate the area of each triangle,
state whether the areas are equal, and check the work by
finding the area of the rectangle and showing that this is
equal to the sum of the areas of the triangles.

4. On a piece of squared paper draw a parallelogram
1.9 in. long and 1 in. high. Draw either diagonal, estimate
the area of each triangle thus formed, and proceed further,
as in Ex. 3.

5. On a piece of squared paper draw a trapezoid 1 in.
high, with lower base 2 in. and upper base 1.2 in. Esti-
mate the area of the trapezoid by the method of Ex. 3.

This method of approximation for estimating areas is sufficient
for many purposes, but the methods of geometry, some of which
we shall now study, are greatly superior to this method.

164

GEOMETRY OF SIZE

1 square inch
1 sq. in.

lin.

Unit of Area. We have seen how we may estimate an
area to a fair degree of accuracy. Whetlier we estimate
or actually measure, we commonly ex-
press the area by means of some unit
square, such as the square inch.

A square inch is not the same as 1 in.
square ; 1 sq. in. is the area of a space that
is 1 in. square. A circle may have this area.

There are also other units of area,
such as the acre (160 sq. rd.).

Area of a Rectangle. A school building has a rectangular
entrance hall 10 ft. long and 4 ft. wide, the floor being made
of marble squares 1 ft. on a
side. What is the easiest
way of finding the area of
the floor ?

There are 10 squares in
each row and there are 4
rows of squares. Since there are 4 x 10 squares, we have:

Area = 4 x 10 sq. ft. = 40 sq. ft.

The area of a rectangle is equal to the 'product of the base
and height.

This means that 10 x 4 = 40, the numher of square feet.

We often express this statement by 2i formula^ using A
for area, h for base, and h for height, thus:

A = h xh,
or, briefly, A = hh.

The absence of a sign between letters in a formula indicates
multiplication.

In this book rooms, boxes, fields, and the like are to be considered
as rectangular unless the contrary is stated.

AREA OF A RECTANGLE 166

Exercise 23. Area of a Rectangle

Using the formula given on page 164, find the areas of the
following rectangles :

1. 181ft. by 26 ft. 5. 36.3 ft. by 142.5 ft.

2. 27 yd. by 421yd. 6. 12.8ft. by 17.3ft.

3. 13^ rd. by 28 rd. 7. 42.3 yd. by 46.8 yd.

4. 71 in. by 9| in. 8. 37^ in. by 62 in.

9. Find the floor area of a room that is 28 ft. 6 in. long
and 32 ft. wide.

10. Find the area of a sheet of paper 31 in. square.
Verify the result by drawing the figure and ruling it
off into ^-inch squares.

11. A garden, 38 ft. by 56 ft., contains a 3-foot walk
laid inside the garden along the four sides. The mid-points
of the long sides are joined by a 2-foot path. Find the
area left for cultivation, and draw a plan to scale.

12. On the floor of a room 32 ft. long and 24 ft. wide
a border 2 ft. wide is to be painted. Find the cost of
painting the border at 300 per square yard.

13. At 16 (f per square foot find the cost of cementing
a walk 6 ft. wide round the outside of a garden that
measures 56 ft. by 82 ft.

14. A student, being asked to measure a rectangle, under-
stated the length 2% and overstated the width 3%. Fmd
the per cent of error in the area computed.

15. Draw a plan of the floor of the basement of a house
to scale as follows : Start at A, go north 14 ft. to B, west
8 ft. to C, north 5 ft. to D, west 8 ft. to E, south 19 ft.
to F, and east to A. Find the cost of cementing the floor
at 15(^ per square foot.

166 GEOMETRY OF SIZE

16. Construct a rectangle and then construct a similar
rectangle the area of which is three times that of the first.

17. The screens A, B, and C are 1 ft, 2 ft., and 3 ft.
respectively from an electric light L. If screen A should be
removed, the quantity of light which fell

on it would fall on B. If screens A and /-j/f-

B should be removed, the same quantity A '/S". "i ^

of light would fall on screen C. How A/t^J^^
would the intensity of light compare for rf^
a given area on each of the screens ?

18. How many paving blocks each 4 in. by 4 in. by
10 in., placed on their sides, will be required to pave a
street 1800 ft. long and 34 ft. 8 in. wide ?

19. Printers usually cut business cards from sheets 22 in.
by 28 in. How many cards 2 in. by 3 in. can be cut from
one sheet and how much will be wasted ? Draw a plan.

20. At 6^ a square foot find the cost of enough wire
screen for the 8 windows of a gymnasium, each being 28 in.
by 64 in. inside the frame. Allowance should be made for
the wire to overlap the frames 1 in. on every side.

21. Find the area of a double tennis court.
A standard double tennis court is 36 ft. by 78 ft.

22. What is the meaning of the statement A=hh?

23. How many acres are there in a football field ?

A standard f ootbaU field is 100 yd. by 53 yd. 1 ft.
An acre is 160 sq. rd., and a rod is 5^ yd. The teacher should give
plenty of practical work in finding the areas of floors and the like.

24. A farm team plowing a field walks at the rate of
2 mi. per hour and is actually plowing ^ of the time. What
area will be plowed from 7 a.m. to noon if a plow is used
which turns a furrow 14 in. wide ?

AEEA OF A PARALLELOGRAM

167

Area of a Parallelogram. It is often convenient or nec-
essary to find the area of a parallelogram.

If from any parallelogram, like ABCD in the first figure,
we cut off the shaded triangle T by a line perpendicular

to DC, and place the triangle at the other end of the paral-
lelogram, as shown in the figure at the right, the resulting
figure is a rectangle.

That is, the area of a parallelogram is equal to the area
of a rectangle of the same base and the same height. But
the formula for the area of a rectangle is ^ = hh.

Therefore the area of a parallelogram is equal to the
product of the base and height.

This may be expressed by the formula

A = hh.

The teacher should make sure that the students understand the
meaning of this formula. The purpose is to introduce algebraic
forms as needed. The students should see that the value of A
depends upon the values of h and A. In the language of more
advanced mathematics, A is called a function of h and h. The
students should see> that all formulas are expressions of functions.

Rectangular pieces of cardboard, as in the figures shown just
above, may be arranged to lead the student to infer that when
the base and height of a rectangle are equal respectively to the
base and height of a parallelogram, the areas are equal.

168 GEOMETRY OF SIZE

Exercise 24. Area of a Parallelogram

1. Draw parallelograms of the shapes and sizes of the
following and show, by cutting off triangles and placing
them as explained on page 167, that each parallelogram
can be transformed into a rectangle of the same area.

OJ Z7

2. On squared paper draw four parallelograms and a
rectangle, all having equal bases and equal heights, but
all of different shapes. By cutting off a triangle from
each parallelogram and moving it to the other side, trans-
form each into a rectangle of the same area. Count the
squares and compare the areas of the resulting rectangles.

Draw the following parallelograms to scale and find the
area of each:

3. Base 30 in. ; other side, 20 in. ; height, 15 in. ; scale ^.

4. Base 12 in. ; other side, 8 in. ; height, 6 in. ; scale ^.

5. On squared paper draw a rectangle and a parallelo-
gram with equal bases and equal heights. Compute the
area of each by counting the included squares, and thus
compare the areas.

6. A floor is paved with six-sided fO^fOYC-^TO^T^
tiles, as here shown. The tiles have ^^^^^^V^^^KT^^
been divided by dotted lines in the ^^^^^'^y?''''^^
picture to suggest a method of

measuring them. What measurements would you take to
find the area of each tile ? What other divisions of the
tiles can you suggest for convenience in finding the area
of each?

AEEA OF A TRIANGLE 169

Exercise 25. Area of a Triangle

1. How is the area of a parallelogram found?

2. In the parallelogram here shown how do the areas
of the triangles ABC and CD A compare? A triangle is
what part of a parallelogram of the same

base and height ? i z:^

The parallelogram should be cut out of paper / ^,-'''' I

and then divided into two congruent triangles 1^ ^

by cutting along one of the diagonals.

3. If the parallelogram in Ex. 2 is 6 in. wide and 3 in.
high, what is its area ? What is the area of each of the
triangles formed by drawing the diagonal AC^.

4. If a parallelogram is 5 ft. wide and 2 ft. high, what
is its area ? What is the area of each of the triangles ?

5. If a parallelogram is 8 yd. wide and 3 yd. high, what
is its area ? What is the area of each of the triangles ?

6. Find the area of a rectangle with base 7 ft. and
height 10 ft.; of a triangle with base 7 ft. and height 10 ft.

Notice that a rectangle is one kind of a parallelogram.

7. Considering the above examples, state a rule for
finding the area of a triangle.

8. Draw to scale a triangle with sides 6 in., 7 in., and
8 in. respectively. Draw lines to show that the area of
the triangle is half the area of a rectangle with the same
base and height.

9. What is the area of a triangular garden with base
32 ft. and height IB ft.?

T'ind the areas of triangles with bases and heights as follows :

10. 4 in., 3.6 in. 12. 8 yd., 9 yd. 14. 9 ft., 4 ft. 4 in.

11. 9 in., 7.4 in. 13. 7.5 in., 8.4 in. 15. 6 ft. 3 in., 8 ft.

170

GEOMETRY OF SIZE

Area of a Triangle. From the illustrations given and
the questions asked on page 169 it is easily seen that

The area of a triangle is equal to half the product of the
base and height.

This may be expressed by the formula

For example, what is the area of a triangle of base 14 in.
and height 9 in. ?

■| of 14 X 9 sq. in. = 63 sq. in.

Exercise 26. Area of a Triangle

Examijles 1 to 9, oral
State the areas of triangles with these bases and heights :

1. 12 in., 9 in. 4. 28 in., 3 in. 7. 32 in., 8 in.

2. 14 in., 11 in. 5. 8 in., 4.5 in. 8. 3.5 in., 4 in.

3. 9 in., 10 in. 6. 3.5 in., 6 in. 9. 8 in., 9.5 in.

10. What fraction of a square yard of bunting is there
in a triangular pennant of base 56 in. and height 2 yd. ?

Find the areas of triangles with these bases and heights:

11. 17 ft., 46 ft.

12. 19.5 ft., 18.3 ft.

13. 22.7 ft., 16.4 ft

17. The span AB of a roof
is 40 ft., the rise MCis 15 ft.,
the slope CB is 25 ft., and the
length BE is 60 ft. Find the
area of each gable end and
the area of the roof.

14. 361ft., 17.6 ft.

15. 18.3 ft., 14.4 ft.

16. 29.7 yd., 24.8 yd.

AREA OF A TRIANGLE

171

18. On squared paper draw a right triangle with the
two sides respectively 1.5 in. and 2.5 in. Estimate the area
by counting the squares, compute the area accurately, and
then find what per cent the first result is of the second.

When we speak of the two sides of a right triangle we always
mean the two perpendicular sides.

19. A field 65 rd. by 140 rd. is cut by a diagonal into
two equal right triangles. A railway runs
along this diagonal and takes 3 A. off
each triangular field. How much is the
rest of the field worth at $140 an acre ?

20. In this figure ABCD represents an 8-inch square,
E, F^ G, and H being the mid-points of the sides. In the
square AEOH, AP=QE=ER = SO ^ ^ ^
= OT=^'.=^AE. Find the area
of each of the small triangles and
also of the octagon PQRSTUVW.

The dots ( • • • ) mean " and so on " and,
in this case, take the place of

= wa:'

An octagon is a figure of eight sides.

UH = HV

21. The triangle ABC is made by driving pins at A
and B^ running a rubber band around them, and stretching
this band to the point C. Now
imagine C to move along CE
parallel to AB, stopping first at
D and then at E. Have ABC,
ABB, and ABE different areas?
State your reasons fully.

Since any field may be cut into triangles by drawing certain
diagonals, the students are now prepared to find the area of any
piece of land that admits of easy measurement.

JMl

172 GEOMETEY OF SIZE

Area of a Trapezoid. If a trapezoid T has its double cut
from paper and turned over and fitted to it, like X>, the two

together make a parallelo- > ^- ,

gram. How does the area / T \ n /

of the whole parallelogram

compare with the area of the trapezoid T? How does the
base of the parallelogram compare with the sum of the
upper and lower bases of the trapezoid? How do you find
the area of the parallelogram ? Then how do you find the
area of the trapezoid ? j, ^,

If from the trapezoid ABCD, here
shown, the shaded portion is cut off and
is fitted into the space marked by the ^ '*

dotted lines, what kind of figure is formed ? How is the
area of the resulting figure found ? ^ ^_,

If the shaded portions of this trapezoid ]/ \

are fitted into the spaces marked by the ^^ ^

dotted lines, what kind of figure is formed ? How is the
area of the resulting figure found ?

From these illustrations we infer the following:

To find the area of a trapezoid, multiple/ the sum of the
parallel sides hy one half the height.

This may be expressed by the formula

A=^hQB + h),

where A stands for the area, h for the height, B for the
lower base, and h for the upper base.

The parentheses show that B and h are to be added before the
sum is multiplied hj ^h.

For example, if A = 4, J? = 7, and 6 = 5, we have
^=:ix4x(7 + 5)
= 2x12 = 24.

AEEA OF A TEAPEZOID 173

Exercise 27. Area of a Trapezoid

Examples 1 to 6, oral

Find the area of each of the trapezoids whose height is first
given helow, followed hy the parallel sides :

1. 6 in.; 9 in., 11 in. 7. 18 in.; 9.5 in., 27.3 in.

2. 8 in.; 14 in., 6 in. 8. 24 in.; Ill in., 93 jn.

3. 12 in.; 41 in., 3^ in. 9. 17 in.; 18 in., 26 in.

4. 11 in.; 8 in., 12 in. 10. 14 ft; 6 ft. 4 in., 9 ft.

5. 9in.; 4^ in., 51 in. 11. 42yd.; 191 yd., 371yd.

6. 13 in.; 11 in., 7 in. 12. 127 ft.; 96|ft., 108^ ft.

13. Find the number of acres in a field in the form of
a trapezoid, the parallel sides being 33^ rd. and 17|^ rd.
and the distance between these parallel sides being 14 rd.

14. If the area of a trapezoid is 396 sq. in. and the
parallel sides are 19 in. and 17 in., what is the height?

15. In this figure show that we may find the area
of the trapezoid by adding the areas of two triangles.

This should be taken up at the blackboard. b

The teacher should show that in this case we P\^
have J A5 + I hh = ^h(B -\- h), just as /

3x2 + 3x7 = 3x(2 + 7) = 3x9. L ^

A little algebra may thus be introduced as necessity requires and
the way made easier for more elaborate algebra later.

16. Suppose that the upper and lower bases of a trape-
zoid are equal, does the formula for the trapezoid still
hold true ? The trapezoid becomes what kind of a poly-
gon ? The formula becomes the formula for what figure ?

Practical outdoor work in measuring fields and in compviting
areas may now be given, or it may be postponed until after page 174
has been studied.

174

GEOMETRY OF SIZE

Area of any Polygon. A polygon like ABCDEF may be
divided into triangles, parallelograms, and trapezoids as
here shown, and the areas of these
parts may be found separately and
then added.

As an exercise the teacher may assign to
the class the finding of the area of the field
here represented, the figure being drawn to
the scale 1 in. = 200 rd.

Area found from Drawing. Suppose that the area of a
field ABC has to be found, and that there is a large
swamp as indicated in the figure. In such a case it is
not easy to find the height of the triangle ; that is, the dis-
tance CD. The lengths of the three c
sides may, however, be measured, and
then the area may be found by draw-
ing the outline to scale and measuring
the height of this triangle.

Only the drawing to scale is here
shown. If the scale is 1 in. = 100 rd., ^ D B

we see that CD is 90 rd., because CD is 0.9 in. If AB is
100 rd. the area of the triangle is ^ X 90 x 100 sq. rd., or
4500 sq. rd., which is equal to 28^ A.

Therefore, to find- the area of a field from a drawing.

Draw the plan to scale; divide the plan into triangles;
from the base and height of each triangle on the plan com-
pute the base and height of each triangle in the field ; from
these results find the areas of the several triangles and thus
find the area of the field.

It must be understood that surveyors have better methods, but
this method is sufficient for our immediate purposes. The immediate
object in view is not to make practical surveyors but to show the
general power of mathematics.

AEEAS

175

Exercise 28. Areas

1. This plan represents a space 150 ft. long and .75 ft,
wide, with two triangular flower beds, in a city park.
Around the inside of the space
is a sidewalk 6^ ft. wide. Meas-
ure the figure, determine the
scale used in drawing the plan,
and find the area of each of the
flower beds in the park.

2. This map is drawn to the scale 1 in. = 520 mi. Care-
fully measure the map and determine approximately the
length of each side of each state, and
then find the approximate area of
each state.

Wyoming

Utah

Colorado

The results obtained will be, of course,
merely approximate, since the map is so
small. The method is, however, the one
which is employed in practical work with
larger maps.

3. Each side of a brick building with a slightly sloping
roof is in the form of a trapezoid, as here shown. The
building is 57 ft. wide, 57 ft. high on the front,
and 52 ft. high on the rear. On this side there
are 4 windows each 4 ft. wide and 9 ft. high
and 4 windows of the same width but 6^ it.
high. If it takes 14 bricks per square foot of
outside surface to lay the wall, how many bricks will be
needed to lay this wall, deducting for the 8 windows?

4. The sides of a triangular city lot are respectively
72 ft., 60 ft, and 48 ft. Draw a plan of the lot to the
scale of 1 in. to 12 ft., measure the altitude of the scale
drawing, and find the altitude and area of the lot.

176

GEOMETRY OF SIZE

12' 20' 11'

20'

///

5. This sketch shows the plan of some small suburban
garden plots which are offered for sale at 20 (f a square foot.
Find the price of each lot.

6. In a certain city Washing-
ton Street runs east and west
and intersects Third Avenue at
right angles. Using the scale
1 in. = 100 ft., draw a plan of
the property on the southeast corner from the following
description ; Beginning at the corner, run south 160 ft.,
then east 75 ft., then north 15 ft., then east 50 ft., then
by a slanting line to a point on Washington Street 100 ft.
from the corner, and then to the corner. Find the area of
the plot and the value at |2.20 per square foot.

7. In order to measure the distance AB across a swamp
some boys measure a line CD, drawn as shown in the
figure, and find it to be 280 ft.
long. They find that DA = 40 it. 40 ff
and CB = 90 ft., DA and CB being
perpendicular to CD. Draw the
plan to some convenient scale and determine the distance
AB. Find also the area of the trapezoid A BCD.

8. A swimming tank is 60 ft. long and 35 ft. wide. DraAV
a plan to the scale 1 in. = 10 ft., determine the length of
the diagonal by measurement, and then compute the num-

*ber of yards that a student will swim in swimming along
the diagonal of the tank eight times.

9. A lot has a frontage of 65 ft. and a depth of 150 ft.,
and a path runs diagonally across it. Draw the plan to
scale and find, by measurement, the distance saved by
using the path instead of walking round the two sides
at a distance of 2 ft. outside the edges of the lot.

AKEAS 177

10. Suppose that yon have 360 ft. of wire screen to
inclose a plot in which to keep chickens. If you wish to
inclose the largest possible area in the form of a parallelo-
gram, triangle, or trapezoid, which form would you use ?
Show by a drawing on squared paper that the form which
you choose incloses a larger area than the others. Remem-
ber that there are several kinds of triangles, several kinds
of parallelograms, and several kinds of trapezoids.

11. Draw three different triangles, each with base 2 in.
and height 1 in. Find the area of each. What do you infer
as to the equality of the areas of triangles having equal
bases and equal heights ? Write the statement in full.

12. Upon the same base of 2 in. draw three different
parallelograms, each having a height 1^ in. Find the area
of each parallelogram. What do you infer as to the equal-
ity of the areas of parallelograms on the same base and
with equal heights? Write the statement in full.

13. For computing the area covered by 1000 ft. of a river,
some boys at 0 wish to find the width OB of the river, as
here shown. They know that the distance AB is 300 ft.
and that the angle at B is 90°. Show
how, by sighting along YB and XA and
by making certain measurements, the
boys can find the distance OB without
crossing the river.

14. Find the area of an equilateral
triangle 3.1 in. on a side.

This may be done by drawing the triangle on squared paper and
counting the squares, or, more accurately, by first approximating
the height by measurement on the squared paper.

15. Find the area of an isosceles triangle with sides
2 in., 2 in., and 1^ in.

178 GEOMETRY OE SIZE

Exercise 29. Optional Outdoor Work

1. Determine the area of your school grounds by care-
fully making the necessary measurements and dividing
the grounds into triangles, if necessary.

2. In Ex. 1 determine the area by drawing the plan
to scale.

3. Drive two stakes in the ground at A and j5, 12 ft.
apart. Fasten one end of a 15-foot line at ^ and one end
of a 9-foot line at B. Draw the loose
ends taut and drive a stake where they
meet, at C. What kind of an angle is
formed at -B?

4. Draw the figure of Ex. 3 to the scale
which is four times the one here used, and determine from
your figure some other measurements which might be
used to lay out the same kind of angle. Try this on the
school grounds.

5. What is the largest scale on which a plan of your
school grounds could be drawn on a piece of paper 12 in.
by 14 in., if you allow for a margin of at least 1 in.?

6. Draw to scale a plan of the lot on which your home
stands and indicate the ground plan of the house.

7. Draw to scale a floor plan of some public building
in your vicinity. Compute the area covered.

8. Lay off on your school grounds an isosceles triangle,
an equilateral triangle, and a right triangle, each with a
perimeter of 30 ft. Compute and compare the areas.

9. Lay off on your school grounds several rectangles,
each of which has a perimeter of 30 ft. Compute and
compare the areas.

ANCIENT PROOFS

179

7hfifi'Boo{e

q-hei.Tbeoreme. Thei^.Tropoftion.

aftTvo right lines cut the one the other: thehtdangksJUU
eaudtheone to the other,

«^ Vppofethaith(fe m right ImesASandCp.docuttkmU
'^^^^^ '^'^^ tUpthtE.ThenIfay,thatthean^UAEC,ve<iujlltoA,
7-1 c-ya C^^ f„r,,nnurh atihe rivht Lnt AE,lland»h)>P<mJv

h.nd^E3:theior,(^h^yr^'"^propofiU.
on)theansks jE'D^'DEf^ Hmllto
W fwUmhs. ^itisfroMd, th^t the angles'
CEA%iAE'D, arealfo equal! to (Jor.ghtan-
ohs Wherfore the angles CEA,and A E'JJ^e
%Lllto tk angles A<D,a»d<DES.Take a-
■aaytheangleAED,i»hich is common to them

ther. ifhiih Tfos required to he demonHrated.
.u. MJLji^ rhaUsMU'fus the Philofopher was the fixft ''^"fJ^'T^^^
t'"'"- ^^M^o^ atall. Portheexpontion ofthe thmg geu«,u fuffioa>t»«?

".,w-,- andirefoaIlcd,b«aufetheheddes of the two inglesare loyorf to|etl»r
Utjn. oaepointe«

Theconrnfe cfthlsfrofofithnafta-'Pelitams.

''^'"- Suppofe,h.,,h.«b.fo.errightlin«ABACAD»dA^^ |

Kgl=DAE,aiidthe^gleBADtotheandeCAE.TT^nIf9^:t?^ g

areondyt»9rightUDe5:tlutis,thetwonghtllnejBAandAEa«arawu. ^ ^

|iiiiiiitiiinniitiiiniiii)iiiiiii|iii)iiniiniiiiiiii|i!i!iiiiiiiii!!iiii)H.i|iiiiiirt!i>iii!M'ti^

cy^Z page from the first English edition of the great geometry
■written by Euclid of <Akxandria^ about 300 S. Q. It sho^s
hoT9 the ancient Greeks pro'ved their statements.

180 GEOMETRY OF SIZE

10. If the street is to be paved in front of your school-
house, what measurements are necessary to determine the
area to be covered ? Make the measurements for the block
in which your schoolhouse stands, draw the plan to scale,
and compute the area.

11. If a sidewalk is to be laid in front of your school-
house, what measurements are necessary and what prices
must be known in order that you may find the cost of the
walk ? Make the measurements, find by inquiry the prices,
and compute the cost of the walk.

Such examples are merely typical of the work which many schools
will wish to have done. It is impossible, however, to anticipate the
practical cases which may arise in any given locality. They may
relate to some building in process of erection, to the laying of a
water main in the street, to the reservoir of the city water supply,
or to the cost of stone steps for a schoolhouse. The important thing
is that the problem should be real and interesting to the class.

12. Compute the number of square feet of the surface of
some building which needs to be painted, find the average
cost per square yard for painting it one coat, and then
compute the cost of painting the building.

13. Suppose that a water main is to be laid in the street
in front of the schoolhouse. Ascertain by inquiry the usual
width of a trench for such a purpose, and draw a plan of
the street to scale, showing the location of the trench and
giving it the proper width to scale on the drawing.

14. In the upper picture on the opposite page can you see
how the height of the tower could be measured by simply
tipping the quadrant over fiat' and making certain measure-
ments on the ground ? Try this plan in measuring the
height of some tree or building.

In this case also it may be noticed that the angle is exactly 45°,
and so there is another and better way of finding the height.

ANCIENT INSTRUMENTS

181

182 GEOMETRY OF SIZE

Ratio. We often hear of the ratio of one number to
another, as when some one speaks of the ratio of the width
of a tennis court to its length, or the ratio of dayhght to
darkness in the winter, or the ratio of a man's expenses
to his income. By the ratio of 3 to 4 we mean 3 -^ 4, or
I", while the ratio of 1 in. to 1 ft. is j^ and the ratio of
1 to I is 1-^|, or f .

The relation of one number to another of the same
kind, as expressed by the division of the first number by
the second, is called the ratio of the first to the second.

A few examples of ratio should be given on the blackboard. Thus
the ratio of $3 to $6 is ^, or, in its simplest form, \ ; the ratio of
1 yd. to 1 ft. is the same as the ratio of 3 ft. to 1 ft., or 3 ; the ratio
of 5 to 2 is -|, or 2i ; and the ratio of any number to itself is 1.

The ratio of 2 to 3 may be written in the fraction forms,
^ or 2/3, or it may be written with a colon between the
numbers ; that is, as 2:3.

The teacher should explain to the class that the ratio of 12 ft. to

12 ft. 12
4 ft., for example, may be written -j-r— * > — > 12 : 4, or simply 3. The

word " ratio " is used for each of these forms. The expression 12 : 4
is read "the ratio of 12 to 4," or "as 12 is to 4," 12 and 4 being
called the terms of the ratio.

Since any number divided by a number of the same
kind, as inches by inches or dollars by dollars, has an
abstract quotient, we see that

A ratio is always abstract, and its terms may therefore
he written as abstract numbers.

That is, instead of labeling our numbers, as in 2 f t. : 4 ft., we may
omit all labels and write simply 2 : 4, or |, or ^.

Teachers should use the familiar fraction form first. Indeed, the
special symbol (:) is slowly going out of use because it is not neces-
sary. We often see 2 : 3 written as 2/3 instead of |. Ratios are little
more than fractions and may be treated accordingly.

EATIO

183

Exercise 30. Ratio

All work oral

1. Expressed in simplest form, what is the ratio of
6 to 12? of 12 to 6?

When a ratio is asked for, the result should always be stated in
the simplest form unless the contrary is expressly stated.

2. What is the ratio of $4 to |12 ? of 4 ft. to 12 ft.?

3. What is the ratio of 41 to 9 ? of 15 to 7^?

4. In the figure below, what is the ratio of ^ to D?
What is the ratio of -E' to C?

When we speak of the ratio of ^ to D we mean the ratio of their
number values ; that is, of 1 to 2, the ratio being ^. When we speak
of the ratio of £* to 2 B we mean the ratio of 1 to 2 X 4. This is J.

5. In the figure below, what is the ratio of E to B?
What is the ratio of ^ to ^ ? of JS to D+ C?

Referring to the figure^ state the following ratios :

6. E to^B. 11. A to E.

7. i) to 2 A.

8. 2 ^ to D.

9. 2 D to B.
10. (7 to %E.

12. C to B.

13. D to C.

14. C to A.

15. A to 2D.

11 ■

III 11

D E

16. What is the ratio of any
number to twice itself ?

17. What is the ratio of a foot

to a yard ? of an inch to a foot ? of 8 oz. to 1 lb. ? of
1 pt. to 1 qt.? of 2 qt. to 1 gal.?

In every such case the measures must be expressed in the same
units before the ratio is found. Thus the ratio of 1 yd. to 7 ft. is the
ratio of 3 ft. to 7 ft., or of 1 yd. to 2^ yd., either of which is f .

184 GEOMETRY OF SIZE

Proportion. An expression of equality between two ratios
is called a pro'portion.

For example, $5 : t8 = 10 ft. : 16 ft. is a proportion. This
proportion is read '' $5 is to |8 as 10 ft. is to 16 ft." or '' the
ratio of $5 to |8 is equal to the ratio of 10 ft. to 16 ft."
It may, of course, be written simply 5:8 = 10:16, or |-= ^.

The first and last terms of a proportion are called the extremes ;
the second and third terms are called the means. These expressions
are unnecessary, however, in the treatment of the subject in the
junior high school.

We often have three terms of a proportion given and
wish to find the fourth. For example, we may have the
proportion ' „:14=27:63,

where n represents some number whose value we wish to
find. We may write the proportion in the more familiar

fraction form, thus : ot

' n _zl

14~63*

If, now, j^ of n is equal to -|-J, we see that n must
be equal to 14 x |-|, or 6.

The teacher should show on the blackboard that we need merely
multiply the two equal ratios by 14, canceling as much as possible,
and we have n = 6.

If we have 4 : w = 12 : 6, we may simply take the ratios the
other way, and have ?i : 4 = 6 :12, and then solve as above.

The old method of solving business problems by ratio and pro-
portion is no longer used to any considerable extent. The subject
of ratio has a value of its own, however, and proportion is peculiarly
useful in geometry.

It is interesting to notice that in any proportion of
abstract numbers the product of the first and fourth terms is
equal to the product of the second and third terms.

PROPOETION . 185

Exercise 31. Proportion

Find the value of n in each of the following proportions:

1. 71:18 = 7:9. 3. 7:n=9:72.

2. n:42=13:14. 4. 15:13 = ?z:65.

5. A certain room is 24 ft. by 32 ft. and the width is
represented on a drawing by a Hne 9 in. long. How long
a line should represent the length ?

6. When a tree 38 ft. high casts a shadow 14 ft. long,
how long is the shadow cast by a tree 64 ft. high ?

In all such cases the trees are supposed to be in the same locality
and perpendicular to a level piece of ground.

7. If a picture 42 in. by 96 in. is reduced photographi-
cally so that the length is 7|- in., what is the width ?

8. By means of a pantograph a student enlarges the
floor plans for a house in the ratio of 8 : 3. If the dining
room in the original plans measures 2^ in. by 3 in., what
are the dimensions in the enlarged drawing?

9. The sides of a triangle are 9 in., 7 in., and 6 in.
Construct a triangle the corresponding sides of which are
to the sides of the given triangle as 3:4.

10. A map is drawn to the scale of 1 in. to 0.8 mi.
How many acres of land are represented by a portion of
the map 1 in. square ?

1 mi. = 320 rd., and 1 A. = 160 sq. rd.

11. The floor of a schoolroom is 24 ft. by 30 ft. The
total window area is to the floor area as 1:5, and the
6 windows have equal areas, each window being 3|- ft.
wide. Determine the height of each window to the nearest
quarter of an inch.

186 GEOMETEY OF SIZE

Proportional Numbers. Numbers which form a propor-
tion are called proportional numbers.

Similar Figures. As stated on page 141, figures which
have the same shape are
called similar figures and
are said to be similar.

For example, these two tri-
angles are similar. Likewise
triangles ABC and XYZ on
page 187 are similar.

Proportional Lines. The lengths of corresponding lines
in similar figures are proportional numbers ; that is, corre-
sponding lines in similar figures are proportional.

For example, in the above triangles XY : YZ = AB : BC. In two
circles the circmnferences and radii are proportional, the circum-
ference of the first being to the circumference of the second as the
radius of the first is to the radius of the second.

Exercise 32. Similar Figures

Examples 1 to 4, oral

1. In the above triangles, if XY is twice as long as AB,
how does ZX compare in length with CA ?

2. In the figure below state two proportions 'that exist
among AB, AD, AC, and AE.

3. In this figure, if ^^ is | of AD, C.
what is the ratio oi AC to AE?

4. In the same figure, if DE repre-
sents the height of a man 6 ft. tall, BC
the height of a boy, DA the length of the shadow cast by
the man, and BA the length of the shadow cast by the
boy, show how to find the height of the boy by measuring
the lengths of the shadows.

PROPOETIONAL LINES 187

5. If a tree BC casts a shadow 35 ft. long at the same
time that a post YZ which is 12 ft. high casts a shadow
15 ft. long, how high is the tree ?

Suppose YZ to be the post, X Y to be its shadow, and A B to be
the shadow of the tree.

Since the triangles ABC and XYZ are similar, we may find h,
the height of the tree, from the proportion

BC ^ YZ
AB~~ XY'

or by writing the values,

85 ~ 15 ' y\^'

whence , ^ 2i2ii2 = 28.

That is, the tree is 28 ft. high.

6. If a tree casts a shadow 58 ft. long at the same time
that a post 8 ft. high casts a shadow 14 ft. 6 in. long, how
high is the tree ? Draw the figure to scale.

7. If a telephone pole casts a shadow 27 ft. long at the
same time that a boy 5 ft. tall casts a shadow 4 ft. 6 in.
long, how high is the pole ? Draw, the figure to scale.

8. A boy threw a ball directly upward and watched its
shadow on the sidewalk. When the ball began to descend,
the shadow of the ball was at a fence post 32 ft. away. The
boy was 4 ft. 6 in. tall and his shadow was 2 ft. 3 in. long.
How high did the boy throw the ball above the level of
the ground ? Draw the figure to scale.

9. A water tower casts a shadow 87 ft. 8 in. long at the
same time that a baseball bat placed vertically upright casts
a shadow twice its own length on a level sidewalk. Find
the height of the water tower. Draw the figure to scale.

188

GEOMETRY OF SIZE

i^=.

10. This man is holding a right triangle AJ3C in which
AB = BC. What is the height of the tree in the pictnre
if the base of the triangle is
5 ft. 3 in. from the gronnd and
HAD is 32ft.?

This is a common way employed
by woodsmen for measuring the
heights of trees. The man backs
away from the tree until, holding the
triangle ABC ^o that ^-B is level, he
can just see the top of the tree along
the side A C.

In all problems involving heights
and distances the student should estimate the result in advance.
This will serve as a check on the accuracy of the work.

11. In Ex. 10 suppose that a triangle is used which
has AB equal to twice BC, that AD is 62 ft., and that the
point i> is 5 ft. 7 in. above the ground ; find the height
of the tree.

12. A woodsman wishes to determine the distance from
the ground to the lowest branch of a tree. He finds that
if he places a stick vertically
in the ground at a distance of
32 ft. from the tree, lies on his
back with his feet against the
stick, and sights over the top
of the stick, the line of sight
will strike the tree at the lowest
limb, as shown in the figure.
The woodsman's eye is 5 in.
above the ground, the distance

EF, as shown in the figure, is 5 ft. 6 in., and the top of
the stick is 4 ft. 9 in. above the ground. Determine the
distance BC from the ground to the lowest branch.

PROPORTIONAL LINES

189

13. A boy whose eye is 15 ft. from the bottom of a
wall sights across the top and bottom of a stick 8 in. long
and just sees the top

and bottom of the wall,
the stick being held
parallel to the wall as
shown. If the bottom
of the stick is 18 in.
from the eye, what is
the height of the wall ?

14. A woodsman steps off a distance of 30 ft. from a tree,
faces the tree, and holds his ax handle at arm's length in
front of him parallel to the tree. His hand is 27 in. from
his eye, and 2 ft. 4 in. of the ax handle just covers the
distance from the ground to the lowest limb of the tree.
How high is the lowest limb of the tree ?

This method suffices for a fair approximation to the height.

15. Wishing to find the length AB of a pond, some
boys choose a point C in line with A and B, and at B and
V draw lines perpendicular to BC, and
draw AI). By measuring they then find
^C to be 84 ft., D£J to be 112 ft., and
UA to be 154 ft. What is the length
of the pond ?

16. In Ex. 15 what other measurements
may be used to find the distance AB ?

17. If £ in. on a map represents a distance of 375 mi.,
how many miles will 2^ in. represent ?

18. If a tree casts a shadow 40 ft. long when a post
5 ft. high casts a shadow 6^ ft. long, how high is the tree ?

19. If 1|^ m. on a map represents a distance of 325 mi.,
how many inches represent a distance of 340 mi. ?

190 GEOMETRY OF SIZE

20. In one of the upper illustrations on the opposite
page suppose the length of the shadow of the post to be
1 ft. 6 in. shorter than the height of the post, and suppose
the shadow of the tower to be 69 ft. 4 in. and the height
of the post to be 5 ft. 2 in. Find the height of the tower.

21. In one of the upper illustrations on the opposite
page there is also shown a very old method of finding the
height by means of a mirror placed level on the ground.
Can you see two similar triangles in the picture? If so,
describe the method by which you could find the height
of a tree in this way.

22. Some members of a class made a right triangle with
one side 9 in. and the other side 12 in. One of them held
the triangle so that the longer side was vertical and then
backed away from a tree until he could just see the top
by sighting along the hypotenuse. The class then measured
and found that the eye of the observer was 45 ft. in a hori-
zontal line from the tree and 4 ft. 10 in. from the ground.
Draw the figure to scale and find the height of the tree.

23. Draw a plan* of the top of your desk to scale, rep-
resenting the length by 3 in. What will be the width of
the drawing, and how can it be found ?

24. The extreme length of a new leaf is 2 in. and the
extreme width is 1 in. After the leaf has grown 1 in.
longer, maintaining the same shape, what is its width?

25. A girl is maknig an enlarged drawing from a photo-
graph of a friend. In the photograph the distance between
the eyes is ||^ in. and the length of the nose is -^^ in.
If the distance between the eyes in the drawing is 50%
more than it is in the original, what is the length of the
nose in the drawing?

PROPORTIONAL LINES

191

192 GEOMETRY OF SIZE

Exercise 33. Optional Outdoor Work

1. Measure the height of any tree, telegraph pole, or
church spire in the vicinity of the school building, using
any convenient method.

In such cases it is desirable to have the class discuss the methods
in the class hour preceding the outdoor work, decidiijg upon the
methods to be used. It is then a good plan to separate the class into
groups, each group using a different method from the others. The
results can then be compared and, if the methods and work are
equally good, an average may be taken as a fair approximation.

2. Measure the distance from one point in the vicinity
of the school, preferably on the school grounds, to another
point so situated that a line cannot be run directly between
them. Use any convenient method.

In case no such points can be found, the distance across the street
may. be measured without actually crossing.

3. Making the necessary measurements, find the area of
the school grounds or tlie area of such a portion as is
decided upon by the teacher and the class.

In suburban or rural communities the areas of fi(dds may be
found. The class should see that it now has mastered enough
mathematics for finding the area of any ordinary field.

4. Ascertain the cost of a concrete sidewalk, per square
foot or square yard, and compute the cost of a good side-
walk in front of the school.

In case any excavations are being made for buildings near the
school, and it is feasible to have the class make the necessary meas-
urements, the amount of earth removed may be computed.

In some schools this optional work may be practicable, while in
others it may not be. Teachers will have to be guided by circum-
stances in assigning the above and similar exercises.

The problems on page 193 are typical of those which may be
considered for outdoor work.

OUTDOOR WORK

193

5. To find the distance across a river measured from
^ to ^, a point C was so chosen by the class that BC was
perpendicular to AB at B, Then a per-
pendicular to i? (7 prolonged was drawn.
The class sighted from C to ^ and
placed a stake at E where the line of
sight from Cto A cut the perpendicular
from D. By measurement BC was then
found to be 168 ft. and CB 290 ft. and BE 125 ft. What
was the distance across the river, measured from B to A?

6. In order to determine the distance from ^ to i? on
opposite sides of a hill, what measurements indicated in
this figure are necessary ? Make a problem
involvitig this principle, with reference to
two points near the school, take the neces-
sary measurements out of doors, solve the
problem, and check the work by measuring
the figure drawn to scale.

7. Wishing to determine the length AB of a pond, a class
placed a stake at S, as shown in the figure. The line BS
was then run on to B\ 121 ft. from S, and
BS was measured and found to be 253 ft.
By sighting from B, the angle B was marked
off on a piece of cardboard, and then the
angle B' was made equal to it, B'A' being
thus drawn to a point A' exactly in line ^' ^^^ ^'
with A and S, By measurement B'A' was found to be 132 ft.
Find the length of AB.

This should be considered at the blackboard before solving.
Notice the advantage of using A' to correspond to A, and B' to B.

8. Make a problem similar to Ex. 7, take the necessary
measurements, and solve.

194 GEOMETRY OF SIZE

Circumference, Diameter, and Radius. The line bounding
a circle is called the circumference. Any line drawn through
the center of a circle and terminated at each end by the
circumference is called a diameter, and any line drawn
from the center to the circumference is called a radius.

Ratio of Circumference to Diameter. Cut from cardboard
several circles with different diameters. Mark a point F on
each circumference, roll the circle along a
straight Ime, and determine the length of
the circumference by measuring the line be-
tween the points where P touches it. In
each case the circumference will be found to
be approximately 3y times the diameter.

The number 3.1416 is a closer approximation, but ^\ should be
used unless otherwise stated.

Since all circles have the same shape,

Any circumference _ Any other circumference _ qi
Diameter of its circle Diameter of its circle ^*

A special name is given to this ratio 3^; it is called
pi (written tt), a Greek letter. That is, c: d = tt,

or c = ird.

Since the diameter is twice the radius, d=2r; therefore

1. Find the circumference of a bicycle wheel the
diameter of which is 28 in.

We have c = ird = Y- x' 28 in. = 22 x 4 in. = 88 in.

2. Find the radius which was used iu constructing the
base of a circular water tank 24.2 ft. in circumference.

Since c = 2 Trr, it follows that r = c -f- 2 tt. Therefore we have
r = 24.2 ft. -4- (2 X ^^) = 3.85 ft.

CIRCLES 195

Exercise 34. Circles

Examples 1 to 8, oral
State the circumferences of circles of the following diameters :
1. 7 in. 2. 21 in. 3. 42 in. 4. 56 in.

State the circumferences of circles of the following radii :
5. 14 in. 6. 7 in. 7. 21 in. 8. 28 in.

Find the circumferences, given the following diameters :
9. 68.6 in. 11. 9.38 ft. 13. 53.9 ft. 15. 128.8 ft.
10. 420 in. 12. 3.01ft. 14. 13 ft. 5 in. 16. 116.2 ft.

Find the diameters, given the following circumferences :
17. 176 in. 18. 770 yd. 19. 3.96 ft. 20. 48.4 ft.

Find the circumferences, given the following radii :

21. 77 in. 22. 105 in. 23. 1.75 in. 24. 126 in.

25. What is the circumference of a 56-inch wheel?

26. Given that the inner circumference of a circular
running track is half a mile, find the diameter.

27. A girl has a bicycle with 26-inch wheels. How many
revolutions will each wheel make in going a mile ?

28. A circular pond 30 ft. in diameter is surrounded
by a path 6 ft. wide. What is the length of the outer
circumference of the path?

29. How many revolutions will an automobile wheel
37 in. in diameter make in going one mile ?

30. If the cylinder of a steam roller is 6.5 ft. in diameter
and 8 ft. long, how many square feet of ground will it
roll at each complete revolution ?

196 GEOMETKY OF SIZE

Area of a Circle. A circle can be separated into figures
which are nearly triangles. The height of each triangle is the
radius, and the sum of the bases is the circumference. If
these figures were exact triangles the area of the circle would

be |- X height x sum of bases ; and, smce they are nearly
triangles whose bases together are the circumference, we
may say that the area is ^ X radius x circumference. It is
proved later in geometry that this is the true area.
We may now express this by a formula, thus:

A = ^ re.

Since c=27rr we may put 2 ttv in place of <?, and have

A = ^r X 2'jTr = 7rr\

The teacher should explain to the class, if necessary, that the square
of a number is the product obtained by multiplying the number by
itself, and that it is customary to write 3^ for the square of 3.

1. A tinsmith in making the bottom of a tin pail draws
the circle with a radius of 5 in. How much tin does he
need, not countmg waste ?

Since A = irr^,

we have A = ^^^- x 5 x 5 sq. in. = 78 1 sq. in.

2. In order to have an iron pillar capable of supporting a
certain weight, the cross section must be 50|- sq. in. What
radius should be used in drawing the pattern ?

Since A = ttt^, we have yl -^ tt = r^, or 50f ^ ^^- = r^, and so 16 = r^.
Because 16 = 4 x 4, we see that r = 4.

AREA OF A CIECLE 197

Exercise 35. Area of a Circle

Exam2Jles 1 and 2, oral
State the areas of circles^ given the radii as follows :
1. Tin. 2. lin. 3. 1.4 in. 4. 2.8 in. 5. 56 in.

6. If the radius of one circle is twice as long as the
radius of another circle, how do the areas of the circles
compare ? How do the circumferences compare ?

7. If one circle has a radius three times as long as
the radius of another circle, how do the areas compare ?
How do the circumferences compare ?

8. A circular mirror is 2 ft. 3 in. in diameter. Find
the cost of resilvering the mirror at 36 (f a square foot.

9. What is the area of the cross section of a 3-inch
water pipe ? ^_^

3.5 ft.

10. Find the entire area of a window the lower
part of which is a rectangle 3.5 ft. wide and 6 ft. ^
high, and the upper part a semicircle, or half circle, "^
as shown in the figure.

11. In a park there is a circular lake of diameter 120 ft.
Find the number of square yards of surface in a walk 5 ft.
wide around the lake.

12. In the lake of Ex. 11 there is a circular island with
radius 25 ft. What is the surface area of the water ?

13. What is the cost, at |2.10 a yard, of erecting a wall
around the lake of Ex. 11 ?

14. A circular tree has a circumference of 12 ft. 3 in.
at a certain height above the ground. What will be the
area of the top of the stump made by sawing horizontally
through the tree at this point ?

198

GEOMETRY OF SIZE

Exercise 36. Volumes

All work oral
1. If C = 1 cu. in., what is the volume oi B? of ^ ?

■ill

—

A BO

2. Find the vokime of a workbox 3 in. by 4 in. by 2 in.

State the volumes of solids of the following dimensions :

3. 4 in., 5 in., 6 in. 5. 2 in., 3 in., 10 in.

4. 3 in., 3 in., 7 in. 6. 6 in., 8 in., 10 in.

Rectangular Solid. A solid having six sides, each side
being a rectangle, is called a 7'ectangular solid. If all the
sides are squares the rectangular solid is called a cube,

A cube may be constructed from
cardboard by cutting out a diagram
like this, bending the cardboard on
the dotted lines, and pasting the
flaps. In a similar way any rectan-
gular solid may be constructed.

Volume of a Rectangular Solid. From the exercise given
above we see that

The volume of a rectangular solid is equal to the product
of the three dimensions.

Expressed as a formula, using initial letters, we have
V=lbt.

1
1
1

!
1

VOLUMES 199

Exercise 37. Volumes

Examples 1 to 9, oral
State the volumes of solids of the following dimensions :

1. 3", 4'', Q". 4. r, 2', 3'. 7. 3yd., 7 yd., 2 yd.

2. 10", 4'', 5". 5. 20', 30', 4'. 8. 4 in., 5 in., 8 in.

3. 3", 8", 10". 6. 9", 3", 2". 9. 9 ft., 4 ft, 3 ft.

10. A cellar 24 ft. by 32 ft. by 6 ft. is to be excavated.
How much will the excavation cost at 45 (^ a load ?

Consider a load as equal to 1 cu. yd.

11. How much will it cost, at 52 (^ a cubic yard, to dig
a ditch 180 ft. long, 3 ft. wide, and 5 ft. deep ?

12. The box of an ordinary farm wagon is 3 ft. by 10 ft.
and the depth is usually 24 in. or 26 in. Find the contents
in cubic feet and cubic inches for each of these depths.

13. The interior of a certain freight car is 36 ft. long,
8 ft. 4 in. wide, and 7 ft. 6 in. high. How many cubic feet
does it contain ? If it is filled with grain to a height of
4|- ft., what is the weight of the grain at 60 lb. to the
bushel, allowing 1^ cu. ft. to the bushel ?

14. To what depth must a tank 5 ft. wide and 6 ft. 8 in.
long be filled with water to contain 120 cu. ft. of water?

15. It is estimated that 2^ cu. ft. of corn in the ear
will produce 1 bu. of shelled corn. How many bushels
of shelled corn can be obtained from a crib 10 ft. by 18 ft.
by 7 ft., filled with corn in the ear ?

16. A cellar 22 ft. by 30 ft. by 7 ft. is to be dug for a
house on a level lot 62 ft. by 140 ft. The dirt taken from
the cellar is to be used on the rest of the lot. To what
depth will the lot be filled if the dirt is evenly distributed ?

200 GEOMETKY OF SIZE

Cylinder. A solid which is bounded by two equal parallel
circular faces and a curved surface is called a cylinder.
Only the right cylinder is considered in this book.

The two parallel circular faces are called
the hases^ and the distance between the bases
is called the Jieight or altitude of the cylinder.

Volume of a Cylinder. Since we can place 1 cu. in. on
each square inch of the lower base if the cylinder is 1 in.
high,^ we see that if the cylinder is 5 in. high we can place
5 times as many cubic inches. Hence we see that

The volume of a cylinder is equal to the area of the base
multiplied by the altitude.

Expressed as a formula, using initial letters, we have

V=bh.
Since the base 5 is a circle, it is equal to 7rr% and so

V= irrVi.

Exercise 38. Volume of a Cylinder

1. What is the capacity of a cylindric gas tank 60 ft.
in diameter and 38 ft. 6 in. high ?

In all such cases the inside dimensions are given.

2. Find the capacity in gallons (231 cu. in.) of a water
tank of diameter 24 ft. 3 in. and height 66 ft. 9 in.

3. A water pipe 16 ft. long has a diameter of 1 ft. 9 in.
How many cubic inches of water will it contain ?

4. A farmer built a silo 24 ft. high and 14 ft. in diameter.
If 1 cu. ft. of silage averages 37 lb., how long will the silage
that fills the silo last 52 cows at 38 lb. per cow per day ?

Such problems should be omitted in places where the students
are not familiar with silos.

CYLmDERS 201

Curved Surface of a Cylinder. If we cut a piece of paper
so that it will just cover the curved surface of a cylinder,
how may we find the area of the paper? What are the
dimensions of a piece of paper that will just cover the
curved surface of a cylinder 6 in. high and having a cir-
cumference of 8 in. ? What is the area of the paper ? What
^is the area of the curved surface of the cylinder?

We see that the area of the curved surface of a cylinder
is equal to the circumference multiplied hy the height.

That is, area = circumference x height,
or area = ^ x diameter x height.

Expressed as formulas the above statements become
A = ch, A = irdh, or A = 2 irrh.

Exercise 39. Curved Surface of a Cylinder

All tcork oral

1. How many square feet in the curved surface of a
wire 1 in. in circumference and 600 ft. in length ?

2. How many square feet of tin in a pipe of length 12 ft.
and circumference 6 in., allowing 1 sq. ft. for the seam ?

3. A tin cup is 4 in. high and 7 in. around, including
allowance for soldering. How many square inches of tin
are needed for the curved surface?

4. A tin water pipe has a circumference of 9 in. and a
length of 10 ft., both measures includmg allowance for
soldering. How many square inches of tin were used ?

State the areas of the curved surfaces of cylinders with
heights and circumferences as folloivs :

5. 36 m., 80 in. 6. 40 in., 50 in. 7. 40 in., 35 in.

202 GEOMETRY OF SIZE

Exercise 40. Volume and Surface of a Cylinder

1. How many square feet of sheet iron will be required
to make a stovepipe 4^ ft. long and 7 in. ill diameter, if we
allow ^ in. for the lap in making the seam ?

2. A certain room in a factory is heated by 246 ft. of
steam pipe of diameter 2 in. Find the radiating surface ;
that is, the area of the curved surface which radiates heat.

3. Compare the surface of a cylinder 12 ft. long, having
a radius of 20 in., and the combined surfaces of 10 cylin-
ders each 12 ft. long and each having a radius of 2 in.

4. A farmer has a solid concrete roller 2 ft. 2 in. in
diameter and 6 ft. wide. How many cubic feet of concrete
are there in the roller ? How many square yards of land
does it roll in going ^ mi. ?

5. At 40^ a square yard, find the cost of painting a
cylindric standpipe 64 ft. high and 9 ft. 3 in. in diameter.

6. State a rule and a formula for finding the circum-
ference of a cylinder, given the radius, and also a rule and
a formula for finding the area of the circular cross section.

7. State a rule and a formula for finding the total
surface of a cylinder, given the radius of the base and
the height.

8. State a rule and a formula for finding the volume
of a cylinder, given the diameter and the height ?

9. If you know the volume of a cylinder and the cir-
cumference of the base, how do you find the height ?

10. A large suspension bridge has 4 cables, each 1942 ft.
long and 1 ft. 3 in. in diameter. In letting the contract
for painting these cables, it is necessary to know their
surface. Compute it.

LATHING AND PLASTERING 203

Lathing and Plastering. Laths are usually 4 ft. long,
1^ in. wide, and J in. thick and are sold in bunches of
100. Since the laths are laid ^ in. apart, 1 lath is required
for a space 2 ft. long and 4 in. wide.

Metal lathing is also commonly used.

Plastering is commonly measured by the square yard,
and there is no uniform practice m regard to the method
of making allowance for openings. The allowance to be
made should be mentioned in the contract.

Exercise 41. Lathing and Plastering

1. How many laths are required to cover a space 16 ft.
long and 10 ft. wide ?

2. How many laths are required for the walls and ceil-
ings of the liviug room, dining room, and kitchen shown
in the plan on page 204 if the rooms are 9 ft. high and
no allowance is made for openings, baseboards, or waste ?

3. At 50 (^ per square yard, compute the cost of plastering
the walls and ceilings of the reception hall, living room,
and dining room shown on page 204, no allowance being
made for openings or baseboards.

4. At 30 (f per square yard, compute the cost of rough
plastering the kitchen shown on page 204, allowing for the
four doors, which are 7 ft. high and 3 ft. wide and for
the two windows, which are 6 ft. high and 3 ft. wide.

5. Find the total cost of the materials required to lath
and plaster a room at the following prices : 20 bu. of
lime at 45 (f per bushel; 3^ cu. yd. of sand at 65 (^ per
cubic yard ; 4 bu. of hair at 45 (j^ per bushel; 200 lb. of
plaster of Paris at 55 (^ per hundred; 2800 laths at $2.90
per thousand; 20 lb. of nails at 16 (^ per pound.

JMl

204 GEOMETRY OF SIZE

Exercise 42. Practical Building

1. Determine the scale to vdiicli the architect drew tlie
plans for a two-story frame honse which are shown below.

FIRST PLOOF? FLAN

SECOND FLOOR FLAN

2. Compute the cost, at 40(/; a cubic -yard, of excavating,
1 ft. beyond the walls, for a 7-foot cellar under the main
part of the house, but not under the pantry and porches.

3. Compute the cost, at 62 (^ a square yard, of cementing
the floor of the cellar if the cellar walls are 1 ft. thick.

4. Compute the cost of flooring with hard wood J in.
thick the reception hall, living room, and dining room at
•$90 per M board feet, allowing 25% for waste.

A board foot (bd. ft.) is the measure of a piece of lumber 1 ft.
long, 1 ft. wide, and 1 in. or less thick. The number of board feet
in a board less than 1 in. thick is the same as in a board of the
same length and breadth but 1 in. thick. A fraction of a board
foot is counted a whole board foot.

5. Draw a plan of the first floor on a scale three times
as large as the one above, using a pantograph if desired.

6. Draw a plan of the second floor on a scale six times
as large as the one above, using a pantograph if desired.

METRIC MEASURES

205

Metric Measures. We are familiar with our common meas-
ures of length, including inches, feet, yards, and miles ; of
weight, including ounces, pounds, and tons ; of capacity,
including quarts, gallons, and bushels ; of surface, including
square inches, square yards, and acres. We need to know
something, however, about the measures used in countries
with which we have extensive business relations. During
the European war the newspapers spoke often
of the 75-millimeter guns, the 305-millimeter
guns, the 700-kilogram shells, and the capture
of 500 meters of trenches. None of this means
much to us unless we know Avhat millimeters,
kilograms, and meters represent in our measures.

Our cities carry on a great deal of business
with foreign countries, and since those countries
buy our goods, we must be able to describe them in
terms of foreign measures, especially because of
the recent remarkable increase in our foreign trade.

The work on pages 205-211 is optional. It is not re-
quired in subsequent work in this book, but it should
be taken if time permits.

Metric Measures of Length. A meter (m.) is
equal to 39.37 in., or about 1 yd. 3 in. One thou-
sandth of a meter is called a millimeter (mm.) ; one
hundredth of a meter, a centimeter (cm.) ; one
thousand meters, a kilometer (km.).

The meter and centimeter should be drawn on the
blackboard.

The meter stick should be used in actual measurement
in the schoolroom. Students should be told that 1 km. is
about ^ mi.

Metric Measures of Weight. A gram (g.) is equal to 15.43
grains, and a kilogram (kg.) is equal to 1000 g., or 2.2 lb.

*"*

•X)

-*l

»1

206 GEOMETRY OF SIZE

Metric Measures of Capacity. A liter (1.) is nea;:ly the
same as a quart. The prefix milli means thousandth, centi
means hundredth, and kilo means thousand, and so we
know what a milliliter, centiliter, and kiloliter mean.

A liter is equal to 0.91 of a dry quart or 1.06 liquid quarts, but
such equivalents need not be taught to students at this time.

Exercise 43. Approximations

All work oral

1. The newspaper says that a 75-millimeter gun was
used by an army. About what was the diameter in inches ?

1 m. = 39.37 in., and 75 mm. = 0.075 of 39.37 in., or about ^^ of
40 in., or about how many inches ?

2. A gun on a battleship has a bore of 300 mm. About
what is this ui inches ?

300 mm. = 0.300 m. = 0.3 m. = about 0.3 of 40 in., or about how-
many inches?

3. A manufacturer shipped a lot of flatirons weighing
3 kg. each. Express this weight in pounds.

4. An order is received in New Orleans for 8000 1. of
molasses. About how many gallons is this?

Exi^ress the followiiig apj^roxiniately in our measures :

5. 2L 11. 2 km. 17. 6 m. 23. ^ m. 29. 40 km.

6. 3 m. 12. I km. 18. 2 cm. 24. 24 L 30. 40 m.

7. 1 kg. 13. 4 km. 19. 20 m. 25. 10 L 31. 40 1.

8. 5L 14. 8 km. 20. 8L 26. 10 m. 32. 0.5 m.

9. 5 kg. 15. 6L 21. 7L 27. 10 kg. 33. 0.5 km.
10. 1 km. 16. 9 L 22. 7 m. 28. 10 cm. 34. 0.5 L

METRIC MEASURES

207

Prefixes in the Metric System. Although we have now
studied the most important measures in the metric system,
we should know thoroughly the meaning of the prefixes
used and learn the tables of length, weight, capacity,
surface, and volume. It will be found that the metric
system is very simple if these prefixes are known.

Just as 1 mill = 0.001 of a dollar,

so 1 millimeter = 0.001 of a meter.

Just as 1 cent = 0.01 of a dollar,

so 1 centimeter = 0.01 of a meter.

Just as the word decimal means tenths,
so 1 decimeter = 0.1 of a meter.

PREFIX

MEANS

AS IN

WHICH

MEANS

myria-

10,000

myriameter

10,000

meters

kilo-

1000

kilogram

1000

grams

hekto-

100

hektoliter

100

liters

deka-

dekameter

meters

deci-

0.1 .

decimeter

0.1

of a meter

centi-

0.01

centigram

0.01

of a gram

milli-

0.001

millimeter

0.001

of a meter

The teacher should show that this system is as much easier than
our common one for measures and weights as the system of United
States money is easier than the English system. This is the reason
why the metric system is so extensively used on the continent of
Europe and in Central America and South America.

It should always be remembered that measures never mean much
to a student unless they are actually used in the schoolroom.

The students should be led to see that metric units may be
changed to units of a higher or a lower denomination by simply
moving the decimal point, exactly as in changing from 205^ to $2.05.

Unless the student is to use the metric system in his measure-
ments in science, the further study of this subject may be omitted.

208 GEOMETKY OF SIZE

Table of Length. The table of length is as follows :

A myriameter = 10,000 meters

A kilometer (km.) = 1000 meters

A hektometer(hm.) = 100 meters

A dekameter = 10 meters

Meter (m.)

A decimeter (dm.) = 0.1 of a meter

A centimeter (cm.) = 0.01 of a meter

A millimeter (mm.) = 0.001 of a meter

The meter is about 39.37 in., 31 ft., or a little over a yard; the
kilometer is about 0.6 of a mile.

The names of the units chiefly used are printed in heavy black
type in the tables.

The abbreviations in this book are recommended by various
scientific associations. Some writers, however, use Km., Hm., cm.
and mm. for kilometer, hektometer, centimeter, and millimeter.

Exercise 44. Length
Express as inches^ taking 39.37 in. as 1 m. :

1. 35 m. 3. 32 m. 5. 275 cm. 7. 5200 mm.

2. 60 m. 4. 47.5 m. 6. 4.64 cm. 8. 8750 mm.

9. Express in inches the diameter of a 6-centimeter gun
and the diameter of a 75-millimeter gun.

10. A certain hill is 425 m. high. Express this height
in feet.

11. A tower is 48.6 m. high. Express this height in feet.

12. A wheel is 2.1 m. in diameter. Express this in inches.

13. The distance from Dieppe to Paris is 209 km. Ex-
press this distance in miles.

14. The distance between two places in Germany is
34.8 km. Express this distance in miles.

METRIC MEASURES 209

Table of Weight. The table of weight is as follows :

A metric ton (t.) = 1,000,000 grams

A quintal (q.) = 100,000 grams

A myriagram = 10,000 grams

A kilogram (kg.) = 1000 grams

A hektogram = 100 grams

A dekagram = 10 grams

Gram (g.)

A decigram = 0.1 of a gram

A centigram (eg.) = 0.01 of a gram

A milligram (mg.) = 0.001 of a gram

A kilogram, commonly called a kilo, is about 2^ lb. A 5-cent piece
weighs 5 g. A metric ton is about 2204. G2 lb.

Table of Capacity. The table of capacity is as follows:

A hektoliter (hi.) = 100 liters
A dekaliter = 10 liters

Liter (1.)
A deciliter = 0.1 of a litfer

A centiliter (cl.) = 0.01 of a liter
A milliliter (ml.) = 0.001 of a liter
A liter is the volume of a cube 1 dm. on an edge.

Exercise 45. Weight and Capacity
Express as kilos :

1. 2441b. 4. 120 oz. 7. 2500 g. 10. 28.46 g.

2. 326 1b. 5. 68.4 1b. 8. 6852 g. 11. 5700 g.

3. 4601b. 6. 400 T. 9. 252.51b. 12. 3268 g.

Express as liters, taklyig 1 I. as 1 qt. :

13. 5 hi. 15. 3.85 hi. 17. 4000 ml. 19. 656 hi.

14. 16 pt. 16. 371 gal. 18. 800 gal. 20. 9.65 hi.

210 GEOMETRY OF SIZE

Table of Square Measure. In measuring surfaces in the
metric system we use the following table : •

A square myriameter == 100,000,000 square meters

A square kilometer (km^.) = 1,000,000 square meters
A square hektometer (hm^.) = 10,000 square meters

A square dekameter = 100 square meters

Square meter (m^.), about 1.2 sq. yd.

A square decimeter (dm^.) = 0.01 of a square meter

A square centimeter (cm^.) =0.0001 of a square meter
A square millimeter (mm^.) =0.000001 of a square meter

The abbreviation sq. m. is often used for nA, sq. cm. for
cm^., and so on.

In measuring land the square dekameter is called an are
(pronounced ar); and since there are 100 square dekame-
ters in 1 hm^., a square hektometer is called a hektare (ha.).
The hektare is equal to 2.47 acres, or nearly 2^ acres.

Exercise 46. Square Measure

1. Find the area of a rectangle 3.2 m. by 12.7 m.

2. Find the area of a parallelogram whose base is 45 cm.
and height 22.4 cm.

3. Find the area of a triangle whose base is 7.3 m. and
height 4.6 m.

4. Find the area of each face of a cube of edge 27.2 cm.

5. A block of granite is*1.2m. long, 0.8 m. wide, and
0.7 m. thick. Find the area of the entire surface.

6. A cylinder has a diameter of 0.75 m. and a height of
0.8 m. Find the area of each base ; the area of the curved
surface ; the total area of the surface.

METRIC MEASURES 211

Table of Cubic Measure. In measuring volumes in the
metric system we use the following table :

A cubic kilometer = 1,000,000,000 cubic meters

A cubic hektometer = 1,000,000 cubic meters

A cubic dekameter = 1000 cubic meters

Cubic meter (m^.), about 1^ cu. yd.
A cubic decimeter (dm^.) =0.001 of a cubic meter

A cubic centimeter (cm^.) = 0.000001 of a cubic meter
A cubic millimeter (mm3.)= 0.000000001 of a cubic meter

In measuring wood a cubic meter is called a stere (st.),
but this unit is not used in our country.

Exercise 47. Cubic Measure

1. A box is 2.3 m. long, 1.7 m. wide, and 0.9 m. deep,
interior measure. Find the volume.

2. What part of a cubic meter is there in a block of
marble 1.3 m. long, 0.8 m. wide, and 0.6 m. tliick ?

3. A cubic centimeter of water weighs 1 g. How much
does a cubic decimeter of water weigh ? How much does
a cubic meter of water weigh ?

4. From Ex. 3 how much does 1 mnr^. of water weigh ?

5. Knowing that gold is 19.26 times as heavy as water,
find from Ex. 4 the weight of 1 mm-^. of gold.

6. A certain pile of wood is 7 m. long and 1.8 m. high,
and the wood is cut in sticks 1 m. long. How many
steres of wood are there in the pile ?

Express the followiyig as cubic meters :

7. 1250 dm3. 9. 2550 cm^. 11. 50,200 dm^.

8. 257,820 mm3. 10. 2700 dm^. 12. 75,000 cm^

212 GEOMETRY OF SIZE

Exercise 48. Miscellaneous Problems

1. Write the formula for finding the volume of a cube,
and also of a rectangular prism.

2. Write the formula for finding the area of each of
the following : a circle ; a trapezoid ; a triangle ; a paral-
lelogram; a rectangle.

3. What must you have given and how do you proceed
to find the area of a trapezoid ? the volume of a cube ?
the area of a circle ? the volume of a cylinder ? the area
of the curved surface of a cylinder ? the area of a tri-
angle ? the area of the surface of a cube ?

4. What must you have given and how do you proceed
to find the number of cubic yards of earth to be removed
in digging a cellar in the shape of a rectangular solid ?

5. . What must you have given and how do you pro-
ceed to compute the cost of polislnng a cylindric marble
monument at 52(f. a square yard?

6. How many loads of gravel averaging 1 cu. yd. each
will be required to grade 2^ mi. of road, the gravel to be
laid 15 ft. wide and 6 in. deep ?

7. A water tank is 7 ft. 6 in. long and 5 ft. 9 in. wide.
Water is flowing through a pipe into the tank at the rate
of 3 cu. ft. in 2 min. How long Avill it take to fdl the
tank to a depth of 3 ft. 8 in. ?

8. At what heiglits on the sides of a cylindric measuring
vessel whose base is 8 in. in diameter should the marks for
1 gal. (231 cu. in.), 1 qt., 2 qt., and 3 qt. be placed ?

9. Water is flowing into a cylindric reservoir 28 ft. in
diameter at the rate of 280 gal. a minute. Find the rate,
that is, the number of inches per minute, at which the
water rises in the reservoir.

OUTDOOE WORK 213

Exercise 49. Optional Outdoor Work

1. Make the necessary measurements; then draw a plan
of your school grounds to a convenient scale. Indicate on
the plan the correct position and the outline of the ground
covered by the school building. ,

2. Draw a plan of the athletic field where your school
games are played. Indicate on the plan the correct positions
of the principal features of the field.

3. Step off distances of 100 ft. in various directions on
your school grounds. Check the distance each time before
stepping off the next. Do the same for distances of 75 ft.,
140 ft, and 60 ft.

4. Estimate the distance between two trees or other
objects on your school grounds, then step off the distance.
Check by measuring the distance.

5. Estimate the height of several objects on or near your
school grounds, and then check the accuracy of your esti-
mate by determining the heights of the objects by some of
the methods previously explained.

6. Estimate the number of square rods or acres in j^our
school grounds. Check your estimate by measurement.

7. Estimate the number of acres in one of the city or
village blocks near your school or near your home. Check
your estimate by measurement.

8. Lay off, by e'stimate, on your school grounds an area
which you believe is a quarter of an acre. Check your
work by measurement.

9. Estimate the extreme length and the height of your
school building. Check your estimates. Which is the more
difficult for you to estimate with a fair degree of accuracy,
length or height ?

214 GEOMETRY OF SIZE

Exercise 50. Problems without Figures

1. If you know the dimensions of a field in rods, how
do you find the area in acres ? If you know the dimensions
in feet, how do you find the area in acres?

2. If you can make all necessary measurements of a
triangular field, how can you find the area ?

3. If you can make all necessary measurements of a
trapezoidal field, how can you find the area ?

4. How can you determine the width of a stream with-
out crossing the stream?

5. How can you determine the height of a church
spire without climbing to the top ?

6. If you have a good map of the state of Colorado,
drawn to a scale which you know, how can you find from
the map the number of square miles in the state ?

For our present purposes we may consider that the state is
rectangular, although this is only an approximation.

7. If you know the circumference of a circuiar water
tank, how can you find the diameter?

8. If you know the circumference of a circle, how can
you find the area?

Consider first the result of Ex. 7.

9. If you know the circumference and height of a
cylindric water tank, how can you find* the capacity ?

Consider Exs. 7 and 8.

10. If you know the length of a water pipe in feet and
the internal diameter of the pipe in inches, how can you
find the number of cubic inches of water it will hold ?
How can you then find the number of gallons and the
weight of the water it will hold ?

GEOMETRY OF POSITION 215

III. GEOMETRY OE POSITION

Position of Objects. We have already seen, on page 111,
that geometry can ask three questions about an object:
(1) What is its shape ? (2) How large is it ? (3) Where
is it ? The first question led us to study those common
forms about which everyone needs to know; the second
led us to find the size of such forms, including not only
areas and volumes but also heights and distances ; and
now we come to the third question and consider position.

For example, when two boys have located the home
plate and second base in laying out a baseball field, how
can they find the positions of first base and third base or
the position of the pitcher's box ? Or if they have located
the home plate and the line to the pitcher's box, how can
they locate the other bases ? Such questions show that
position is an important subject in geometry.

It has often happened in war that people have buried
their valuables. They did not dare to mark the spot,
because then their enemies might find the hiding place
and dig up the valuables ; and so it became necessary to
be able to locate the spot in some other way.

This may be done in various ways. For example, a
man may select two trees, A and B, 160 ft. apart, and
measure the distance from each tree to
F, the point where he buried his valu- ^YV

ables, say 80 ft. and 120 ft., remember-
ing that V is north of a line drawn . •
from A to B. Then when he returns
he may take two pieces of rope 80 ft. and 120 ft. long,
besides what is needed for tying, and by stretching these
from the trees A and B respectively he can fiiiil Fin just the
same way that the triangle on page 116 was constructed.

216 GEOMETRY OF P0!SIT10:^i

Exercise 51. Fixing Positions

1. During a war a man buried some valuables. He re-
membered that they were buried south of an east-and-west
line joining two trees 60 ft. apart, and that the point was
50 ft. from the eastern tree and 70 ft. from the western tree.
Draw a plan to the scale of 10 ft. — 1 in. and indicate the
point where the valuables were buried.

2. In olden times, before there were many banks, it was
the common plan to bury treasures for safe-keeping. Sup-
pose that a man buried a treasure 120 ft. from one tree
and 160 ft. from another one, but that during his absence
the second tree is entirely destroyed. Draw a plan showing
where he should dig a trench so as to strike the treasure.

3. Two points, A and B, are 3 in. apart. Is it possible
to find a point which is 1 in. from each? l^in. from
each ? 2 in. from each ? 3 in. from each ? Draw a figure
illustrating eacli case and consider whether, in any of the
cases, there is more than one point.

4. To keep the water clean a farmer covers a spring
with a large flat stone and pipes the water to his house.
He then covers the stone and piping with soil, and seeds
it all down. Before doing this he takes the necessary
measurements for locating the spring, using as the known
points two corners of the field in which it is. Draw three
plans showing how to take the measurernents.

5. Two streets, each 66 ft. wide, intersect at right angles.
Two water mains meet 40 ft. from a certain corner and
45 ft. from the diagonally opposite corner. Some workmen
wish to dig to find where the mains meet. Draw a plan
to scale and show the two possible places where they
should dig.

DRAWING MAPS 217

Positions on Maps. Knowing the shortest distance from
Cliicago to MinneapoUs to be approximately 354 mi., that
from Chicago to St. Louis 263 mi., that from Chicago to
Kansas City 415 mi., that from IVIinneapohs to St. Louis
466 mi., and that from MinneapoUs to Kansas City 411 mi.,
we can easily make a map showing the location of all four
of these cities.

If we draw the map to the scale of 1 in. = 252 mi., the
map distance from Chicago to Mmneapolis will be -||^|^in.,
or 1.40 in., and similarly for the other distances. That is,

Chicago to Minneapolis %j^ in. = 1.4 in.

Chicago to St. Louis ^|-| in. = 1.0 in.

Chicago to Kansas City |^ in. = 1.6 in.
Minneapolis to St. Louis -||^|^ hi. = 1.8 in.
Minneapolis to Kansas City -|^^ hi. = 1.6 in.

With the aid of a ruler draw the line CM, making it
1.4 in. long, C representing Chicago and 31 representing
Minneapolis. Next place the ^

compasses with one point at
C and draw an arc with a
radius 1.0 in., representing
the distance to St. Louis ;
also draw an arc with radius
1.6 in., representing the dis- . ^

tance to Kansas City. Simi-
larly, with center M and radii
1.8 in. and 1.6 in. describe \
arcs intersecting the other ^"^ c ^^ --

arcs, thus locating St. Louis '' "^

and Kansas City. We can now approximate the distance
from Kansas City to St. Louis, for if it is 0.95 in. on the
map, it is really 0.95 of 252 mi., or 239 mi.

218 GEOMETRY OF POSITION

Exercise 52. Map Drawing

1. The distance from Cincinnati to Cleveland is 222 mi.,
from Cincinnati to Pittsburgh 258 mi., and from Cleveland
to Pittsburgh 115 mi. Draw a Ihie to represent the dis-
tance from Cincinnati to Cleveland and then mark the
position of Pittsburgh. Use the scale of 1 in. to 50 mi.

It is to be understood that the distances stated in this and the
following problems are only approximate, and that they are meas-
ured in a straight line and not along roads or railways.

2. The distance from Philadelphia to Harrisburg is
95 mi., from Harrisburg to Baltimore 70 mi., and from
Baltimore to Philadelphia 90 mi. Indicate the position of
the three cities, using the scale of 1 in. to 75 mi.

3. On a map drawn to the scale of 1 in. to 108 mi. the
distance from Atlanta to Raleigh is 3^- in., that to Savannah
2 in., and that to Jacksonville 2^ in. What is the dis-
tance in. miles from Atlanta to each of the other cities ?

4. On a map of scale 1 in. = 145 mi. the distance from
San Francisco to Portland is 3|^ in., that to Seattle 4|- in.,
and that to Los Angeles 2i in. What is the distance in
miles from San Francisco to each of the other cities ?

5. On a map drawn to the scale of 1 in. to 95 mi. the
extreme length of Pennsylvania is 3 in. and the extreme
width is 1-| in. What are the dimensions of the state ?
What is the approximate area of Pennsylvania ?

6. The distance from Nashville to Memphis is 2^g- in.
measured on a map drawn to the scale of 1 in. to 90 mi.
The distance from Nashville to Louisville is 1 j^ in., from
Nashville to Mobile 4-|- in., and from Nashville to Chatta-
nooga 1^ in. What is the distance in miles from NashviUe
to each of the other cities ?

LOCATING POINTS 219

Points Equidistant from Two Points. We shall now con-
sider again the case of a spring of water that has been
covered by a stone slab and earth so that its position is
not visible. If the farmer who owns the land has lost the
measurements and remembers only that the spring was just
as far from one corner A of the field as from the diagonally
opposite corner B, how shall he dig to find the spring ?

If A and B are 540 ft. apart he might try digging at
M, the mid-point between them, 270 ft. from both A and
B. Failing here, he might try a point P,
which is 280 ft. from both A ond B. He
might then try a point Q, which is also
280 ft. from both A and P, and so on.
He might thus try a large number of
points, and yet miss the spring. It would be better for
him to run a line along the ground and know that if he
should dig along this line he would certainly strike the
spring. Let us see how this could be done.

If you connect P, 3f, and Q, what kind of line do you
seem to have ? What does this suggest as to how the farmer
should run the line ?

Take two points A and B on paper ; find four points such
that each is equidistant from A and P, and then connect
these points and see if you have the kind of line you expect.
Was it necessary to take as many as four points ? How
many points need be taken ?

From the above work we see the truth of the following :

To find all the points which are equidistant from two given
points, find any two such points and draw a straight line
through them. All the required points lie on this line or on
the prolongation of this line.

Instead of using paper or blackboard for the above work, points
may be taken on the schooh-oom floor or on the school grounds.

JMl

220 GEOMETRY OF POSITION

Exercise 53. Points Equidistant from Two Points

1. Two water pipes are known to join at a place equi-
distant from two trees which are IGO ft. apart. One tree
is exactly north of the other, and the pipes join west of
the line connecting them. Draw a plan on the scale of
1 in. to 40 ft. and show the line along which a workman
should dig to find where the pipes join.

2. The residences of Mr. Anderson and Mr. Williams
are 300 ft. apart and on the same side of a straight water
main. Mr. Anderson's residence is 50 ft. and Mr. Williams's
residence is 65 ft. from the water main. They decide to
have trenches dug so as to strike the main at a point
equidistant from the two houses. Draw a plan on the
scale 1 in. = 60 ft. showing the position of the trenches.

3. It is planned to place a circular flower bed in a park
so that its center shall be' 90 ft. distant from each of two
trees which ape 80 ft. apart. The radius of the flower bed
is to be 20 ft. Draw a plan on the scale of 1 in. to 20 ft.

4. How can you find a point on one side of your school-
room which is equidistant from two of the diagonally
opposite corners ? Draw a plan of your schoolroom on
the scale of 1 in. to 10 ft. and locate the point on the side
of the room.

5. A landscape gardener wishes to plant a tree which
shall be equally distant from three trees forming the
vertices of a triangle, as shown in the figure. c
How can he find where to plant the fourth ^>
tree ? Suppose it is known that ^5= 140 ft.,
^C-110 ft., and C^ = 130 ft., draw a plan ^' m
on the scale of 1 in. to 20 ft. and determine the location of
the fourth tree and its distance from each of the other trees.

J.OCATING POINTS

221

'^/A

"AV

6. Does a point equally distant from the three vertices
of a triangle always lie inside the triangle ? Consider these
triangles -. AB = \ in., BC=1 in., CA = ^ in. -, AB=Z in.,
BC=4t in., CA = b in. ; ^7^ = 2 in., ^C = 3 in., CA = 1.5 in.
Construct the triangles and locate the points.

7. In a park there are three walks which meet as
shown in the figure, and ^5 = 180 ft., BC=12^ ft., and
CA — 145 ft., inside measure. It is
planned to place a cluster of lights
which shall be equally distant from
the three corners A^ B, and C. Draw a
plan to some convenient scale, indicate
the position at which the cluster of lights is to be placed,
and determine its distance from each of the three corners.

8. In an athletic park are three trees A, B, and C\
so situated that AB = 150 ft., BC = 11^ ft, and CA = 90 ft.
A running track is planned, as shown in
the figure, so that an arc of a circle shall
pass close to the trees. Draw a plan to
any convenient scale and determine the
radius of the circle.

9. In the ruins of Pompeii a portion of the tire of
an old chariot wheel was found. The curator of a
museum wished a drawing of the original wheel.
How was it possible to determine the center and to
reproduce the tire of the wheel in a drawing ?

10. Three points. A, B, and (7, are so situated that
AB = 3.5 in., BC= 2.5 in., and CA = 4.5 in. By drawing
to scale and measuring, determine the length of the radius
of the circle which passes through the three points.

11. Solve Ex. 10 for three points which are so situated
that AB = 4 ft. 2 in., BC=^ ft. 8 m., and CA= 5 ft. 1 m.

222

GEOMETRY OF POSITION

Road

Distance of a Point from a Line. In the figure below,
if H represents a house and AB a straight road, how far
is it to the house from the road ?

It is obvious that the answer to the question depends
on how we are to go. If there are several straight paths
such as HA, HB, HC, and HD leading
to the house from the road, the lengtli
of each path is an answer to tlie ques-
tion. We see at once, however, that
the shortest of the paths shown in the
figure is HB, and when we speak of the distance from the
house to the road or from a pomt to a line, we mean the
shortest distance.

If a stone P is hanging by a string and the other end of
the string is held at 0, and if the stone swings so as just
to graze the line AB, it is evident that OP
represents the shortest distance from 0 to
the line and also that OP makes two right
angles with AB. These two right angles are
BPO and OP A, and, of course, are equal. ^ p B

Perpendicular. There is a special name for the straight
line which represents the shortest distance from a point
to a line. In the above figure OP is said to be perpen-
dicular to AB or to be the perpendicular from 0 to AB,

The perpendicular from a point to a line makes right angles
with the line and is the shortest path from the point to the line.

Perpendicular must not be confused with vertical. A i)lumb line
(a line with a weight at the

end) hangs vertically, but a line
may be perpendicular to another
line and still not be vertical. In ^
each of tliese figures OP is per-
pendicular to AB, but in only the first of the figures is OP vertical.

'BAP

DISTANCE FROM A LINE 223

Exercise 54. Distance of a Point from a Line

1. Draw a perpendicular to the line AB from the point
0 outside the line, proceeding as follows: Set one point
of the compasses at 0, then adjust the compasses until the
other point just grazes the line AB at F. Draw OF ; then
OF is evidently perpendicular to AB.

See the second figure on page 222.

This method of drawing the perpendicular shows clearly what
the perpendicular is, but it is not frequently used in practice. Usually
the best method is that which is described on page 121 c

or that on page 122.

2. In this figure AB is 45 ft., and AC and
BC are each 5 2, ft. Find by measurement the
perpendicular distance CM.

3. A straight gas main runs from one street light A
to another light B, the distance AB being 220 ft. It is
desired to place a third lamp C in a park fronting on
the street, so that the distance AC shall be 260 ft. and the
distance i? (7 shall be 190 ft. The gas main AB is to be
tapped at the point nearest to C.

Draw the figure to the scale of 1 in. to 40 ft. ; find by
measurement the approximate distance of C from AB and
the distance from B to the point where the main is tapped.

The figure is somewhat like that of Ex. 4.

4. In this figure ABC is an equilateral triangle each of
whose sides is 14 in. Draw the triangle to a convenient
scale and determine the length of the c
perpendiculars drawn from ^ to ^C and
from B to AC. Does the perpendicular
drawn from C to AB appear to pass
through the point in which the other two
perpendiculars intersect ?

224 GEOMETRY OF POSITION

5. A stone is tied to a string 15 in. long. The string is
held at the other end, and the stone is then swung so that
it just grazes the ground. When
the stone is at the point B, 12 in.
from the perpendicular Qui, how
far is it from the ground ? Draw
the figure carefully to some con-
venient scale and thus find tliis distance by measurement.

6-. Some boys stretcli cords from the top of a flagstaff
to two points on level ground 40 ft. apart.
The first cord is 90 ft. long and the second
is 65 ft. long. Draw the figure to scale and
thus find the heiglit of the flagstaff.

7. Two boys observe a bird on the top of a 90-foot flag-
staff. One boy is 65 ft. from the foot of the staff and the
other is 130 ft", from its foot. Draw the figure to scale and
thus find the distance of each boy from the bird.

8. Two trees on the same bank of a straight riA^er are
285 ft. apart. The distance from one of the trees to a point
on the opposite bank is 200 ft. and the distance from the
other tree to the same point is 260 ft. Draw the figure
to scale and thus find the width of the river.

9. An airplane is 2800 ft. above a level railway line
connectmg two villages A and i?, which are 1.5 mi. apart.
The airplane is above a point which is ^ of the distance
from A to B. Draw the figure to scale and thus find the
distance of the airplane from each of the villages.

10. A pyramid has a square base 120 ft. on a side. The
distance from the vertex to the mid-point of one side is
90 ft. Draw the figure to scale and thus find the height
of the pyramid.

DISTANCE FEOM A LINE 225

Points at a Stated Distance from a Line. If a man wishes
to build a liouse at a distance of 100 ft. from a straight
road, he can build it in any one of a great many places,
for he can stand anywhere on the road, lay off a per-
pendicular on either side of it, and then measure a dis-
tance of 100 ft. along this perpendicular, any point thus
found being 100 ft. from the road.

Pages 121 and 122 may profitably be reviewed at this time.

To illustrate this further we may draw a straight line
XY, using a carpenter' s square or a right triangle to lay
off four points on one side of the line
and ^ in. from it and to lay off four

points on the other side of the line and x Y

^ in. from it. It is evident that the

points lie in two lines each of which is

J in. from XY. All points on the paper ^ in. above or

below XY evidently lie on one of these two lines.

In geometric and mechanical drawing it is necessary to
be able to locate such points quickly and accurately.

To draw a line perpendicular to a given line XY, we
may, as stated on page 121, place a right triangle ABC so
that one side JW lies along XY and the hypotenuse AC
along a ruler MX. Then AB is perpendicular to XY, and
as we slide the triangle along MX, as shown in the figure,
AB will remain perpendicular to XY and the line can be
drawn with accuracy right across XY.

After we have drawn two such perpendiculars on each
side of XY, and the required distance has been laid off, we
can draw the two lines which contain all the points at the
given distance from the line. It is evident that we need
to find only two points of a straight line in order to draw
the line with a ruler.

226 GEOMETEY OF POSITION

Exercise 55. Points at a Given Distance from a Line

1. Draw a plan showing 40 ft. of a straight railroad
track 4 ft. 8|- in. wide, using the scale of 1 in. to 8 ft. and
being careful to make the two rails everywhere equally
distant from each other.

2. Using a convenient scale, draw a plan of the straight-
away of a running track showing 100 yd. of a straight
track 16 ft. wide.

3. Draw a plan of the floor of your schoolroom, being
careful to make the sides perpendicular at each corner.
If the opposite sides are parallel you can test whether
your schoolroom is a perfect rectangle by measuring the
diagonals to find whether they are equal.

4. An old description of a corner lot states that there
is a covered well 30 ft. from one of two streets which cross
at right angles (but it does not state which street) and
50 ft. from the other street. Draw a plan to a convenient
scale and indicate all possible positions of the well.

5. The residence of Mr. Weber is 130 ft. from a straight
road. He wishes to build a garage 80 ft. from his house

♦and 120 ft. from the road. Indicate on a plan the position
of the house and the garage.

6. Two straight driveways in a park meet as shown in
the figure. A fountain is to be placed 80 ft. from driveway
AB and 60 ft. from driveway CD. Draw a
figure to scale showing the two possible

positions of the fountain and determine

the distance between these positions. First

copy the figure accurately, enlarging it on any convenient
scale, extending it as may be necessary, and taking the
width of driveway AB to be 40 ft.

POSITION FIXED BY TWO LINES 22T

Position fixed by Two Lines. Having already seen how
the position of an object can be found if we know its dis-
tance from each of two points, let us see if we can find the
position if we know its distance from each of two lines.

Take, for example, your desk. If anyone was told, that
your desk is 6 ft. to the west of the east wall and 8 ft.
north of the south wall, could he find which
desk is yours ? How would he do it ? In
this plan he might run a line 6 ft. from BC -^^ —

and parallel to it, and another line 8 ft. from
AB and parallel to it, and where these lines
crossed he would find the desk D. . ^

This is the way we locate a place on a map of the
world. We say it is so many degrees north or south of
the equator and so many degrees east or west of the
meridian of Greenwich. Thus we say that a place is
40° N. and 70° W., meaning that it is 40° north of the
equator and 70° west of the prime meridian, the one
which passes through Greenwich.

The principle applies whether we use a Mercator's i^rojection or a
globe, except that the curvature of the lines is seen in the latter case.

The lines we use need not meet at right angles. For
example, if we know that two water mains join at a point
60 ft. south of the road AB^ as
here shown, and 90 ft. from the
road CD and on the side towards
i>, we can locate the point F by aZ
simply drawing two lines. How
are these lines drawn ?

If we know that P is 60 ft. south oi AB and 90 ft. from CD,
but do not know on which side, there would be two possible points.
Under what circumstances would there be three possible points,
or four possible points?

228 GEOMETRY OF POSITION

Exercise 56. Position fixed by Two Lines

1. Suppose that we do not know the side of the road CD
on which the two water mains mentioned on page 227 join,
but we do know that it is on the ^d.

south side of the road AB. Draw
a figure showing all the possible
positions of the water mains. ^-

2. Suppose that we know that ^

the water mains join to the west of CD, but do not know
whether the point is to the north or to the south of AB,
Draw a figure showing all the possible positions.

3. Suppose that we are uncertain as to the side of each
road on which the water mains join. Draw a figure show-
ing all the possible positions.

4. Each side of a square park ABCD is 150 yd. long.
A monument is to be erected in the park at a point 130 ft.
from AB and 170 ft. from BC. Draw the figure to scale
and determine the distance of the monument from each
corner of the park.

5. There are three survivors of a shipwreck. The first
says that the ship lies between 2 mi. and 2^ mi. from a
straight coast line which runs from a lighthouse L to the
west ; the second says that the ship lies between 1^ mi.
and 2 mi. from a straight coast line which runs from L to
the north ; and the third says that it lies 2^ mi. from L.
Can they all be right? If so, draw the figure to scale
and indicate where to dredge for the wreck.

6. In a rectangular field ABCD, 90 rd. by 130 rd., there
is a water trough which is 290 ft. from the side AB and
320 ft. from BC. Draw the figure to scale and thus find
the distance of the water trough from the point D,

POSITION FIXED BY TWO LINES 229

Points Equidistant from Two Lines. Suppose that two
roads AB and CD intersect at 0 and that it is desired
to place a street lamp at a point
equidistant from the two streets. \ // ^.^^
How many possible positions are \//-

there for the lamp ? 7^/\

If we think of ourselves as walk- ,,^-"'' // \
ing in such a direction as to be ^

always equidistant from OB and OD, we see that we shall
be walking along OiV". Similarly, to be equidistant from
OD and OA we must walk along OP. In general, any
point on any of the dotted lines in the figure is equi-
distant from AB and CD. Therefore we may locate the
lamp anywhere on either line, these lines evidently bisect-
ing the angles formed by the roads.

Exercise 57. Points Equidistant from Two Lines

1. By using this figure draw a line containing points
equidistant from two lines. . p

The dotted lines are each Jin. from AB a / ^/q -B

and CD respectively. They intersect at P, v^^'/'^^

and OP is drawn and prolonged to M. —~^;^^'—r-

M / /

2. Draw a line containing points ' c

equidistant from two lines by using a figure somewhat like
the one used in Ex. 1, page 228.

3. In a park P, which lies between two streets M and
iV, as shown in the figure, an electric

light is to be placed so as to be equidis-
tant from M and N and 80 ft. from the j~
corner C. Copy the figure and show how
to find the position of the light.

In this case a circle intersects a straight line.

230 GEOMETRY OF POSITION

4. The manager of an amusement park decides to place
six lights at intervals of 80 ft., so that each light shall
be equidistant from two intersecting drive- \^

ways M and iV, as shown in the figure. The "--4{^"^ ■

first light is to be placed 15 ft. from the inter- yiy

section of the driveways. Copy the figure to

scale and indicate the position of each of the six lights.

5. A fountain is to be erected in the space S between
the two streets M and iV. The contract

provides that the fountain shall be 50 ft. ^-"^

from the nearest side of each of these
streets. Copy the figure and indicate
the position of the fountain.

6. A monument is to be so placed in a triangular city
park that it shall be equally distant from the three sides.
Draw a plan on any convenient scale and
show on the plan all the lines necessary to
find the point 0, Is O equidistant from the
sides AB and BC^. Is it equidistant from
AB and CA ? Is it necessary to draw a line from C ?

7. A flagpole is to be so placed that it shall be equally
distant from the three sides of a triangular park whose
sides are 380 ft., 270 ft., and 300 ft. respectively. Draw
a plan on a convenient scale and find the distance of the
flagpole from each side of the park.

8. A water main has a gate located at a point 7 ft.
from a certain lamp-post which stands on the edge of a
straight sidewalk. The gate is placed 3 ft. from the edge
of the walk, towards the street. Draw a plan showing
every possible position of the gate.

9. Consider Ex. 8 when the gate is located at a point
3 ft. from the lamp-post.

USES OF ANGLES 231

Use of Angles. There is still another method by which
a man may locate valuables which he has buried. Suppose
as before, that there are two trees, A and x

B, and that he has buried his valuables ^^^

at X. If he knows the exact direction A'^ b

and distance from A to X, he can easily find the place
where the valuables are buried.

Both the distance and direction can be recorded on paper.
For example, the man might write : '' 30° north of line joining
the trees, 150 ft. from the west tree," and this would recall
to his mind that the angle of 30° is to be measured at J,
namely, the angle BAX^ and that X will be found 150 ft.
from A. The man can lay off the angle on a piece of
paper by the aid of a protractor like the one described on
page 115, and can then sight along the arms of this angle.

The use of the protractor, as given on page 115, may be reviewed
at this time if necessary.

Exercise 58. Uses of Angles

1. Estimate the number of degrees in each of the following
angles, then check your estimate by use of a protractor.

2. Without using a protractor draw several angles of as
near 45° as you can, and in different positions. Check the ac-
curacy of your drawings by measuring the angles with a
protractor. Similarly, draw angles of 10°, 80°, 75°, 15°, 50°,
and 30°. Check the accuracy of your drawing in each case.

You will find that practice will enable you to estimate lengths
and the size of angles with a fair degree of accuracy.

232 GEOMETRY OF POSITION

3. Indicate on paper a direction 30° west of south of the
school ; 20° east of north ; 20° east of south.

By 30° west of south is meant a direction making an angle of 30°
with a line running directly south, and to the west of that line.

4. It is found that a submarine cable is broken 5 mi. from
a certain lighthouse and 30° west of south. Draw a plan,
using the scale of 1 in. to 1 mi., and show where the repair
ship must grapple to bring up the ends for splicing.

5. A boy starts to Avalk on a straight road which runs
20° east of north. After he has gone 3 mi. he turns on an-
other straight road and walks 2 mi. due west. Is he then
east or west of his starting point, and how many degrees ?

6. A flagstaff CD stands on the top of a mound the
height BC of which is known to be 30 ft. ad
From the point A the angle BAD is ob-
served to be 50° and the angle CAD to be
20°. Draw a diagram to scale and find ap-
proximately the height of the flagstaff CD. j^

7. Wishing to find the length of a pond, some boys staked
out a right triangle as shown in the figure. Using a pro-
tractor they found that the angle A was 40°,
and they measured AB, finding it to be 500 ft.
Draw the figure to the scale of 1 in. to 200 ft.
and thus find the length of the pond.

8. Some men buried a chest 150 yd. from a tree A and
250 yd. from another tree B, which was 200 yd. due north
of A. On returning some years later they found that the
tree B had disappeared. Draw a plan to scale and show
how they can find the buried chest, provided that they
have a compass and remember on which side of AB they
hid the chest.

USES OF ANGLES 233

9. A seaport is on a straight coast which runs due north
and south. A steamer sails from it in a direction 30° west
of north at the rate of 15 mi. an hour. When will she be
25 mi. from the coast, and how far will she be from the
seaport at that time ?

10. Some boys wished to determine the height of a cliff.
They found the distance AC to be 90 ft. and c
the angle BAC to be 45°. They then drew the ^/W^
figure to scale and determined the height of y^ ff
the cliff. What is the height of the cliff ? a^ — \

11. While a ship is steaming due east at the rate of
20 knots an hour, the lookout observes a light. At 9 p.m.
the light is due north, at 9.15 p.m. it is 10° west of north,
and at 9.30 p.m. it is 20° west of north. Determine by a
drawing whether the light observed is stationary.

A knot is generally taken as 6080 ft., this being approximately
tlie sea mile ; the statute mile used on land is 5280 ft.

12. There is a seaport A on a straight coast running east
and west. A rock B lies 3000 yd. from A and 30° north of
east from the coast line. A ship steams from A in a direction
20° north of east from the coast line. What is the nearest
approach of the ship to the rock? How fdr is the ship
from the coast at that time ?

13. A man knows a tree across a stream to be 60 ft. high ;
he finds the angle at his eye as shown m the figure to be 30°,
and he knows that his eye is 5 ft. from the
ground. Make a drawing to scale and de- ^y^
termine the width of the stream.

14. Two towers of equal height are
300 ft. apart. From the foot of each tower
the top of the other tower makes an angle of 30° with a
horizontal line. Determine the height of the towers.

234 GEOMETRY OF POSITION

Exercise 59. Miscellaneous Problems

1. A ship steams from a seaport yi in a direction 50°
east of north. A dangerous rock hes northeast from A and
6000 yd. from the coast running east and west through A.
Find how near the ship approaches to the rock.

2. Two sides of a rectangular field ABCD are 80 rd.
and 95 r.d. A tree in the field stands 120 ft. from AB and
160 ft. from BC. Draw the figure to scale and show the
position of the tree.

3. A man computes that he should buy 16 T. of soft
coal for his winter supply. His cellar is so arranged that
he can make a coal bin 18 ft. long, and the height of the
cellar is 8^ ft. How wide a bin should be constructed if
the top of the coal is to be a foot from the ceiling when
the bin contains 16 T. of coal ?

Allow 35 cu. ft. of coal to the ton.

4. A boy standing due south of a flagpole finds its
angle of elevation, tliat is, tlie angle to the top, to be 20°.
After he has walked 280 ft. to the northwest on level
ground, he sees the flagpole to the northeast. Draw the
figure to scafe and determine the height of the flagpole.

5. How long a shadow will the flagpole mentioned in
Ex. 4 cast when the sun's angle of elevation is 40° ?

6. A tree 90 ft. high casts a shadow 140 ft. long. Draw
the figure to scale and find the siin's angle of elevation.

7. A military commander standing 1000 ft. from a fort
finds its angle of elevation to be 30°. Draw the figure to
scale and determine the- height of the fort.

8. In order to check his work in Ex. 7 the commander
took the angle of elevation from a point 1200 ft. from
the fort. What did he find this angle to be ?

OUTDOOR WORK 235

Exercise 60. Optional Outdoor Work

1. Determine the height of a tree on or near your school
grounds by finding the angle of elevation of the top from
two different positions and measuring the distance of each
position from the tree.

2. Two boys wishing to determine which of two smoke-
stacks is the taller computed the height of each stack by
three different methods and then took the average of the
results as the correct height. What three methods might
they have used ? Use three methods to determine the
height of some high object near your school.

3. A graduating class decides to present a drinking-
fountain to the school. The fountain is to be placed 30 ft.
from a straight street which runs in front of the school
grounds and 15 ft. from the front door of the school build-
ing. How could the members of the class determine the
desired position ? Locate, if possible, such a position on
your school grounds or on some lot in the vicinity.

4. Suggest two methods of determining the distance
between two points when the distance cannot be measured
directly. Use both methods to determine the distance be-
tween two easily accessible points on your school ground.
Check the accuracy of each method by actually measuring
the required distance.

5. A hawk's nest is observed in a high tree and some
distance below the top of the tree. Suggest two methods
by which the height of the nest may be determined other
than by direct measurement. Check the accuracy of the
two methods by applying them to a similar situation where
the required distance can be actually measured as a check
upon the accuracy of the methods.

236 GEOMETRY OF POSITION

Exercise 61. Problems without Figures

1. A workman has a circular disk of metal and wishes
to find its exact center. How should he proceed ?

2. A man wishes to set out a tree so that it shall be
equally distant from three trees which are not in the same
straight Ime. How should he proceed to find the position?

3. A contractor wishes to tap a water main at a point
equidistant from two hydrants. How should he proceed
to find the required point?

4. A city engineer is asked to place an electric-light pole
at a point equidistant from two intersecting streets and at a
given distance from the corner. How does he do it ?

5. A man wishes to build on a corner lot a house at a
given distance from one street and at another given distance
from the other street. How does he lay out the plan ?

6. A drinking-fountain is to be placed in a park at a
given distance from a hydrant which is at the side of the
street in front of the park, and at a given shorter distance
back from the street. Show that there are two possible
points, and show how to find them.

7. If you know that two water mains join somewhere
under a certain road, but you do not know where, what
measurements could be given you with respect to one or
more trees along the side of the road that would tell you
where to dig to find the point?

8. A straight electric-light wire runs under the floor of
a room. The distances of one point of the wire from the
northeast corner and from the north wall are known, and
also the distances of another point from the southwest
corner and from the south wall. Show how to mark on
the floor the course of the wire.

SUPPLEMEKTARY WORK

237

IV. SUPPLEMENTARY WORK

Squares and Square Roots. If a square has a side 4
units long, it has an area of IG square units. Therefore
16 is called the square of 4, and 4 is called the
square root of 16.

Square Roots of Areas. Therefore, consider-
ing only the abstract numbers which represent
the sides and area,

The side of a square is equal to the square root of the area.

Symbols. The square of 4 is written 4'-^, and the square
root of 16 is written VI6.

Perfect Squares. Such a number as 16 is called a j^er-
fect square^ but 10 is not a perfect square. We may say,
however, that VlO is equal to 3.16 +, because 3.16'"^ is
very nearly equal to 10.

Square Roots of Perfect Squares. Square roots of perfect
squares may often be found by simply fac-
toring the numbers.

For example, V441 =V3x3x7x7
^ V3 X 7 X 3~>r7 -
= V21x21=21.

That is, we separate 441 into its factors,

and then separate these factors into two

equal groups, 3x7 and 3x7. Hence we see that 3 x 7,

or 21, is the square root of 441.

We prove this by seeing that 21 x 21= 441.

To find the square root of a perfect square^ separate it
into two equal factors.

The work in square root may be omitted if therQ is not time for i,t.,

238

SUPPLEMENTARY WORK

Square of the Sum of Two Numbers. Since 47 ^
the square of 47 may be obtamed as follows:

40 + 7,

40 + 7

(40 >

c7) + 72

402 +

(40 X 7)

402+2

x(40)

<7)+72

1600 + 2

x280

+ 49

1600 + 560

+ 49

2209.

280

1600
40

280

+ M

This relationship is conveniently seen in the above figure,
in which the side of the square is 40+7.

Every number consisting of two or more figures may be
regarded as composed of tens and units. Therefore

The square of a number contains the square of the tens,
plus twice the product of the tens and unifSj plus the square
of the units.

This important principle in square root should be clearly under-
stood, both from the multiplication and from the illustration.

Separating into^ Periods. The first step in extracting the
square root of a number is to separate the figures of the
number into groups of two figures each, called periods.

Show the class that 1 = 1^, 100 = 10^, 10,000 = 100^, and so it is
evident that the square root of any number between 1 and 100 lies
between 1 and 10, and the square root of any number between 100
and 10,000 lies between 10 and 100. In other words, the square root
of any integral number expressed by one figure or Uvo figures is a
ijumber of 07ie figure ; the square root of any integral number ex-
pressed by three or four figures is a number of two figures ; and so on.

Therefore, if an integral number is separated into periods of two
figures each, from the right to the left, the number of figures in the
square root is equal to the number of the periods of figures. The
last period at the left may have one figure or two figures.

22 09(47
16

6 09
6 09

SQUAEE EOOT 239

Extracting the Square Root. The process of extract-
mg the square root of a number will now be considered,
although in practice such roots are usually found by tables.

For example, required the square root of 2209.

Show the class that if we separate the figures of the number into
periods of two figures each, beginning at the right, we see that there
will be two integral places in the square root of the number.

The first period, 22, contains the square of the tens' number of
the root. Since the greatest square in 22 is 16, then 4, the square
root of 16, is the tens' figure of the root.

Subtracting the square of the tens, the re-
mainder contains twice the tens x the units,
plus the square of the units. If we divide by
twice the tens (that is, by 80, which is 2 x 4
tens), we shall find approximately the units.
Dividing 609 by 80 (or 60 by 8), we have 7
as the units' figure.

Since twice the tens x the units, plus the
square of the units, is equal to (twice the tens + the units) x the
units, that is, since 2 x 40 x 7 + 7^ = (2 X 40 + 7) x 7, we add 7 to
80 and multiply the sum by 7. The product is 609, thus completing
the square of 47. Checking the work, 47- = 2209.

Exercise 62. Square Root

Find the square roots of the following numbers :
1. 3249. 2. 3721. 3. 3969. 4. 5041.

Find the sides of squares^ given the following areas :

5. 6724 sq. ft. 7. 9025 sq. ft. 9. 7921 sq. yd.

6. 7569 sq. ft. 8. 9409 sq. ft. 10. 6889 sq. ft.

Find the square roots of the following fractions by taking
the square root of each term of each fraction :

11 144 io 961 lo 1089 i^ 1936
li. 5 2 9- ^^' TF24- ^^' TXtt- ^*- 5 0TT-

240 SUPPLEMENTARY \yORK

Square Root with Decimals. Find the value of V151.29.

Show the class that the greatest square of the tens in 151.29 is
100, and that the square root of 100 is 10.

Then 51.29 contains 2 x 10 x the units' number of the root, plus
the square of the units' number. Ask why this is the case.

Dividing- 51 by 2 x 10, or 20, we find that the next figure of the
root is 2.

We have now found 12, the square being 100 + 44 = 144.

Then 7.29 contains 2 x 12 x the tenths' number of the root, plus
the square of the tenths' number, because we have subtracted 144,
which is the square of 12.

Dividing by 24, we find that the
tenths' figure of the root is 3.

Hence the square root of 151.29
is 12.3.

If the number is not a perfect
square, we may annex pairs of zeros
at the right of the decimal point and
find the root to as many decimal places
as we choose.

151.29(12.3
1

240

7 29

243

7 29

Summary of Square Root. We now see that the following
are the steps to l)e taken in extracting square root :

Separate the number into periods of two fic/ures each, he-
ginning at the decimal point.

Find the greatest square in the left-hand period and write
its root for the first figure of the required root.

Square this root, subtract the result from the left-hand
period, and to the remainder annex the next period for a
dividend.

Divide the new dividend thus obtained by twice the part
of the root already found. Annex to this divisor the figure
thus found and multiply by the number of this figure.

Subtract this result, bring down the next period, and pro-
ceed as before until all the periods have been thus annexed.

The result is the square root required.

SQUARE EOOT

241

9. 63,001.

10. 21,224,449.

11. 49,112,064.

12. 96,275,344.

Exercise 63. Square Root
* Find the square roots of the following :

1. 12,321. 5. 19.4481.

2. 54,756. 6. 0.2809.

3. 110.25. 7. 1176.49.

4. 8046.09. 8. 82.2649.

In Exs. 13-17 give the square roots to two decimal places only.

13. 2. 14. 5. 15. 7. 16. 8. 17. 11.

18. Find, to the nearest hundredth of an inch, the side
of a square whose area is 3 sq. in.

Square on the Hypotenuse. As we learned on page 112,
in a right triangle the side opposite the right angle is called
the hypotenuse.

If a floor is made up of triangu-
lar tiles like this, it is easy to mark
out a right triangle. In the figure
it is seen that the square on the
hypotenuse contains eight small tri-
angles, while the square on each
side contains four such triangles.
Hence we see that

The square on the hypotenuse is equal to the sum of the
squares on the other two sides.

This remarkable fact is proved in geometry for all right triangles.

Given that AB = 12 and AC = 9, find BC.

Since

BC^ = AB" + AC^,

(

therefore

5C' = 122 + 92,

BC^ = U4: 4- 81 = 225,
BC = V225 = 15.

and

12 JB

242

SUPPLEMENTAKY WORK

Exercise 64. Square Root

1. How long is the diagonal of a floor 48 ft. by 75 ft. ?
On this page state results to two decimal places only.

2. Find the length of the diagonal of a square that
contains 9 sq. ft.

3. The two sides of a right triangle are 40 in. and 60 in.
Find the length of the hypotenuse.

4. What is the length of a wire drawn taut from the
top of a 75-foot building to a spot 40 ft. from the foot ?

5. A telegraph pole is set perpendicular to the ground,
and a taut wire, fastened to it 20 ft. above the ground,
leads to a stake 15 ft. 6 in. from the foot of the pole, so
as to hold it in place. How long is the wire ?

6. A derrick for hoisting coal has its
arm 27 ft. 6 in. long. It swings over an
opening 22 ^t. from the base of the arm.
How far is the top of the derrick above
the opening?

Reversing the procedure on page 241, the square on either side is
equal to the difference between what two squares ?

7. The foot of a 45-foot ladder is 27 ft. from the wall

of a building against which the top rests. How high does
the ladder reach on the wall ?

8. To find the length of this pond a
class laid off tlie right triangle ACB as
shown. It was found that ^6'= 428 ft.,
BC = 321 ft., and AD = 75 ft. Find DB. a,

9. How far from the wall of a house must the foot of
a 36-foot ladder be placed so that the top may touch a
window sill 32 ft. from the ground ?

PYRAMIDS

243

Prism. A solid in which the bases are equal polygons
and the other faces are rectangles is called a prism.

Volume of a Prism. It is evident that we
may find the volume of a prism in the same
way that we found the vojume of a cylinder
(page 200). That is,

The volume of a prism is equal to the product of the base
and height.

Pyramid. A solid of this shape in which the base is
any polygon and the other faces are triangles
meeting at a point is called a pyramid. The
point at which the triangular faces meet is
called the vertex of the pyramid, and the
distance from the vertex to the base is called
the altitude of the pyramid. The faces not
including the base are called lateral faces.

Volume of a Pyramid. If we fill a hollow prism with
water and then pour the water into a
hollow pyramid of the same base and
the same height, as here shown, it will
be found that the pyramid has been
filled exactly three times with the water
that filled the prism. Therefore,

Tlie volume of a pyramid is equal to one third the product
of the base and height.

Lateral Surface of a Pyramid. The height of a lateral
face of a pyramid is called the slant height of the pyramid.

Since the area of each lateral face of a pyramid is equal
to half the product of the base and altitude,

TJie area of the lateral surface of a pyramid is equal to
the perimeter of the base multiplied by half the slant height.

244 SUPPLEMENTARY WORK

Lateral Surface of a Cone. If we should slit the surface
of a cone and flatten it out, we would have part of a circle.

The terms "lateral surface " and "slant height" will
be understood from the study of the pyramid.

From our study of the circle* we infer that
The lateral surface of a cone is equal to the

circumference of the base multiplied hy half the

slant height

Volume of a Cone. In the way that we found the volume
of a pyramid we may find the volume of a cone. Then

The volume of a cone is equal to one third the product of
the base and height.

Find the volume of a cone of height 5 in. and radius 2 in.

Area of base is ^^^ x 4 sq. in.

Volume is ^ X 5 X \2 x 4 cu. in. = 20.95 cu. in.

This method of calculation gives 20.95 2^ cu. in. as the volume, but
20.95 cu. in. is a much more practical form for the answer.

Teachers will observe that only the simplest forms of prisms,
pyramids, and cones have been considered in this book.

Exercise 65. Lateral Surfaces and Volumes

Find the volumes of prisms and also the volumes of pyra-
mids with the following bases a7id heights :

1. 36sq.in., 7in. 2. 48 sq.in.,5|-in. 3. 5.7sq.in.,4.8 in.

Find the lateral surfaces of pyramids ivith the following
perimeters of bases and slant heights :

4. 30 in., 18 in. 5. 3 ft. 3 in., 8 in. 6. 5 ft. 9 in., 10 in.

Find the volumes of cones with the following radii of bases
and heights :

7. 14 in., 6 in. 8. 5.6 in., 15 in. 9. 49 in., 15 in.

CONES AND SPHERES 245

Surface of a Sphere. If we wind half of the surface of
a sphere with a cord as here shown, and then wind with
exactly the same length of the cord the surface of a

cylinder whose radius is equal to the radius of the sphere
and whose height is equal to the diameter, we find that
the cord covers half the curved surface of. the cylinder.

Therefore the surface of a sphere is equal to the curved
surface of a cylinder of the same radius and height.

We can now easily show that

surface of sphere = ^-^ x 2 x radius x 2 x radius.

Hence tlie surface of a sphere is equal to 4 times y times
the square of the radius.

Volume of a Sphere. It is shown in geometry that

The volume of a sphere is equal to j times ^ times the
cube of the radius.

The cube of r means r x ?- x r and is written r^.

1. Considering the earth as a sphere of 4000 mi. radius,
find the surface.

4 X \2_ X 40002 = 4 X -2^? X 16,000,000 = 201,142,857^.
Therefore the surface is about 201,143,000 sq. mi.

2. If a ball is 4 ft. in diameter, find the volume.

The radius is I- of 4 ft., or 2 ft.

The volume is | x ^t?. x 2 x 2 x 2 cu. ft., or 33.52 cu. ft.

246 SUPPLEMENTARY WORK

Exercise 66. Surfaces and Volumes

1. If a ball has a radius of 1^ in., find the surface.

2. Find the surface of a tennis ball of diameter 2§ in.

3. If a cubic foot of granite weighs 165 lb., find the
weight of a sphere of granite 4 ft. in diameter.

4. A bowl is in the form of a hemisphere 4.9 in. in
diameter. How many cubic inches does it contain ?

5. A ball 4' 6" in diameter for the top of a tower is
to be gilded. How many square inches are to be gilded ?

6. A pyramid has a lateral surface of 400 sq. in. The
slant height is 16 in. Find the perimeter of the base.

7. A conic spire has a slant height of 34 ft. and the
circumference of the base is 30 ft. Find the lateral surface.

8. What is the entire surface of a cone whose slant
height is 6 ft. and the diameter of whose base is 6 ft. ?

9. What is the weight of a sphere of marble 3 ft. in
circumference, marble being 2.7 times as heavy as water
and 1 cu. ft. of water weighing 1000 oz. ?

10. Taking the earth as an exact sphere with radius
4000 mi., find the volume to the nearest 1000 cu. mi.

11. If 1 cu. ft. of a certain quality of marble weighs
173 lb., what is the weight of a cylindric marble column
that is 12ft. high and 18 in. in diameter?

12. How many cubic yards of earth must be removed
in digging a canal 8 mi. 900 ft. long, 180 ft. wide, and
18ft. deep?

13. A marble 1|- in. in diameter is dropped into a
cylindric jar 5 in. high and 4 in. in diameter, half full of
water. How much does the marble cause the water to rise?

TABLES FOR REFBEENCE

Length

12 inches (in.) =1 foot (ft.)
3 feet = 1 yard (yd.)
5| yards, or 16| feet = 1 rod (rd.)
320 rods, or 5280 feet = 1 mile (mi.)

Square Measure

144 square inches (sq. in.) = 1 square foot (sq. ft.)
9 square feet = 1 square yard (sq. yd.)
30| square yards = 1 square rod (sq. rd.)
160 square rods = 1 acre (A.)

640 acres = 1 square mile (sq. mi.)

Cubic Measure

1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.)
27 cubic feet = 1 cubic yard (cu. yd.)
128 cubic feet = 1 cord (cd.)

Weight

16 ounces (oz.) = 1 pound (lb.)
2000 pounds = 1 ton (T.)

Liquid Measure

4 gills (gi.) = 1 pint (pt.)
2 pints = 1 quart (qt.)

4 quarts = 1 gallon (gal.)
3l| gallons =1 barrel (bbl.)

2 barrels =1 hogshead (hhd.)

247

248 TABLES FOR REFERENCE

Dry Measure

2 pints (pt.) = 1 quart (qt.)
8 quarts = 1 peck (pk.)
4 pecks = 1 bushel (bu.)

Time

60 seconds (sec.) = 1 minute (min.)
60 minutes = 1 hour (hr.)
24 hours = 1 day (da.)
7 days = 1 week (wk.)
12 months (mo.) = 1 year (yr.)

365 days = 1 common year

366 days = 1 leap year

Value

10 mills = 1 cent (^ or ct.)
10 cents = 1 dime (d.)
10 dimes = 1 dollar ($)

Angles and Arcs

60 seconds (60'^) =1 minute (1')
60 minutes = 1 degree (1°)

Counting

12 units = 1 dozen (doz.)
12 dozen, or 144 units = 1 gross (gr.)
12 gross, or 1728 units = 1 great gross

Paper

24 sheets = 1 quire
500 sheets = 1 ream

Formerly 480 sheets of paper were called a ?eam. The word "quire "
is now used only for folded note paper, other paper being usually
sold by the pound.

INDEX

PAGE

Account .... 2, 4, 23, 57, 87
Accurate proportions . . . 140

Acute angle 112

triangle 112

Addition 33

Aliquot parts 42

Altitude 150, 200, 243

Amount of a note ..... 94

Angle 112

Arc 115

Area 162,164,167,169,172,174,190

Balance 2

Bank 79, 86, 90, 97

Base 114, 150, 200

Bill 48

Bisection 124

Cash check 43

Center 149, 150

Check 2,43,92, 155

Circle .... 115, 149, 194, 196
Circumference . . 115, 149, 194

Commercial paper 99

Compound interest .... 86

Cone 150, 244

Congruent figures 115

Constructing triangles . . . 116

Creditor 49

Cube 198

Cylinder 150,200

Debtor 49

Deposit slip 90

249

PAGE

Diameter . . 115, 149, 150, 194

Discount 44, 46, 97

Distances 222

Dividing a line 130

Drawing instruments . . . 115
to scale 136, 204

Ellipse 149

Equilateral triangle . . 112, 118

Face of a note 94

Formulas 164, 167, 170, 172, 194,

196, 200, 201

Fractions . . ^ 67

Geometric figures 112

measurement 155

patterns 132

Height 150,200

Hypotenuse 112, 241

Indorsement 92, 95

Instruments, drawing . . . 115

Interest 81

Invoice 50

Isosceles triangle . . . 112, 118

Lateral surface .... 243, 244

Lathing 203

Length 155

Line 114

List price 44

Locating points 219

250

INDEX

PAGE

Maker of a note 95

Map drawing 217

Marked price 44

Material for daily drill . . . 105

Metric measures 205

Miscellaneous problems . . 30, 73,

102, 212, 234

Multiplication 37

Net price 44

Note 94

Oblique angle 112

Obtuse angle 112

triangle 112

Outdoor work 153, 156, 178, 192

213, 235
Overhead charges 74

Pantograph . 145

Parallel lines 128

Parallelogram .... 114, 167

Payee 92, 95

Per cent 6, 7

Percentage problems . . . . ' 17

Perimeter 112, 114

Perpendicular . . . . 121, 222

Photograph 143

Plane figures 149

Plastering 203

Polygon 114, 174

Position . . . 215, 219, 227, 229
Postal savings bank .... 89

Price list 24, 47

Principal 81, 94

Problems without figures . . 154

214, 236
without numbers 32, 54, 66,

76, 104
Proceeds 97

PAGE

Promissory note 94

Proportion 184

Proportional 186

Protractor 115

Pyramid 243

Quadrilateral 114

Radius ... 115, 149, 150, 194

Rate of interest 81, 95

Ratio 145, 182

Receipted bill 49

Rectangle 114, 164

Rectangular solid 198

Review drill 31, 53, 65, 75, 103, 152

Right angle 112

triangle . 112

Savings bank 79, 86

Several discounts 46

Short cuts in multiplication . 38
Similar figures .... 141, 186
Six per cent method .... 100

Size 155

Sphere 150, 245

Square 114, 196, 237

roots 237

Squared paper 162

Subtraction 36

Symmetry 147

Tables for reference .... 247

Trapezoid 114, 172

Triangle ..'.... 112, 169

Units of area 164

Uses of angles 231

Vertex ... 112, 114, 150, 243
Volume . 198, 200, 243, 244, 245

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