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SAMPLE^ COPY. New Rudiments of Arithmetic.

PRICE.

Fop Introduction, _______ ^O cts

Allowance for old book in use of similar grade, "when

given in exchange, --__-_- TO ct».

Books ordered for introduction will be delivered at above named prices in any part of the United States. Sample copies for examination, with a view to introduction, will be sent by mail to teachers or school ofjksrs on receipt of the introduction price. Address,

Clark & Maynard,

5 Barclay St., New York.

(P. O. Box 1«19.)

THOMSON'S NEW GRADED SERIES.

NEW

RUDIMENTS

OF

AKITHMETIC

COMBINING

MENTAL AND SLATE EXERCISES

FOR

INTERMEDIATE DEPARTMENTS.

By JAMES B. THOMSON, LL. D.,

AUTHOR OF DAY & THOMSON'S ARITHMETICAL SERIES ; EDITOR OF DAY'i SCHOOL ALGEBRA, LEGENDRE'S GEOMETRY, ETC.

NEW YORK:

CLAEK & MAYNARD, PUBLISHERS,

5 barclay street.

Chicago: 46 Madison Street.

• wn<oa,

Thomson's" Mathematical Series. — ~~ ^ft.

I. A Graded Series of Arithmetics, in three Books, viz. :

New Illustrated Table Book, or Juvenile Arithmetic. With oral and slate exercises. (For beginners.) 128 pp.

New Rudiments of Arithmetic. Combining Mental with Writ- ten Arithmetic. (For Intermediate Classes.) 224 pp.

New Practical Arithmetic. Adapted to a complete business education. (For Grammar Departments.) 384 pp.

II. Independent Books.

Key to New Practical Arithmetic. Containing many valuable suggestions. (For teachers only.) 168 pp.

New Mental Arithmetic. Containing the Simple and Com- pound Tables. (For Primary Schools.) 144 pp.

Complete Intellectual Arithmetic. Specially adapted to Classes in Grammar Schools and Academies. 168 pp.

III. Supplementary Course.

New Practical Algebra. Adapted to High Schools and Acad- emies. 312 pp.

Key to New Practical Algebra. With full solutions. (For teachers only.) 224 pp.

New Collegiate Algebra. (In preparation.)

Complete Higher Arithmetic. (In preparation.)

##* Each book of the Series is complete in itself.

Copyright, 1872, by James B. Thomson.

EDUCATION DEPT.

PREFACE.

The "New Graded Series," of which this is the second book, is divided into three parts. The object of this arrangement is convenience and economy.

While there may be objections to an " indeterminate series of school-books," it must be admitted that exer- cises in reading, arithmetic, etc., which are adapted to the capacity of beginners, are totally unfit for advanced classes. In view of this fact, it requires no arguments to show that a "limited series," adapted to the different capacities of learners, is a dictate of common sense.

Each book in this Series is complete in itself. The defini- tions and principles, so far as each extends, are expressed in the same language, but the examples are all different.

The present work consists of a course of Mental and Written Exercises combined. It is designed :

ist. To develop the elementary principles of the science by oral examples.

2d. To familiarize the pupil with the application of these principles to the solution of problems requiring the use of the slate.

3d. To lead him to generalize the principles thus developed, and to put the steps of particular solutions into a concise statement, or General Rule.

4th. To secure accuracy and rapidity in the combina- tion of numbers.

Finally, the work is specially adapted to intermediate classes, who are beginning to " cipher." The New Rudi- ments, it is hoped, may facilitate the progress of pupils, and merit the approval of teachers.

James B. Thomson. New York, July. 1872.

SUGGESTIONS.

i. Pakticular attention should be paid to the assignment of Lessons. They should be neither too long, nor too short ; but adapted to the capacity of the class, and the time they have for preparation.

2. Thoroughness should be insisted on, at every step. The acceptance of an imperfect lesson, whether from sympathy, or inattention, is a positive injury to the pupil.

3. The most effective auxiliaries of thoroughness are frequent reviews and Tabular drills. " Practice makes perfect."

4. A perfect recitation implies both promptitude and correctness. In reciting problems, the analysis should be logical, and the lan- guage correct.

5. Pupils should be encouraged to study out different solutions of the same problem, and to exercise their judgment in selecting the most simple, logical, and concise.

6. Care should also be taken to prevent the Jidbit of adding by counting the fingers. Counting is not addition. Pupils should be taught to add numbers as a whole, and be able to name the sum of any two given digits, instantly.

7. The definitions should be carefully explained, and thoroughly committed to memory. Each principle and rule should be dwelt upon until the pupil comprehends it, and is able to give a correct account of it, in his own language, or that of the author.

8. Cultivate the habit of self-reliance in the solution of problems. It is better for the pupil to solve one example, independent of the answer and all extraneous aid, than a dozen by the help of a teacher, or a key.

9. Special pains should also be taken to cultivate the perceptive faculties, and correct the erroneous ideas of learners as to distance, surface, weight, etc.

10. In developing the idea of Fractions, and the Units of Weights and Measures, let the pupil divide some object into halves, thirds, etc., and, if possible, let him see and handle the actual stiutdardx of length, surface, capacity, and weight. These simple acts will give him a more exact idea of Fractions, and of Weights and Measures, than a score of pictures, or a talk an hour long.

CONTENTS.

PAGE

Number, .. ....--7

dotation, ------- 8

Arabic Notation, ------ 8

Roman Notation, - - - - - ' 13

Numeration, - - - - - 14

Addition, - - - - - - 17

When the sum of a column is less than 10, - - 24

What numbers can be added together, - - - 25

When the sum of a column is 10 or more, - - 26

Carrying illustrated, - - - - - 27

General Rule for Addition, - - - - 23

Subtraction, - - - - - - 32

When each figure of the subtrahend is less than the one

above it, ------ 37

What numbers can be subtracted one from another, - 38 When a figure in the lower number is larger than the

one above it, - - - - 41

General Rule for Subtraction, - - - - 42

Drill in Rapid Combinations, - - - 44

Multiplication, - - - - - - 45

When the Multiplier has but one figure, - - 49

When the Multiplier has two or more figures, - - 53

General Rule for Multiplication, - - - - 54

Contractions, - - - - - * 5°

Questions for Review, - - - - 59

Division, ------- 60

Objects of Division illustrated, - - - 63

Short Division, ------ 64

Long Division, - . - - - - 70

Contractions, ------ 73

Drill in Rapid Combinations, - "75

Questions for Review, ..... 76

Factoring, • - - • - -79

CONTENTS.
	PAGH
Cancellation, -	81
Greatest Common Divisors, -	- 83
Least Common Multiple, -	86
Fractions,	- 88
Notation of Fractions,	90
General Principles of Fractions,	- 93
Reduction of Fractions, -	93
Common Denominator,	- 101
Least Common Denominator	102
Addition of Fractions,	- 104
Subtraction of Fractions, -	107
Multiplication of Fractions, -	- no
Division of Fractions,	116
Reducing Complex Fractions to	Simple ones, - - 122
Questions for Review,	124
Fractional Relation of Numbers,	- 126
Decimal Fractions, -	129
Notation of Decimals,	- 130
Reduction of Decimals, -	133
Addition of Decimals,	- 135
Subtraction of Decimals, -	136
Multiplication of Decimals, -	- 138
Division of Decimals,	140
United States Money, -	- 143
Addition of U. S. Money,	147
Subtraction of U. S. Money, -	- 149
Multiplication of U. S. Money,	150
Division of U. S. Money,	- 152
Applications of U. S. Money,	155
Compound Tables,	- 158
Reduction,	i73
Measurement of Rectangular Su	rfaces, • - - 179
Measurement of Rectangular So^	ids, - - - 180
Denominate Fractions,	- 1S2
Compound Rules,	- 185-191
Percentage, -	- 193
Profit and Loss, -	201
Interest,	- 204

RUDIMENTS.

DEFINITIONS

1. What is Arithmetic? Arithmetic is the science of numbers.

2. What is a single thing called ?

A Unit, or One.

3. If another is put with it, what ?

Two.

4. If another, and another, etc., what ?

Three, four, five, six, etc.

5. What is number ?

Number is a unit, or a collection of units.

6. When a number is not applied to any object, what is it called?

An Abstract Number.

7. When it is applied to some object, what ?

A Concrete Number. Give examples.

8. When numbers express units of the same kind, as, 3 apples and 4 apples, 5 and 7, etc., what are they called ?

Like Numbers.

9. When they express units of different kinds, as, 4 books and 6 pencils, what ?

Unlike Numbers.

NOTATION

10. What is Notation ?

Notation is the art of expressing numbers by figures, letters, or other numeral characters.

11. What are the two principal methods in use? The Arabic and the Roman.

ARABIC NOTATION.

1. What i9 the Arabic Notation, and why so called ?

The Arabic Notation is the method of express- ing numbers by figures ; and is so called because it was introduced into Europe from Arabia.

2. How many figures does it employ, and what called ?

Ten, called —

i, 2, 3, 4, 5, 6, 7, -8, 9, o.

One, two, three, four, five, six, seven, eight, nine, naught.

3. What are the first nine called, and why ?

The first nine are called significant figures, because each of them always expresses a number.

They are also called digits, from digitus, a finger, because the ancients used to reckon upon their fingers.

4. What is the last called, and why ?

The last is called naught, because when standing alone it has no value, and when connected with sig- nificant figures, it denotes the absence of the orders iu whose place it stands.

It is also called zero, or cipher.

Note. — The pupil should learn to distinguish and write tho Arabic figures with readiness before proceeding further.

5. How is each of the first nine numbers expressed ?

By a single figure.

jB. What are these numbers called ?

Units of the first order, or simply units.

NOTATION. 9

7. What is the greatest number expressed by one figure?

Nine*

8. How is ten expressed?

Ten is expressed by writing i in the second place, with a cipher on the right ; as, io.

9. What is the I called, standing in the second place ?

A unit of the second order.

i. Explain and write each of the numbers from ten to twenty.

Eleven is composed of one ten and one unit, and is expressed by writing i in the second place to denote the ten, and i in the first or right hand place to denote the unit ; as, n.

Twelve is composed of one ten and two units, and is expressed by writing i in the second place, and 2 in the first ; as, 12.

Tliirteen is composed of one ten and three units, and is expressed by writing 1 in the second place, and 3 in the first; as, 13, etc.

Twenty is two tens, and is expressed by placing 2 in the second place, and o in the first ; as, 20.

2. Explain in like manner, and write each of the numbers from twenty to thirty.

3. From thirty to forty. From forty to fifty.

4. From fifty to sixty. From sixty to seventy ; and so on to one hundred.

10. What is the greatest number that can be expressed by tioo figures?

Ninety-nine.

11. How is a hundred expressed?

A hundred is expressed by writing 1 in the third place, with tivo ciphers on the right ; as, 100.

12. What is the 1 called, standing in the third place?

A unit of the third order.

10 NOTATION.

5. Explain and write each of the numbers from one hundred to one hundred and ten.

One hundred and one equals one hundred, no tens, and one unit, and is expressed by writing 1 in the third place, o in the second, and 1 in the first ; as, 101.

One hundred and two is expressed by 102 ; one hun dred and three by 103, etc. ,

6. Explain and write each number from one hundred and ten to one hundred and twenty.

One hundred and ten is composed of one hundred, one ten, and no units, and is expressed by writing 1 in the third place, 1 in the second, and o in the first; as, no.

One hundred and eleven by in; one hundred and twelve by 112, etc.

7. Explain in like manner, and write the numbers from one hundred and twenty to one hundred and thirty.

8. Write the numbers from one hundred and thirty to one hundred and fifty. From one hundred and fifty to two hundred.

9. Write two hundred. Three hundred. Four hun- dred. Five hundred.

Write the following numbers in figures :

10. One hundred and twenty-three.

1 1. Two hundred and thirty-seven.

1 2. Three hundred and forty-five.

13. Four hundred and ten.

14. Six hundred and seven.

15. Five hundred and sixty-three.

16. Six hundred and five.

1 7. Seven hundred and thirty.

18. Six hundred and seventy-five.

19. Eight hundred and forty-three.

20. Nine hundred and ninety-nine.

NOTATION. 11

13. What is the largest number that can be expressed by three figures ?

Nine hundred and ninety-nine.

Note. — The preceding exercises should be repeated and sup- plemented by dictation, until the class become perfectly familiar with writing numbers less than a thousand.

1 -4. How are numbers larger than 999 expressed ?

By Other Orders, called thousands, tens of thou* Bands, hundreds of thousands, millions, tens of millions, ate, each succeeding order having ten times the value of the preceding.

1 5. What is the general law by which the orders of units Increase ?

TJiey increase from right to left by the scale of ten ; that is,

Ten simple units make one ten ;

Ten tens make one hundred ;

Ten hundreds make one thousand ; and, universally, ten of any loiver order make one of the next higher.

16. What places do the different orders occupy? Simple units occupy the right hand place ; Tens, the second place ;

Hundreds, the third place ; Thousands, the fourth place ; Tens of thousands, the fifth place ; Hundreds of thousands, the sixth place ; Millions, the seventh place, etc. ; the order of units corresponding with the place which the figure occupies.

17. What is the effect of moving a figure from right to left, or from left to right.

Its value is increased tenfold for every place it is moved from right to left ; and is diminished tenfold for every place it is moved from left to right.

18. What are the different values of a figure called ? The simple and local values.

12 NOTATION-.

1 9. What is the simple value of a figure ?

The simple value of a figure is the number of units it expresses when it stands alone.

20. The local value of a figure ? Illustrate both.

The local value is the number it expresses when connected with other figures, and is determined by the place it occupies, counting from the right

21. What is the rule for expressing numbers by figures ? Begin at the left hand, and write the figures of the

given orders in their successive places toicard the right.

If any intermediate orders are omitted, supply their places ivith ciphers.

Write the following numbers in figures :

20. One thousand, three hundred, and sixty.

2i. Five thousand, seven hundred, and thirty-five.

22. Seven thousand, three hundred, and sixty-two.

23. Twenty-six thousand and seventy-five.

24. Thirty-seven thousand, one hundred, and six.

25. Ninety-five thousand and seventeen.

26. One hundred and twenty-three thousand and two hundred.

27. Three hundred and forty-eight thousand and two hundred.

28. Four hundred and ten thousand, three hundred, and forty.

29. Five hundred and forty thousand, six hundred, and thirty.

30. Six hundred thousand, two hundred and forty.

31. Seven hundred and fifty-five thousand, two hun- dred, and three.

32. Eight hundred and fifty thousand and three hundred.

33. Nine hundred and thirty-eight thousand and sixty-eight.

NOTATION.

13

ROMAN NOTATION.

22. What is the Roman Notation ; and why so called?

The Roman Notation is the method of express- ing numbers by letters ; and is so called because it was employed by the Romans.

23. How many, and what letters are used ? Seven capitals, viz, : I, V, X, L, C, D, and M.

24. What does each of these letters express ? * The letter I expresses one ; V, five ; X, ten; L, fifty,

C, one hundred ; D, five hundred ; and. M, one thou* sand.

		TABLE.
I	denotes one.	XXIV	denotes twenty-four.
II	m	two.	XXV	" twenty-five.
III	"	three.	XXVI	" twenty-six.
IV	u	four.	XXVII	" twenty-seven.
V	a	five.	XXVIII	" twenty-eight.
VI	«	six.	XXIX	twenty-nine.
VII	**	seven.	XXX	" thirty.
VIII	M	eight.	XL	forty.
IX	«	nine.	L	« fifty.
X	M	ten.	LX	" sixty.
XI	M	eleven.	LXX	" seventy.
XII	«	twelve.	LXXX	" eighty.
XIII	t(	thirteen.	XC	" ninety.
XIV	«	fourteen.	C	" one hundred.
XV	«	fifteen.	CC	f two hundred.
XVI	<i	sixteen.	CCC	" three hundred.
XVII	"	seventeen.	cccc	M four hundred.
XVIII	U	eighteen.	D	" five hundred.
XIX	t(	nineteen.	DC	" six hundred.
XX	«	twenty.	DCC	" seven hundred.
XXI	m	twenty-one.	DCCC,	" eight hundred.
XXII	u	twenty-two.	DCCCC	" nine hundred.
XXIII	11	twenty-three.	M	" one thousand.
MDCCCLXXII, one thousand,	eight hundred, and seventy-two.

14

NUMERATION.

Notes.— i. Repeating a letter repeats its value. Thus, 1 de- notes one ; II, two ; III, three; X, ten ; XX, twenty, etc.

2. Placing a letter of less value fo/<?re one of greater value, diminishes the value of the greater by that of the less; placing the less after the greater increases the value of the greater by that of the less. Thus, I denotes one, and V five ; but IV is four and VI six.

3. Placing a Iwrizontal line over a letter increases its value a thousand times. Thus, I denotes a thousand ; X, ten thousand ; C, a hundred thousand ; M, a million.

4. Four was formerly denoted by IIII ; nine, by Villi ; forty, by XXXX ; and ninety, by LXXXX.

Express the following numbers by letters :

I. 17.	7. 48.	13. 98.	19. 564.
2. 26.	8. 53-	14. in.	20. 896.
3. 24.	9. 67.	15. 109.	21. 1116.
4- 37-	10. 78.	16. 114.	22. 1320.
5. 29.	11. 84.	17. 118.	23- 1536.
6. 44.	12. 89.	18. 377-	24. 1876.

NUMERATION

25. What is Numeration ?

Numeration is the art of reading numbers ex- pressed by figures, letters, or other numeral characters.

26. How read numbers expressed by figures?

Divide them into periods of three figures each, count- ing from the right.

Beginning at the left hand, read the periods in suc- cession, and add the name to each,except the last.

27. Why is the name of the last period omitted ? Because the right hand period always denotes simple

units, therefore its name need not be mentioned.

2* U M E R A T I 0 :N\

15

Repeat the Numeration Table, beginning with units. NUMERATION TABLE.

5 7

2 3:

1 » S

3 a S

& £ e

8 7 4,

I

W H b 267.

Period V. Trillions.

Period IV. Billions.

Period IH. Millions.

Period II. Thousands.

Period I. Units.

The periods in the Table are thus read : 298 trillions, 570 billions, 923 millions, 874 thousand, two hundred and sixty-seven.

Note. — This method of reading numbers is commonly ascribed to the French, and is thence called the French Numeration.

Others ascribe it to the Italians, and thence call it the Italia . Method.

Copy and read the following :

I. 107.	13-	10315-	25-	342^201
2. 260.	14.	12065.	26.	23C0400
3- 3i5-	*5-	24308.	27.	5408025
4. 809.	16.	13020.	28.	4532°265
5. 1020.	17-	20460.	29.	63205308
6. 2405.	18.	35°°7-	30-	265310275
7. 3200.	19.	40800.	3i-	8045007
8. 5007.	20.	85408.	32-	925604300.
9. 6080.	21.	115326.	33-	4260345809
10. 7650.	22.	208065.	34-	61204400000
11. 8075.	23.	400304.	35-	160240030045
12. 9364.	24.	803025.	36.	407008300416,

L6	NUMERATION.
Copy and read the following:
i. III.	II. LIX.	21. CIV.
2. VI.	12. LVIII.	22. cvin.
3- iv.	13. LXIX.	23. CXII.
4. VIII.	14. LIV.	24. CIX.
5- VII.	15. LXXIII.	25. CXL.
6. IX.	16. XLIX.	26. CLXXIX.
7. XIII.	17. LXXXIV.	27. ccxc.
8. XXXII.	18. LXXXVIII.	28. DXIX.
9. XXVIII.	19. xov.	29. MDCCCXI.
0. XLII.	20. XCIX.	30. MDCCCLXXV.

Express by figures, and read the following numbers :

1. Two hundred and five thousand, six hundred, and ninety-one.

2. Eight hundred and forty thousand, five hundred, and nine.

3. Two millions, four hundred thousand, and seventy.

4. Forty-five millions, sixty thousand, two hundred, and sixty.

5. Three hundred and ninety millions, four thousand, a_ ^ seventy-two.

C Six hundred millions, forty -eight thousand, and ten.

7. Five billions, six hundred and ten millions, and three lundred.

8. One hundred and forty billions, and thirty-five mil- lions. ,

9. Forty-five millions, seven hundred and sixty thou- sand.

10. Three hundred and twenty-nine trillions, six hundred and thirty-seven billions, three hundred and forty millions, four hundred and nineteen thousand, two hundred and eighty-four.

Note. — Dictation exercises in reading and writing numbers should be continued, till the class is perfectly familiar with both.

ADDITION.

MENTAL EXERCISES.

To Teachebs.— The object of this Exercise is to teach beginners tha process of adding two digits together. If young, let them illustrate the examples by counters or unit marks.

i. If you have i apple, and I give you i apple more, how many apples will you have ?

" One apple and i apple more are 2 apples/,

2. If you have 2 cents, and you find 1 more, how many cents will you have ?

" Two cents and 1 cent more are 3 cents."

3. How many are 3 marbles and 2 marbles ?

4. Show this by your fingers.

5. Sarah has 3 red roses, and 3 white ones : how many roses has she of both kinds ? Show it.

6. If an orange costs 5 cents, and a lemon 4 cents, how much will both cost ? Show it.

7. William earned 6 cents in the morning, and 4 in the afternoon : how much did he earn in both ?

8. Sanford obtained 6 credit marks, and his sister 6 : how many did both obtain ? Show it.

9. If I gather 6 quarts of cherries, and buy 8 quarts, how many quarts shall I have ? Show it.

10. Joseph picked 5 quarts of blackberries, and his brother 7 quarts: how many quarts did both pick ?

11. If I pay 8 dollars for a barrel of flour, and 7 dol- iars for a ton of coal, what shall I pay for both ?

12. A teacher received two bouquets, one containing 8 flowers and the other 10': how many flowers were there in both ?

13. George picked 12 peaches from one tree, and 3 from another : how many did he pick from both ?

18

ADDITION.

ADDITION TABLE

2 and	3	and	4 and	5	and
i are 3	1	are 4	1 are 5	1	are 6
2 " 4	2	" 5	2 " 6	2	? 7
3 " 5	3	" 6	3 " 7	3	" 8
4 " 6	4	« 7	4 " 8	4	" 9
5 " 7	5	" 8	5 " 9	5	" 10
6 " 8	6	" 9	6 " 10	6	" 11
7 " 9	7	" 10	7 " 11	7	" 12
8 * io	8	" 11	8 " 12	8	u I3
9 " ii	9	" 12	9 " J3	9	« 14
IO " 12	10	" 13	10 " 14	10	" 15
6 and	7	and	8 and	9	and
i are 7	1	are 8	1 are 9	1	are 10
2 " 8	2	" 9	2 " 10	2	" 11
3 * 9	3	" 10	3 " 11	3	« 12
4 " 10	4	" 11	4 " 12	4	" 13
5 " ll	5	u 12	5 " i3	5	" 14
6 "12	6	" 13	6 " 14	6	" 15
7 " 13	7	" 14	7 " 15	7	« 16
8 " 14	8	" i5	8 " 16	8	" i7
9 " 15	9	" 16	9 " 17	9	« 18
10 u 16	10	« I7	10 " 18	10	" 19

1. Show by counters or unit marks how many 2 added to 3 will make.

2. Show how many 2 added to 4 will make.

3. Show how many 2 added to 5 will make.

4. Show how many 3 added to 4 will make.

5. Show how many 3 added to 5 will make.

6. Show how many 4 added to 5 will make, etc.

*** Particular care should be taken to see that beginners understand how the Addition Table is constructed; that they fully comprehend the results of adding two digits together, be- fore they are requirod to commit them to memory.

ADDITION. 19

DEFINITIONS.

1. What is Addition ?

Addition is uniting two or more numbers in one.

2. What is the number obtained by addition called ?

The Sum or Amount ' Note. — The sum or amount contains as many units or ones as all the numbers added.

3. When the numbers to be added are the same denomination, what is the operation called ?

Simple Addition.

4. How is Addition denoted ?

By a perpendicular cross called plus ( + ), placed between the numbers to be added. Thus, 5+3 shows that 5 and 3 are to be added together, and is read, " 5 plus 3," " 5 and 3," or " 5 added to 3."

Note. — The term plus is a Latin word, signifying more, or added to.

5. How is the equality between two numbers or sets of num bers denoted ?

By two short parallel lines, called the sign of equality (=). The expression 5+3 = 8, shows that 5 increased by 3 equals 8, and is read, " 5 plus 3 equal 8," or the sum of " 5 plus 3 equals 8."

Read the following expressions : 7 + 2 = 9; 6 + 4 = 7+3; 17 +3 + 5 = I2 + 7 + 6; 21+8 + 9 = 12 + 20 + 6 ; 30 + 3 + 20 = 40 + 13.

MENTAL EXERCISES. To Teachers.— The Mental and Slate Exercises are intended to be com- bined in each recitation. Hence, no more examples should be assigned to a lesson, than the class can thoroughly master.

i. Henry gave 4 cents for an orange, 3 cents for a pear, and 2 cents for an apple : how many cents did he give for all ?

Analysis. — 4 cents and 3 cents are 7 cents, and 2 are 9 cents. Therefore, be gave 9 cents for all.

20 ADDITION.

2. John paid 5 cents for a writing-book, and 2 cents for a pen : how many cents did he pay for both ? 5 and what make 7 ?

3. If an orange costs 4 cents, a pear 3 cents, and a peach 2 cents, what will all three cost ?

4. George gave 4 apricots to one of his sisters, 3 to another, and 5 to another: how many did he give to all?

5. If you pick up 4 apples under one tree, 3 under another, and 5 under another, how many apples will you have ?

6. A man paid 3 dollars for a cane, 5 dollars for an umbrella, and 4 dollars for a hat : how much did he pay for all ?

7. How many are 5 cents, and 3 cents, and 2 cents, and 1 cent?

8. Sarah paid 9 dollars for a hat, and 4 dollars for a parasol : what did she pay for both ?

9. Count by twos rapidly to 60. Thus, two, four, six, eight, ten, twelve, etc.

10. Count by threes in like manner to 60.

11. Count by fours to 100.

SLATE EXERCISES. Columns of Single Figures.

1. Write and add 2, 4, 5, 3, 6, and 1, upward and downward four times.

2 Explanation. — Write the numbers in a column,

and draw a line under it.

First. Beginning at the bottom, add upward ; as, ->

one, seven, ten, fifteen, nineteen, twenty-one. Set 21 ~ under the column.

Second. Begin at the top, and add downward ; as, _J_

two, six, eleven, etc. 2 1 A ns.

ADDITION. 21

Copy and add the following upward and down ward orally till rapidity is attained :

(»•) (3-) (4-) (5-) (6.) (7-) (8.) (9.)

2	3	4	5	2	3	4	5
4	2	3	2	I	2	i	3
3	4	5	3	5	4	3	2
i	2	i	2	2	3	4	5
2	I	3	3	I	i	2	2
2	2	2	i	3	2	5	4
I	3	4	2	2	3	3	3
2	2	3	3	4	4	3	4
3	I	2	2	i	2	4	5
4	5	5	4	_5	3	5	4

%* Care should be taken to write figures with neatness and symmetry, and set them in perpendicular columns.

MENTAL EXERCISES, i. There are 5 ducks in one pond, 6 in another, and 7 in another : how many are there in all the ponds ?

2. A man picked 8 quarts of blackberries, and his son 6 quarts : how many quarts did both pick ?

3. William has 9 chickens in one coop, and 7 in another : how many chickens has he in both ?

4. 7 and what make 16 ? 9 from 16 leaves what ?

5. How many are 7, and 6, and 5, and 4, and 8 ?

6. If there are 9 pears in one dish, and 8, in another, Aow many are there in both dishes ?

7. 8 and what make 17 ? 9 from 17 leaves what ?

8. Henry solved 6 examples, George 9, and Samuel 8 : how many did they all solve ?

9. Lucy gave her teacher 7 roses, Julia 9, and Hattie 10 : how many roses had the teacher?

10. Count by fives to 100, with rapidity.

11. Count by sixes to 100, in like manner.

22 ADDITION.

SLATE EXERCISES, Copy and add the following as before :

!•)	(2.)	(3.)	(4.)	(50	(6.)	(7.)	(8.)
6	5	4	7	8	7	4	9
3	4	5	4	6	5	7	5
5	2	6	7	2	8	9	4
4	5	7	3	8	3	i	9
2	3	4	5	4	4	3	6
5	4	7	7	8	8	9	9
4	6	3	4	5	6	7	8
6	4	7	8	9	5	6	4
3	7	6	9	4	8	4	3
7	5	3	5	6	7	8	9

MENTAL EXERCISES.

i. Henry had 5 cents, and earned 6 more: how many cents had he then ? 5 and what make 1 1 ?

2. William picked 7 quarts of cherries, and his brother 6 quarts : how many quarts did both pick ?

3. 5 and what make 12 ? 6 and what make 13 ?

4. A farmer had 8 cows, and bought 6 more : how many had he then ?

5. If you pay 7 dollars for a velocipede, and 8 dollars for an overcoat, what will both cost you ?

6. 6 and what make 14? 7 and what make 15 ?

7. If Helen pays 8 dollars for a music-box, and 9 dol- lars for a fur cape, what will both cost her ?

8. How many are 9 dollars, and 6 dollars, and 7 dol- lars?

9. How many are 15 rods, 6 rods, and 8 rods ?

10. Count by sevens to 100, with rapidity.

11. Count by eights to 100, in like manner.

ADDITION. 23

SLATE EXERCISES. Copy and cAd, the following upward and downward as before :

(I.)	(»■)■	(3-)	(4.)	(5-)	(6.)	(7-)	(8.)
4	3	2	9	5	6	8	9
3	4	3	8	3	8	8	8
6	2	5	4	8	9	5	4
8	3	7	3	4	7	6	7
I	i	6	i	2	4	7	5
2	6	i	7	3	5	3	6
7	7	3	6	9	3	5	7
3	2	4	5	6	7	9	8
5	7	6	8	7	6	5	7
8	6	7	4	6	7	8	9

Write in columns, and add the following:

9. 3 pounds, 4 pounds, 5 pounds, 2 pounds, 7 pounds, and 1 pound.

10. 6 yards, 5 yards, 3 yards, 4 yards, 2 yards, 8 yards, and 7 yards.

MENTAL EXERCISES.

1. How many are 3 and 10 ? 13 and 10 ? 23 and 10 ? 33 and 10 ? 43 and 10 ? 53 and 10 ? 63 and 10 ? 73 and 10 ? 83 and 10 ? 93 and 10 ?

2. 7 and 10 ? 17 and 10 ? 27 and 10 ? 37 and 10 ? 47 and 10? 57 and 10? 67 and 10? 77 and 10? 87 and 10 ? 97 and 10?

3. 3 and 4? 13 and 4? 23 and 4? 33 and 4? 53 and 4 ? 43 and 4 ? 63 and 4 ? 83 and 4 ? 93 and 4 ?

4. 5 and 3 ? 15 and 3 ? 25 and 3 ? 45 and 3 ? 35 and 3 ? 55 and 3 ? 65 and 3 ? 85 and 3 ? 75 and 3 ? 95 and 3 ?

24 ADDITION.

5. 17 and 5 t 37 and 5 ? 27 and 5 ? 57 and 5 ? 47 and 5 ? 67 and 5 ? 87 and 5 ? 77 and 5 ? 97 and 5 ?

6. 15 and 6 ? 35 and 6 ? 25 and 6 ? 45 and 6 ? 65 and 6 ? 55 and 6 ? 75 and 6 ? 95 and 6 ? 85 and 6 ?

7. 18 and 7 ? 28 ivnd 7 ? 38 and 7 ? 48 and 7 ? 58 and 7? 68 and 7? 78 and 7? 88 and 7 ? 98 and 7 ?

8. 16 and 8 ? 26 and 8 ? 36 and 8 ? 46 and 8 ? 56 and 8? 66 and 8? 76 and 8? 86 and 8? 96 and 8?

9.* 14 and 9 ? 24 and 9 ? 34 and 9 ? 44 and 9 ? 54 and 9 ? 64 and 9 ? 74 and 9 ? 84 and 9 ? 94 and 9 ?

10. Count by nines to 100, with rapidity.

11. Count by tens to 100, in like manner.

SLATE EXERCISES.

When the Sum of a Column is Less than 10.

1. What is the sum of 234 dollars, 423 dollars, and 132 dollars ?

Analysis. — Wri te the numbers one under another, Operation. the units figures in one column, the tens in the next, 234 and so on. Begin at the right and add : 2 units 423 and 3 units are 5 units, and 4 are 9 units. Set the i<,2

9 under the units column, because it is units.

Next, 3 tens and 2 tens are 5 tens, and 3 are 8 tens. 7°9 tlols. Set the 8 under the tens column, because it is tens. Finally, 1 hundred and 4 hundred are 5 hundred, and 2 are 7 hun- dred. Set the 7 under the hundreds, because it is hundreds.

Note.— In practice, it is better simply to pronounce the results ; as, two, five, nine, etc.

Copy and add the following, in like manner:

(2.) (3.) (4.) (5.) (6.) (70 (8.) (90

21 32 14 43 32 321 124 434

34 11 23 2I 2° i32 53i 243

12 45 42 34 45 434 233 322

ADDITION. 25

Write in columns, and add the following :

10. 25 + 12+30. 13. 34+21+40.

11. 31 + 22 + 24. 14.46 + 30 + 12. 12.40 + 13 + 36. 15. 51 + 25 + 23.

6. How do you write numbers to be added ? Write units under units, tens under tens, etc.

7. Where begin to add, and how proceed ?

Begin at the right, and add each column separately.

8. When the sum of a column is less than 10, what is done with it ; and why ?

Set it under the column added; because it is the same order as that column.

9. What two principles are necessary to be observed in addition ?

1 st. The numbers must be Like numbers. 2d. Units of the same order must be added, each to each.

Can 3 books be added to 5 pencils ? Why ?

Do 4 units and 3 tens make 7 units, or 7 tens ? Why ?

MENTAL EXERCISES.

1. A certain school had 40 girls and 30 boys in attendance : how many pupils did it contain ?

Analysis. — 40 is equal to 4 tens, and 30 is equal to 3 tens. Now 4 tens and 3 tens are 7 tens, or 70. The school, therefore, contained 70 pupils.

2. 50 is how many tens? 40? 60? 70? 80? 100?

3. 5 tens are how many ? 7 tens? 6 tens? 9 tens? 8 tens? 10 tens?

4. How many are 3 tens and 5 tens ? 6 tens and 4 tens ? 5 tens and 7 tens ?

5. How many are 20 and 30 ? 50 and 80 ?

6. How many are 70 and 50 ? 80 and 90 ?

2

26 ADDITION.

7. In a certain grove there are 89 sugar-maples and 46 elms : how many trees are there in the groye ?

Analysis. — 89 is 8 tens and 9 units, and 46 is 4 tens and 6 units. Now 8 tens and 4 tens are 12 tens, or 120 ; and 9 units and 6 units are 15 units, which, added to 120, make 135. There- fore, there are 135 trees in the grove.

Note. — When numbers to be added mentally are large, it is advisable to separate them into the units, tens, etc., of which they are composed, and begin to add with the highest order.

8. How many are 34 and 43 ? 26 and 51 ?

9. How many are 45 and 62 ? 71 and 46 ?

10. A man paid 68 dollars for a cow. and 55 dollars for a colt : what did he pay for both ?

SLATE EXERCISES. When the Sum of a Column is 10, op More.

1. A farmer had 3 flocks of sheep ; one of 325, another 436, the other 541 : how many sheep had he ?

Analysis. — We write units under units, tens Operation.

under tens, etc., and begin at the right hand 325 8*

as before. Thus, 1 unit and 6 units are 7 units, 436 s.

and 5 are 12 units, or 1 ten and 2 units. We set ^41 s.

the 2 units under the column added, because it is

the same order as this column, and add the 1 ten -c,■/'0• xov °« to the next column, because it is the same order as that column. Now 1 ten added to 4 tens makes 5 tens, and 3 are 8 tens, and 2 are 10 tens, or 1 hundred and o tens. \V« sot the o, or right hand figure, under the column added, because there are no units of this order, and add the 1 hundred to the next column, because it is the same order as that column. Add- ing the 1 hundred to the next column, the sum is 13 hundred, or 1 thousand and 3 hundred. This being the last column, we set down the whole sum, putting the 3 under the column added, be- cause it is the same order as this column, and the 1 in the next, 01 thousands place, because it is thousands. Therefore, etc.

ADDITION.

27

10. When the sum of a column exceeds 9, what do you do ? Write the units figure under the column, and add the

tens to the next higher order.

11. What do you do with the last column? Set down the whole sum.

- 12. What is adding the tens to the next order called ? Carrying the tens.

	Thou.	HUND.	Tens.	Units.
		Ill	11	Hill
		llll	III	llllll
		Mill	llll	1
	M.	H.	T.
A	.m't 1	Ill	0	li

Carrying Illustrated by Unit Marks.

Analysis. — 325 = 3 hun- dred+2 tens +5 units; 436 = 4 hundred + 3 tens + 6 units; and 541 = 5 hundred + 4 tens + 1 unit. Now, to represent the first number, we place three counters in the column of hundreds, two in the column of tens, and Jive in the column of units. The other numbers are represented in a similar manner. Beginning at the right hand, we find there are 12 counters in units column. With- drawing ten of them, two will be left, which we put under the column added. Now 10 units make 1 ten ; hence, to represent the ten units withdrawn, we put a single counter (T.), denoting a unit of the 2d order, in the column of tens. Again, adding the column of tens, we find there are ten counters, and withd.-L.wing ten tens from this number, no tens are left. To express the absence of tens, we put a cipher in tens place. But ten tens make 1 hundred ; hence, to represent the ten tens withdrawn, we put a single counter (H.), denoting a unit of the 3d order, in the column of hundreds. Finally, in the hundreds column, there are 13 counters, and withdrawing ten of them leaves three. We therefore put three counters in the column of hundreds, and to represent the ten withdrawn, we put a single counter (M.), de- noting a unit of the 4th order, in the column of thousands. The amount is 1 thousand, 3 hundred, o tens, and 2 units. Hence, carrying the tens is simply taking a part from a lower order and adding it to the next higher, which can no more affect the amount than it will affect the amount of money a man has, if he changes 10 cents for a dime.

28

		ADDITION.
M	(30	(4.)	(50	(6.)
4683	3605	5072	7304	8097
3427	7036	38l6	6og4	3845
5°32	5783	730I	3752	5132

13. The preceding principles may be summed up in the following

GENERAL RULE.

I. Place the numbers one under another, units under units, etc., and beginning at the right, add each column

II. If the sum of a column does not exceed nine, write it under the column added.

If the sum exceeds nine, write the units figure under the column, and carry the tens to the next higher order.

Finally, set down the whole sum of the last column.

Proof. — Begin at the top and add each column down- ward. If the two results agree, the work is right.

Note. — This method of proof depends upon the principle that reversing the order of the figures will be likely to detect any error that may have occurred in the operation. The learner should prove every answer.

EXAMPLES FOR PRACTICE.

(».)	(2.)	(3.)	(4.)	(5.)
Yards.	Pounda	Rods.	Dollars.	Acres.
135	333	496	542	604
26$	664	175	37	160
786	548	586	764	489
182	345	257	343	853
348	563	845	577	348

	ADDITION.
(6.)	(7-)	(8.)	(9.)	(10.)
684	103	496	840	965
937	85	37	6	4
685	967	4	28	382
129	49	132	394	4i
845	732	563	825	985

29

11. Herbert read 235 pages one day, 264 the next, *nd 362 the next : how many pages did he read in all ?

12. What is the sum of 2362 days, 375 days, and 27 days?

13. If a yoke of oxen cost 250 dollars, and a cart 119 dollars, what will both cost ?

14. A man paid 67 dollars for his coat, 16 for his vest, 23 for his pants, and 13 dollars for his boots: what did he pay for his suit ?

1 5. A farmer has four flocks of sheep, one flock con- taining 256 sheep, the second 320, the third 195, and the fourth 168: how many sheep had he ?

16. What is the sum of five hundred and sixty-one, two hundred and seven, and nine hundred and fourteen?

1 7. What is the sum of twelve thousand and twelve, six thousand and two, and ninety-five hundred ?

18. A man paid 2250 dollars for his farm, 1600 dol- lars for stock, and had 168 dollars left: how much money had he at first ?

19. Washington was born in the year 1732, and lived 67 years: what year did he die ?

20. A man paid 6270 dollars for a horse and sold it for 1565 dollars more than he paid for it: how much did he get for it?

21. In what year will a person born in 1865, be 21 years old ?

30 ADDITION.

22. What is the sum of 643 yards, 820 yards, 605 yards, and 319 yards?

23. If I pay 925 dollars for house rent, 430 dollars for clothing, and 768 dollars for other expenses, how much shall I spend in a year ?

24. The age of four brothers is 89, 84, 78, and 67 years respectively : what is their united age ?

25. John has 63 marbles, Henry 41, and William as many as both the others : how many did they all have ?

26. What is the sum of 453 dols., 269 dols., 804 dols., 1000 dols. ?

27. In a certain army there are 28260 infantry, 16325 cavalry, and 1328 artillery : how many men did the army contain ?

28. A man bequeathed his wife 23260 dols., his son 17380 dols., and his daughter the same as his son : how much did he leave them all ?

29. What is the sum of 365 days, 873 days, 219 days, and 35 days?

30. If a vessel sails 235 miles a day on three succes- sive days, how far will she be from port ?

31. My neighbor's farm contains 563 acres, and my own 435 acres : how many acres do they both contain ?

32. Eequired the sum of 1725 years + 1007 years -+- 8520 years.

^. Required the sum of 1308 ounces + 710 ounces + 353 ounces and 42 ounces.

34. Required the sum of 2103 pounds + 106 pounds + 26 pounds + 89 pounds + 645 pounds.

35. If a man annually receives 1350 dollars salary, and 350 dollars interest, what is his annual income ?

36. A man lias four farms ; one containing 340 acres, another 235 acres, another 250 acres, and the other 178 acres : how many acres were there in all ?

		ADDITION.		3:
(370 (38.)	(390		(40.)		(4i.)	(42.)
Dols. Dols.		Dols.	Dols. Cts.	Dole. Cts.	Dols. Cts.
35 82		423	3	45	.	4 26	5 75
42 40		607	2	76	,	3 75	3 81
37 61		440	6	08	.	4 9°	1 09
61 43		851	4	30	;	5 7i	6 75
70 17		760	7	05	;	2 43	2 33
25 28		978	8	26	,	3 78	8 45
38 73		465	7	40	1	9 25	9 67
47 68		886	2	61	,	4 08	7 30
63 94		529	8	35	1	6 25	8 05
85 87		735	3	42	'	4 16	9 35
(430	(44.)	(450	(46.)	(470
Dols. Cts. Dols.	Cts.	Dols.	Cts.	Dols.	Cts.	Dols. Cts.
29 13	40	53	49	31	52	60	467 53
3° 5°	24	47	12	53	29	35	613 64
42 35	63	15	64	31	42	18	87 02
10 78	70	40	52	49	60	20	75* 34
2<? 42	34	^5	38	24	73	00	872 60
72 96	62	12	63	19	28	67	493 °4
26 04	73	68	72	43	49	28	81 73
50 30	58	76	30	04	83	00	757 02
29 l7	82	94	72	85	16	27	563 4o
36 23	64	47	37	23	40	23	80 33
47 58	87	28	52	92	7i	19	1 94
53 22	92	86	46	25	83	24	693 03
84 35	73	52	56	83	85	^	903 48

*** Exercises in adding long columns upon the slate or black, board are highly useful in acquiring accuracy and rapidity, and should be supplemented by the teacher. Columns of single figures are preferable for beginners.

SUBTRACTION

MENTAL EXERCISES.

To Teachers.— The design of this Exercise is to teach the pupil how to illustrate the 'process and the result of taking one number from another.

i. If you have 2 apples, and a boy takes 1 of them away, how many will you have left ?

" One apple taken from 2 apples leaves 1 apple."

2. Show this by your fingers or unit marks.

3. If you have 3 peaches, and give 1 of them to your sister, how many will you then have ? Show this.

4. If you have 5 cents, and lose 2 of them, how many will you have ? Show this.

5. George had 4 pears, and sold 2 of them : how many did he then have ? Show this.

6. Jennie had 6 roses, and gave 3 of them to her teacher: how many had she left ? Show this.

7. James had 6 cents, and spent 4 of them for candy: how many cents did he have left ? Show this.

8. A schoolboy had 10 marbles, and lost 5 of tliem: how many had he left ? Show this.

9. If you buy a slate for 8 cents, and sell it for 5 cents, how much will you lose ? Show this.

10. William gained 7 credit-marks, but lost 3 by bad conduct : how many did he then have ?

1 1. The price of a hat is 6 dollars, and that of a cap 2 dollars : what is the difference in their price ?

12. In a certain class there are 9 girls and 6 boys : how many more girls than boys in the class ?

13. If you pay 5 cents for an orange, and sell it for 10 cents, how much will you gain ?

14. 4 from 10 leaves how many ? 4 from 12 ?

SUBTRACTION.

33

SUBTRACTION TABLE.

2 from		2	; from		4 from		5 from
2	leaves	0	3	leaves	O	4 leaves	0	5 leaves 0
3	a	I	4	tit	I	5 "	1	6 u 1
4	a	2	5	a	2	6 "	2	7 " 2
5	u	3	6	a	3	7 "	3	8 " 3
6	ii	4	7	a	4	8 "	4	9 " 4 I
7	a	5	8	a	5	9 *	5	10 " 5
8	a	6	9	a	6	10 "	6	11 " 6
9	a	7	10	a	7	11 "	7	12 " 7
i IO	a	8	11	a	8	12 "	8	13 " 8
ii	a	9	12	a	9	13 "	9	14 " 9
12	a	10	13	a	10	14 "	10	15 " 10
6 from		7 from		8 from		9 from
6	leaves	0	7	leaves	0	8 leaves	0	9 leaves 0
7	u	1	8	a	1	9 "	1	10 " 1
8	ii	2	9	n	2	10 "	2	11 " 2
9	ii	3	10	a	3	11 "	3	12 " 3
IO	it	4	11	a	4	12 "	4	13 " 4
ii	ii	5	12	a	5	13	5	H " 5
12	li	6	13	it	6	14 "	6	15 " 6
13	ii	7	14	a	7	15 «	7	16 " 7
14	ii	8	15	a	8	16 "	8	17 " 8
15	ii	9	16	tt	9	17 "	9	18 " 9
16	ii	10	17	a	10	18 "	10	19 " 10

1. Show by counters or unit marks how many 2 takea from 3 will leave ?

2. Show how many 2 taken from 4 will leave.

3. Show "how many 2 taken from 5 will leave.

4. Show how many 3 taken from 7 will leave.

5. Show how many 3 taken from 8 will leave, etc.

Note. — It is advisable to let young pupils verify the resvlts of the Subtraction Table by counters or unit marks, before they are required to commit it to memory.

34 SUBTRACTIOK.

DEFINITIONS.

1 . What is Subtraction ?

Subtraction is taking one number from another,

2. What is the number to be subtracted called ?

The Subtrahend.

3. The number from which the subtraction is made ?

The Minuend.

4. What is the number obtained by subtraction called ?

The Difference, or remainder.

i. When it is said that 5 taken from 9 leaves 4, which is the minuend ? The subtrahend ? The remainder ?

2. When it is said that 6 taken from 14 leaves 8, what is the 6 called ? The 14 ? The 8 ?

5. When both numbers are the same denomination, what is the operation called ?

Simple Subtraction,

6. How is Subtraction denoted ?

By a short horizontal line, called minus (— ). When placed between two numbers, this sign shows that the number after it is to be taken from the one before it. Thus, 6 — 4, shows that 4 is to be taken from 6, and is read " 6 minus 4," or " 6 less 4."

Note. — The term minus is a Latin word, signifying lest.

Read the following expressions : 1. 12— 5 = 14 — 7. 4. 20 — 6 = 8 + 6.

5. 50-12 = 30+8.

6- 75 + 25 = 105 - 10 + 5c

2- I5~ 3 = 1° + 2-

3- 35-i° = 3<> — 5-

MENTAL EXERCISES.

1. If you pay 9 cents for a sponge, and sell it for 5 cents, how much will you lose ?

Analysis.— Five cents from 9 cents leave 4 cents. Therefore, you will lose 4 cents.

SUBTRACTION. 35

2. A farmer having 10 cows, sold 4 of them: how many had he left ? 6 and what make ten ?

3. Homer is 1 1 years old, and his sister 5 years : what is the difference in their ages ?

4. Susan had 8 pinks, and gave 3 to one of her school-- mates : how many had she left ? 3 and what make 8 ?

5. If you have 9 doves, and 3 of them fly away, how many will you have left ?

6. George having 12 dollars, gave 4 dollars for a pair of skates : how much had he left ?

7. The price of a vest is 8 dollars, and that of a coat 14 dollars: what is the difference in the price?

8. A farmer had 13 cows, and sold 4 of them: how many did lie then have ? 4 and what make 13 ?

9. Henry is 11 years old, and Charles 6 years: what is the difference in their ages ? 6 and what make 11?

10. Count backward by twos from 50 to o, with rapid- ity. Thus, fifty, forty-eight, forty-six, forty-four, etc.

11. Count backward by threes from 60 to o.

12. Count backward by fours from 60 to o.

SLATE EXERCISES. Copy the following, and subtracting the lower num' ber from the upper, set the result under the figure subtracted. Eepeat the operation till it can be per- formed without hesitation.

From 7 Take 4	8 5	(30 7 6	(4.) 9 5	(5-) 8 6	(6.) 8 7	(7.) 6 6	(8.) 9 5
(9.) From 8 Take 6	(10.) 7 5	(11.) 11 7	(12.) 9 7	(13.) 8 - 8	(i4.) 10 7	(15.) 9	(16.) 9 8

36 SUBTRACTION.

MENTAL EXERCISES, i. John bought 10 peaches, and gave 6 of them to his brother : how many did he have left ? 6 and what make io?

2. A man had 16 horses, and sold 7 of them: how many had he left? 7 and what make 16 ?

3. Julia solved 17 examples, and Harriet 8 examples: how many more did Julia solve than Harriet ?

4. 8 from 1 1 leaves how many ? 8 from 16 ? 8 from 14? 8 from 15 ?

5. If a man earns 15 dollars a month, and spends 9 dollars, how many dollars will he have left ?

6. 9 from 15 leaves how many? 9 from 17? 9 from 14?

7. A market-boy had 18 eggs in his basket, and let- ting it fall broke 8 of them : how many whole ones did he have left ?

8. Frank having 8 cents, wishes to buy a slate which costs 1 2 cents : how many cents more does he need, to pay for the slate ?

9. Count backward by fives from 70 to o with rapidity.

10. Count backward by sixes from 72 to o, in like manner.

SLATE EXERCISES. Copy and sulitract th 3 following as above :

(1.) (2.) (3.) (4.) (50 (6.) (7.) (8.) From 9 11 10 12 14 13 16 17 Take 67879879

(9-)	(10.)	(11.)	(12.)	(13.)	(MO	(15.) (16.)
From 14	16	*5	13	12	16 .	17 19
Talce 6	7	8	5	8	7	1 _?

SUBTRACTION. 37

MENTAL EXERCISES. i. George had 12 apples, and gave 5 of them away: how many had he left? 12 less 7 are how many ?

2. Horace is 14 years old, and his sister is 9: what is the difference in their ages ? 14 less 5 are how many ?

3. A person having 20 acres of land, sold 10 acres: how much did he then have ?

4. 18 less 8 are how many ? 17 less 5 ? 15 less 7 ?

5. If from a piece of silk containing 19 yards, 10 yards are cut, how much will be left ?

6. What is the difference between 17 dols, and 9 dols.?

7. If you pay 1 7 dollars for a goat and sell it for 9 dols., what will be your loss ?

8. If you buy a calf for 8 dollars, and sell it for 14 dols., what will be your gain ?

9. A gardener set out 1 8 peach trees, i-nd 9 of them died : how many lived ?

10. Count backward by sevens from 70 to o, as before.

11. Count backward by eights from 80 to o.

SLATE EXERCISES.

When each Figure in the Lower Number is Less than the one above it.

1. What is the difference between 465 dolla: .. and 123 dollars ?

Analysis. — Write the less number unde~. the Operation. greater, placing the units under units, the tent, under 465 dols. Uns, and the hundreds under hundreds. Begin 123 dols.

at the right, and proceed thus: 3 units from 5

units leave 2 units. Set the 2 in units place, under 342 QO*S. the figure subtracted, because it is units. Next, 2 tens from 6 tens leave 4 tens. Set the 4 in tens place, under the figure subtracted, because it is tens. Finally, 1 hundred from 4 hundreds leaves 3 hundreds. Set the 3 under the hundreds column because it is hundreds. The difference is 342 dollars.

6$ SUBTRACTION.

Solve the following examples in the same manner:

(*♦) (3.) (4.) (5.) (6.)

From 629 745 846 4382 7468

Take 416 421 526 2150 3405

7. How do you write numbers for subtraction ? Write the less number under the greater, units under units, tens under tens, etc.

§. Where do you begin to subtract, and where put the result ?

Begin at the right hand, and set the result under the figure subtracted.

9. What two principles are necessary to be observed in sub- traction ?

1 st. The numbers must be Like numbers. 2d. Units of the same order must be sub- tracted, one from the other.

Can 3 pears be taken from 5 inkstands ?

Explain the reason.

Do 3 units from 7 tens leave 4 units, or 4 tens ?

MENTAL EXERCISES.

1. 10 from 16 leaves how many? 10 from 26? ro from 46 ? 10 from 76 ? 10 from 86? 10 from 96 ?

2. 10 from 27? 10 from 37? 10 from 57? 10 from 47? 10 from 67? 10 from 87? 10 from 77?* 10 from 97 ?

3. 10 from 24? From 35? From 48? From 57: From 63? From 76? From 83? From 92?

4. Take 4 from 7. 4 from 17. 4 from 27. 4 from 57. 4 from 47. 4 from 37. 4 from 67. 4 from 87.

4 from 97. 4 from 77.

5. Take 5 from 8. 5 from 18. 5 from 28. 5 from 48. 5 from 38. 5 from 58. 5 from 78. 5 from 68.

5 from 88. 5 from 98.

SUBTRACTION. 39

6. Take 6 from 19. 6 from 29. 6 from 69. 6 from 4.9. 6 from 59. 6 from 79. 6 from 99. 6 from 89.

7. Take 7 from 16. 7 from 26. 7 from 46. 7 from 2,6. 7 from 56. 7 from 76. 7 from. 66. 7 from S6. 7 from 96.

8. 2 from 11. 2 from 21. 2 from 31. 2 from 41. 2 from 51. 2 from 61. 2 from 71. 2 from 81. 2 from 91.

9. 4 from 22. 4 from 32. 4 from 62. 4 from 52. 4 from 42. 4 from 82. 4 from 72. 4 from 92.

10. 5 from 13. 5 from 23. 5 from 43. 5 from 53.

5 from $s- 5 fr°m 63- 5 fr°m 83- 5 from 73. 5 from 93.

11. 6 from 23. 6 from 43. 6 from ^3- 6 from 53. 6

6 from 83. 6 from 93. 6 from 73.

12. 7 from 12. 7 from 22. 7 from 52. 7 from 72.

7 from 62. 7 from 42. 7 from 82. 7 from 92.

13. 8 from 94. 8 from 84. 8 from 74. 8 from 64.

8 from 54. 8 from 44. 8 from 34.

14. 9 from 85. 9 from 75. 9 from 65. 9 from 55.

9 from 45. 9 from 35. 9 from 25.

15. Count backward by nines from 90 to o, as above.

16. Count backward by tens from 100 to o.

SLATE EXERCISES.

Copy and subtract the following as above :

('•) (»•) (3-) (4.)

From 736 pounds 674 yards 8567 hats 9678 dols. lake 513 pounds 411 yards 4251 hats 8567 dols.

(7-)	(8.)	(9-)	(10.)	en.)
From 5876 in.	6341 oz.	7043 weeks	8672	9000
Take 2314 in.	1240 oz.	4043 weeks	5461	5000

40

SUBTRACTION.

MENTAL EXERCISES. i. The age of a father is 50 years, and that of his son 20 years : how much older is the father than the son ?

Analysis. — 50 is 5 tens, and 20 is 2 tens ; now 2 tens from 5 tens leave 3 tens, or 30. Therefore, etc.

2. 30 from 40 leaves how many ?

3. 40 from 70 leaves how many ?

4. 24 from 47 leaves how many ?

5. 24 from 68 leaves how many ?

6. 32 from 65 leaves how many ?

20 from 40 ? 23 from 75 ? 35 from 47 ? 45 from 76 ?

SLATE EXERCISES.

When a figure in the Lower Number is Larger than the one

above it.

1. Find the difference between 723 dols. and 476 dols. ?

ist Method. — Set down the numbers and begin at the right hand as before. Since 6 units cannot be taken from '3 units, we borrow 1 of the 2 tens and add it to the 3, making 13 units. Now 6 from 13 leaves 7, which we set under the figure subtracted. As we borrowed 1 of the 2 tens there is but 1 left ; and 7 tens cannot be taken from 1 ten. We therefore borrow 1 of the 7 hundred and add it to the 1 ten, making 11 tens; and 7 from n leaves 4, which we set in tens' place. As we borrowed 1 of the 7 hundred, there are but 6 hundred left ; and 4 from 6 leaves 2, which we set in hundreds' place.

Borrowing Illustrated by Unit Marks.

Operation.

723 dols.

476 dols.

Ans. 247 dols.

Hundreds.	Tens.	Units.
Tens borrowed, Minuend, 723= | | | 1 1 \% Subtrahend,476= | | | |	10 tens. 1* mini	10 units. Ill MIMI
Remainder, 247= | |	MM	linn;

Analysis. — Let the 7 hundreds of the minuend be representor! by 7 marks, the 2 tens by 2 marks, and the 3 units by 3 marks Let the subtrahend be represented in like manner.

SUBTRACTION. 41

Since we cannot take 6 units from 3 units, we borrow one of the 2 tens, which reduced to units, Ave add to the 3 units, making 13 units ; and 6 from 13 leaves 7. Next, 7 tens cannot be taken from 1 ten (1 ten being erased and transferred to the units), we therefore borrow one of the hundreds, and add it to the 1 ten, making II tens; then 7 from 11 leaves 4. Finally, 4 hundreds from 6 hundreds (1 hundred being erased and transferred to the tens), leave 2 hundred. The result is 247 dollars.

2D Method. — As 6 units cannot be taken from 3 units, we add 10 to the 3, making 13 ; and 6 from 13 leaves 7, which we set under the figure subtracted. To balance the 10 added to the 3, instead of considering the next upper figure 1 less than it is, we add 1 ten to the 7 tens, the next figure in the lower number, making 8 tens. But 8 tens cannot be taken from 2 tens ; we again add 10 to the 2, making 12 tens, and 8 from 12 leaves 4, which we set in tens' place. Finally, to balance the 10 added to 2, we add 1 to the next figure in the lower number, making 5 hundred, and 5 from 7 leaves 2, which we set under the figure subtracted. The result is 247 dols., the same as before.

10. What is adding 10 to the upper figure called ?

Borrowing ten.

11. Why does not borrowing 10 affect the difference between the two numbers ?

The First Method simply transfers a unit from a higher to the next lower order of the minuend ; therefore its value is not altered.

By the Seeond Method the two numbers are equally increased j and when two numbers are equally increased, their difference is not altered.

Note. — This method is the less liable to mistakes, and is more generally practiced by business men.

1 2. How proceed by the second method, when the figure in the lower number is larger than the one above it ?

Add 10 to the upper figure, then subtract, and add i to the next figure in the lower number.

43 SUBTRACTION.

(2.) (3-) (40 (50 (6.)

From 4363 5830 7406 8738 9847

Take 2172 3517 5183 7329 8043

15. The preceding principles may be summed up in the following

GENERAL RULE.

I. Place the less number under the greater, units under units, tens under tens, etc.

II. Begin at the right, and subtract each figure in the lower number from the one above it, setting the remainder under the figure subtracted.

III. If a figure in the lower number is larger than the one above it, add 10 to the upper figure ; then subtract, and add 1 to the next figure in the lower number.

Proof. — Add the remainder to the subtrahend; if the sum is equal to the minuend, the work is right.

Note. — This proof depends upon the Axiom that the whole is equal to the sum of all its parts.

	EXAMPLES FOR	PRACTICE.
	(I.)	(*•)	(30	(4-)
From	465	6253	7464	629O
Take	230	3145	4273	6146
	(50	(6.)	(70	(8.)
From	5434	8670	7202	629O
Take	4260	3452	4101	4062

9. From 6435 quarts, take 4268 quarts.

10. From 265045 barrels, take 120328 barrels.

SUBTRACTION. 43

ii. A farmer having 2568 bushels of corn, sold 1830 bushels : how many bushels had he left ?

12. A?s income is 2345 dollars, B's 3068 dollars: what is the difference between their incomes ?

t^. A man paid 1730 dollars for his horses, and 2135 dollars for his carriage : what was the difference in their cost?

14. What is the difference between nineteen hundred and nine, and nine hundred and nineteen ?

15. What is the difference between two thousand and four, and one thousand and fourteen ?

16. Find the difference between eight hundred and eight, and eight thousand and eighty.

17. 7800461 — 4560231. 18. 8000030 — 6234521. 19. 7930451 —4000459. 20. 9603245 — 2896750. 21. 6235672 — 4000563. 22. 1900000 — 899996.

23. If a farmer has 738 sheep, how many more must he buy to make up 1320 ?

24. A man bought goods for 1943 dollars, and sold the same for 2365 dollars : what did he gain :

25. A man born in 1783, died in 1866: how old was he?

26. A person bought a drove of cattle for 5263 dollars, and sold them for 4675 dollars: how much did he lose ?

27. The difference between the ages of two persons is 15 years, and the older is 79 years: how old is the younger ?

28. 1463 and what number make 3185 ?

29. A has 765 dollars, B 1695 dollars, and C's money was equal to the difference between A's and B's : how much money had C ?

30. The Pilgrim Fathers landed at Plymouth Eock in 1620, and the independence of the colonies was declared in 1776: how many years between these two events?

44 SUBT It ACTION.

DRILL FOR RAPID COMBINATIONS.

To Tbacheks.— These exercises, if properly conducted, will secure tw< objects : First, the habit of fixing the attention ; Second, rapidity in the com- bination of numbers. They should be dictated slowly at first, increasing hi speed as the class acquire ability to follow. The answers may be given individually, or by the class simultaneously.

! Oral. — i. From 12, subtract 5; add 6; subtract 4; add 3 ; add 4 ; subtract 10 ; add 9 ; subtract 4 ; add 2 ; subtract 5 : what is the result ?

Explanation. — The teacher says, " from 12 subtract 5," the class think 7 ; " add 6," the class think 13 ; " subtract 4," the class think 9 ; " add 3," the class think 12, *aid so on.

2. To 9, add 4 ; subtract 2 ; add 5 ; subtract 6 ; add 3 ; subtract 5 ; add 7 ; add 3 ; mbtiact 5 : the result ?

3. From 17 take 15 ; add 10 ; take 2 , add 9; add 6 ; take 5 ; take 10 ; add 7 ; take 2 : result ? •

4. To 23 add 5 ; take 6 ; add 9 ; add 4 ; take 10 ; add 9; take 4; add 7 ; add 10: take 9; add 4: result ?

5. From 24 take 6; add 8; take 10; add 5; add 7; take 3 ; add 7 ; take 6 ; add 5 ; add 3 : result ?

6. To 35 add 4 ; take 6 ; take 3 ; add 8 ; take 7 ; add 5 ; add 3 ; take 9 ; add 7 ; take 8 ; add 10 : result ?

Slate.— 1. To 375 add 123; subtract 47; add 23; add 47 ; subtract 36 ; add 87 ; subtract 68 : the result ? 5 2. From 62 take 34 ; add 76 ; take 40 ; add 78 ; take 99 ; add 76 ; add 24 ; take 43 : result ?

3. Add 344 to 65 ; take 64; add 784; take 678; add 407 ; take 309 ; add 860 : result ?

4. From 780 take 607 ; add 788; add 28; take 19; add 976 ; take 306 ; add 1000 : result ?

5. To 4678 add 6246 ; take 4004; add 5020; take 50S ; add 1700; take 468; add 2500: result?

6. From 8640 take 3476; add 4578; take 5065; add 87; take 1000; add 608; take 47 : result?

MULTIPLICATION.

MENTAL EXERCISES.

To Teachers.— The object of this Exercise is to develop the idea of h times,''' as used in Multiplication, preparatory to learning the Table.

i. If your father gives you 3 books at one time, and 3 at another, how many books will you have ? " 3 books and 3 books are 6 books."

2. How many times 3 books will you have ? " Two times."

3. How many are 2 times 3 books ? "6 books."

4. How many are 2 times 2 pencils ?

5. Show this by counters or unit marks.

6. How many are 2 cents, and 2 cents, and 2 cents ?

7. How many are 3 times 2 cents ? Show it.

8. If you have 4 fingers on each hand, how many have you on both hands ?

9. How many are 2 times 4 ? Show it.

10. John has 5 apples, and Henry has 2 times as many : how many has Henry ? Show it.

11. If 1 orange costs 6 cents, what will 2 oranges cost?

Analysis. — If 1 orange costs 6 cents, 2 oranges will cost 2 times 6 cents ; and 2 times 6* cents are 12 cents. Therefore, 2 oranges will cost 12 cents. *

12. How many are 7 slates and 7 slates ? Show it.

13. If 1 yard of braid costs 7 cents, what will 2 yards cost ? Show it.

14. At 8 cents each, what will be the cost of 2 ink- stands ? Show it.

15. If 1 writing-book costs 9 cents, what will 2 writing- books cost? Show it.

16. How many are 2 times 10 dollars ? Show it

46

MULTIPLICATION.

MULTIPLICATION TABLE

once	2	times	3	times	4	times
i is i	1	are 2	1	are 3	1	are 4
2 " 2	2	" 4	2	" 6	2	" 8
3 « 3	3	" 6	3	" 9	3	« 12
4 " 4	4	" 8	4	" 12	4	" 16
5 " 5	5	" 10	5	" 15	5	" 20
6 " 6	'5	" 12	6	" 18	6	" 24
7 " 7	7	" 14	7	" 21	7	" 28
8 " 8	8	" 16	8	" 24	8	« 32
9 " 9	9	" 18	9	" 27	9	" 36
IO " IO	10	" 20	10	« 30	10	" 40
ii " ii	11	" 22	n	" 33	n	" 4T~
12 " 12	12	" 24	12	" 36	12	" 48
5 times	6 times	7	times	8 times
i are 5	1	are 6	1	are 7	1	are 8
2 " 10	2	" 12	2	" 14	2	" 16
3 " 15	3	" 18	3	" 21	3	" 24
4 " 20	4	" 24	4	" 28	4	a 32
5 " 25	5	" 30	5	" 35	5	" 40
6 " 30	6	« 36	6	" 42	6	* 48
7 " 35	7	" 42	7	" 49	7	« 56
8 " 40	8	" 48	8	* 5(>	8	" 64
9 " 45	9	" 54	9	" 63	9	" 72
.12— 1LJ°	10	" 60	10	" 70	10	t( oQ
^i"3s—		" 66	n	" 77	11	« 88
12 " 60	12	" 72	12	" 84	12	" 96
9 times	IC	times	II	times	12	times
1 are 9	I	are 10	1	are n	1	are 12
2 " 18	2	" 20	2	" 22	2	" 24
3 " 27	3	" 30	3	" 33	3	" 36
4 " 36	4	" 40	4	" 44	4	« 48
5 " .45	5	" 50	5	" 55	5	" 60
6 " 54	6	" 60	6	« 66	6	u 72
7 " 63	7	" 70	7	" 77	7	" 84
8 « 72	8	" 80	8	« 88	8	" 96
9 " 81	9	" 90	9	" 99	9	" 108
10 " 90	10	" 100	10	" no	10	" 120
1 1 " 99	11	" no	11	" 121	n	" 132
12 " 108	12	" 120	12	" 132	12	* i44

MULTIPLICATION. 4?

DEFINITIONS.

1 . What is Multiplication ?

Multiplication is finding the amount of a num- ber taken or added to itself a given number of times.

2. What is the number to be multiplied called?

The Multiplicand,

3. What the number by which you multiply ?

The Multiplier ; and shows how many times the multiplicand is to be taken.

4. What is the number obtained by multiplication called ?

The Product.

When it is said that 3 times 4 are 1 2, which is the multiplicand ? The multiplier ? The product ?

When it is said that 4 times 3 are 12, what is the 4 ? The 3? The 12?

5. What else are the multiplier and multiplicand called ? Factors ; for they make or produce the product.

The number 12 is made up of four 3s, or three 4s; hence, 3 and 4 are factors of 1 2.

Remark. — The product is the same in whatsoever order the factors are multiplied. Thus, if 4 be represented by a horizontal row of unit marks upon the blackboard, ** ** ■* * and 3 by a perpendicular row of 3 unit marks, it is S* 13 & J3 plain that the horizontal row taken 3 times, is equal ^ 0 0 ^ to the perpendicular row taken 4 times.

6. When the multiplicand contains only one denomination, what is the operation called ?

Simple Multip I i cation.

7. How is Multiplication denoted?

By an oblique cross, called the sign of multiplica- tion ( x ). Thus, 6x4 shows that 6 and 4 are to be multiplied together, and is read "6 times 4," "6 into 4," or "6 multiplied by 4."

Read the following expressions : 2x6 = 3x4; 4x5

= 10x2; 2x2x6-12x2,

48 MULTIPLICATION.

MENTAL EXERCISES.

i. What will 4 pears cost, at 3 cents apiece ?

Analysis. — Since 1 pear costs 3 cents, 4 pears will cost 4 times 3 cents ; and 4 times 3 cents are 12 cents. Therefore, 4 pears will cost 12 cents.

*** It is important for the pupil to analyze every concrete ex- ample in a concise, distinct, and scholarly manner.

2. What will 5 oranges cost, at 6 cents apiece ?

3. If 1 hat costs 6 dollars, what will 4 hats cost ?

4. At 7 dollars a barrel, how much will 3 barrels of flour come to ?

5. If you obtain 6 credit-marks each da)7 for 5 days in succession, how many will you have ?

6. In 1 week there are seven days : how many days are there in 6 weeks ?

7. George has 7 marbles, and Henry has 4 times as many : how many marbles has Henry ?

8. At 8 cents apiece, what will 6 tops cost ?

9. At 9 dollars each, what will 4 trunks cost ?

SLATE EXERCISES.

Copy and multiply the following, setting each result nnder the figure multiplied :

(1.) (2.) (3.) (4.) (50 (6.) (7.) (8.) Mult. 67897589

ByAAAA — 1 — 2.

Prod. 30 45 6$

(9.) (10.) (11.) (12.) (13.) (14.) (15.) (16.)

Mult. 8 9 7 9 8 9 10 11

2fy 78969789

MULTIPLICATION. 49

MENTAL EXERCISES.

i. Bought 7 barrels of flour, at 6 dollars a barrel: what did the flour come to ?

2. Sold 8 silk umbrellas, at 7 dollars each : what was the amount of the bill ?

3. If a man earns 9 dollars a week, how much will he earn in 8 weeks ?

4. What must be paid for 6 quarts of cherries, at 12 cents a quart ?

5. In a certain orchard there are 8 rows of trees, and 12 trees in a row: how many trees does it contain ?

6. If 1 table costs 8 dollars, what will be the cost of 10 tables ?

7. What must I pay for 11 yards of muslin, which is 12 cents a yard?

8. How many quarts in 1 1 pecks, allowing 8 quarts to a peck ?

9. In 1 dime there are 10 cents: how many cents are there in 1 1 dimes ?

10. In 1 year there are 12 months: how many months are there in 10 years?

SLATE EXERCISES.

When the Multiplier has but one figure, and the Product of each figure in the Multiplicand is Less than 10.

1. Multiply 1232 by 3.

Analysis. — Write the multiplier under the mul- Operation.

fciplicand, and begin at the right, 3 times 2 1232 units are 6 units. Set the 6 in units place, under ~

the figure multiplied, because it is units. 3 times

3 tens are 9 tens. Set the 9 in tens place, because 3696 Am, it is tens. 3 times 2 hundreds are 6 hundreds. Set the 6 in hundreds place, because, etc. 3 times 1 thousand are 3 thousands. Set the 3 in thousands place.

50 MULTIPLICATION.

Copy and multiply the following in like manner:

(«•) (3-) (4-) (5-)

Mult. 42414 22321 12212 iiiii

By 2 3 4 r

	(6.)	(70	(8.)	(9.)
Mult. By	IOIIOI 6	332032 3	IIOIOI 7	111111 8

MENTAL EXERCISES, i. At 9 dollars a barrel, what will 8 barrels of cran- berries come to ?

2. What will be the cost of 6 flutes, at 12 dollars each ?

3. A farmer sold 1 1 calves, at 9 dollars apiece : what did he receive for them ?

4. How many are 9 times 7 ? 8 times 9 ?

5. Bought 10 accordions, at 12 dollars each : what was the amount of the bill ?

6. How many are 8 times 7 ? 9 times 8 ?

7. If 1 plough cost 11 dollars, what will be the cost of 12 ploughs, at the same rate ?

8. How many are 11 times 10? 12 times 11 ?

WRITTEN EXERCISES. When the Product of the respective figures is 10, or More.

1. If 1 horse costs 435 dollars, how much will 3 horses cost?

Analysis. — Since 1 horse costs 435 dollars, Operation.

3 horses will cost 3 times as much. Write the 435 mult'd.

multiplier under the multiplicand, and bepinnintf - mult

at the right, proceed thus : 3 times 5 units are

15 units; we set the 5 in units place, under the I3°5 dols.

figure by which we multiply, and carry the 1 to

the product of the nt'^t figure, as in addition. Next, 3 times 3 tens

MULTIPLICATION. 51

are 9 ten?, and 1 (to carry) makes 10 tens ; we set the o in tens place, and carry the 1 to the product of the next figure. Finally, 3 times 4 hundred are 12 hundred, and 1 (to carry) makes 13 hun- dred. Therefore, 3 horses will cost 1305 dollars.

7. How write numbers for multiplication?

Write the multiplier under the multiplicand, units under units, etc.

§. How proceed when the multiplier contains but one figure ?

Begin at the right hand, and multijoly each figure in tlie multiplicand by the multiplier, separately.

9. What do you do with the partial results, when 10, or more ?

Set the units figure under the figure multiplied, and carry the tens to the product of the next figure.

10. When the multiplier is units, what order is the product ? The same order as the figure multiplied.

11. What are the principles as to the nature of the multiplier, the multiplicand, and the product ?

1st. The J&lultipli&r must always be considered an abstract number.

2d. The Multiplicand may be an abstract, or con- crete number.

3d. The Product is always the same name or hind as the true multiplicand; for, repeating a number does not change its nature.

12. Which of the factors is the true multiplicand?

The true multiplicand is that number, which added to itself the given number of times, will produce the required product.

Remark. — Neither a concrete nor abstract number can properly be said to be repeated as many times as another is long, or heavy. Hence, money can not be multiplied by yards, pounds, etc. ; but any given sum can be multiplied by a number of units equal to the number of yards, pounds, etc., in the given quantity, and tht. product will be money.

62 MULTIPLICATION.

2. In i year there are 365 days : how many days are there in 5 years ?

3. If 1 piano costs 750 dollars, what will 6 pianos cost?

4. If 1 farm contains 875 acres, how many acres will 8 farms of the same size contain ?

Mult. By	(50 2136 2	(6.) 7345 3	(7.) 28536 4	(8.) 65043 5
Mult By	(9.) 701230 6	(10.) 635728 7	(11.) 830405 • 8	(12.) 973080 9

MENTAL EXERCISES.

1. What will 3 tables cost, at 45 dollars apiece ?

Analysis. — 3 tables will cost 3 times as much as 1 table. But 45 is equal to 4 tens and 5 units. Now 3 times 4 tens are 12 tens, or 120; 3 times 5 units are 15 uuits, or 1 ten and 5 units; and 1 ten and 5 units added to 120 make 135. Therefore, 3 tables will cost 135 dollars.

Note.— When tie numbers to be multiplied mentally are large, it is advisable to separate them into the units, tens, etc., of which they are composed, and multiply the highest order first, then the next lower, etc., adding the results as we proceed. (P. 26, N.)

2. How much can a man earn in 2 months, if he earns 36 dollars a month ?

3. In a peach orchard there are 5 rows of trees, and 27 trees in a row: how many peach trees does the orchard contain ?

4. How many are 3 times 54? 4 times 37 ?

MULTIPLICATION". 53

5. If a man can earn 56 dollars a month, how much can he earn in 7 months ?

6. If 1 hogshead contains 63 gallons of molasses, how many gallons will 5 hogsheads contain ?

7. If 1 melodeon can be had for 75 dollars, what will be the cost of 6 melodeons ?

8. If 1 sofa is worth 83 dollars, how much are 9 sofas worth ?

WRITTEN EXERCISES. When the Multiplier has two or more Figures.

1. What will 106 buggies cost, at 268 dollars apiece?

Analysis.— 106 buggies will cost 106 Operation. times as much as 1 buggy. We write ^g rn,rilt'd

the multiplier under the multiplicand, as , u

1^ ^1 • • A xt. • i x 1 j 106 mult.

before, and beginning at the right hand,

proceed thus: 6 times 8 units are 48 1 608

units. The 8 is set in units place, under 268 •

the figure which produced it, because it

is units; and the 4 is carried to the pro- AflS. 28408 dols. duct of the next figure, because it is the

same order as that figure. The other figures of the multipli- cand are multiplied by 6, and the results set down in a similar manner. Next, the product by o tens is o ; we therefore omit it. Again, 1 hundred times 8 units are 8 hundreds. The 8 is set in hundreds place, under the figure which produced it, because it is hundreds. The other figures of the multiplicand are multiplied •by 1 in the same manner. Finally, adding these partial products together, the result, 28408 dollars, is the whole product required.

1 3. When the multiplier has two or more figures, how pro- ceed?

Beginning at the rigid hand, multiply the multipli- cand by each figure of the multiplier separately, and set the first figure of each partial product under the multi- plying figure.

54 MULTIPLICATION.

14. What js meant by partial products?

They are the several results which arise from multi- plying the multiplicand by the separate figures of the multiplier, and are so called because they are parts of the whole product.

1 5. What is done with the partial products, and why ?

We add them together, because the whole product is equal to the sum of all its parts.

	(*•)	(30	(4.)	(5.)	(6.)
Mult.	3724	4103	5378	6037	8734
By	25	34	46	57	78

1 6. The preceding principles may be summed up in the following

GENERAL RULE.

I. Place the multiplier under the multiplicand, units under units, tens under tens, etc.

II. When the multiplier has tut one figure, beginning at the right, multiply each figure of the multiplicand by it, and set down the result as in addition.

III. If the multiplier has two or more figures, multiply the multiplicand by each figure of the multiplier sepa- rately, and set the first figure of each partial produd under the multiplying figure.

Finally, the sum of the partial products will be the ansiver required.

Proof. — Multiply the multiplier by the multiplicand ; if this result agrees with the first, the work is right.

Note. — This proof is based upon the principle, that the retail will bo the same whichever of the given numbers is taken as the multiplicand. (P. 47, Q. 5.)

MULTIPLICATION.

55

EXAMPLES FOR PRACTICE. i. Multiply 78 by 43, and prove the operation ?

Operation.

Multiplicand 78

Multiplier 43

Proof.

The given multiplier 4; " " multiplicand 7^

	234		344
	3*2		301
Product	3354	The same as	the first 335h.
(2.)	(30	(4.)	(5.)
27356	40256	57189	70203
21	27	32	47
(6.)	(70	(8.)	(9.)
63I42O	507060	81367O	973848
158	249	365	1476

10. There are 24 hours in a day : how many hours £re there in 365 days ?

11. There are 320 rods in a mile: how many rods are in 150 miles?

12. What will 265 acres of land cost at 87 dollars per acre ?

13. What cost 97 melodeons, at 250 dollars apiece ?

14. Multiply 43846 by 123.

15. Multiply 57028 by 321.

16. Multiply 604326 by 237.

17. Multiply 673862 by 250.

18. Multiply 703562 by 304. '

19. Multiply 570031 by 402.

20. Multiply 439275 by 425- 2 j. Multiply 789426 by 521.

56 MULTIPLICATION.

22. What will be the cost of 85 pianos, at 650 dollars apiece ?

23. If a ship sails 115 miles in one day, how far will she sail in 198 days?

24. If there are 63 yards in' 1 piece of cloth, how many are there in 268 pieces ?

25. At 320 dollars a yoke, what will 500 yoke of oxen

30St?

26. What will no wagons cost, at 175 dollars apiece ?

27. What cost 350 suits of clothes, at 115 dollars a suit ?

28. What cost 1645 saddles, at 75 dollars apiece?

29. What cost 3250 tons of iron, at 87 dollars a ton ?

30. What does the President's salary amount to in 8 years, at 25000 dollars a year ?

31. What is the expense of furnishing an army of 1 1500 men with uniforms which cost 57 dollars apiece?

32. If 1 ox weighs n 63 pounds will 100 oxen weigh? ^ What cost 465 velvet cloaks, at 129 dollars apiece ? 34. What cost 1567 tons of lead, at 120 dollars per ton?

CONTRACTIONS.

I. When the Multiplier is 10, 100, 1000, etc.

17. What is the effect of annexing a cipher to a number ? Annexing one cipher to a number multiplies it by 10 5 annexing tivo ciphers multiplies it by 100, and so on.

Remark. — The learner will observe that each cipher annexed to a number, removes each preceding figure in the number to the next higher order, which has ten times the value of the order from which it has been removed. (Page IX, Q. 17.)

2. What will 10 sofas cost, at 56 dollars apiece?

Solution.— ^Annexing a cipher to 56 dollars, the result i-» 560 dollars, which is the cost required.

MULTIPLICATION. 57

3. What will 100 acres of land cost, at 75 dollars an acre ?

Solution. — Annexing 2 ciphers to 75 dollars, the result is 7500 dollars, which is the answer required.

1§. How then do you multiply by 10, 100, 1000, etc. ?

Annex as many ciphers to the multiplicand as there are ciphers in the multiplier, and the result ivill be the pro- duct.

4. What is the product of 361 multiplied by 100?

5. Multiply 453 by 100.

6. Multiply 2045 by 1000.

7. Multiply 46208 by 1000.

8. Multiply 58241 by 1000.

9. Multiply 326072 by 10000.

10. Multiply 4007289 by 1 00000.

11. What cost 10 cows, at 51 dollars apiece ?

12. At 265 dollars apiece, what will 100 buggies come to ?

13. What cost 100 acres of land, at 205 dollars per acre ?

14. If 1 bushel of apples is worth 6^ cents, what will be the price of 1000 bushels?

II. When one op both Factors have Ciphers on the right.

15. If 1 railroad car costs 2700 dollars, what will 50 cars cost ?

Analysis. — If 1 car cost 2700 dollars, Operation.

50 cars will cost 50 times as much. We 2700

resolve the multiplicand into the factors ^

27 and 100 ; and the multiplier into 5 and

10. Now, as the product is the same in Ans. 135000 dols. whatever order the factors are taken,

omitting the ciphers on the right of the multiplicand and multi- plier, we multiply the significant figures together as before, and annex the ciphers omitted to the product. The result is 135000 d.

58 MULTIPLICATION-.

19. How proceed when one or both factors have ciphers on the right ?

Multiply the significant figures together ; and to the result annex as many ciphers as are found on the right of both factors.

(16.) (17.) (18.)

Mult, i860 25000 4°53

By 300 • 7 2000

Prod. 558000 175000 8106000

19. Multiply 37000 by 31.

20. Multiply 52300 by 65.

21. Multiply 42721 by 2000.

22. Multiply 60045 by 3100.

23. Multiply 85000 by 2300.

24. Multiply 375000 by 57000.

25. Multiply 204200 by 20500.

26. Multiply 800400 by 600300.

27. What will 200 acres- of land cost, at 70 dollars per acre ?

28. What cost 21000 bushels of oats, at 60 cents a bushel ?

29. If a man travels 120 miles a day, how far can he travel in 300 days ?

30. If 1 acre produces 50 bushels of corn, what will 3000 acres produce ?

31. Multiply 25 thousand by 25 hundred.

32. Multiply two hundred and forty-five thousand by 16 thousand.

33. Multiply 65 thousand and seventy by 21 thou- sand seven hundred.

34. Multiply one million, one hundred and ten thou- sand, by 26 thousand.

MULTIPLICATION. 59

QUESTIONS FOR REVIEW.

Oral.— i. If 9 men can bnild a wall in 12 days, how long will it take 1 man to build it ?

Analysis. — It will take 1 man 9 times as long as 9 men, and 9 times 12 days are 108 days. Therefore, it will take I man 108 days.

2. If a jar of butter will last a family of 8 persons 6 weeks, how long will it last 1 person ?

3. Henry can read a book through in 1 1 days by read- ing 6 hours each day : how long will it take him if he reads 1 hour a day ?

4. If 12 men can frame a house in 8 days, how long will it take 1 man to frame it ?

5. If I buy 4 barrels of apples at 3 dollars a barrel, and 4 barrels of pears at 5 dollars, what will be the cost of both ?

6. A farmer haying 15 bushels of wheat, sold 9 bush- els at 2 dollars a bushel, and the remainder at 3 dollars a bushel : how much did he get for his wheat ?

Written. — 1. If it takes 285 laborers 18 months to build a railroad, how long would it take 1 man to build it ?

2. A ship of war has provisions to last a crew of 625 men 90 days : how long would they last 1 man ?

3. If a clerk has 36 dollars a month for the first 4 months ; 48 dollars a month for the next 4 ; and 6c dollars a month for the next 4 ; what will he receive for the year ?

4. A man having 1000 dollars in his pocket, gave 45 dollars each to 1 2 poor persons : how much had he left ?

5. If I receive 150 dollars a month, how much shall I have at the end of the year, after deducting 28 dollars a month for board?

DIVISION.

MENTAL EXERCISES.

10 TEA.CHBHS.— The object of this preliminary Exercise is to develop Ute idea of " times,'" as used in Division, preparatory to learning the Table.

i. If I have 9 pencils, how many boys can I supply with 3 pencils each ?

Analysis. — If I give one boy 3 pencils, how many will be left ?

" Six pencils."

If I give another boy 3, how many pencils will be left?

"Three."

If I give another 3, how many will be left ? " None." How many boys have I supplied with 3 pencils ? " Three." How many times are 3 pencils contained in 9 pencils? " Three times."

2. How many peaches, at 2 cents each, can you buy for 8 cents ? Show this by counters.

3. How many oranges, at 4 cents each, can you buy for 12 cents? How many times 4 make 12 ? Show this.

4. In 1 gallon there are 4 quarts: how many gallons are there in 8 quarts ? How many times 4 make 8 ? Show this by unit marks.

5. If a lad earns 5 dollars a week, how long will it take him to earn 25 dollars? How many times 5 make 25 ? Show this.

6. At 6 cents an ounce, how many ounces of candy can you buy for 18 cents? Show this by unit marks.

7. How many lambs, at 2 dollars apiece, can be bad for 20 dollars ? Show this.

8. At 4 dollars a pair, how many pair of boots can I buy for 16 dollars? Show this.

9. If I have 20 pounds of flour, how many poor per- sons can I supply with 5 pounds each ?

Division.

61

DIVISION TABLE.

i is in	2 is in	3 is in	4 is in
i, once.*	2, once.	3, once.	4, once.
2, 2	4, 2	6, 2	8, 2
3, 3	6, 3	9, 3	12, 3
4, 4	8, 4 ,	12, 4	16, 4
5, 5	10, 5	15, 5	20, 5
6, 6	12, 6	18, 6	24, 6
7, 7	14, 7	21, 7	28, 7
8, 8	16, 8	24, 8	32, 8
9, 9	18, 9	27, 9	36> 9
IO, IO	20, 10	30, 10	40, 10
5 is in	6 is in	7 is in	8 is in
5, once.	6, once.	7, once.	8, once.
IO, . 2	12, 2	14, 2	16, 2
15, 3	18, 3	21, 3	24, 3
20, 4	24, 4	28, 4	32, 4
25, 5	30, 5	35, 5	40, 5
30, 6	36, 6	42, 6	48, 6
35, 7	42, 7	49, 7	56, 7
40, 8	48, 8	56, 8	64, 8
45, 9	54, 9	63, 9	72, 9
50, 10	60, 10	70, 10	80, 10
9 is in	10 is in	11 is in	12 is in
9, once.	10, once.	11, once.	12, once.
18, 2	20, 2	22, 2	24, 2
27, 3	3°, 3	33> 3	36, 3
36, 4	40, 4	44, 4	48, 4
45, 5	5o, 5	55, 5	60, 5
54, 6	60, 6	66, 6	72, 6
63, 7	70, 7	77, 7	84, 7
72, 8	80, 8	88, 8	96, 8
81, 9	90, 9	99, 9	108, 9
90, 10	IOO, 10	no, 10	120, 10

* After 2, 3, 4, etc., in the second column, " times " is under- stood.

62 DIVISION.

DEFINITIONS.

1. What is Division ?

Division is finding how many times one number is contained in another.

2. What is the number to be divided called?

The Dividend.

3. The number to divide by ?

The Divisor.

4. What is the number obtained by division called ?

The Quotient.

5. What is the number left called ?

The Remainder*

When it is said that 3 is contained in 13, 4 times and 1 over, which is the dividend ? The divisor ? The quo- tient ? The remainder ?

Remakks. — 1. The remainder is always the same denomination as the dividend; for, it is a part of the dividend not yet divided. 2. A proper remainder is always less than the divisor.

6. When the dividend contains only one denomination, what is the operation called ?

Simple Division.

7. How is Division denoted?

By a short horizontal line between two dots (—), called the Sign of division.

8. When placed between two numbers what does it show ?

It shows that the number before it is to be divided by the one after it. Thus, 21-^-3, shows that 21 is to be divided by 3, and is read "21 divided by 3."

9. How else is division denoted?

By writing the divisor under the dividend with a short line between them ; as %k

Read the following : 9-7-3 = 3; 24-7-4 = 5 + 1; 39 + 3 = " + 3; 5+4 = 36 + 4; V = 7> ¥ = 4 + 3-

division. 63

OBJECTS OF DIVISION.

i. A lad having 6 cents wishes to buy pears, which are 2 cents apiece : how many can he buy?

Analysis. — He can buy as many pears as there are times 2 cents in 6 cents. The Illustbatiow.

object then is to find how many times 2 JS 0 I 0 51 I J3 J3 is contained in 6 ; and 2 is in 6, 3 times.

2. A lad has 6 pears, which he wishes to divide equally between 2 companions : how many can he give to each ?

Analysis. — The object of this example is to divide 6 pears into 2 equal parts. Illttstbatxon.

Dividing 6 by 2, the quotient is 3, which SJ 21 & | & fil 8 shows that there are 3 pears in each part.

10. What is the object or office of Division ?

Its object or office is twofold : First, To find lioio many times one number is contained in another. (Ex. 1.)

Second, To divide a number into equal parts. (Ex. 2.)

Remark. — The two preceding examples are representatives of the two classes of problems to which Division is applied. In the first class, the divisor and dividend are always of the same denom- ination, and the quotient is times, or an abstract number.

In the second, the divisor and dividend are of different denom- inations, and the quotient is always of the same denomination as the dividend. This class involves the idea of Fractions, and will receive further attention under that branch of the science.

Note. — The process of reasoning in the solution of these two classes of examples is somewhat different ; but the practical oper- t:ion is the same, viz. : to find how many times one number is con- tained in another, which accords with the definition of Division.

11. How divide a number into two, three, four, etc., equal parts ? Divide the number by 2, 3, 4, 5, etc., respectively.

12. When a thing is divided into 2, 3, 4, etc., equal parts, what are the parts called ?

If divided into two equal parts, the parts are called halves ; into three, the parts are called thirds ; into four, they are called fourths ; into five, fifths ; etc.

64 division.

13. When a thiug is divided into equal parts, from what do the parts take their name ?

From the number of parts into which the thing is divided.

3. What is a half of 10 ? A third of 12 ? A fourth of 16? A fifth of 20? A sixth of 30?

4. What is a seventh of 35 ? An eighth of 56 ? A ninth of 45 ? A tenth of 60 ? A twelfth of 84 ?

SHORT DIVISION. MENTAL EXERCISES.

1. How many lemons, at 2 cents apiece, can George buy for 10 cents ?

Analysis. — Since 2 cents will buy 1 lemon, 10 cents will buy as many lemons as 2 is contained times in 10 ; and 2 is in 10, 5 times. Therefore, he can buy 5 lemons.

2. At 4 cents each, how many bananas can you buy for 12 cents?

3. How many yards of tape, at 6 cents a yard, can be had for 18 cents ?

4. At 4 dollars a yard, how many yards of cloth can you buy for 28 dollars ?

5. When the fare on the city railroads is 5 cents a ride, how many rides can you take for 30 cents?

6. If 3 oranges cost 12 cents, what will 1 cost?

Analysis. — If 3 oranges cost 12 cts., 1 orange will cost 1 third of 12 cts. ; and 1 third of 12 cts. is 4 cts. (P. 63, Q. 12.

7. If 5 slates cost 60 cents, what will 1 cost?

8. A baker divided 28 loaves of bread equally among 7 beggars : how many loaves did he give to each ?

9. A grocer sold 9 barrels of flour for 72 dollars: \t Lat waa thai a barrel ?

division. 65

SLATE EXERCISES.

When the Divisor is exactly contained in each figure of the Dividend.

i. How many times is 2 contained in 6402 ?

Analysis. — Write the divisor on the left of the Operation. dividend, with a curve line between them, and pro- 2)6402

ceed thus : 2 is contained in 6, 3 times ; write the 3 .

under the figure divided, for it is the same order as «3

that figure. Next, 2 is contained in 4, 2 times ; write the 2 under the figure divided, for the same reason as before. 2 is contained in o, no times ; write a cipher in the quotient. Finally, 2 is in 2, 1 time ; set the 1 under the figure divided.

10. How write numbers for division ?

Place the divisor on the left of the dividend, with a curve line between them.

11. How proceed when the divisor is contained exactly in each figure of the dividend ?

Begin at the left of the dividend, and divide each figure by the divisor ; placing the result wider the figure divided.

12. What order is each quotient figure ?

The same order as the figure divided. Copy and divide the following in like manner : M (3.) (4.) (5.)

3)6393 2)4062 4)8404 5)50505

(6.) (7.) (8.) (9.) f

4)8084 6)6606 7)7070 8)80808

MENTAL EXERCISES.

1. At 4 dollars a head, how many sheep can a man buy with 35 dollars, and what will he have left?

Analysis. — 4 dollars are contained in 35 dollars 8 times, and 3 over. Therefore, he can buy 8 sheep, and have 3 dollars left.

66 division.

2. How many times is 3 contained in 1 7, and ho\* many over ?

3. In 24 how many times 5, and how many over?

4. In 39 how many times 4? 5? 6? 7 ? 8? 9?

5. How many boxes, each containing 6 quarts, can be filled with 40 quarts of blue-berries ?

6. Horace has 38 marbles, which he wishes to distn bute equally among his 3 brothers : how many can he give to each ; and how many over ?

7. How many times 7 in 29, and how many over?

8. How many times 8 in 5 7 ? In 63 ? In 74 ? In 83 ?

SLATE EXERCISES.

When the Divisor is not contained exactly in ecch figure of the Dividend.

1. How many barrels of flour, at 5 dollars a barrel, can be bought for 157034 dollars ?

Analysis. — As the divisor is not contained in Operation. the first figure of the dividend, we m ust find how 5 ) x 5 7 °34

many times it is contained in the first two fig-

ures, which is 3 times, and set the 3 under the -A-IIS. 31400^ right hand figure divided. Again, 5 is contained in 7, once and 2 remainder. Set the 1 under the figure divided, and prefixing the 2 remainder mentally to the next figure of the dividend makes 20. "Now 5 is in 20, 4 times. Set the 4 under the figure divided. Next, 5 is not contained in 3, the next figure of the dividend ; we therefore put a cipher in the quotient, and prefixing the 3 men- tally to the next figure of the dividend, makes 34. Now, 5 is in 34, 6 times and 4 remainder. We set the 6 under the figure divided, and as there are no more figures in the dividend, wc write this last remainder over the divisor, and annex it to tho quotient

13. When the divisor is not contained in the first figure of the dividend, how proceed?

Find how many times it is contained in the first two figures.

division. 67

14. When it is npt contained in a subsequent figure of the dividend, how ?

Put a cipher in the quotient, and find how many times the divisor is contained in this and the next figure. setting the result under the right hand figure divided.

3 5. When there is a remainder after dividing a figure, how ?

Prefix it- mentally to the next figure of the dividend, and divide this number as before. *

16. If there is a remainder after dividing the last figure of the dividend, what is to be done ?

Write it over the divisor, and annex it to the quotient ?

Note. — To prefix signifies to place before; to annex, to place after.

1 7. What are the principles as to the nature of the divisor and dividend, the quotient and remainder ?

ist. The divisor and dividend may be abstract or concrete numbers.

2d. When they are the same denomination, the quo- tient denotes times, and is an abstract number.

3d. When they are different denominations, the quo- tient denotes equal parts, and is the same denomination as the dividend.

4th. The remainder is always the same denomination as the dividend ; for, it is an undivided part of it.

18. What is Short Division?

Short Division is the method of dividing, when the results of the several steps are carried in the mind, and the quotient only is set down.

Copy and divide the following by Short Division : (2) . (3-) (4.) (50

4)l2568 3)60429 6)l8728 7)84079

68 division.

10. The preceding principles may be summed up in the following

RULE FOR SHORT DIVISION.

I. Place the divisor on the left of the dividend, and beginning at the left, divide each figure by it, setting the result under the figure divided,

11. If the divisor is not contained in a figure of the, dividend, put a cipher hi the quotient, and find how many times the divisor is contained in this and the next figure, setting the result under the right hand figure divided.

III. If a remainder arises from any figure before the last, prefix it mentally to the next figure, and divide as before. If from the last, place it over the divisor, and annex it to the quotient

Pkoof. — Multiply the divisor and quotient together, and to the product add the remainder. If the result is equal to the dividend, the worlc is right.

Note. — This proof depends upon the principle, that Division is the reverse of Multiplication ; the dividend answering to the pro- duct, the divisor to one of the factors, and the quotient to the other.

EXAMPLES FOR PRACTICE.

i. At 8 dollars apiece, how many hats can be bought ffor 23243 dollars ?

Operation. Proof.

8)23243 2905 x 8 = 23240

~ . Add the remainder, 3

Quot. 2905, 3 over. _

Ans. 2905 hats, and 3 dols over. Dividend 23243

(2.) (3.) (40 (50

2)23416 3)34169 4)48016 5)9°3IQ

DIVISION. 69

(6.) (70 (8.) (9-)

6)67419 7)75008 8)89619 9)93048

10. How many barrels of apples, at 3 dollars a barrel, can you buy for 846 dollars ?

11. At 5 dollars apiece, how many hats can be bought for 2300 dollars ?

12. At 6 dollars a barrel, how many barrels of flour will 3522 dollars buy?

13. At 7 days each, how many weeks in 365 days?

14. If a man travels 8 miles per hour, how long will it take him to travel 1000 miles?

15. If 1 boat will carry 9 persons over a river, how many boats will it require to carry 468 persons over?

16. If a man lays up 12 dollars a week, how long will it take him to lay up 288 dollars?

17. At 11 dollars a barrel, how many barrels of cran- berries can be bought for 770 dollars?

18. How many boxes will it require to pack 1530 pounds of butter, allowing 9 pounds to a box ?

19. A man left 31265 dollars to be divided equally among his 5 children : how much did each receive ?

20. A grocer sold oranges at 8 dollars a box, and re- ceived 22464 dollars: how many boxes did he sell?

21. If a man has 26436 acres of land, how many acres can he give to each of his 12 children ?

22. A company of 11 men took a prize worth 1 16633 dollars, which was equally divided among them : what did each receive ?

23. If a ship sails 10 miles an hour, how many hours will it be in sailing 25000 miles?

24. If 1 stage will sea* 12 passengers, how many stages will be required to seat 1500 passengers ?

70 DIVISION.

LONG DIVISION.

i. Divide 22431 by 4, by Long Division.

Analysis. — Write the divisor on the left Operation.

of the dividend, as in Short Division, and *>>v- Dividend. o_uot. proceed thus: First, the divisor 4 is con- 4) 22431 ( 5607 J tained in 22, 5 times ; set the 5 on the right 20 ' ' '

of the dividend with a curve line between —

them. Second, multiply the divisor by this 24

quotient figure, and set the product 20, under 24

the figure divided. Third, subtract the pro- '

duct from the figures divided, and the re- ^

mainder is 2. Fourth, bring down and annex to the remainder the next figure of the divi- ^

dend, making 24 for the next partial divi- dend. Divide this partial dividend, and the quotient figure is b. Multiply and subtract as before, and the remainder is o. Bring down the next figure 3 for a new partial dividend. But the divi- sor 4 is not contained in 5 ; we therefore put a cipher in the quo- tient, and bringing down the next figure, we have 31 for a partial dividend, which we divide as before. As there are no more figures to be divided, we place the last remainder over the divisor, and annex it to the quotient. The answer is 5607I.

Note. — To prevent mistakes, it is customary to place a mark under the several figures of the dividend, when brought down.

20. What is Long Division ?

Long Division is the method of dividing when the results of the several steps and the quotient are both. Bet down.

21. How write numbers for Long Division ?

Place the divisor on the left of the dividend, and the quotient on the right, with a curve line between them.

22. How many steps in long division ? "Four." 2:5. The first?

Find hoiv many times the divisor is contained in the fewest figures on the left of the dividend that will contain it.

DIVISION. 71

21. The second?

Multiply the divisor by the quotient figure, and eet the product under the figures divided.

25. The. third?

Subtract the product from the figures divided.

26. The fourth?

Annex to the remainder the next figure of the divi dend, for a new partial dividend ; then divide as before.

Remark. — The quotient figure in Long and in Short Division, is the same order as the right hand figure of the partial dividend.

2. Divide 5463 by 4. Ans. 1365 J.

3. Divide 17382 by 5. 4. Divide 43652 by 6.

5. At 45 dollars an acre, how much land can be bought with 6750 dollars? Ans. 150 acres.

6. How many suits of clothes, at 63 dollars a suit, can be had for 7686 dollars? Ans. 122 suits.

27. The preceding principles may be summed up in the following

RULE FOR LONG DIVISION,

I. Find how many times the divisor is contained in the fewest figures on the left of the dividend that will contain it, and set the quotient on the right.

II. Multiply the divisor by this quotient figure, and subtract the product from the figures divided.

III. To the right of the remainder, bring down the next figure of the dividend, and divide as before.

IV. If the divisor is not contained in a partial divi- dend, place a cipher in the quotient, bring down another figure, and thus continue the operation.

If there is a remainder after dividing the last figure* net it over the divisor, and annex it to the quotient.

72 DIVISION".

Notes. — i. Long Division, is the same in principle as Short. The only difference is, in one the results of the several steps are carried in the mind, and in the other they are set down.

Short Division is the more expeditious, and should be employed when the divisor does not exceed 12.

2. If the product of the divisor into the figure placed in the quotient is greater than the partial dividend, it is plain the quo- tient figure is too large, and therefore must be diminlslied.

3. If the remainder is equal to or greater than the divisor, the quotient figure is too small, and must be increased.

EXAMPLES FOR PRACTICE.

1. How many times is 24 contained in 1963 ?

2. Divide 40369 by 18. 6. Divide 85345 by 53.

3. Divide 45683 by 21. 7. Divide 906530 by 68.

4. Divide 614897 by 35. 8. Divide 990046 by 74.

5. Divide 598061 by 47. 9. Divide 867604 by 84.

10. Eequired the quotient of 9134669 divided by 92.

11. How many cows, at 35 dollars apiece, can be* bought for 7140 dollars?

12. How much land, at 28 dollars per acre, can be bought for 5611 dollars.

* 3. If a man earns 45 dollars a month, how long will it take him to earn 1620 dollars ?

14. How many stoves, at 38 dollars each, can be bought for 6840 dollars ?

15. If there is 1 year in 52 weeks, how many yean are there in 8202 weeks?

16. If a man's expenses are 6^ dollars a month, how long can he live on 5260 dollars?

17. If a man pay 70 dollars a hogshead for molasses, how many hogsheads can he buy for 6940 dollars ?

18. At 87 dollars per yoke, how many yoke of qxqu can be bought for 6525 dollars?

DIVISION. 73

To find the Quotient Figure, when the Divisor is large,

19. Divide 12451 by 382.

Analysis. — Taking 3 for a trial divisor, it 382 ) 1245 1(32

is contained in 12, 4 times. But in multiply- _ j .^

ing the 8 by 4, we have 3 to carry, and 3 added

to 4 times 3, make 15, which is larger than - 99 1

the partial dividend 12. Hence, 4 is too large 764

for the quotient figure. We therefore place

3 in the quotient, and proceed as before. liem. 227

2§. How find the quotient figure, when the divisor is large?

Take the first figtire of the divisor for a trial divisor, and find how many times it is contained in the first or first two figures of the dividend, making dtie allowance for carrying the tens of the product of the second figure of the divisor into the quotient figure,

20. Divide 8732409 by 657. 22. 10342675 -f- 3435.

21. Divide 9753102 by 950. 23. 23046750-^7625.

24. A certain fort has provisions sufficient to last 1 man 15360 days : how long will it last 256 men ?

25. The president's salary is 25000 dollars a }<ear : how much is that per day ?

26. At a certain auction, 498 pictures were sold for 13944 dollars: what was the average price ?

CONTRACTIONS. I. When the Divisor is 10, 100, 1000, etc.

1. At 100 dollars a set, how many sets of fur can be bought for 1935 dollars, and how much over ?

Analysis. — Annexing a cipher to Opekation. a number, multiplies it by 10. (P. tIoq^ Tn|-C 56, Q. 17.) ' ; 9I35

Conversely, removing a cipher or Quot. 19, and 35 Rem. figure from the right of a number,

divides it by 10; for, each figure in the number is removed one place to the right. (P. 11, Q. 17.) 4

71 DIVISION.

In like manner, cutting off two figures from the right of a num. ber, divides it by ioo ; cutting off three, by iooo, etc.

Now as the divisor is ioo, it is only necessary to cut off two figures on the right of the dividend : those left, viz., 19, are the quotient, and those cut off, viz., 35, the remainder.

29. How proceed when the divisor is 10, 100, 1000, etc. ? From the right of the dividend cut off as many figures

as there are ciphers in the divisor. TJie figures left icid he the quotient; those cut off, the remainder.

2. Divide 8564 hyj 100. 6. 39467 by 10000.

3. Divide 46531 by 1000. 7. 272364 by 100000.

4. Divide 48000 by 1000. 8. 1 000000 by 1 00000.

5. Divide 4375681 by 10000. 9. 85325764 by 1000000.

II. When there ar*e Ciphers on the right of the Divisor.

10. At 20 dollars apiece, hovr many bureaus can be bought for 3453 dollars ?

Analysis. — The divisor, 20, is com- Operation.

posed of the factors 2 and 10. In the 2]°) 345J3

operation, we first divide by io, by cut- A T" .

x. ^.i . v^v j * r*i a- • Ans. 172 b. 13 rem.

ting off the right-hand figure of the divi- ' °

dend ; then divide the remaining figures by 2, the other factor of the divisor. The result 172 is the quotient; and 3, the figure cut off, being annexed to the remainder, forms the true remainder.

30. How proceed, when there are ciphers on the right of the divisor ?

I. Cut off the ciphers on the right of the divisor, and as many figures on the right of the dividend.

11. Divide the remaining part of the dividend by the remaining paH of the divisor for the quotient.

III. Annex the figures cut off to the remainder, and (lie result will he the true remainder.

II. Divide 8534 by 20. 14. Divide 23681 by 300.

12. Divide 12345 by 30. 15. Divide 40642 by 130Q.

13. 163045-^1900. 16. 264168 — 31000.

division. 75

DRILL FOR RAPID COMBINATIONS.

To Teachers.— These and other drill exercises, should he continued hut a few minutes at a time. If spirited and frequent, better results will he obtained from them, though short, than from scores of examples recited in an indifferent, sluggish manner.

Oral. — i. To 4 add 8 ; subtract 2 ; multiply by 3 ; divide by 5; add 4; multiply by 3 ; add 10 ; result ?

2. From 1 2 subtract 5 ; add 2 ; multiply by 4 ; divide by 6 ; add 5 ; multiply by 3 ; result ?

3. Multiply 3 by 6 ; add 4 ; subtract 2 ; divide by 5 ; multiply by 6 ; add 8 ; divide by 4 ; result ?

4. Divide 42 by 7 ; multiply by 4 ; subtract 6 ; divide by by 3 ; add 5 ; add 9 ; divide by 5 ; multiply by 1 1 ; result ?

5. To 14 add 8; take 4; divide by 9; multiply by 8; add 8 ; divide by 8 ; multiply by 9 ; result ?

6. From 27 take 9; divide by 9; multiply by 9; add 9; take 7 ; multiply by 2 ; divide by 10; result ?

7. Multiply 9 by 7; subtract 7 ; divide by 8; add 12 ; subtract 4; divide by 5; multiply by 12; divide by 9; add 20 ; divide by 6, multiply by 1 1 ; result ?

8. Divide 54 by 9 ; multiply by 7 ; subtract 6 ; divide by 9 ; multiply by 8 ; add 7 ; subtract 4 ; divide by 7 ; add 30 ; result ?

9. Add 7 to 15; divide by 11; multiply by 9; add 10; divide by 7; multiply by 12; add 11; subtract 4; divide by 5 ; multiply by 8 ; result ?

10. Multiply 8 by 7; subtract 6; divide by 10; mul- tiply by 9; add n; divide by 8; multiply by 9; sub- tract 3 ; add 30 ; subtract 7 ; result ?

Slate. — 1. To 36 add 45 ; subtract 37 ; multiply by 6 ; divide by 8 ; multiply by 9 ; add 99 ; divide by 9 ; add 200 ; result ?

2. From 87 take 33 ; multiply by 7 ; divide by 6 ; add 233 ; take 48; divide by 8; multiply by 25 ; result?

76 DIVISION".

3. Multiply 348 by 9; add 556; divide by 8; multi- ply Dy 48 j divide by 24; add 545 ; take 378 ; result?

4. Divide 576 by 24 ; multiply by 35 ; add 1200 ; di- vide by 20 ; multiply by 45 ; divide by 9 ; take 375 ; add 2375 ; result?

5. To 785 add 357; take 571; add 629; divide by 24; multiply by 64 ; divide by 32 ; add 873 ; take 367 ; result ?

6. From 3256 take 840 ; divide by 302 ; add 78 ; mul- tiply by 56 ; divide by 28 ; add 1575 ; result ?

7. Multiply 456 by 28; divide by 7; add 256; take 1200; divide by 44; multiply by 325 ; result?

QUESTIONS FOR REVIEW.

Oral. — 1. If 1 man can do a job of work in 72 days, how long will it take 9 men to do it?

Analysis. — 9 men can do 9 days work in 1 day : therefore, to do 72 days work, it will take them as many days as 9 is con- tained times in 72, which is 8. Ans. 8 days.

2. If a barrel of apples will last 1 person 56 days, bow long will it last a family of 7 persons ?

3. How many weeks in 8 times 9 days ?

4. How many 4-quart cans can be filled from three 8-quart pails ?

5. A farmer bought 4 pair of boots, at 5 dollars a pair, and paid for them in wheat, at 2 dollars a bushel: how many bushels did the boots come to ?

6. How many times 8 in 7 times 12 ?

7. A market-woman sold 6 dozen eggs, at 10 cents a dozen, and took her pay in muslin, at 12 cents a yard: how many yards did she receive ?

8. How many 2-gallon measures can be filled from 6 ten -gallon casks of water ?

division. 77

9. Herbert bought 20 marbles at one time, and 16 at another; meantime he lost 12 : how many had he then ?

10. If you earn 9 dollars a week, and pay 3 dollars for board, and 2 dollars for incidentals, how much will you lay up in 9 weeks ?

11. In 40 less 12, how many times 7 ?

12. A trader bought 12 pair of shoes at 2 dollars, and 6 huts at 5 dollars : how much did he pay for both ?

13. Three men gave a poor person 75 dollars; one gave 30 dollars, and another 25 dollars : how much did the other give ?

14. Three lads, counting their money, found A had 25 cents, B twice as much as A, and C as much as both the others : how much had all ?

Slate. — 1. A fort has provisions sufficient to last 1 man 365 days: how long will it last a company of 73 men?

2. Bought 100 hogs, weighing 300 pounds each, at 7 cents a pound, and sold them at 10 cents a pound: what was the profit ?

3. A grocer bought 15 hogsheads of molasses, at 35 dollars per hogshead; 151 boxes of oranges, at 6 dollars a box ; and 91 sacks of coffee, at 20 dollars a sack ; and Bold the whole for 4856 dollars: what did he make by the operation ?

4. A farmer having 115 dollars, paid 40 dollars for a cow, and the remainder for 15 sheep: what did the sheep cost him apiece ?

5. A shoe-dealer sold 87 pair of overshoes, at 2 dollars a pair; no pair of boots, at 9 dollars a pair; and took his pay in coal, at n dollars a ton: how much coal ought he to receive ?

6. A teacher was engaged at 1260 dollars a year; at

78 division.

the end of 9 months his health failed and he left: how much should he receive ?

7. A grocer bought 455 barrels of flour for 3185 dol- lars; he afterward bought another lot at the same rate for 1 6 10 dollars: how many barrels were there in both lots; and what did it cost him per barrel ?

8. A man left 6528 dollars to his wife and 3 children; to the latter he gave 1265 dollars apiece: what wras the portion of his wife ?

9. A young man's salary amounted to 1208 dollars a year for 3 years; his expenses the first year were 375 dollars; the second, 420 dollars; and the third, 519 dollars : how much did he lay up in the 3 years ?

10. If I earn 1350 dollars a year, and spend 1785 dol- lars a year, how much shall I be in debt in 3 years ?

1 1. What number taken from 25973 4- 8230 will leave 8768 ?

12. What number taken from 41260 — 32S1 will leave 8600?

13. What number taken from 62135 — 161 2 will leave 21500 — 2861 ?

14. If one man can perform a piece of work in 750 days, how long will it take 25 men to do it ?

15. If a man earns 1645 dollars a year, and his ex- penses are 517 dollars a year : how long will it take him to lay up 4512 dollars ?

1 6. Three lads, talking of their money, the first said he had 187 cents; the second said he had as much as the first minus 23 cents ; and the third said if he had 40 cents more, he should have as many as the other two : how many cents had the second ? The third ?

17. Two men being 1950 miles apart, traveled towards each other at the rate of 35 and 43 miles a day respec- tively: how long before they met?

FACTOEISG.

*:!> Teachers who prefer to have pupils study United States Money before Douimou and Decimal Fractions, are referred to p. 143.

i. What two numbers multiplied together make 6?

2. What then are the factors of 6 ? (P. 47, Q. 5.)

3. What are the factors of 10?

4. What are the factors of 8 ? Of 1 2 ?

5. What are the factors of 14 ? Of 15 ? Of 21 ?

6. Name two factors of 16. Of 18. Of 20.

7. Name two factors of 24. Of 35. Of 48.

8. Name two factors of 54. Of 63. Of 72.

9. Name two factors of 84. Of 96. Of 108.

DEFINITIONS.

1. What is a Factor ?

A Factor of a number is one of the numbers, which multiplied together, produce that number. (P. 47, Q. 5.)

2. What is a Composite Number ?

A Composite Number is the product of two or more factors, each of which is greater than 1. Thusy when it is said that 3 x 5 = 15, fifteen is a composite number, and 3 and 5 are its factors.

it. What is a Prime Number ?

A Prime Number is one which cannot be pro- duced by multiplying any two numbers together, except 1 unit and itself.

4. What are Prime Factors?

The Prime Factors of a number are the prime numbers which, multiplied together, produce that number.

5. What is an Odd Number?

An Odd Number is- one which cannot be divided by 2, without a remainder ; as, 1, 3, 5, 7, etc.

80

FACTORING.

6. What is an Even Number ?

An Even Number is one which can be divided by 2, without a remainder; as, 2, 4, 6, 8, etc.

Note. — All even numbers except 2 are composite numbers

7. What is meant by Factoring a number ?

Factoring a Number is finding two or more factors which multiplied together, produce that number

MENTAL EXERCISES.

r. Name the odd numbers under 30.

2. Name the even numbers under 30.

3. Name all the composite numbers under 30.

4. Namo all the prime numbers under 30.

5. What are the prime factors of 30 ?

Analysis. — By inspection we perceive that 30 is divisible by the prime number 2, giving the factors 2 and 15. Again, dividing 15 by 3 we have the factors 3 and 5, both of which are prime Therefore, 2, 3, and 5, are the prime factors required.

6. What are the prime factors of 12? 15? 18?

7. What are the prime factors of 20 ? 28 ? 30 ?

8. What are the prime factors of 35 ? • 40 ? 42 ?

SLATE EXERCISES. 1. What are the prime factors of 105 ?

OPERATION.

Analysis. — Dividing 105 by the prime number 3, we have the factors 3 and 35. Again, dividing the 1st quo- tient 35, by the prime number 5, we have the factors 5 an J 7. Finally, dividing the 2d quotient 7 by 7. we have 7 and 1. But the divisors 3. 5, and 7 are all prime numbers, and therefore arc the prime factors required.

1st divisor, 3 2d " 5 3* " 7 105 = 3x5*7

105, given.

35, 1st quot 7, 2d " i,3d ■

CANCELLATION. 81

§. How is a composite number resolved into prime factors ?

Divide the given number by any prime number that will divide it without a remainder. Again, divide this quotient by a prime number, and so on till the quotient is i. The several divisors are the prime factors required.

Note. — The least divisor of every number is a prime factor; hence, to avoid mistakes, it is advisable for beginners to take for the divisor, the least number that will divide the several dividends without a remainder.

Find the prime factors of the following numbers :

2. 42.	6. 100.	10. 200.	14. 625.
3. 48.	7. 125.	11. 256.	15. 1000.
4. 60.	8. 132.	12. 325.	16. 1728.
5- 72.	9- 175-	13- 45o- — ♦	17. 1872.

CANCELLATION

i. What is the quotient of 3x3x5 divided by 3 x 5 ?

Analysis. — 3 X3X 5=45, and 3X 5 = 15 ; now 45-^-15=3. But it will be seen by inspection that the factors 3 and 5 are common tc the dividend and the divisor. If we cancel or cross out the 3 in each, we have 3x5-5-5, or 15-5-5, which equals 3, the same as before. Again, if we cancel or cross out the 5 in each, we have 3-7-1, which equals 3, as before.

Note. — To cancel a factor of a number means to erase or reject it.

2. What is the quotient of 2x3x7-^-2x3x5?

Solution. — Cancelling the common factors 2 and 3, we have 7-*-5 = if A?is.

1 . What is the effect of cancelling a factor from a number ?

It divides the number by that factor.

82 CANCELLATION.

10. What is Cancellation ?

Cancellation is the method of abbreviating opera- tions by rejecting equal factors from the divisor and rividend.

Remakk. — When the factor cancelled is equal to the number itself, i is always left in its place ; for, dividing a number by itself, the quotient is i. When the i stands in the dividend, it i^ast be retained ; when in the divisor, it may be disregarded.

3. What is the quotient of 2x5x7-^2x3 x-7 ?

Analysis. — Writing the Operation.

divisor under the dividend, 1 I

and cancelling the factors 2 2 X 5 X ^. I X 5 X I _ s

and 7, which are common to 2> X 3 X % 1x3x1 ^

both, we have 1x5x1-^-1x3 j 1

x 1. Now the product of 1 x 5 x 1 = 5, that of 1 x 3 x 1=3, and 5-^-3=1! Ana.

11. What is the rule for Cancellation?

Cancel all the factors common to the divisor and divi* de?id, and divide the product of those remaining in tin dividend by the product of those remaining in the divisor.

4. What is the quotient of 77 divided by 21 ?

Solution. — By inspection, we per- Operation.

ceive that 7 is a factor common to the %%y j 1

divisor and the dividend. Cancelling ^1 ~~=II~3>0Y 3i this factor, we have V"* or 3$, Ans.

Perform the following divisions by cancellation.

5. 4x5x7-^5x4x3. 8. 28x13x114-11 x 13x7c

6. 7x3x11-4-8x3x7. 9. 63x39x2-4-13x9x3.

7. 23x5x9-1-5x7x9. 10. 96x7x11-4-12x8x7.

1 1. How many yards of cloth, at 8 dollars a yard, can be bought for 25 pair of boots, at 4 dollars a pair ?

1 2. How many barrels of flour, at 7 dollars a barrel, must be given for 18 tons of hay, at 14 dollars a ton ?

13. How long must a man work, at 3 dollars a day to pay his rent for a year, at 1 1 dollars a month ?

COMMON DIVISORS

MENTAL EXERCISES.

i. What will divide 9 and 15 without a remainder?

2. What will divide 14 and 24 without a remainder ?

3. What will divide 16 and 20 without a remainder ?

4. What will divide 42 and 18 without a remainder ?

5. What is the greatest divisor of 18 and 27 ?

6. What is the greatest divisor of 12 and 36 ?

DEFINITIONS.

12. What is a Common Divisor?

A Common Divisor is a number which will Jivide tivo or more numbers without a remainder.

13. What is the Greatest Common Divisor?

The Greatest Common Divisor of two or

more numbers, is the greatest number that will divide each of them without a remainder.

Remarks. — 1. A common divisor of two or more numbers is always a common factor of those numbers ; and the greatest com- mon divisor of them is their greatest common factor.

2. A common divisor is often called a common measure.

3. The greatest common divisor of two or more numbers is equal to the product of all the prime factors common to those numbers.

i. What is the greatest common divisor of 16 and 20 ?

1st Method.— Dividing the greater by 1st Operation. the less, the quotient is 1, and 12 remain- 16)28(1 der. Again, dividing the first divisor by 16 the first remainder 12, the quotient is 1, T^Wfif t and 4 remainder. Next, dividing the sec- ond divisor by the second remainder 4, the ■ — quotient is 3, and o remainder. The last 4 ) T 2 ( 3 divisor 4, is the greatest common divisor. \^_

8-1 COMMON DIVISORS.

2d Method. — Setting the numbers in a Iwr- 2d Operation.

izontal line, divide by any prime number, as z, 2)16 28

that will divide each of them without a remain- ~7~z

der, and set the quotients under the correspond- 2 / T4

ing numbers. Dividing each of these quotients 4 7

by 2 again, the new quotients 4 and 7 have no Ans. 2x2=4, common factor. Hence, the product of the common divisors 2 into 2, or 4, is the greatest common divisor.

14. How find the greatest common divisor of two or more numbers ?

Divide the greater number by the less, the first divisor by the first remainder, the second divisor by the second remainder, and so on until the remainder is nothing ; the last divisor will be the greatest common divisor.

Or, write the numbers in a horizorital line, and divide by any prime number that will divide each without a remainder ; setting the quotients in a line below.

Divide these quotients as before, and thus proceed, till no number can be found that will divide all the quotients without a remainder. Tfie product of all the divisors will be the greatest common divisor.

Notes. — 1. If there are more than two numbers, and the first method is used, first find the greatest common divizor of two of them, then of this divisor and a third number, and so on, until all the numbers have been used.

2. When there are three or more numbers, the second method has the advantage both in simplicity and facility of application. f

SLATE EXERCISES.

2. Kequired the greatest com. divisor of 15, 45, and 60 ?

Solution. — Divide the given numbers by 3)15 45 60

3, and the quotients thence arising by 5 ; ~c

the next quotients, 1, 3, and 4, have no com- *> ' * 5

mon factor. Hence, the product of 3 into 5, 134

or 15, is the answer. 3x5=15, AtlS.

COMMON MULTIPLES. 85

Find the greatest common divisor of the following numbers :

2. 27 and $6. 8. 120 and 148.

3. 32 and 48. 9. 256 and 512.

4. 45 and 60. 10. 36, 84, and 108.

5. 72 and 24. 11. 45, 60, and 135.

6. 75 and 105. 12. 30, 75, and 225.

7. 81 and 108. 13. 48, 144, and 288.

14. What is the greatest number by which 128, 160, and 192 can be exactly divided ?

15. What is the longest pole by which 108, 132, and 144 feet can be exactly measured?

16. A shopkeeper has three balls of twine, containing 120, 100, and 200 yards, which he wishes to cut into kite-lines of equal length : what is the greatest length he can make them 9

COMMON MULTIPLES.

MENTAL EXERCISES.

1. WHat numbers under 12 can be divided by 2 with- out a remainder ?

2. By what numbers can 12 be exactly divided?

3. What numbers under 20 can be exactly divided by 4?

4. By what numbers can 15 be exactly divided ?

5. By how many numbers can 18 be exactly divided ?

6. By what numbers can 24 be exactly divided ?

7. By what two factors can 33 be exactly divided ?

8. By what two factors can 35 be exactly divided ?

9. What 4 numbers will exactly divide 42 ?

10. Name two numbers that can be exactly divided by 4, 5, and 6.

Ob COMMON MULTIPLES.

DEFINITIONS.

15. What is a Multiple ?

A Multiple is a number which can be divided by another number without a remainder.

1 6. What is a Common Multiple ?

A Common Multiple is a number which can be divided by two or more numbers without a remainder. Thus, 15 is a common multiple of 3 and 5.

Remark. — A common multiple of two or more numbers, con- tains all the prime factors of those numbers.

17. What is the Least Com. Multiple of two or more numbers ?

The Least Common Multiple of two or more numbers, is the least number which can be divided by each of them without a remainder. Thus, 12 is the least common multiple of 2, 3, and 4.

1. What is the least com. multiple of 12, 18, and 21 ?

1ST Method. — Writing the numbers 1st Operation.

in a horizontal line, we divide by any 3)12 18 21

prime number 3, which will divide two

or more of them without a remain- 2 ) 4 6 7

der, and set the quotients in the line "^ I ~~Z

below. Again, dividing these quotients _

by the prime number 2, which will di- ^ .57—5

vide two of them without a remainder, we set the quotients and undivided number 7 in a line below as before. As the numbers 2, 3, and 7, are prime factors, the division can be carried no fur- ther. Finally, the continued product of the divisors and numbers in the last line, 3x2x2x3* 7=252, is the least com multiple.

2d Method. .— Resolve the given 2d Operation.

numbers into their prime factors, as in 12 = 2x2x3

the margin. But we have seen that a l8=2x$X3

common multiple of two or more num- 2 1 =$ X 7

bers contains all the prime factors of 2X2X-X-X,_ 2 c 2 those numbers. Hence, it must contain

the prime factors of 12. which are 2x2x3; we therefore retain

these factors. Again, it must contain the prime factors of 18,

COMMON MULTIPLES. 87

which are 2x3x3. But we already have two 2s and one 3 ; we may therefore cancel the 2, and one of the 3s, retaining the other 3. Finally, it must contain the factors of 21, which are 3x7. But since we have retained two 3s, we may cancel this 3, and re- tain the 7. The continued product of the uncancelled factors 2X2X3X3X 7=252, the same as before.

1§. How find the least common multiple of two or mon numbers ?

Write the numbers in a horizontal line, and divide by any prime number that ivill divide tiuo or more of them without a remainder, placing the quotients and numbers undivided in a line below.

Next divide this line as before, and thus proceed till no two numbers are divisible by any number greater than 1. Tlie continued product of the divisors and numbers in the last line ivill be the answer.

Or, resolve the given numbers into their prime factors ; multiply these factors together, talcing each the greatest number of times it occurs in either of the given numbers, and the product ivill be the answer.

Remark. — Both of these methods are based upon the principle, that the least common multiple of two or more numbers is the least number which contains all their prime factors, each factor being taken as many times as it occurs in either of the given num- bers.

SLATE EXERCISES. Find the least common multiple of the following

2. 4, 8, 12. 8. 39, 52, 13.

3. l6, 12, 24. 9. 8l, I08, 72.

4- i5> 3°> 45- IO- 24, 12, 48, 60.

5. 36, 48, 84. 11. 14, 42, 28, 56.

6. 40, 45> 75- I2- 54, 81, 96, 120.

7. 20, 60, 55. 13. 72, 144, 288, 432.

FRACTIONS

INTRODUCTORY EXERCISES.

To Teachers.— The object Of this Exercise is to develop the idea of Fractional parts. The best way to secure this end is, to let beginners divide some object, as a sheet of paper, or an apple, into halves, thirds, fourths, etc. ; then put the parts together and form the whole again.

i. If you divide a sheet of paper into tivo equal parts, what is each part called ? One half.

2. Draw a line an inch long upon your slate or black- board, and divide it into halves.

3. If you divide an apple into three equal parts, what is one of the parts called ?

One third.

4. Two of the parts ? Two thirds.

5. Into how many halves can you divide an apple? Into how many thirds ?

6. Draw a line a foot long, and divide it into halves. Into thirds.

7. How many thirds make a whole one ?

8. If a sheet of paper is divided into four equal parts, what is one of the parts called ?

A Fourth, or quarter. \ 9. What are two of the parts called ? Three of the ♦parts ? How many fourths in a whole sheet ? 1 10. When a thing is divided in five equal parts, what are the parts called ?

Fifths.

11. When a thing is divided in six equal parts, what are the parts called ? If divided in seven, what ? Into eight, what ? Into nine, what ? Into ten, what ? Into twenty, what ? Into fifty, what ?

FRACTIONS. 89

12. Which are greater, halves or thirds ? Thirds or fourths? Sixths or fifths ? Tenths or eighths ?

DEFINITIONS.

1. What is an Integer ?

An Integer is a number which contains one or more entire units only; as i, 3, 5, 8, 12, etc.

2. What is a Fraction ?

A Fraction is one or more of the equal parts into which a unit is divided.

3. What is meant by one-half?

One of the two equal parts into which a unit is divided.

What is meant by a third ? Two thirds ? A fourth ? Fifth ? Seventh ? Tenth ?

4. From what do these parts take their name ?

The Number of equal parts into which the unit or thing is divided.

5. Upon what does their value depend ?

First. Upon the magnitude of the unit or tldng divided.

Second. Upon the number of parts into which it is divided.

Illustrate these two points :

1st. If a large and a small sheet of paper are each divided into halves, thirds, fourths, etc., it is plain that the parts of the former will be larger than the corresponding parts of the latter.

2d. If one of two equal sheets of paper is divided into two equal parts, and the other into four, the parts of the first will be twice as large as those of the second ; if one is divided into two equal parts, the other into six, one part of the first will be equal to three of the second, etc. Hence,

Note. — 1. A half is twice as large as a fourth, three times as large as a sixth, four times as large as an eighth, etc. ; and generally,

90 FRACTIONS.

2. The greater the number of equal parts into which the unit is divided, the less will be the value of each part. Conversely,

3. The less the number of equal parts, the greater will be the value of each part.

FINDING FRACTIONAL PARTS.

1. What is 1 half of 10 dollars?

Analysis. — If 10 dollars are divided into two equal parts, one o. these parts is 5 dollars. Therefore, &c. (P. 63, Q. II.)

2. What is 1 half of 8 peaches ?

3. What is a third of 9 ? Of 15 ? Of 1 8 ? Of 24 ?

4. What is a fourth of 12 ? Of 16 ? Of 28 ? Of 40 ?

5. What is a fifth of 20 ? Of 45 ? Of 35 ? Of 60 ?

6. What is an eighth of 32 ? Of 56 ? Of 64? Of 72 ?

7. What is 2 thirds of 18 yards ?

Analysis. — 2 thirds are twice 1 third ; now 1 third of 18 yards is 6 yards, and 2 times 6 yards are 12 yards. Therefore, etc.

8. What is 3 fourths of 32 ?

9. What is 3 eighths of 48 ?

10. What is 4 fifths of 35 ?

11. What is 7 tenths of 60 ?

NOTATION OF FRACTIONS,

6. Into what two classes are fractions divided ?

Into Common and Decimal.

7. What is a Common Fraction?

A Common Fraction is one in which the unit is divided into any number of equal parts.

8. How are common fractions usually expressed ?

By Figures written above and below a line, called the numerator and denominator; as, f , J, -fa.

FRACTIONS. 91

9. Where is the Denominator placed, and what does it show ? The Denominator is written lelow the line, and

shows into liow many equal parts the unit is divided.

10. Where the Numerator, and what does it show ?

The Numerator is written above the line, and shows how many parts are expressed by the fraction.

Notes.— i. The denominator is so called because it names the parts ; as, halves, thirds, etc.

2. The numerator is so called because it numbers the parts taken. Thus, in the fraction §, 3 is the denominator, and shows that the unit is divided into 3 equal parts ; 2 is the numerator, and shows that 2 of the parts are taken.

11. What are the Terms of a fraction?

The Terms of a Fraction are the Numerator and Denominator.

WRITTEN EXERCISES.

1. How express one-half, one-third, three-fourths, etc., by figures ? Ans. \, ± f .

Write the following fractions in figures :

2. Two thirds. 9. Ten twelfths.

3. Four fifths. 10. Eight twenty- thirds.

4. Three sevenths. 11. Mne thirty-firsts.

5. Seven eighths. 12. Twenty-three fortieths.

6. Five ninths. 13. Nineteen seventy-fifths.

7. Seven thirds. 14. Seventy-four hundredths.

8. Four fourths. 15. Ninety-nine thousandths.

Copy and read the following :

16." -J. ii.Jf. 26. |J, 31. i3|,

17. tV 22. j}. 27. ** 32. m-

*«• A- 23. if. 28. -v- 33. «*.

19. ^. 24. «. 29. -v-. 34. iff

20. ^ 25. Jf 30. ^. 35, .£00..

9- FRACTIONS.

DEFINITIONS.

12. Into what are common fractions divided?

Into proper, improper, simple, compound, complex frac- tions, and mixed numbers,

13. Explain each.

A Proper Ft*action is one whose numerator is less than the denominator ; as, |, j.

An Improper Fraction is one whose numerator equals or exceeds the denominator ; as, f , § .

A Simple Fraction is one having but one numer- ator and one denominator, each of which is a whole number, and may be propter or improper ; as, J, £.

A Compound Fraction is a fraction of a frac- tion ; as \ of \.

A Complex Fraction is one which has a frac- tional numerator, and an integral denominator: as, | £l* 3' A

A Mixed Number is a whole number and a frac- tion expressed together; as, 5$, 342].

1. What kind of fractions are f , $ , and J ? Why ?

2. What kind are f and J ? j and | ? Why ?

3. What kind are J of J ? f of f ? Why ?

4. What do you call 4h ih 9s ? Why ?

5. What do you call J% t ? Why ?

4 5

11. What is the value of a fraction ?

The Value of a Fraction is the quotient of the numerator divided by the denominator. Thus, the value of 1 half is i-i-2; of 2 thirds, is 2-7-3; °f 4 fourths, is 4-1-4, or 1 ; of 6 thirds, is 6^-3, or 2, etc.

* See New Practical Arithmetic, Rem. p. 101.

FRACTIONS. 93

GENERAL PRINCIPLES OF FRACTIONS.

1 5. What are some of the principles upon which the opera- tions in fractions depend ?

1. Multiplying the numerator by any number, multi- plies the fraction by that number.

II. Dividing the numerator, divides the fraction.

III. Multiplying the denominator, divides the frac- tion.

IV. Dividing the denominator, multiplies the fraction.

V. Multiplying or dividing both the numerator and denominator by the same number, does not alter the value of the fraction.

REDUCTION OF FRACTIONS.

MENTAL EXERCISES.

i. If I divide an apple into halves, how can I express one of these parts by figures ?

By \. (The pupil writes it upon the blackboard.)

2. If you multiply both terms of \ by 2, what will it become ?

It will become j.

3. If you multiply both terms of \ by 3, 4. 5, etc., what ? It will become f , f , T5^, -^, and so on.

4. How many fourths in \ ? Two fourths.

5. How many sixths in \ ? How many eighths ? How many tenths ? Twelfths ?

6. If you multiply both terms of \ by 2, what will it become ?

It will become f .

7. If you multiply both terms of \ by 3, 4, 5, etc., what will it become ?

It will become % , ^, *fc, etc.

94 BEDUCTIONOF

8. How many thirds in } ? In ■& ? T6? ? -J % ?

9. If I divide an orange into 4 equal parts, how can I express one-half of these parts by figures ?

By J. (The pupil writes it upon the blackboard.)

10. If you divide both terms of j by 2, what will it become ?

It will become £.

11. To how many halves are f equal ? -^9 -fa ?

1 2. To how many thirds are £ equal ? f ? T8j ?

13. How many fourths equal £ ? T6^ ? -J^ ?

DEFINITIONS.

16. TOdtf is Reduction of Fractions?

Reduction of Fractions is changing the terms, without altering the value of the fractions.

17. What is meant by reducing a fraction to higher terms?

It is changing its numerator and denominator to larger numbers, without altering its value,

18. What by reducing a fraction to loxcer terms ?

It is changing its numerator and denominator to smaller numbers, without altering its value.

19. What is the principle upon which these changes are made ?

The principle that multiplying or dividing both he numerator and denominator by the same number ioes not alter the value of the fraction. (P. 93, Pr. V.)

CASE L* To reduce a Fraction to higher terms.

1. Keduce f to twentieths.

Analysis. — The required denominator 20, Operation.

contains 5, the given denominator, 4 times. 20-^5=4.

But if both terms of a fraction are multiplied 3x4 by the same number, its value is not altered. - X4 = a& AnS* Therefore, multiplying both terms of \ by 4, we have %=$%, the fraction required. (P. 93, Prin. V.)

FRACTIONS. 95

20. How reduce a fraction to higher terms ?

Multiply both terms of the fraction by such a number as will make the given denominator equal to the required denominator.

Note. — The multiplier is found by dividing the proposed de- nominator by the given denominator.

2. Eeduce f to thirtieths.

3. Reduce £ to fortieths.

4. Reduce | to sixty-thirds;

5. Reduce ^ to fifty-fifths.

6. Reduce ^ to seventy-fifths.

7. Reduce ■£? to seyenty-sixths.

8. Reduce %} to one-hundred-and- thirty-fifths.

9. Reduce ff to two-hundred-and-sixty-eighths. 10. Reduce -^T to one-thousandths.

CASE II. To reduce a Fraction to lower terms.

11. To how many tenths are |£ equal ? Analysis. — The given denominator 20 Operation.

contains 10, the required denominator, 2 20-j-lo=2.

times. But, if both terms of a fraction are 12-^2

divided by the same number, its value is _;_ — TU> Ans.

not altered ; therefore, dividing both terms

of \% by 2, it becomes -,%, the fraction required. (P. 93, Prin. V.)

21 . How is a fraction reduced to lower terms ? Divide both terms by such a number as will make the yiven denominator equal to the required denominator.

12. Reduce ^ to fourths. 17. Reduce 37T to fifths.

13. Reduce ^ to thirds. 18. Reduce Jf to ninths.

14. Reduce ■£$ to eighths. 19. Reduce ff to sixths.

15. Reduce ■£$ to sixths. 20. Reduce -£f to eighths.

16. Reduce Jf to fourths. 21. Reduce T8o\ to twelfths.

96 REDUCTION OF

CASE III. To reduce a Fraction to its lowest term9.

22. What are the lowest terms of a fraction ?

The Lowest Terms of a fraction are the smallest numbers in which its numerator and denominator can be expressed.

i. What are the lowest terms to which -ff can be reduced ?

Analysis. — Dividing both terms of a frac- 1st Operation. tion by the same number, does not alter its 2 )-J-|-=-| and value. (Prin. V.) Now if we divide both 2 \ 6 — 3 A?IS. terms of \% by 2, we have $ . Again, dividing both terms of the new fraction f by 2, the result is f , which are the lowest terms in which || can be expressed.

Or, we may divide both terms of the frac- 2d Operation. tion by their greatest common divisor, which 4)^2=^ Ans. is 4, and obtain the same result. (P. 84, Q. 14.)

23. How reduce a fraction to its lowest terms ?

Divide the numerator and denominator continually by any number that tvill divide both tvithout a remainder, until no number greater than 1 will divide them.

Or> divide both terms of the fraction by their greatest common divisor. (P. 84, Q. 14.)

2. What are the lowest terms to which £§ can be re- duced ? Ans. f .

Keduce the following fractions to their lowest terms :

3. IS-

4. ft-

5- it-

6- » 7. » & if

n 42 9- T2'	«s»	21. -,?8v.
10. «.	16. T%.	». m-
11. If-	■7- m-	23. H*
12- Tin-	18. T<&.	24. «f
13. AV	19- j*.	*s-m
•4- tt*	29. T^.	»«■ A'iV

FRACTIONS. 97

CASE IV.

To reduce an Improper Fraction to a Whole or Mixed

Number.

i. In 7 half dimes, how many whole dimes ?

Analysis. — Since in 2 halves there is 1 whole dime, in 7 halves there are as many dimes as 2 is contained times in 7, which is 3 and 1 half over, or 3^ times. Therefore, in 7 half dimes there are 3$ dimes.

2. In 10 half dollars, how many dollars ?

3. To what whole number is ty equal ?

4. To what mixed number is -^ equal ?

5. Seduce ^f- to a whole or mixed number.

6. Eeduce Afl to a whole or mixed number.

SLATE EXERCISES.

1. Reduce -^ to a whole or mixed number. Analysts. — Since in 3 thirds there is a unit, or one, operation.

in 25 thirds there are as many units as 3 is contained 3 ) 2 5

times in 25. Dividing the numerator 25 by the de- 7

nominator 3, the quotient is 8 and 1 over, or 8£. Alls. o-£ Therefore, %5 = 8$.

21. How reduce an improper fraction to a whole or mixed number 1

Divide the numerator by the denominator, and the quotient to-ill be the number required.

Reduce the following to whole or mixed numbers :

2. 4f. 6. ¥• i°- W« H. -¥i2--

3. ¥• 7. ***• »-W* J5- W-

4. *£. 8. -W- 12. ^. 16. W.O,

5- ¥• 9. -W *3- -5#- i7- ^c?-

i3. How many sheets of paper shall I require, to give

5 pupils half a sheet apiece ?

19. A man meeting 15 beggars, gave each a quarter

of a dollar : how many dollars did he give to all ? 5

98 REDUCTION OF

CASE V.

To reduce a Whole or Mixed Number to an Improper Fraction.

i. How many halves in 2 pears ?

Analysis. — In 1 pear there are 2 halves ; therefore, in 2 pears there are 2 times 2 halves, which are 4 halves.

2. How many halves in 3 whole ones ? In 4 ? In 5 ?

3. In 5 how many thirds ? In 6 ? In 7 ? In 10 ?

4. How many fourths in 5 ? In 7 ? In 1 1 ? In 12 ?

5. How mnny fifths in 8 ? In 9 ? In 12 ?

6. How many tenths in 7 ? In 9 ? In 10 ?

7. How many thirds in 4! ?

Analysts. — Since in 1 there are 3 thirds, in 4 there must be 3 times 4, or 12 thirds, and 2 thirds are 14 thirds. Therefore, in 4! there are LaA.

8. How many fourths in 5 J ? In 6J ?

9. How many sevenths in 5$ ? In 6§ ? In 8^?

10. Eeduce 8 J to an improper fraction.

11. Reduce iof to an improper fraction.

12. Reduce i2f to an improper fraction.

SLATE EXERCISES. »

i. Reduce 17 to fifths.

Analysis. — Since there are 5 fifths in a unit, Operation. there must be 5 times as many fifths in a num- 17 X 5 — &5*

ber as there are units in that number; and 5 Ans. ^5. times 17 are 85. Therefore, 17= V»

2. Reduce 13^ to an improper fraction ; that is, to fifths.

Analysis. — Reasoning as before, in 13 units thoro 133.

are 5 times 13, or 65 fifths, and 4 fifths are 69 fifths. ,-

We therefore multiply the whole Dumber 13, by the

denominator 5, and adding the 4 fifths to the product, 69

wo have ^. Ans. -V-

FRACTIONS. 99

25. How reduce a whole or mixed number to an improper fraction ?

Multiply the tvhole number by the given denominator ; to the product add the numerator, and place the sum over the denominator.

Note. — A whole number may be reduced to an improper frac- tion, by making i its denominator. Thus, 3=f; for multiplying or dividing a number by i, does not alter its value.

3. Eeduce 25 to thirds. 5. Seduce 31 to fourths.

4. Eeduce 43 to fifths. 6. Eeduce 65 to sevenths.

Eeduce the following to improper fractions :

7- iif-	12.	5<5f	17.	100-2V	22.	iooof.
8. i4f.	13.	m	18.	1 17 -h-	23-	I064I.
9. i7l-	14.	89fV	19,	245iV-	24.	2207J.
10. 3 Ji-	15.	73tV	20.	430yV	25-	3°46|.
ii. 44-£-	16.	97tt-	21.	5°7tw	26.	523*tV

27. A gentleman having 25 J dollars, divided it equally among a company of beggars, giving each 1 fourth of a dollar : how many were there in the company ?

CASE VI. To reduce a Compound Fraction to a Simple one.

1. To what is § of -} equal ?

Analysis. — ^ of £ is equal to -,'5- ; for multiply- Operation. ing the denominator divides the fraction. Now, 2 0f I— 2 if ^ of £ is iV, 2 thirds will be twice as much, and 2 times iL5 are -,%>. In the operation, we multiply the numerators together for the new numerator, and the denominators together for the new denominator.

2. Eeduce | of J to a simple fraction? Solution. — f x i=2%, or -,'„, Ans.

3. Eeduce J of } of f to a simple fraction.

100 DEDUCTION OF

4. Reduce J of £ of -J to a simple fraction.

Analysis. — We have seen that Operation.

dividing the numerator and de- I S 4

nominator by the same number y 0I ? 01 's^=^XilXrZ^, ioes not alter the value of the

fraction ; also that cancelling a factor divides a number by that factor. We therefore cancel the common factors 3 and 4, then multiplying the factors remaining in the numerators together for the new numerator, and those remaining in the denominators for the new denominator, we have £ for the simple fraction required.

26. How reduce a compound fraction to a simple one ?

Cancel the common factors, and place the product of the factors remaining in the numerators over the product of those remaining in the denominators.

Note. — The object of cancelling the common factors is two- fold ; it shortens the operation, and reduces the result to the lowest terms.

5. Eeduce £ of J of % to a simple fraction. Ans. -fa.

Reduce the following to simple fractions :

6. I of } of f. 11. J off off 16. I off of 11.

7. f of i of |. 12. i of J of 4|. 17. i of i of 6J.

8. J off of i. 13. f of I of A. 18. I off of 9.

9. f of i off. 14. i of * off 19. J off of 2*. 10. I of i off. 15. fofifofi. 20. J off of 10*.

CASE VII. To reduce a Fraction to any acquired Denominator.

1. Reduce % to twelfths.

Analysis. — The given denominator 4 is Operation.

contained in 12, the required denominator, 3 12-^4 = 3.

times ; therefore, multiplying both terms of 3 x 3

5 by 3, it becomes rif and is the fraction re- 7 x ~ = TJ> -^W* quired.

FRACTIONS. • IGkl

27. How reduce a fraction to any required denominator ? Multiply both terms of the fraction by such a number

as will make the given denominator equal to the required denominator. (P. 95, Q. 20, n.)

Beduce the following fractions to the denominators indicated :

2. f to 3oths. 5. -fa to 72ds. 8. ff to i7ists.

3. I to 4oths. 6. H to 99ths. 9. f§ to 2 76ths.

4. f to 35ths. 7. H to i44ths. 10. -Jgg to ioooths.

CASE VIII. To reduce Fractions to a Common Denominator.

28. What is a Common Denominator ?

A Common Denominator is one that belongs equally to two or more fractions ; as, f , f, f .

1. It is required to reduce \, \, and }, to equivalent fractions, having a common denominator.

Analysts. — If we multiply each denom- Operation.

inator by all the other denominators, the 1x3x4

several products will be the same ; for, 1 2x3x4 — ^ 4' each is composed of the same factors, 2, 3,

and 4, the product of which is 24. Again, i — 7— _^..

if we multiply each numerator by all the 3 x 2 x 4

denominators except its own, it follows x 1x2x3 6

that the terms of each fraction will be mul- * 4x2x3 ~^*'

tiplied by the same number ; therefore,

the value of the fractions is not altered. (P. 93, Prin. V.)

20. How reduce fractions to a common denominator ?

Multiply the terms of each fraction by all the denomi- nators except its oivn.

Notes. — 1. Mixed numbers must be reduced to improper frac- tions, and compound fractions to simple ones, before applying the rule. (Ex. 12.)

10£ ItE'DUCTION OP

2. It should be observed that the value of the given fractions is not altered by reducing them to a common denominator. The reason is, that the terms of each fraction are multiplied by the eame numbers. (P. 93, Prin. V.)

2. Eeduce -f, J, ^, to a common denominator. Reduce the following to a common denominator :

3. f and £.	6. 1, i i	9- TT> & f
4. f and \.	7. f, i f	10. H, }*, **
5. | and f .	Q 2 2 7 °* "9"' ~3~> ~5'	TT 19 SO 7 XI' TT? T(JOJ TH

12. Find a common denominator of | of f, 3f and 5.

Analysis. — \ of. f=f; 3%—Lt, and 5=4. Now, f, V", and \ by the rule, become -2a4-, f |, J^1.

13. Reduce *j, J of f , and 3 to a common denomi- nator.

14. Reduce -f of f, 7, and 51 to a common denomi- nator.

CASE IX.

To reduce Fractions to the Least Common Denominator.

SO. What is the Least Common Denominator of two or more fractions.

The Least Common Denominator of two or

more fractions is the least common multiple of their de- nominators. (P. 86, Q. 17.)

1. What is the least common denominator of -f, fa and T\ ?

Analysis. — The solution of this , \, Tn TC

and similar Examples requires two

steps : 1st. To find the least com. mul- 5 ) J> IO> 5

tiplc of the denominators for the re- "^ ~2 Y.

quired denominator. 2d.To reduce the »/-•*>« r n is

given fractions to this denominator. u J

FRACTIONS. 103

The least com. multiple of 3, 10, and 2X10

15 is 30. (P. 87.) To reduce f, A, 3 x io^T°'

and -,A5 to thirtieths, we multiply

both terms of the given fractions by — =^. f-.^?w.

such a number as will reduce them ■•'

to thirtieths. Now multiplying both _4_ x 2__ . 8

terms of f by 10, the fraction becomes 15x2

IS ; multiplying both terms of *fo by

3, the fraction becomes & ; and multiplying both terms of W b, 2, we have 3a0-. (P. 101, Q. 27.) Therefore, the required fractions are §#, -,90-, and -380-.

31. How reduce fractions to the least common denominator ?

Find the least common multiple of all the denomina- tors; then multiply both terms of each fraction by such a number as will reduce it to this denominator. (P. 10 1, Q.27.)

Note. — Mixed numbers must be reduced to improper fractions, compound fractions to simple ones, and all fractions to their lowest terms, before applying the rule. (Ex. 12.)

2. Eeduce -J, -J, and J to the least com. denominator. Solution. — The least com. multiple of 2, 3, and 4 is 12. Now i=fV ; *=-& ; and |_=-&. Ans. -&, -fr, -,%-.

Eeduce the following to the least com. denominator: 2 2 3 I 6 3 2. 7 o -4 A ?3

4. h h tV 7- A> i A- i°- T& i y> £•

r- 7 5 I Q 8 6 2 tt A JL .5 ?-

1 2. Find the least com. denominator of 2J, £ of ^ J and 5.

Analysis.— 2*=V- ; £ of ft-ft^rVj f=i *-* 5=t- The least com. multiple of 5, 10, 4, and 1, is 20. Now L6L—tt'> iV=Ai

2 — 15.. on(l 3_ljQ.Il JOTo 4.1 JL 15. Ilia

13. What is the least com. denominator of §, 5 J, and I ?

14. What is the least com. denominator of J of f, 4}, I, and 4 ?

i.04 ADDITION OF FRACTION'S.

ADDITION OF FRACTIONS.

To add Fractions which have a Common Denominator.

Remark. — When two or more fractions express parts of the same kind of unit, and have a common denominator, their nu- merators are like numbers ; hence, they may be added, subtracted, and divided as whole numbers.

MENTAL EXERCISES. i. "What is the sum of f- dollar, f- dollar, and -f dollar ?

Analysis.— 3 fifths dollar and 2 fifths are 5 fifths, and 4 are 9 fifths, which are equal to i£ dollar.

2. What is the sum of -J, f, and \ ?

3. What is the sum of f , J, f, and £ ?

4. What is the sum of |, f , f , and -f ?

5. What is the sum of f, -J, |, and | ?

6. What is the sum of T33-, -f^, yV, and T8T ?

. SLATE EXERCISES.

32. When fractions have a common denominator, what is true of their numerators ?

Their numerators express like parts of a unit, and therefore are like numbers.

1. What is the sum of J$, -££, and \ % ?

Analysis. — As these fractions Operation.

have a common denominator, their ~zis + ^u + To" — "2To~> or 2"2o~* numerators are like numbers.

Hence, they may be added as whole numbers. (P. 25. Q. 9.) Thus, the sum of 13 twentieths + 16 twentieths + 14 twentieths= $1, or 2fij, Ans.

33. How add fractions which have a common denominator? Add the numerators, and place the sum over the common

denominator.

Note. — The answers shculd be reduced to the l-zoest terms, and improper fractions to whole or mixed number*.

ADDITION OF FRACTIONS. 105

2. A man sold f of an acre of land to one customer, J to another, f to another, and £ to another : how much did he sell to all ?

3. What is the sum of ^ pound, -^ pound, jj pound, and ^ pound ?

4. What is the sum of -^ -JJ, if, £J ?

5. What is the sum of Jf, A> tt ft ?

6. What is the sum of ■&, f J, j|, jj ?

To add Fractions which have Different Denominators.

1. What is the sum of J dollar, £ dollar, and £ dollar ?

Analysis. — Since these frac- Operation.

tions have different denominators, 2 X 4 X 6 = 48 Cow. Z).

tlieir numerators denote unlike , v, . v^ < _ „ . T „/ at

1x4x0 = 24, 1 Sf Jy. parts of a unit; consequently, a— a 7 aj

they cannot be added, any more 3X2x6-3°> 2« ^ than units of different orders. 5 x 2 X 4=40, 3 J JV.

We therefore reduce them to a 24 -_|_ 3_6 _|_4o_ J£-0-f or 2-^. common denominator, which is 48, and add the numerators, as above.

Or, we may reduce the fractions to the least common denomi- nator, which is 12, and then add the numerators. Thus, i= £> ; $ = ft ; and .§=!§• : now ft + & +if =li, or 2-^, the same as before.

84. What then is the general rule for adding fractions ?

Reduce tlmn to a common denominator, and place the sum of the numerators over it.

Or, reduce them to the least common denominator, and over this, place the sum of the numerators.

Note. — The integral and fractional parts of mixed numbers should be added separately, and the results be united. (Ex. 12.)

Or, mixed numbers may be reduced to improper fractions, and compound fractions to simple ones, and then be added. (Ex. 18.)

2. Henry paid $ dollar for an arithmetic, } dollar for a slate, and £ dollar for a geography : what did he pay for all ? Ans. 1 ^ dol.

106 ADDITION OF FRACTIONS.

3. What is the sum of | pound, £ pound, and ^ pound ?

4. Add f, J, and |. 8. Add TVf, and f.

5. Add f, f, and i 9. Add ^, Jj, and J-J.

6. Add ^ -J, and }f 10. Add Jfc ^ and f J.

7. Add A, ^, and ft 1 1. Add Jf, JJ, and f f .

12. What is the sum of 10J pounds, 17 J, and 2oS pounds ?

Analysis. — Reducing the fractional parts to ioi^io-1-0-.

the least common denominator, which is 40, we I73_I724

add the fractions and integers separately. 5 JY"

The sum of the fractions is $f=lif. Adding 23?— 23?tr-

the 1 to the whole number, the sum is 51^0, Ans. Ans. 5 1 3(7.

13. How many pounds of tea are there in 2 chests, containing 45-J and 56-^ pounds respectively ?

14. How much cloth in 3 pieces, containing 12 J, 17}, and 2 if yards?

15. If a man walks 2of miles in one day, 25 J the next, and 31! the next, how far will he travel in all ?

16. If a housekeeper buys 3 J dollars worth of sugar, 5! dollars worth of coffee, and 15I dollars worth of flour, what is the amount of her bill ?

17. Three men buying a sail-boat, put in 2 7 J, 23I, and 2of dollars respectively: what was the cost of the boat?

18. What is the sum of £ of f, \ of ■&, and J of 2 J ?

Solution.— Reducing the compound fractions to simple ones, we have \ of £=i, \ of /o=ViT, and £ of 2£=£. Reducing these fractions to the least common denominator, 20, they become A,, 2Lif, and fft; and the sum tfr+^+M=Hi or i\, Ans.

19. What is the sum of f of J, J of ft, and 5 J ?

20. Add f of 45, tt of A> and ^ of 3}.

21. Add 2 2|, J off, and £ of ~fa.

22. What is the sum of -f of 4. 28 J, and 15} ?

SUBTRACTION OF FRACTIONS. 107

SUBTRACTION OF FRACTIONS. To Subtract Fractions which have a Common Denominator.

i. If Frank has f of a pound of maple sugar, and gives away f of it, how much will he have left ?

Analysis.— 3 fifths from 4 fifths leaves 1 fifth. Therefore, he will have 1 fifth of a pound left.

2. From i yard, take .$■ yard ?

3. What is the difference between ^ of a dime and •^ of a dime ?

4. What is the difference between ■& and ^ ?

5. What is the difference between ^ of a week and f of a week ?

6. What is the difference between T5^- and -fj ?

SLATE EXERCISES.

1. What is the difference between ~fe of a foot and -f-J of a foot ?

Analysis. — Since these fractions have a Operation. common denominator, their numerators are TI TZ— TZ It. like numbers, and may be subtracted as whole numbers. (P. 104, Rem.) || minus -fe equal -fe foot, Ans.

35. How subtract fractions which have a common denominator? Take the less numerator from the greater, and place the difference over the common denominator.

2. From H of a day, subtract \\ of a day. Ans. -} d.

3. From ! J °f a ton, subtract Jf of a ton.

4. From |-| of a bushel, subtract f-| of a bushel.

5. From iJf, subtract ^.

6. From AtV? subtract T3^rV

7. What is the difference between ^- and ^/o" -?

8. What is the difference between |ffi and fff-f ?

108 SUBTRACTION OF FRACTIONS.

To Subtract Fractions which have Different Denominators.

i. It is required to find the difference between | and j$.

Analysis. — Since these fractions have Operation.

not a common denominator, their nu- 8x12 = 96, CD. n levators are unlike numbers; conse- 3_ 36. an(j 10 — 8o# quently one cannot be taken directly 8o_3 6 — 44 . iV from the other. Hence, we reduce them ^ ^ ^^, *** to a common denominator, and subtract as above.

36. What then is the general Rule for subtracting fractions ? Reduce them to a common denominator, and over it place the difference of the numerators.

Notes. — 1. WJwle and mixed numbers should be reduced to improper fractions, and compound fractions to simple ones ; then proceed as above. (Ex. 15, 23.)

2. If both are mixed numbers, it is sometimes more expeditious to reduce the fractions to a common denominator ; then subtract the fractional and integral parts separately. (Ex. 17.)

3. The operation may often be shortened by reducing the frac- tions to the least common denominator.

2. Bought a cargo of corn, at | of a dollar a busheb and sold it at f of a dollar: what was the gain \ei' bushel ?

' 3. A man owning -J-f of a ship, sold J of her : wl .at part had he left ?

4. From -J, take f 8. From f J, take f .

5. From \\, take f. 9. From fj, take §.

6. From Jf, take £J. 10. From ££, take £{,

7. From 3f, take ■&. 1 1. From ^, take $}

1 2. What is the difference between ^ff and -f£T ?

13. What is the difference between %%% and %£6 ?

14. What is the difference between £ Jg and f-J2 ?

SUBTKACTIOtf OF FEACTIOKS. 109

15. Subtract 14-f hogsheads from 36 hogsheads ?

Analysis. — Reducing 36 and 14^ 1st Operation.

to improper fractions, we have Ha 36=— f2-.

and x^a. Now 252 minus 103 equals j^s—LQ^

x?- ; and -4a equals aif« -4.7W. 2if 2 - 2 _ f Q _ 1 49. or 2 j a hogsheads.

2d Operation.

36

Or, borrowing 1, which equals \9 we have \ minus f, equal to f . Then 1 to carry to 14 makes 15 ; *4y

and 36 minus 15=21. Ans. 2if. Ans. 21^ h.

16. A farmer having 85 bushels of wheat, sold 63I bushels : how much had he left ?

17. What is the difference between 21 j tons and i6f tons ?

Analysis. — Reducing the fractions to the Operation.

common denominator 15, the minuend 2i| 2lf=2i^f. =2i}£; and the subtrahend 16?;= i6}f. Now .gi_. ,51a

j§ is larger than j§, the fraction above it ; 5 _5

hence we borrow 1, or \§, and add it to |f, Ans. 4^-f tons, making fjf; and ff— ||=|f ; carrying 1 to 16 makes 17, and 21 — 17=4. Ans. 4}! tons.

18. From a cask of molasses, containing 56! gallons, 20 } gallons were drawn : how many remained ?

19. Take 18 j from 37 \. 21. Take 62 } from 83-J-.

20. Take 31J from 66%. 22. Take 106J from 135^.

23. Required the difference between J of f of 5, and

i off of 3|.

Solution. — Reducing the compound fractions to simple ones, cancelling, etc., the minuend becomes §, and the subtrahend f. Again, reducing f and f to the common denominator 15, we ob- tain if, and fr; and fg- -&=-&, 4««.

24. From f of J, take -J of f .

25. From |- of T\, take -^ of T3T.

110 MULTIPLICATION OF FBACTIONS.

MULTIPLICATION OF FRACTIONS.

CASE I.

To Multiply a Fraction by a Whole Number.

i. At \ cent apiece, what will 3 plums cost ?

Analysis. — Since 1 plum costs \ cent, 3 plums will cost 3 times as much ; and 3 times 1 half are 3 halves, equal to i^ cent. Therefore, 3 plums will cost 1 \ cent.

2. At J of a dollar a pound, what will 3 pounds of tea come to ?

3. If a lad earn j of a dollar per day, how much will he earn in 5 days ?

4. What cost 6 bushels of apples, at | of a dollar a bushel ?

5. What is the product of 5 times T6T ? Of 7 times J ?

6. What cost 12 photographs, at § dollar apiece ?

7. What cost 1 1 rabbits, at f dollar apiece ?

8. What cost 5 oranges, at d\ cents each ? Analysis. — If 1 orange costs 6^ cents, 5 will cost 5 times 6^

cents. Now, 5 times 6 cents are 30 cents, and 5 times \ are $ , equal to i^, which added to 30 make 31^ cents. Therefore, etc.

9. At 2>\ dimes each, what will 9 melons cost ? 10. What cost 12 gold pens, at 4! dollars apiece ?

SLATE EXERCISES. 1. At I dollar a box, what will 3 boxes of starch cost ?

Analysis. — Since 1 box costs {j- dollar, 1st Operation

3 boxes will cost 3 times {j- dol. ; and 3 |. x 3^=-^, or 2 \ d times |= Lb£, or 2$ dollars, the cost required.

Or, if we divide the denominator by 3, 2d Operation.

the result will be the same; for, diviling J— ^-3 = ^ or 2\ d. the denominator multiplies the fraction.

37. How multiply a fraction by a whole number?

Multiply the numerator by the whole number.

Or, divide the denominator by it. (P. 93, Prin. IV.)

MULTIPLICATION OF FRACTIONS. Ill

Notes. — i. The second method is preferable, when the denom- inator can be divided by the whole number without a remainder.

2. If the multiplicand is a mixed number, multiply the frac- tional and integral parts separately, and unite the products.

Or, reduce the mixed number to an improper fraction ; then apply the rule. (Ex. 15.)

2. What will 1 7 pounds of honey cost, at £ of a dollar a pound ?

3. What cost 25 bushels of potatoes, at T5^ of a dollar a bushel ?

4. At 4^ dollars apiece, what will 3$ straw hats come to ?

5. Multiply jj by 15. 10. Multiply \\ by 26.

6. Multiply \% by 17. 11. Multiply || by 42.

7. Multiply |J by 9. 12. Multiply fj by 50.

8. Multiply ff by 11. 13. Multiply ffo by 83.

9. Multiply If by 18. 14. Multiply ^ by no.

15. What will 5 hundred weight of sugar cost, at 6 J dollars per hundred ?

Analysis. — Multiplying the fraction and inte- Operation. ger separately by the whole number, we have J x 6|. dols.

5=-^, or Jf j and 6x5=30. Now 30+3l=33i 5 dols. Therefore, etc. ~y , ,

Or, the mixed number 6}=*£, and ¥x$=*f* 33* =334 dollars, the cost required.

16. What cost 23 yards of muslin, at 12} cents a yard ?

17. What cost 45 yearlings, at 18 J dollars apiece ?

18. Multiply 31 J by 25. 21. Multiply 62^- by 57.

19. Multiply 37^ by 42. 22. Multiply 66f by 75.

20. Multiply 40! by 61. 23. Multiply 87^ by 100.

24. What cost 24 bureaus, at 27 J dollars apiece ?

25. What cost 12 melodeons, at 62-J dollars apiece ?

26. What cost 31 sofas, at 71 J dollars apiece ?

112 MULTIPLICATION OB Fit ACTIONS.

CASE II. To Multiply a Whole Number by a Fraction.

i. What will J yard of edging cost, at 10 cents a yard ? Analysis. — If 2 halves, or a whole yard, cost 10 cents, 1 half yard will cost 1 half of 10 cts. ; which is 5 cts. Therefore, etc.

.2. If a dozen eggs cost 16 cts., what will \ dozen cost?

3. If a melon is worth 12 cents, what is \ of it worth ?

4. If a pie is worth 20 cents, what is J of it worth?

5. What cost f pound of grapes, at 14 cents a pound ? Analysis. — Since 1 pound is worth 14 cents, § of a pound are

worth % of 14 cents. But 1 third of 14 cents is 45- cents, and 2 thirds are 2 times 4I, or 9^ cents. Therefore, etc.

6. If a cake costs 80 cents, what will J of it cost ?

SLATE EXERCISES. 3§. What is meant by multiplying by a fraction?

Multiplying by a Fraction is taking a certain part of the multiplicand as many times as there are Ufa parts of a unit in the multiplier.

39. How find a fractional part of a number ?

Divide the number into as many equal parts as there are units in the denominator, and then take as many of these parts as there are units in the numerator. That is,

To multiply a number by J, divide it by 2.

To multiply a number by J, divide it by 3.

To multiply a number by J, divide it by 4 for J, and multiply this quotient by 3 for J, etc.

Remarks. — 1. Multiplying a whole number by a fraction is the same as taking a corresponding; fractional part of the number.

2. When the multiplier is 1, the product is equal to the mulfi- plicand; when the multiplier is greater than 1, the product is greater than the multiplicand ; when the multiplier is less than 1, the product is less than the multiplicand.

MULTIPLICATION OF FRACTIONS. 113

:. What will f of a gallon of cider cost, at 38 cents a gallon ?

Analysis. — Since 1 gallon costs 38 cents, § of a gallon must cost § times 38. or f of 38 cts. Now ^ of 38 cts. is 12 J cts., and 2 thirds are 2 times I2f cts. Multiplying I2| by 2, we have 2 times f=4, or 1^. 2 times 12 are 24, and 1^ make 25^ cts., the cost required.

Or, thus: § of a gallon will cost i of 2 times the cost of 1 gallon. Now 2 times 38 cts. are 76 cts., and ^ of 76 cts. equals 76-7-3, or 25^ cts., the same as before.

1st Operation.

3)38 "I

2

Ans. 25J cts.

2d Operation. 38 X 2 = 76.

76-^3=25^ cts.

40. How multiply a wlwle number by a fraction?

Divide tlie wlwle number by the denominator of tlie fraction, and multiply by the numerator.

Or, multijrty the whole number by the numerator of the fraction, and divide by the denominator.

Notes. — 1. The fraction may be taken for the multiplicand, and the whole number for the multiplier, at pleasure, without affecting the result. (P. 47, Rem.)

2. Mixed numbers, when multipliers, may be reduced to im- proper fractions ; then proceed according to the rule.

Or, multiply by the fractional and integral parts separately, and unite the results.

2. If a bushel of barley is worth 75 cts., what is f of a bushel worth ?

3. If an acre of land is worth $100, what is f of aq acre worth ?

4. Multiply 45 by %.

5. Multiply 61 by f.

6. Multiply 78 by f

7. Multiply 87 by f.

8. Multiply no by T3T.

9. Multiply 238 by ^. 10. Multiply 378 by {f n. Multiply 500 by -fi^.

12. In 1 year there are 365 days: how many days are there iu f of a year ?

Operation.
)42 •	dols.	I	ton,
H
252	M	6	a
8f	«	t	u
25i	U	J	a

114 MULTIPLICATION OF FRACTIONS.

13. What cost 6f tons of iron, at 42 dollars a ton?

Analysis. — If 1 ton costs 42 dol- lars, 6} tons will cost 6i times 42 dols. We first multiply by the whole num- ber 6, and the product is 252. In mul- tiplying by the fraction £, we take £ of the multiplicand, and setting it under the product of the integral part, mul- tiply it by 3; for, fc=i+f. We now have the partial products of 6, of £, and 285! dols. 6f tons,

f, and their sum, 285! dols., is the

answer required.

14. What cost 8 J yards of alpaca, at 80 cts. a yard ?

15. If a man can walk 45 miles a day, how far can he walk in iof days?

16. Multiply 52 by 6J. 19. Multiply 101 by 10-i.

17. Multiply 57 by 7 J. 20. Multiply 365 by n|.

18. Multiply 78 by 8 J. at. Multiply 500 by 12 J.

CASE III. To Multiply a Fraction by a Fraction.

1. What will A of a gallon of syrup cost, at £ of a dollar a gallon ?

Analysis. — 1 tenth of a gallon will Operation.

cost fa the price of 1 gal. ; and -fa of £ 5 4* 2 _ dol. is ft dol. (P. 93, Prin. III.) 3,&Xis>~~T

Again, -fa gal. will cost 4 times as much as -,1,, ; and 4 times fa are gft, or $ dol. In the operation we cancel the common factors, and multiply the numerators together, and then the denominators.

41. How multiply a fraction by a fraction? Cancel the common factors ; then multiply the numera- tors together for the new numerator, and the denominators

for the new denominator.

MULTIPLICATION OF FEACTIOKS. 115

Notes. — i. Compound fractions are multiplied like simple ones ; the word of being equivalent to the sign x .

2. Reduce mixed numbers to improper fractions, and then multiply them according to the rule. (Ex. 16.)

3. The object in cancelling the common factors is twofold : it x?iortens the operation, and gives the answer in the lowest terms.

2. What will f of a pound of sugar cost, at -J of a dollar a pound ?

3. What cost J of a yard of muslin, at t2q of a dollar a yard ?

4. Multiply f by J.	10. Multiply f by J x f.
5. Multiply f by TV	11. Multiply f by £x^.
6. Multiply T\ by J.	12. Multiply f by J x -ft.
7. Multiply tV by f	13. Multiply TV x A x -2T-
8. Multiply }f by fj.	14. Multiply Jxfx f Xf.

9. Multiply If by f|. 15. Multiply J x J x ^ x }f 16. What cost 8£ yards of calico, at 1 2\ cents a yard ? Solution— 81=^ ; i*i=¥. Now> ¥ x ¥=&iA, or 109J cts.

4.£. The preceding principles may be summed up in the following

GENERAL RULE.

Reduce wlwle arid mixed numbers to improper frac- tions; tlien cancel the common factors, and place the pro- duct of the numerators over the product of the denom- inators.

EXAMPLES FOR PRACTICE.

1. What is the product of fxfxfxlf?

2. Multiply j of i of 1 J by f of T%.

3. Multiply f of f of ^ by § of 4J.

4. Multiply f of 6\ by | of || of 8.

5. Multiply f of J of 18 by f of 25.

6. What is the product of 6£ multiplied by 2f ?

116 DIVISION OF FRACTIONS.

7. What cost 10 } pounds of beef, at 15^ cents a pound ?

8. At 1 if dollars a barrel, what will 20 \ barrels of vin- egar come to ?

9. Multiply i6| by g£. 12. Multiply 45! by 31 J.

10. Multiply 31 J by 1 8 J. 13. Multiply 66| by 3 7 J.

11. Multiply 37 J by i6|. 14. Multiply 110J by 60^.

DIVISION OF FRACTIONS.

CASE I.

To Divide a Fraction by a Whole Number.

1. If 2 citrons cost T\ of a dollar, what will 1 citron cost?

Analysis. — If 2 citrons cost -fo of a dollar, 1 will cost 1 half of /tf of a dollar ; and 1 half of 4 tenths is -fc, or £ of a dollar.

2. If 3 apples cost -^ of a dime, what will 1 apple cost?

3. If 4 peaches cost T8^ of a shilling, what will 1 cost ?

4. If 5 yards of calico cost -f £ of a dollar, what will 1 yard cost ?

5. If 3 doves cost | dollar, what will 1 dove cost ?

Analysis. — 1 dove is ^ of 3 doves ; therefore, 1 dove will cost £ of i dollar, and £ of £ dollar equals £ dollar. (P. 63, Q. 12.)

6. If 4 balls cost -J dollar, what will 1 ball cost ?

7. If J bushel of oats are equally divided among 5 horses, how many will each horse receive ?

8. If I of a barrel of apples are divided equally among 7 persons, what part of a barrel will each receive ?

9. If f-$ of an acre of land are divided into 4 equal lots, how much will there be in each lot ?

10. If \% of a ton of hay are divided into 3 equal loads, how much will there be in each load ?

DIVISION" OF FRACTIONS

117

1st Operation. 2d Operation.

3=A-

Ans.

A=

|dol.

SLATE EXERCISES.

i. If 3 pounds of raisins cost f dollar, what will pound cost ?

1st Method.— i pound is £ of 3 pounds, therefore 1 pound will cost £ of f dol. Di- viding the numerator into 3 equal parts, we have f dol.-f-3 = f dollar. (P. 93, Prin. II.)

2d METnOD. — Since multiplying the de- nominator divides a fraction, it follows that $ dol.-^-3=/r, or f dol., the same as before. (P. 93, Prin. III.)

Remahk. — The solution of this and similar examples is an application of the second office of Division. (P. 63, Q. 10.)

43. How divide & fraction by a whole number? Divide the numerator by the whole number. Or* multiply the denominator by it.

Notes. — 1. When the dividend is a mixed number, reduce it to an improper fraction ; then apply the rule. (Ex. 13.)

2. This rule depends upon the principle that a part of a unit may be divided into other parts, as well as a wJwle unit.

2. A lad paid |f dollar for 6 balls: what was that

Ans. ^y dol.

apiece ? Ans. -fo d(

3. Divide -fj by 2. 8. Divide if by 31.

4. Divide fj by 3. 9. Divide -^°/ by 50.

r»« •!. tf 1 /- _ - T\ • • ,1 - T •> 7 1 /T .

4. Divide || by #

5. Divide -ff by 6,

6. Divide f^ by 7.

7. Divide Jf by 11.

9. j_/iYiut; --gy- uj ^u.

10. Divide ^j- by 64. 11

Divide ffj by 85.

11. j^iviut; -j-j-j uj

12. Divide f££ by

100.

13. A drover paid 18J dollars for 5 sheep: how much was that per head ?

Analysis. — Reducing the mixed num- Operation.

ber 1 8 i: to an improper fraction , it becomes 1 8 J — 3J-.

V ; and ¥+5=ft» or 3tt dols. *£+ 5=4$, or 3JJ.

118 DIVISION OF FRACTIONS.

14. If 5 barrels of flour cost 37-J dollars, what will 1 barrel cost ?

15. Divide 15J by 3. 18. Divide 65^ by 23.

16. Divide 22 x by 6. 19. Divide 100-ff by 40.

17. Divide 41-J by 11. 20. Divide 225^ by 50.

CASE II. To Divide a Whole Number by a Fraction.

1. At \ dollar apiece, how many chickens can be bought for 3 dollars ?

Analysis.— Since 1 half dollar will buy 1 chicken, 3 dollars will buy as many as there are halves in 3 dols., whicli are 6. Therefore, 3 dols. will buy 6 chickens.

2. At I of a dollar a quart, how many quarts of cher- ries can you buy for 5 dollars ?

3. If you divide 8 apples equally among 4 boys, what part, and how many will each receive ?

Analysts. — 1 is \ of 4 ; therefore, each boy will receive \ part. Again, if 8 apples are divided into 4 equal parts, 1 part will be \ of 8, which is 2. Therefore, etc.

4. A teacher distributed 16 pounds of figs equally among 5 pupils, what part, and how many did each receive ?

5. At J of a dollar a yard, how many yards of poplin can be bought for 5 dollars?

Analysis. — In 5 dollars there are 15 thirds, and 2 thirds are contained in 15 thirds, 7^ times. Therefore, etc.

6. At J of a cent apiece, how many apples can I buy for 8 cents ?

7. At $ of a dollar a pound, how many pounds of cinnamon can I buy for 10 dollars ?

8. At J of a dollar a box, how many boxes of white grapes can be bought for 6 dollars ?

DIVISION OF FRACTIONS. 119

SLATE EXERCISES.

i. At f of a dollar a pound, how many pounds of tea can I bay for 20 dollars ?

Analysis. — At 1 third dollar a Operation.

pound, I can buy as many pounds 2od.-7-£=:(20 x 3) — 2. as there are thirds in 20 dollars, or (20 x o\ _^_ 2 __6 0 or IQ „ So pounds. But the price is 2 ~ • ' T*

thirds dollar a pound; therefore, v/i, -so x 2— 2 — 30 p. I can buy only 1 half of 60 or 30 pounds.

In the operation we multiply the whole number 20 by the denominator 3, and divide the product by the numerator 2. But this is the same as inverting the fractional divisor, and then multiplying the dividend by it. (P. 113, Q. 40.)

44. How divide a whole number by a fraction?

Multiply the whole number by the fraction inverted.

Notes — 1. The reason of the rule is this: multiplying the whole number by the given denominator reduces it to a fraction having the same denominator as the given fraction. Hence, the numerators are like numbers, and one may be divided by the other, is whole numbers. (P. 104, Rem.)

2. To divide a whole by a mixed number, reduce the mixed number to an improper fraction. (Ex. 2.)

3. A fraction is inverted when its terms are made to exchange places. Thus, ~ inverted becomes |.

2. At 1 2\ dollars apiece, how many ploughs can a man buy for 75 dollars? Ans. 6.

3. Divide 40 by f. 7. Divide 96 by i8|.

4. Divide 55 by f. 8. Divide 100 by 2cf.

5. Divide 68 by ■&. 9. Divide 250 by 37-J.

6. Divide 75 by fa 10. Divide 560 by 6 6 J.

11. A lady paid 62 dollars for 15 -J yards of silk: what was the silk a yard ?

12. If a horse travels 75 miles in i8| hours, how far will he go in 1 hour ?

120 DIVISION OF FRACTIONS.

CASE III.

To Divide a Fraction by a Fraction when they have a

Common Denominator.

i. At | of a dollar a pound, how many pounds of pepper can be boug i'~ for -J- of a dollar ?

Analysis. — If 2 thirds of a dollar will buy 1 pound, 7 thirds will buy as many pounds as 2 is contained times in 7, and 2 is contained in 7, 3^ times. Therefore, 3 dollar will buy 3J pounds.

2. How many needles, at f cent apiece, can you buy for I cent ?

3. How many pen-knives, at J of a dollar, can be had for J32- of a dollar ?

4. At J of a dollar a yard, how many yards of ribbon can be purchased for £ of a dollar ?

. 5. At I of a dollar a yard, how many yards of ging- ham will -^ of a dollar buy ?

SLATE EXERCISES.

Remark. — When fractions have a common denominator, their numerators are like numbers. Hence, one numerator may bo divided by the other, as whole numbers.

1. If f of a dollar will buy 1 pound of coffee, how many pounds can be bought for -^ of a dollar ?

Analysis. — If & dollar will buy Operation.

1 pound, 35X dollar will buy as many 2 7_^2_2^^2_ » jpounds as f- are contained times in V, which is 134 times. Therefore, etc.

45. How divide one fraction by another when they have a common denominator?

Divide the numerator of the dividend by that of the divisor.

2. Divide f& by ^. 5. Divide §£ by J£.

3. Divide ft by •&. 6. Divide f| by £f.

4. Divide f f by ^. 7. Divide JJ by ft

DIVISION OF FRACTIONS. 121

To Divide a Fraction by a Fraction, when they have Different Denominators.

i. At -J of a dime apiece, how many pears can be pur- chased for j of a dime ?

Analysis. — \ dime will buy as many opebation.

pears as \ dime is contained times in f 1X4 4

dime. Reducing \ and f to a common 3X5 '

denominator, they become jV and -,22-. , x ~

Their numerators are now like numbers, =t9;? >

and one may be divided by the other. 7 4

Thus, 9-5-4=2! pears, the answer re- T2~T2 — 9~4 >

quired. (P. 120, Eem.) 9-^4 = 2iP-

By inspecting the operation of redu- Or, |xf=|, or 2^. cing the fractions to a common denomi- nator, it will be seen that the numerator of each is multiplied into the denominator of the other. This produces the same com- bination of terms and the same results as inverting the divisor and multiplying the terms of the dividend by it. Thus, |-*ii= £ x $=$, or z\ pears, the same as before.

Remark. — In dividing, no use is made of the common denomi- nator ; hence, in practice, multiplying the denominators together may be omitted.

46. How divide a fraction by a fraction, when they have different denominators?

Reduce the fractions to a com. denominator, and di- vide the numerator of the dividend by that of the divisor.

Or, multiply the dividend by the divisor inverted. Note. — Mixed numbers must be reduced to improper fraction^ and compound fractions to simple ones. (Ex. 17, 24.)

2. At f of a dollar a pound, how much tea can be had for f of a dollar ?

3. How many pineapples, at -^ of a dollar each, can be had for f of a dollar ?

4. At | of a dollar a pound, how much sugar can be had for f of a dollar ?

6

122 DIVISION OF FRACTIONS.

Perform fche following divisions :

5. Divide §■ by J. 11. Divide x| by T7T.

6. Divide f by f. 12. Divide fj by £$.

7. Divide f by f. 13. Divide Jf by ^-.

8. Divide T7<j by TV 14. Divide ^ by TVo-

9. Divide T\ by T\. 15. Divide Jff by ^ 10. Divide ^ by ^. 16. Divide -Jf f by ^f.

17. How many bushels of apples, at 2 J dollars a bushel can be purchased with 8f dollars ?

Analysis. — Reducing the Operation.

mixed numbers to improper 24=JJt, and 84=^;

fractions, we have 2|=14i> 85 44_^._ii_-44x 4 .

= 16i« Inverting the divisor, ^ we cancel the common factor ^4-xT4T- — x — =1$-, or 3^.

11, and proceed as before. 5 ** Alia. 2>\ bushels.

18. Divide 3! by 2J. 21. Divide i8| by 5 -J.

19. Divide 8f by 3 J. 22. Divide 27 J by n|.

20. Divide 13 J by 5^ 23. Divide 55^ by 2 if

24. What is the quotient of -J of f of 3-J divided by

fof J?

Analysis. — Reducing 2>\ to ¥i an(* inverting the divisor, we have \ of J of -L/-*-J of !=:$ xfx^xf x |=|, or ij, ^Ins.

25. Divide f of J by r of §,

26. Divide f of f of |- by $ of |>

27. Divide | of 10 \ by -J of J of f.

28. Divide \ of -J- of £ by | of 6 J.

To reduce a Complex Fraction to a Simple one,

21 1. Reduce the complex fraction -3 to a simple one.

Analysis. — The given complex fraction 1st Opebation.

is equivalent to 2^-4-4. Reducing the nu- %\ #

mcrator 2\ to a simple fraction, it becomes 4 T • 4 >

5, and 3-*-4=&> the answer required. (P. 2£— 4.=X— 4 ■

DIVISION OF FRACTIONS. 123

Or, reducing both the numerator and 2d Operation.

denominator to a simple fraction, they be- 2 j-r4— f-i-f 5

come I and ± Now, $-*-f=*$ x £=-&, the 7_r_4_7vi— ?

, ,. T • T — T A ? — XT' same as before.

I

2. Reduce the complex fraction - to a simple one. Solution. — Performing the division indicated, ^-r-3=£, Ans.

47. How reduce a complex fraction to a simple one? Reduce the numerator to a simple fraction, and divide

it by the denominator. (P. 1 1 7, Q. 43.)

Remarks. — 1. Complex Fractions when reduced to simple ones, are added, subtracted, etc., like other Simple Fractions.

2. The expressions -~, -^, etc., indicate a division of one frac-

tional number by another.

Such expressions are reduced to simple fractions in the samo manner as one fraction is divided by another. (Ex. 3.)

2*

3. Seduce the expression -| to a simple fraction.

5 3

ANALYSIS. — The given expression is Operation.

equivalent to 2J-4-53-. Reducing the di- 2~2__. 1 . 2 .

visor and dividend to simple fractions, 5 J 2 • 53?

they become $ and ^ ; and 1+^—1 x 2}=$, and 5j=^;

*** (P. 99, Q- 250 |H.¥=|xA^ii

Reduce the following complex fractions to simple ones,

4* 6.M. 8.f. 10.£*

8 6 9 10

5. — . 7. — . 9. -— . 11. —^5

7 20 35 42

48. The preceding principles may be reduced to the following

GENERAL RULE.

Reduce whole and mixed numbers to improper frac- tions, compound and complex fractions to simple ones, and multiply the dividend by the divisor inverted.

124 QUESTIONS FOR REVIEW.

QUESTIONS FOR REVIEW.

i. If you pay 3 J dollars a week for board, what will it cost you to board 1 1 weeks ?

2. If a ton of hay is worth 17 dollars, what is £ of a ton worth ?

3. What will I of a yard of ribbon cost, at f of a dol- lar a yard ?

4. What cost 10J pounds of butter, at \ of a dollar a pound ?

5. What cost 16 J yards of silk, at 2% dollars per yard?

6. At f of a dollar a pound, how many pounds of tea can be purchased for 30 dollars ?

7. How many pen-knives can I buy for 60 dollars, if I pay £ of a dollar apiece ?

8. At f of a dollar a pound, how many pounds of almonds can be bought for 58! dollars ?

9. How much maple sugar, at \ of a dollar a pound, can be purchased for ^ of a dollar ?

10. At 2 J dollars a cord, how much wood can be had for 18 dollars?

1 1. How much flour, at 7} dollars a barrel, can be had for 37 J dollars ?

1 2. Required the sum of J and TV Their difference. Their product. The quotient of the former divided by the latter.

13. How many days can you hire a laborer for 37] dollars, if you pay him 1 j dollar a day ?

14. A planter raised 60 bales of cotton, sold \ of them to one merchant, and f to another: how many bales had he left?

15. A speculator bought a quantity of apples for 162J dollars, and sold them for 210J dollars: what was his profit ?

QUESTIONS FOE REVIEW. 125

1 6. Bought 15 pounds of butter, at £ dol. a pound; and 10 gal. of molasses, at f dol. a gal.: what was the cost of both ?

17. At } of a dollar a pound, how many raisins can be bought for ^ of a dollar ?

18. What is the quotient of ^ of f of 5^ divided by fof J?

19. If I pay \ of f of 20 dollars for a ton of coal, what must I pay for \ of 4f tons ?

20. A man having 500 dollars, laid out \ of it in cot- ton, which was \ of £ of a dollar a pound : how much cotton did he have ?

21. A man owniug -fj of a ship, sold f of his share of her : what part of the ship did he sell, and what part had he left ?

22. How long will 150 pounds of coffee last a family, if they use 3 \ pounds a week ?

23. A and B drew a prize amounting to 256^ dollars; A took i6of dollars : how much did B have ?

24. What will it cost to build -J of f of 16J rods of stone wrall, at if of a dollar a rod ?

25. If 2\ of a yard of velvet can be bought for 12 dollars, what part of a yard can be bought for 1 dollar ?

26. Divide f of J by f of f

27. Divide f of 32 by § of f.

28. Divide -fc of i6£ by f of 10.

29. Divide J of f of 18J by f of 3$.

30. Which will cost more, 8 barrels of flour, at 7^ dollars a barrel, or 16 barrels of potatoes, at 3 \ dollars a barrel ?

31. If oranges are 6\ cents apiece, how many can be bought for 87 J cents ?

32. At i6f cents a pound, how much lard can be bought for 83 J cents ?

126 FRACTIONAL RELATION

FRACTIONAL RELATION OF NUMBERS.

To find what part one number is of another.

Remark. — That numbers may be compared with each other. they must be so far of the same nature that one may properly be said to be apart of the other. Thus, a. foot may be compared with a yard; for, one is a third part of the other. But it can not be said that afoot is a part of a pound; therefore the former can nc * be compared with the latter.

i. What part of 3 is i ?

Analysis. — If 3 is divided in 3 equal parts, one of these parts is 1 third. Therefore, 1 is ^ part of 3.

2. What part of 3 is 2 ?

Analysis. — 2 is 2 times 1 third part of 3, or 2 thirds of 3.

3. In 1 gallon there are 4 quarts : what part of a gal- lon is 1 quart ? What part is 3 quarts ?

4. What part of 5 is 3 ? Is 4 ? Is 2 ? Is 1 ?

5. What part of 6 is 2 ? Is 3 ? Is 4? Is 5 ?

6. What part of 4 apples are 5 apples ?

Analysis. — 1 apple is 1 fourth part of 4 apples; therefore, 5 apples must be 5 times 1 fourth, or 5 fourths of 4 apples.

7. What part of 8 pounds is 9 pounds ? Is 1 1 pounds ?

SLATE EXERCISES.

1. What part of 5 cents is 3 cents ? Analysis. — 3 cents are 3 times £, or | of 5 cents. 49. How find what part one number is of another J

Make the number denoting the part the numerator, and that with which it is compared the denominator.

Note. — The fraction thus found should be reduced to its lowest terms.

2. What part of 48 is 1 2 ? Of 63 is 28 ?

3. What part of 81 is 27 ? Of 90 is 63 ? Of 100 is 40?

OF NUMBERS. 127

4. What part of 35 dollars is 19 dollars ?

5. If I divide a bushel of plums equally among 15 boys, what part of a bushel will 1 boy receive ? What part will 9 boys receive ?

6. Helen's age is 18 years, and her brother's 14: what part of her age is her brother's ?

7. Henry has 91 marbles, and Charles 70: Charles' marbles are equal to what part of Henry's ?

8. If 5 pencils cost 17 cents, what will 4 pencils cost?

Analysis. — 4 pencils are f of 5 pencils ; 5)17 cts.

therefore, 4 pencils will cost f of 17 cents.

Now, i of 17 cents is 3^ cents, and 4 fifths are 3y

4 times 3! cents, or 13! cents, the cost re- 4

quired. . 7 ,

^ Am. 13 J cts.

9. If 8 oranges cost 32 cents, what will 6 cost ?

10. If 20 cows cost 625 dollars, what will 35 cost?

11. If 13 sofas cost 572 dollars, what will 6 cost ?

12. What part of 4 pears is f of a pear ?

Analysis. — 1 pear is i part of 4 pears ; Operation.

hence, f of a pear must be f of $=-ft-, or £ 2

of a pear. Therefore, etc. 7 T~4?

Making the fraction which denotes the 2 _ 2 .1

part the numerator, and the tcfo>& number the T • 4 — T2"> 3"*

3.

denominator, we have the complex fraction -, to be reduced to a

simple one. (P. 123, Q. 47.)

Or, what is the same thing, a fraction to he divided by a whole lumber ; and f-M=A, or |. (P. 117, Q. 43.)

13. What part of 15 is f ? 15. What part of 45 is -ft-?

14. What part of 26 is £ ? 16. What part of 63 is -ft ?

17. If 15 barrels of flour cost 100 dollars, what will J of a barrel come to ?

18. If 20 acres of land jMd 250 bushels of corn, what will I of an acre yield ?

128 FRACTIONAL RELATION

To find a number when a part of it is given.

i. 5 is | of what number?

Analysis. — If 5 is £, 3 thirds, or the whole number, must be 3 times 5, or 15. Therefore, 5 is £ of 15.

2. 6 is \ of what number ? 7 is £ of what number ?

3. 8 is f of what number ?

Analysis. — Since 8 is f of a certain number, 1 third is £ of 8„ which is 4, and 3 thirds must be 3 times 4, or 12. Therefore, etc.

5. George has 12 apples, which are f of the number which William has : how many has William ?

SLATE EXERCISES.

I

1. 16 is f of what number?

1st Analysis. — Since f of a number 1st Operation.

is 16, $ or the whole number, must be as i6-t-t=i6x4;

many units as | is contained times in 16; ^ 3_ 48

and i6-s-f=i6 x f , or 24. ? ~Z >

2d Analysis. — Since 16 is § of a certain . _

,.,._, , , , 2d Operatiow. number, 1 third of that number must be ■$• T _ ,. ~

of 16, which is 8. 3 thirds, or the whole ^~

number must be 3 times 8, or 24. "3"— ° x 3 — 24«

50. How find a number when a part of it is given ? Divide the number denoting the part by the fraction. Or, find one part as indicated by the. numerator of the fraction, and multiply this by the denominator.

2. 32 is \ of what? 5. 100 is § of what?

3. 45 is -f of what? 6. 144 is \% of what?

4. 72 is f of what ? 7. 250 is \\ of what ?

8. If -J of an acre of land is worth 35 dollars, what is a whole acre worth ?

9. A man paid 75 dollars toward a horse, which was T7T of the price: what did he give for the horse?

10. A man being asked how old he was, replied that r72 of his age equaled 49 years : what was his age ?

DECIMAL FRACTIONS

PRELIMINARY EXERCISES.

1 . If a sheet of paper is divided into 10 equal parts, what part of a sheet is i of these parts ?

One of these parts is TV of a sheet.

2. If one of these tenths is divided into 10 other equal parts, what part of a sheet is i of these parts 1

One of these parts is -^ of ^, or y-J ^ of a sheet.

3. If one of these hundredths is divided into io other equa^ parts, what part of a sheet is i of these parts ?

Each part is -^ of ^ of -fa, or yoVtf 0I> a sheet

4. What is meant by a tenth, a hundredth, a thousandth, etc. V A tenth is one of the ten equal parts into which a

number or thing may be divided, etc. ?

5. How much greater are tens than units; hundreds than tens ; thousands than hundreds, etc. ?

Tens are io times greater than units, and each suc- ceeding order is io times greater than the preceding.

6. How much less are tenths than units; hundredths than tenths ; thousandths than hundredths, etc. ?

Tenths are io times less than units ; hundredths are io times less than tenths; and so on, each succeeding order being io times less than the preceding.

7. What places do tens, hundreds, thousands, etc., occupy ? Tens occupy the first place on the left of units ; hun- dreds, the second ; thousands, the third, etc.

8. Following this analogy, what place should tenths, hun- dredths, thousandths, etc., occupy ?

Tenths in the decreasing scale correspond with tens in the increasing scale ; hence they should occupy the first place on the right of units. In like manner, hun- dredths, which correspond with hundreds, should occupy the second place ; thousandths, the, third place, etc.

130 NOTATION OF DECIMALS.

9. How many units make a ten, tens a hundred, etc. ?

10. How do the orders of whole numbers increase ? They increase from right to left by the scale of io.

11. How many tenths make a unit; hundredths a tenth: thousandths a hundredth, etc.

Ten - tenths make a unit ; ten hundredths make a tenth ; ten thousandths make a hundredth, etc.

12. How do the orders of these fractions decrease ? They decrease from left to right by the scale of io.

NOTATION OF DECIMALS.

13. What are Decimal Fractions ?

Decimal Fractions are those in which the unit is divided into tenths, hundredths, thousandths, etc.

They arise from dividing a unit into ten equal parts, or tenths ; then subdividing one of these tenths into ten other equal parts, or hundredths ; and so on, the successive orders decreasing regularly by the scale of io.

14. How, and upon what principle are they expressed ?

By placing a point before the numerator, and assign- ing to each figure a value according to the place it occu- pies, as in whole numbers. Thus, -fa is expressed by writing 3 in the first place on the right of units ; as, .3 ; ■jjj-Q by writing 3 in the second place ; as, .03 ; t^ by writing 3 in the third place ; as, .003.

15. What do figures standing in the first, second, third, etc., places on the right of units denote ?

When standing in the first place on the right of uni they denote tenths; in the second place, they denote hundredths ; in the third place, thousandths, etc.

Notes. — 1. The point used to distinguish decimals from ibholt number*, is called the decimal 'point.

2. 'i'heso fractions are called decimals from the Latin decern, ten, which indicates their origin and stale of decrease.

NOTATION OF DECIMALS. 13 1

16. What is the denominator of a decimal fraction ? It is always 10, ioo, iooo, etc., or i with as many ciphers annexed as there are decimals in the numerator. Name the orders of integers, beginning at units. Name the orders "of decimals, beginning at units place.

TABLE.

		1					/" N				CO M	■a S3
2	o- CO	p 1	•g	OQ			I		1	00 "8	s 1 CO	1 O 3	00
OS* 1 1	| S P	"3 02 s	I	1	03 B CD	i	-3 1 CD	QQ* i—i p s	B	1	P 5 p 1	g p	i
6	5	2	3	4	7	3	•	5	2	8	7	3	5
		In	itegers.						v Decimals,

1 7. What is the effect of prefixing ciphers to decimals ? Each cipher prefixed to a decimal, diminishes its

value ten times, or divides it by io.

18. What is the effect of annexing ciphers to decimals? The value is not altered. Thus, .3 =.30 =.300, etc.

19. How write decimals ?

Write the figures of the numerator in their order, as- signing to each its proper place Mow units, and prefix to them the decimal point.

If the numerator has not- as many figures as required, supply the deficiency oy prefixing ciphers.

Note. — A decimal and integer written together, are called a mixed number ; as, 35.263. (P. 92, Q. 13.)

i. On which side of units are tens ? Tenths ? Thou- sands ? Hundredths ? Hundreds ? Thousandths ?

2. What is the name of the second place on the right of units ? The fourth ? The third ? The fifth ?

132 NOTATION OF DECIMALS.

3. How many decimal places are required to express tenths ? Thousandths ? Hundredths ? Millionths ?

SLATE EXERCISES.

Write the following fractions decimally :

1. \z hundredths. 7. 9 thousandths.

2. 25 hundredths. 8. 13 thousandths.

3. 5 hundredths. 9. TWA-

4. 49 hundredths. 10. yf^.

5. 119 thousandths. 11. ttrrttt*

6. 27 thousandths. 12. x^^.

13. Write 6 hundredths. 41 thousandths. 7 thou- sandths.

14. Write 201 ten-thousandths. 752 hundred-thou- sandths.

15. Write 5 millionths. 63 millionths. 98 mil- lionths. 375 millionths.

20. How read decimals ?

Bead the decimals as whole numbers, and apply to them the name of the lowest order.

Remark. — The unit's place should always be the starting point both in reading and writing decimals.

Copy and read the following :

15. .7. 21. 2.35. 27. 21.251. S3- I2I.4502-

16. .75. 22. 3.236. 28. 30.4312. 34. 240.4023. t

17. .06. 23. 5.078. 29. 44.0643.' 35. 306.46531.

18. .121. 24. 6.2356. 30. 53.21034. 36. 500.00729.

19. .065. 25. 7.3062. 31. 72.05213. 37. 607.329267.

20. .008. 26. 8.5602. 32. 84.00605. 38. 730.004308.

*** Dictation exercises in reading and writing decimals should be practiced till the class is perfectly familiar with them.

REDUCTION OF DECIMALS. 133

REDUCTION OF DECIMALS.

CASE I.

To Reduce Decimals to Common Fractions.

i. Reduce .27 to a common fraction.

Analysis. — Since .27 has two decimal fig- Operation.

tires, its denominator must be 100. Hence, '^—ToV? -4w& aj=-fifa. In the operation we omit the deci- mal point, and place the denominator 100 under the 27.

91. How reduce decimals to common fractions? Erase the decimal point, and place the denominator under the numerator, (P. 131, Q. 16.)

Note. — After decimals are reduced to common fractions, they may be reduced to lower terms, to a common denominator, etc., and then be treated in all respects like other common fractions.

2. Reduce .35 to a common fraction, and to its lowest terms. Am. .35 =fw> and TVo = 27o-

Reduce . the following decimals to common fractions in their lowest terms :

3. .24. 7. .04. 11. .4032. 15. .00045.

4. .135. 8. .025. 12. .0005. 16. .00328.

5. .404. 9. .204. 13. .0106. 17. .01032.

6. .675. 10. .1025. 14. .7524. 18. .123456.

CASE II. To Reduce Common Fractions to Decimals. 1. Reduce J to a decimal.

Analysis. — \ is equivalent to 1 divided by 4. Operation. But 1 cannot be divided by 4 ; we therefore reduce 4 ) 1.00

it to tenths by annexing a cipher to it, making 10 ~^Z

tenths. (P. 57, Q. 18.) Now \ of 10 tenths=2 tenths and 2 tenths over. Reducing the 2 tenths to hundredths by annexing a cipher, we have 20 hundredths; and £ of 20 hundred ths= 5 hundredths. Therefore, \ equals .25.

134 REDUCTION OF DECIMALS.

22. How reduce common fractions to decimals ?

Annex ciphers to the numerator, and divide by tlie de- nominator.

Finally, point off as many decimal figures in the result as there are ciphers annexed to the numerator.

Reduce the following fractions to decimals :

2. \. 6. £. io. -ft- 1 4. ^.

3- I- 7- I- . ii. U- i5- Tib-

4- I- a A- 12. TV 1 6. j&.

5- £• 9- tV 13- & 17- iU>

18. Keduce J to the form of a decimal. Analysis. — Annexing ciphers to the nu- Operation.

merator, and dividing by the denominator as 3 ) I'.oooo before, the quotient is 3 repeated continually, ^VVV3 etc.

and the remainder is always 1. Hence, & can- not be exactly expressed by decimals.

1 9. Reduce J J to the form of a decimal.

Analysis.-— Annexing ciphers and divid- Operation.

ing as before, the quotient is 45 repeated 33 ) 15.0000 continually, and the remainder is alternate- 4545 etc.

ly 18 and 15, the latter being the given nu- merator. Therefore, ^§ cannot be exactly expressed by decimals.

23. When the numerator with ciphers annexed is exactly iivisible by the denominator, what is the decimal called ?

It is called a Terminate decimal.

24. When it is not exactly divisible, and the same figure or set of figures continually recurs in the quotient, what is the decimal called ?

It is called an Inlerminate, or Circulating decimal.

25. What are the figure or figures repeated called ? They are called the Repetend.

Note. — When the quotient has been carried as far as desirable, fhe sign ( 4-) is annexed to it, to indicate there is still a remainder.

A D D I T I-ON OP DECIMALS. 135

ADDITION OF DECIMALS.

i. If an arithmetic costs 6 tenths dollar, and a gram- liar 8 tenths dollar, what will both cost ?

Analysis. — Since each of these decimals expresses tentlis they have a common denominator, viz., 10 ; therefore, they are like numbers, and may be added, as whole numbers. Now £ tenths and 8 tenths are 14 tenths ; equal to 1 and 4 tenths dollar

25, a. When have decimals a common denominator ?

Decimals have a common denominator when their nu- merators have the same numher of decimal figures. As .05 and .07, whose denominator is 100. (P. 131, Q. 16.)

25, b. How reduce decimals .to a common denominator ?

Make the number of decimal figures the same in each, by annexing ciphers. (P. 131, Q. 18.) Thus, .3 and .05 re- duced to a common denominator become .30 and .05. • 1. What is the sum of 42.136 ; 6.35 ; 13.7 ; and .245 ?

Analysis. — Reduce the decimals to a com- Operation. mon denominator by annexing ciphers, or, which 42,I3"

is the same, write units under units, tenths °-35°

under tenths, etc., the decimal points being in a I3*7°°

perpendicular line. Beginning at the right, add *245

as in whole numbers, and place the decimal A?IS. 62.431 point in the amount under those in the numbers added. (P. 25, Q. 9.)

2 a. How add decimals?

I. Write the numbers so that the decimal points shal stand one under another, with tenths tinder tenths, etc.

II. Beginning at the right, add as in tohole numbers, , and place the decimal point in the amount under those

in the numbers added. (P. 28, Q. 13.)

Rem.— Placing tenths under tenths, hundredths under hun- dredths, etc., in effect reduces decimals to a common denominator / hence, the ciphers on the right may be omitted in the operation. (P. C31, Q. 18.)

16	ADDITION OF DECIMALS.
N	(30	(4.)	(5.)
26.176	8.65	206.451	3-7056
2.5	.372	40.45	.045
4.38	i.6	3-6	.06
.023	5405	23-75	2.841

6. What is the sum of seventeen and four tenths; si.. and two hundredths ; eight and forty-five thousandths ?

7. What is the sum of 13.71 yards; 21.2 yards; and 10.75 yards ?

8. How many dollars in 3 purses ; the first contain- ing 26.5 dollars ; the second, 1 7.25 dollars ; and the third, 30.625 dollars ?

9. How many acres in four lots, which respectively contain 19.275 acres; 30.41 acres; 23.261 acres; and 31.027 acres?

10. What is the sum of 42.07 gallons + 50.128 g; Is. 4- 1.625 gals- + l6.oi8 gals. ?

11. What is the sum of 28.16 rods + 45.025 rods f 85.7 rods + 17.265 rods.

SUBTRACTION OF DECIMALS.

1. A lad having 8 tenths of a dollar, paid 3 tenth, of dollar for his lunch : how much had he left ? Analysis. — 3 tenths from 8 tenths leave 5 tenths. Therefore itc.

2. Take 7 tenths from 9 tenths.

3. What is the difference between .19 and .17 ?

4. Paid .37 dol. for a bushel of apples, and .60 rtoL ' for a bushel of corn : what was the difference in price ?

5. A man having .75 of an acre of land, sold .48 of an acre: how much land did he have left?

6. What is the difference between .93 and .62 ?

SUBTRACTION OF DECIMALS. l37

SLATE EXERCISES.

I. What is the difference between 2.34 and .543 ? Analysis. — Reduce the decimals to a common de- Operation.

nominator by annexing ciphers, or by writing units- 2«34

under units, tenths under tenths, etc., the decimal «543

points being in a perpendicular line. (P. 38, Q. 9.) i«797 Beginning at the right, we see that 3 thousandths

can not be taken from o thousandths ; hence we borrow 10, and proceed as in whole numbers.

27. How subtract decimals ?

L Write the less number under the greater, so that the decimal points shall stand one under the other, with tenths under tenths, etc.

II. Beginning at the right, subtract as in tvhole num- bers, and place the decimal point in the remainder under that in the subtrahend. (P. 42, Q. 15.)

	Ml	(3.)	(4.)	(50
From	6.432	13.206	28.3607	1.00042
Take	3-i7	7.0378 •	.981	.236

Perform the subtractions indicated in the following :

6. 63.025 — 13.5. 11. 60.001 — 45.008.

7. 7.46 — 3.678. 12. 1.0006 — 0.37.

8. 100.007 — 0.845. 13. 0.05 — 0.005.

9. 275 — 60.75. 14. 0.006 — 0.0006.

10. 17.4 — 10.0008. 15. 0.0001 — .00001.

16. Sold 2 pieces of cloth, one 37.5 yards long, the other 31 \ yards: what was the difference in their length ?

17. A man owning 7 tenths of a ship, sold 25 hun- dredths of her : how much had he left ?

18. If from 150.05 acres of land, 87-J acres arc taken, how much will be left ?

138 MULTIPLICATION OP DECIMALS.

MULTIPLICATION .OF DECIMALS.

i. At .5 of a cent apiece, what will 7 apples come to ?

Analysis. — Since 1 apple costs 5 tenths of a cent, 7 apples will cost 7 times 5 tenths, or 35 tenths of a cent ; and 35 tenths &ro equal to 3.5 cents. Therefore, etc.

2. What cost 4 oranges, at .6 of a dime apiece ?

3. At .3 dollar apiece, what will 6 melons cost ? •

4. How many tenths are 5 times 7 tenths ? 6 times 4 tenths ?

5. How many hundredths are 3 times .15 ?

6. How many tenths in 4 times .25 ?

SLATE EXERCISES.

1. If 1 yard of muslin costs .25 dollar, what will 7 yards amount to ?

Analysis. — .25 are equal to 2 tenths and 5 hun- Operation.

dredths. Now 7 times 5 hundredths are 35 hun- t2K dol.

dredths, equal to 3 tenths and 5 hundredths. Set -

the 5 in hundredths place, and carry the 3 to the pro-

duct of tenths. 7 times 2 tenths are 14 tenths, and 3 1.75 dol. are 17 tenths, equal to 1 unit and 7 tenths. Write the 7 in tenths place, and the 1 in units place.

2. Multiply .375 dollar by 5. Ans. 1.875 dol.

3. At .75 dollar a yard, what cost .5 yard of delaine?

ANALYSIS.— .75 =11oAj, and .5 =-^r. (P. I33> Operation. Q. 21.) Now ^ times -ftfo=-ftftfr= 375 -*■ 1000, -75 dol.

ar . 3 7 5 , A ns. Instead of multiply i ng -^ by &> *5

in the operation we multiply the decimals as Ans. .375 dol. whole numbers ; consequently the product is as many times too large as there are units in the product of their denominators, viz., 1000. To correct this, we point off 3 figures on the right of the product, which divides it by 1000.

By inspecting these operations, it will be seen that each product has as many decimal figures as both its factors.

In like manner it may be shown, that the product of any two decimals must have as many decimal figures as both factors.

MULTIPLICATION OF DECIMALS. 139

28. How multiply decimals?

Mulliply as in whole numbers, and from the right .of the product, point off as many figures for decimals as there are decimal jrtaces in both factors.

Remakes. — i. If the product has not as many figures as there are decimals in both factors, the deficiency must be supplied by prefixing ciphers. (Ex 4.)

2. To multiply a decimal by 10, 100, 1000, etc., remove the deci- mal point as many figures to the right as there are ciphers in the multiplier. For, each removal of the decimal point one place to the right, multiplies the number by 10.

4. What is the product of .004 multiplied by .03 ?

Analysis. — The product of the significant Operation. figures 4 x 3=12. But there are five decimals in .004

the given factors ; therefore the product must have five decimals. Prefixing three ciphers to

•03

12, it becomes .00012. .00012 Ans.

(5-) (6-) (7-) (8.)

Mult. .127 3-°25 .0046 250.07

By .03 .012 .23 3.04

Perform the following multiplications :

.9. 8.02x3.2. 13. 38.065 x. 003. 17. 25.012 x 2.15.

10. 3.51 x. 09. 14. 506.12 x. 016. 18. ioooox.007.

11. 9.027x13. 15. 407.01 x.i 23. 19. 000.01x300.1

12. 365 x .05. 16. 1.0004 x. 006. 2°« 0.0004x2.01;

21. Allowing 5.5 yards to a rod, how many yards are there in 20.25 rods?

22. If a man earns 1.25 dol. a day, how much will he earn in 19.5 days ?

23. How many pounds of coffee in 10^ sacks, allow- ing 37-5 pounds to a sack ?

140 DIVISION OF DECIMALS.

24. If a gallon of molasses is worth .54 dol., how much are 18.75 gallons worth ? . 25. What is the product of 1.005 multiplied by .008 ?

26. What is the product of one thousandth into seven hundredths ?

27. What is the product of five ten-thousandths into seven tenths ?

DIVISION OF DECIMALS. MENTAL EXERCISES.

1. At 2 tenths of a dime apiece, how many oranges can* a lad buy for 8 tenths of a dime ?

Analysis. — 2 tenths dime are contained in 8 tenths dime, 4 times. Therefore, etc. (P. 63, Q. 10.)

2. How many times 3 tenths in 6 tenths of a dollar ?

3. How many times are 7 hundredths contained in 35 hundredths ? .4 in .8 ? yf^ in -fifa ?

4. If a man pays 6 tenths of a dollar for 2 tenths of a barrel of apples, what must he pay for 1 tenth of a barrel ?

Analysis. — The object is to divide 6 into 2 equal parts. (P. 63, Q. 10.) Since these fractions have a common denominator, one numerator may be divided by the other like whole numbers.

5. How many times are .08 contained in .64 ?

6. Divide .4 by .2 ; .6 by .3 ; .8 by .4.

7. Divide .08 by .02; .16 by .04.

SLATE EXERCISES.

1. How many times .2 in .6 ?

Analysis. — Since these decimals have a opekatiow. common denominator, they are like numbers; ,2).6

hence, one can be divided by the other as Ans. % times! common fractions, and the quotient is a whole number. (Page 121, Q. 46.)

DIVISION 0* DECIMALS. 14l

2. How many times .04 in .3 ?

Analysis. — Reducing the given decimals to a operation. common denominator, we have .04 and .30. Now, ,c>4).30 .04 is in .30, 7 times and .02 remainder. We put the Ans 7~T 7 in units' place, because it is units. Annexing a cipher to the remainder, .04 is in .020, .5 times and o remainder. We set the 5 in tentM place. Ans. 7.5 times.

3. How many times .4 in .012 ?

Analysis. — Reducing these decimals to a com- operation. mon denominator, we have .4 = .400 and .012. .4oo).oi2 Now, as .400 is not contained in .012, we put a a ZZ ^7^>

cipher in units' place. Annexing a cipher to the dividend, we find .400 is not contained in .0120 ; we therefore put a cipher in tenths' place. Annexing another cipher, .400 is in .01200, 3 hundredths time and o remainder. Ans. .03 times.

29. How are decimals divided?

Reduce the decimals to a common denominator, and divide the numerator of the dividend by that of the divi- sor, placing a decimal point on the right of the quotient.

Annex ciphers to the remainder, and divide as before. The figures on the left of the decimal point denote whole numbers ; those on the right, decimals. .

Or, divide as in whole numbers, and from the right of the quotient, point off as many decimals as the deci- mal places in the dividend exceed those in the divisor.

Remarks. — 1. If there are not figures enough in the quotient for the decimals required by the second method, prefix ciphers.

2. To divide a decimal by 10, 100, 1000, etc.,

Remove the decimal point in the dividend as many places to the left as there are ciphers in the divisor.

3. If there is a remainder after carrying the work as far as de- sired, the sign ( + ) is annexed to the quotient to show it is not exact.

EXAMPLES FOR PRACTICE.

1. What is the quotient of .028 divided by 7 ? Ans. .004.

2. What is the quotient of .432 divided by .144 ? Ans. 3,

3. What is the quotient of 5 15 divided by 1.03 ? Ans. .500.

142 DIVISION OF DECIMALS.

4. Divide 2.37 by 9. Ans. 0.2633+ ; or, .2633^.

5. At .25 dol. a pound, how much honey can be bought for 2.75 dollars ?

6. How many building-lots can be made from 12.75 acres of land, allowing .25 of an acre to a lot?

7.- Divide 43.12 by 10. 9. Divide .2806 by 1000.

8. Divide 7.312 by 100. 10. Divide 734.201 by 10000.

Perform the following divisions :

n. 57-5-4* 16.36.54-10. ■ 21. IO-+.OI.

12. 194-.25. 17. 3.854-100. 22. 11 4-. 11.

13- ■675-^- -33- iS. .0564-. 112. . 23. .114-11.

14. .0342 -T-. O7. I9. 39 I. O4-MOOO. 24. .OOO5-V-5.

15. .00394-. 26. 20. 246.7514-85. 25. .00003 4- .00004.

26. A farmer sold 75 sheep for 187.5 dollars: what was that apiece ?

27. If you travel 40.75 miles in a day, how long will it take to travel 195.6 miles ?

28. If 34.5 bushels of apples cost 17.25 dollars, what will 1 bushel cost* ?

29. If 18.75 tons of hay cost 196.875 dollars, what will 1 ton cost ?

30. How many revolutions will a wheel 9.4 ft. in cir- cumference make in going 5280 feet?

31. If 1 acre of land produces 25.6 bushels of corn, how many acres will be required to produce 4635 .bushels ?

32. How many times are five thousandths contained in 37 hundredths ?

^. How many times are seventy-five ten-thousandths contained in eight thousandths.

34. How many times are seven millionths contained in three hundred-thousandths ?

UNITED STATES MONEY.

1 . What is Money ?

Money is the standard of value, and -is often called Currency.

2. What is United States Money ?

United States Honey is the national cnrrency of the United States. It is also called Federal Money.

3. What are its denominations ?

Eagles, dollars, dimes, cents, and mills.

TABLE.

io mills (m.) make i cent, cL io cents " i dime, d.

io d., or ioo cts. " i dollar, $, or dot io dollars " i eagle, E.

5octs.=^dol.; 33jcts.=|dol. ; 25 cts.=£dol. ; 2octs.=|dol. ; i2icts.=|dol.; iocts-^-^dol.

Notes. — 1. It will be observed that the denominations of U. S. money, like the orders of whole numbers, increase and decrease by the scale of 10. It is thence called Decimal Currency.

2. The sign of U. S. money is the character ($), called the dol- lar mark, placed before the sum to be expressed.

4 . How is U. S. Money written ?

Hollars are written as whole numbers, with the ign ($) prefixed to them.

Cents are written in the first two places on the right of the decimal point ; because they are hundredths of a dollar. Thus 13 dollars 25 cents are written $13.25.

Mills are written in the third place on the right; be- cause they are thousandths of a dollar; as $25,038.

Remarks. — 1. Eacrles are exrresjjpd by tens of dollars ; dimes by tens of cents. TauB, 15 eagles are $150, and 6 dimes are 60 cuts.

144 seduction of u. s. money.

2. As cents occupy two places, if the number to be expreb^ed is less than 10, a cipher must be prefixed to the figure denoting them.

3. In business calculations, if the mills in the insult are 5 or more, they are considered a cent; if less than 5, they are omitted.

1. Write 17 dollars and 5 cents. Ans. $17.05.

2. Write 20 dollars, 10 cents, and 3 mills. Ans. $20,103.

3. Express 3 eagles and 4 dimes, in dollars and cents. Analysis. — Since in 1 eagle there are 10 dollars, in 3 E. there

are 3 times 10, or $30. Again, in 1 dime there are 10 cents, and in 4 dimes 4 times 10, or 40 cents. Ans. $30.40.

4. Write 43 dollars, 1 2 cents and 5 mills. . 5. Write 100 dollars and 8 cents.

6. Write 2 1 9 dollars, 3 cents and 4 mills.

7. Write a thousand and ten 'dollars and five cents.

EXERCISES IN READING U. S. MONEY-

5. How read U. S. Money ?

Read the figures on the left of the decimal point, as dollars ; those in the first two places on the right, as cents; the next one, as mills ; the others, as decimals of a mill.

Copy and read the following sums of U. S. Money ?

1. $17,213. 6. $100. 11. $1000.043.

2. $30,105. 7. $107. 12. $2100.05.

3. $42.60. 8. $110.50. 13. $1006.40.

4. $0,437. 9. $230,061. 14. $3050.10.

5. $0,805. • 10. $500,007. 15. $4100.01

REDUCTION OF U. S. MONEY. CASE I. To Reduce Dollars to Cents and Mills. 1. How many cents are there in $4 ?

Analysis. — Since there are 100 cents in a dollar, there must bo 100 times as many cents as dollars, or 400 cts. (P. 56, Q. 17.)

SEDUCTION OF U. S. MONET. 145

2. In $6, how many cents? In $7 ? In $10? In 12?

3. How many mills in $6 ?

Analysis. — There are iooo mills in a dollar; hence, in $6 there must be iooo times as many mills as dols., or 6000 mills.

4. How many mills in 15 cents ? In 52 cents ?

5. In $7, how many mills ? In $1 1 ? In $20 ?

SLATE EXERCISES.

1. How many cents in 18 dollars ?

Analysis. — Since there are 100 cents in a Operation.

dollar, there must be 100 times as many 18 dollars,

cents as dollars ; and 100 times 18 are 1800. IO°

Therefore, etc. Ans. ^oo cts.

2. In 87 cents, how many mills ?

Analysis.— Since there are 10 mills in 87 Cents,

a cent, there must be 10 times as many mills 10

as cents; and 10 times 87 are 870. Ans 870 mills.

6. How reduce dollars to cents, etc. ?

To reduce dollars to cents, multiply them by 100. To reduce dollars to mills, multiply them by iooo. To reduce cents to mills, multiply them by 10.

Note. — To reduce dollars, cents, and mills, to cents and mills, erase the sign of dollars ($) and the decimal point

3. In $40.75, how many cents ? Ans. 4075 cents.

4. In $51,073, how many mills? Ans. 51073 mills.

Reduce the following to the denominations indicated?

5. $67 to cents. 10. $85.38 to cents.

6. $125 to cents. 11. $7,375 to mills.

7. $95 to mills. 12. $9.87! to mills.

8. $216 to mills. 13. 8537 to cents.

9. $46.10 to cents. 14. $1385 to mills.

146 REDUCTION OF U. S. MONEY.

CASE II. To reduce Cents and Mills to Dollars, i. How many dollars in 212 cents ?

Analysis. — Since in 100 cents there is $1, in 212 cents there are as many dollars as 100 is contained times in 212 ; and 100 is contained in 212, 2 times and 12 cents over. Therefore, etc.

2. How many dollars in 500 cents ? In 625 cents ?

3. How many dollars in 700 cents ? In 865 cents ?

4. How many dollars in 3000 mills ?

Solution. — As many as 1000 is contained times in 3000, which is 3 times.

5. How many dollars in 5256 mills ? In 7341 mills ?

6. How many cents in 327 mills ? In 432 mills ?

SLATE EXERCISES.

1. How many dollars in 348 cents ?

Analysis. — Since in 100 cents there is 1 dollar, Operation.

in 348 cents there are as many dollars as there are I loo) 3148 times 100 cents in 348 cents ; and ioo is in 348 cents, 3 times and 48 cents over. Therefore, etc. $3.40

2. How many dollars in 4285 mills ?

Solution. — We divide the given mills by 11000)41285

iooo, or what is the same tiling, cut off 3 fig-

ures on the right of the dividend. Ans. $4*285

■7. How reduce cents and mills to dollars ? To reduce cents to dollars, divide them by 100. To reduce mills to dollars, divide them by 1000. To reduce mills to cents, divide them by 10. '

3. In 235 cents, how many dollars ? Ans. $2.35.

Reduce the following to the denominations indicated :

4. 563 cents to dollars. 7. 5770 cents to dollars.

5. 895 cents to dollars. 8. 268 mills to dollars.

6. 1263 cents to dollars. 9. 3275 mills to cents.

ADDITION OF U. S. MONET. 147

ADDITION OF U.S. MONEY.

i. George paid $2.45 for a sled, and $1.63 for a pair of skates : what was the cost of both ?

Analysis. — $2 and $1 are $3 ; 45 cts. and 63 cts. are 108 cts., . or $1.08, which added to $3, make $4.08. Therefore, etc.

\ 2. What is the sum of $5.17 and $12.30 ?

3. If a hat costs $5.50, and a vest $9.75, what is the cost of both ?

4. A farmer sold a sheep for $5.50, and a. calf for $7.30 : what did he receive for both ?

5. The price of a reader is 87 cts., and an arithmetic 63 cts. : what is the price of both ?

6. If a man pays $3.25 a day for board, and 85 cents for cigars, what are his daily expenses for both ?

SLATE EXERCISES. . 8. Upon what principle is U. S. Money founded ? ' - It is founded upon the Decimal Notation.

9. How are its operations performed ?

Its operations are the same as the corresponding opera- tions in whole numbers and Decimal Fractions. 1. What is the sum of $10,625; $16,078; $28?

Analysis. — We write dollars under dollars, cents $10,625 under cents, etc., and beginning at the right, add as 16.078

in simple numbers, placing the decimal point in the 2 8.00

amount under those in the numbers added. (P. 25, $^4.70"? Q. 9.)

10. How add United States money?

Write dollars under dollars, cents under cents, etc., and add as in simple numbers, placing the decimal point in the amount under those in the numbers added.

Note. — If any of the given numbers have no cents, their place should be be supplied by ciphers.

148 ADDITION OF U. S. MONEY.

(2.)	(3.)	(4.)	(5-)
$430,451	$641,375	$890.40	$2056.625
205.06	80.06	708.00	140.50
I28.OO7	65.007	25-56	68.08
ns. 763-5l8	240.25	7-07 .	9-3*5

6. What is the sum of $85.10 ; $164.07 ; and $35.20 ?

7. What is the sum of $207.56 ; $500.65 ; aud $61.52 ?

8. Paid $8.75 for a barrel of flour; $5.25 for 2 barrels of apples ; and $7 for a ton of coal : what was the amount of my bill ?

9. A farmer bought a horse for $120,875 ; a yoke of oxen for $95 ; md a cart for $68.50 : what did he pay >br all ?

10. A merchant sold goods amounting to $150.35 to one customer; to another $96.40; to another $110; and to another $200.68 : what amount did he sell to all?

11. Add 120 dollars, 5 cents, and 3 mills; 45 dollars, and 7 mills ;' 78 cents, and 6 mills.

12. Add 7 dollars, and 7 cents; 10 dollars, and 5 mills; 217 dollars, and 45 cents; and- 31 dollars.

13. Add 371 dollars, 40 cents, and 8 mills; 710 dol- lars ; 90 dollars, and 35 cents ; and 219 dollars.

14. Add 1000 dollars; 100 dollars, and 10 cents; 93 cents ; 860 dollars, and 8 cents ; 5 dollars and 95 cents.

15. What is the sum of 1500 dollars and 8 cents + 807 dollars, 60 cents, and 7' mills +763 dollars, 3 cents and 5 mills + 85 cents and 8 mills ?

16. A lady bought a dress for $45.63; a shawl for $87,625 ; a collar for $15,375 ; a pocket-handkerchief for $7.50 ; what was the amount of her bill ?

17. Bought an overcoat for $35.75 ; a dress-coat for $28.62^ ; and a vest for $9.12^ : required the amount.

SUBTRACTION OF U. S. MONEY. 149

SUBTRACTION OF U.S. MONEY.

i. William having $7.62, gave $2.50 for a cap : how much did he have left ?

Analysis. — $2 from $7 leave $5 ; and 50 cents from 62 cents leave 12 cents. Therefore, etc.

2. Henry gave a 10 dollar bill to pay for a hat, the price being $7.50 : how much change should he receive ?

3. The price of a grammar is 85 cts.. and that of a geography $1.30 : what is the difference in their prices?

4. A father earns $10.75 a week, and his son $8.50: how much more does the former earn than the latter ?

5. A man paid $12.60 for a gal. of brandy, and $8.25 for a bar. of flour : required the difference in cost ?

SLATE EXERCISES.

I. A person having $356.07, paid $109,625 for a horse:' how much did he have left ?

Analysis. — We write the less number under the Operation.

greater, dollars under dollars, cents under cents, etc $35^*°7 Subtract as in simple numbers, and place the deci- 109.625

malpoint in the remainder under that in the subtra- $246,445 hend. (P. 38, Q. 9.)

II. How subtract United States money?

Write the less number under the greater, dollars under dollars, cents under cents, etc., and subtract as in simple numbers, placing the decimal point in the remainder under that in the subtrahend.

Note, — If either of the given numbers has no cents, their place should be supplied by. ciphers.

(*•) (3-) (4.) (S-)

From $65,875 $110.46 $68,004 $ico.oo Take 46.29 95-375 19.086 0.875

150 MULTIPLICATION OF U. S. MONET.

6. George gave $1.75 for his geography, and $0,875 for his arithmetic : what was the difference in cost ?

7. A lady bought articles amounting to $29,375, and gave the clerk a 50 dollar bill: how much change ought she to receive ?

8. Bought a coat for $25.75 ; pants for $14; vest for $11.50; and sold- wood to the tailor amounting to $50: how much am I indebted to him ?

9. If you have $407 on deposit, and check out $219,625, how much will you have left in bank ?

10. Find the difference between $117.45 and $201.03 ?

11. Find the difference between $1000 and 1000 cts. ?

1 2. From two hundred dollars and seven cents, take forty dollars and 5 mills.

13. From one hundred dollars and 6 cents, take five dollars and 20 cents.

14. From $300, take 3 dol., 3 cts., and 3 mills.

15. A father gave one daughter a music-box worth $75,375, the other a sewing-machine worth $55.67 : what was the difference in their cost ?

MULTIPLICATION OF U. S. MONEY.

1. What will 3 chairs cost, at $7.50 each ?

Analysis. — 3 chairs will cost 3 times as much as 1 chair. Now 3 times $7 are $21, and 3 times 50 cts. are 150 cts., equal to $1.50, which added to $21, make $22.50. Therefore, etc. (P. 52, Note.)

2. What cost 4 fruit-knives, at $2.12 apiece ?

3. What cost 5 bouquets, at $3.50 apiece ?

4. What cost 6 paper-folders, at 75 cents apiece ?

5. At $4.10 a box, what will 8 boxes of lemons come to ?

. 6. At $11.20 apiece, what will 10 dresses cost?

MULTIPLICATION OF U. S. MONEY. 151

SLATE EXERCISES.

i. What will 1 8 ploughs cost, at $13,125 apiece ?

Analysis.— If 1 plough costs $13,125, 18 ploughs $13,125

will cost 18 times as much. We multiply in the jg

usual way, and from the right of the product point

off three figures for cents and mills ; because there 5

are three places of cents and mills in both factors. _Jl__~

{P. 139, Q. 28.) $236,250

12. How multiply United States money ?

Multiply as in simple numbers, and on the right of tile product, point off as many figures for cents and mills as there are decimal places in loth factors.

Note. — In United States Money, as in simple numbers, the multiplier must be considered an abstract number.

2. If you spend 87 J cents a day, what will you spend

in 7 days ?

Solution.— 87I cts.=$o.875, and $0,875 x 7=16.125, Arts.

	(3.)	(4.)	(S-)	(6.)
mat	$39-35	$60,075	$100,008	$82650
By	11	•iS	6.5	•75

7. What will $6 chickens come to, at 62J cts» each ?

8. If a man earns $9.50 a week, what will be his wages for 5 2 weeks ?

9. At $1.37! per yard, what will a dress containing 20.5 yards of grenadine come to ?

10. At $7.50 a ton, what will 100.5 tons of coal cost ?

11. What cost 18 pianos, at $750 apiece ?

12. What is the amount of a man's expenses for 12 months, if he spends $86.50 a month ?

13. What cost 25 building-lots, at $1250.50 a lot?

t

152 DIVISION OF U. 8. MONET.

14. At $4.50 each per day, what will be the hotel expenses of 6 persons for 4 weeks ?

15. What is the sum of 12 times 87 J cents, and 15 times 62-J cents ?

16. What is the difference between 20 times $17.65, and 17 times $25.40 ?

17. A farmer bought 12 calves, at $7.60 each ; and 20 sheep, at $4.75 each : how much did he pay for both lots ; and what is the difference in their cost ?

DIVISION OF U. S. MONEY.

1. If 9 oranges cost 6$ cents, what will 1 orange cost ? Analysis.— If 9 oranges cost 63 cents, 1 orange will cost 1

ninth of 63 cts., which is 7 cts. Therefore, etc. (P. 63, Q. 10.)

2. If 7 sheep cost $35, what will one cost ?

3. If 8 yards of velvet cost $72, what will 1 cost ?

4. A man laid out $50 in vests, which were $5 apiece: how many did he buy ?

5. How many hand-carts, at $6, can be bought for $300 ?

SLATE EXERCISES.

1. Sold 6 hats for $42.75 : what was that apiece ?

ANALYSIS. — 1 hat is 1 sixth of 6 hats ; hence 1 Operation. hat is worth 1 sixth of $42.75, which is $7,125. 6)$42-75°

The object of this example is to divide the sum Ans. $7.1 25 O' $42.75 into 6 equal parts. (P. 63, Q. 10.) Dividing as in simple numbers, there is a Temainder of 3 cents . which we reduce to mills, and dividing as before, point off threa figures for cents and mills.

13. How divide United States money by an abstract number ?

Reduce the dividend to mills, and divide as in simple numbers. The quotient will he mills, which must be re- duced to dollars and cents.

Note. — If there is a romaindor, write tho sign + after tho quotient.

DIVISION OF U. S. MONEY. 153

2. How many hats, at $7,125, can be bought for

$42.75 ?

Analysis.— The object here is to find Operation.

how many times one sum of money is con- $7. 1 2 5) $42.7 5 0(6 tained in another. But the divisor contains 42.750

dollars, cents, anr? mills, while the dividend contains dollars and cenU only. We there- fore reduce the latter to mills, and then divide as in simple num- bers. (P. 67, Q. 10.)

13, a. How divide U. S. Money by U. S. Money ?

Reduce the divisor and dividend to the same denomina- tion, and divide as in simple numbers. Tlie quotient will be times, or an abstract number. (P. 63, Q. 10.)

Notes.— 1. In business matters it is rarely necessary to carry the quotient beyond mills.

2. If there is a remainder after all the figures are divided, annex ciphers, and continue the division as. far as desirable, con- sidering the ciphers annexed as decimals of the dividend.

(30 (4.) (50 (6.)

7)$q2.6q4 8)$ii4 9) $13,791 6)$o.8o4

$13,242 $14.25 $1,421+ $0,134

7. How many times are $8 contained in $90.47 ?

8. How many times are $75 contained in $900 ?

9. How many melons, at $0.25 each, can be purchased

/3r $15.75 ?

10. How many ponnds of butter, at $0.30, can be had for $25.65?

11. If I pay $14,875 for 7 baskets of peaches, how much is that a basket ?

12. A stationer sold 5 slates for $0,625: what was that apiece ?

154 QUESTIONS FOR REVIEW.

13. At $20 apiece, how many yearlings can be bought for $280 ?

14. How many acres of land, at $12.50 per acre, can you buy for $1000 ?

15. Paid $43 for 8 excursion-tickets: how much was that for each ticket ?

16. A clerk agreed to work 12 months for $427.56: what was that per month ?

17. If $1600.75 are divided equally among 25 per- sons, how much will each receive ?

18. Sold 280 sheep for $658 : what was that per head ?

19. Sold 35 doz. eggs for $8.75 : what was that a doz. ?

20. Paid $2675.75 for 278 tons of coal: what was the cost per ton ?

QUESTIONS FOR REVIEW.

1. A man bought 6 cords of wood at $4.17, and 5 tons of coal at $7,375 : what did he pay for both ?

2. If you buy 10 pen-knives for $6.75, and sell them at 85 cents apiece, how much will you make or lose ?

3. What is the difference between 7 times $8.50, and 9 times $7,625 ?

4. What is the difference between 11 times $17.65, and 8 times $19.48?

5. Bought 8 boxes of raisins at $6.40, and sold them at $9.63 a box : how much was made on them ?

6. A traveler was robbed of $375, and had $159.60 left: how much money had he before the robbery?

7. If you buy 12 melons for $3.96, and sell them at 50 cents each, how much will you make on each ?

8. A grocer bought 12 bags of coffee for $121.92, but finding it damaged sold it for $30.20 less than cost: for what did he sell it per bag ?

APPLICATIONS OF IT. S. MONEY. 155

9. Bought in barrels of flour for $897.50: for how much must I sell it per barrel to make $300 ?

io. Paid $1162.50 for 25 acres of land: what is that ^>er acre ?

n. Paid $6785 for 100 oxen: what was that apiece ?

12. How many horses, at $150, will $10650 buy ?

13. What is the sum and difference of $567,625 and

*945-5° t

14. How many tubs of butter, at $16.50 each, can be bought for $206.25 ?

15. Exchanged 75 barrels of apples worth $150, for 25 barrels of cider : what did the cider cost per barrel ?

16. How many pair of shoes, at $1 J, can be had for 5000 pounds of rice, at 12^ cents a pound ?

APPLICATIONS OF U. S. MONEY. BILLS.

14. What is a bill?

A Sill is a written statement of goods sold, services rendered, etc., and should include the various items, the price of each, the date, and the place of the transaction.

15. How are bills receipted ?

A Hill is receipted when it contains the words, "Received payment," and is signed by the person to whom it is due, or his agent.

16. What is the meaning of the terms Debtor and Creditor ? A debtor is a person who owes a debt.

A creditor is one to whom a debt is owed.

Notes. — 1. The abbreviation Dr. denotes debit or debtor ; Cb., credit or creditor ; 'per signifies by, and the character @, at.

2. To familiarize the learner with the form of bills, the manner of receipting them, etc., it is advisable for him to copy the follow- ing, in a neat hand, upon his slate or paper.

156 APPLICATIONS OF U. S. MONEY.

Find the amount of the' following bills :

(i.) New YonK,June 3d, 1871.

Hon. GrEORGE Peabody,

Bought of Horace Webster. 7 lbs. coffee, @ $0.38 .... $2.66

1.92

.66

4-35

12 " sugar, " .16

6 " cornstarch, " .11 5 « tea, " .87 .. . .

Amour* Received Payment, .

Hoeace Webster.

(2.) Mobile, Feb. 21st, 1871.

George Walker, Esq.,

To Daniel Kingsbury & Co, Dr. To 15 yds. silk, @ $2.35 "11 * muslin, " .29 " T2 pair hose, " .42 "12 " gloves, " 1.50 " 6 parasols, " 3.50

Amt, Redd Paft by Note,

D. Kingsbury & Co.,

By S. Barret,

(3.) Chicago, May 21st, 1871.

John Murdoch, Esq., in acct. tvith David Joy & Co.

Dr.

For 12 pair shoes, @ $1.62

" 6 " thick boots, * 2.75

" 10 " slippers, " .88

" 24 " woollen hose, " .30

Carried forward, $51.94

APPLICATIONS OF U. S. MONEY. 157

Brought forward,				§51.94
Credit,
By 6 bushels wheat,	@	$1.50
"14 " oats,	a	.60
" 3 barrels cider,	a	2.75
" 10 barrels potatoes,	a	I.87
What is the balance ?			Balance^	44.35 , $7.59

(4.)

Messrs. J. H. Burtis & Co.,

Bought Of SCHERMERHORN & WlLSON.

12 slates @ $.13 ; 3 blackboards @ $9.50; 6 boxes of crayons @ $.68 ; 36 inkstands @ $.12 ; 2 small globes @ $5.25. Eequired the amount.

(5.)

James Barber in acct, with W. C. Young.

Dr.

For 10 shovels % $1.67 ; 12 hoes @ 1.25 ; 6 rakes @

$1.50; 4 axes ® $2.63. ;

Credit.

By 12 days work, man, @ $2.00; 10 days work, self and team, @ $3.60; 20 cords wood @ $3.10; 15 shade- trees @ $1.75.

What is the balance due on the above ?

6. George Morris of Chicago bought of A. T. Stewart & Co. 18 yds. of silk, at $2.63 ; 14 yds. Empress cloth, at $1.75 ; 12 yds. of poplin, at $2,375 ; 15 yds. of French lawn, at $0.65; 6 pair of gloves, at $1.85; 6 pocket- handkerchiefs, at $1.25; 3 parasols, at $3.60. Eequired the amount.

COMPOUND NUMBERS.

1. What are Simple Numbers?

Simple Numbers are those which contain units of one denomination only; " as, two, four, 3 apples, 4 quarts,- etc.

2. What are Compound Numbers ?

Compound Numbers are those which contain two or more denominations of the same natter e ; as, 4 bushels and 3 pecks ; 3 days and 5 hours.

Illustration. — Suppose, for example, we apply the inch as a unit of measure to the side of a table, and find it equal to 30 such measures. Again, if we employ the foot (12 in.) as the unit, it is equal to two such measures, and 6 in. oyer. Now as 6 inches — \ foot, we may call its length 2\ feet, or 2 feet, 6 inches. The former expression contains units of but one denomination, viz., feet ; therefore, it is a simple number. The latter contains units of two different denominations, which are of the same nature, viz., feet and inches ; therefore it is a compound number.

But the expression 2 feet and 4 pounds is not a compound number; for the units are of unlike natures.

Note. — Compound numbers are often called Denominate Numbers.

3. When it is said that a cane is 3 feet long, what is the kind of number used ?

A Simple Number ; because it contains but one denomination, viz., feet.

4. If we say that a cane is 2 feet and 10 inches long, what kind is the number ?

A Compound Number ; because it contains tivo denominations of the same nature, viz., feet and inches.

What kind of a number is 6 days ? Why ?

What kind is 5 pounds 2 shillings and 6 pence? Why?

What kind is each of the following: Two? Three? 12 oranges? 7 pounds and 5 ounces? 10 dollars and 25 .cents? 10 oxen?

COMPOUND NUMBERS. 159

UNITED STATES MONEY. 5. What is United States Money ?

United States Money is the national currency of the United States, and is often called Federal Money. J. What are its denominations ?

Eagles, dollars, dimes, cents, and mills*

TABLE. 10 mills (m.) are i cent, ct

10 cents " i dime, d.

10 d., or ioo cts. " i dollar, $, or dot 10 dollars " i eagle, E.

7. Of how many kinds is U. S. money ? Two, Paper and Metallic.

8. What is the 'paper money of the U. S. ?

The Paper Money of the U.S. consists of Treasury- notes issued by the Government, known as Greenbacks, and Bank-notes issued by Banks.

Note. — Paper money is called Paper Currency.

Treasury-notes less than $i are called Fractional Currency.

9. What is metallic money ?

3fetallic Money consists of stamped pieces of metal, called coins. It is also called specie, or specie currency.

Note. — For exercises in U. S. money see pp. 143-157.

10. Of what do the coins of the United' States consist ? Gold coins, silver coins, and the minor coins.

11. Name the coins of each.

The gold coins are the double-eagle, eagle, half -eagle, quarter-eagle, three-dollar, and dollar piece.

The silver coins are the trade dollar, half-dollar, quarter-dollar, twenty-cent piece, and dime.

The minor coins arc the nickel $-cent and ycent pieces, and the bronze -cent.

160 COMPOUND NUMBERS.

ENGLISH MONEY.

12. What is English money?

English Money is the national currency of Great Britain, and is often called Sterling Money.

13. What are the denominations? Pounds, shillings, pence, and farthings.

TABLE.

4 farthings (qr. or far.) are i penny, d.

\2 pence " i shilling, s.

20 shillings " i pound or sovereign, £.

21 shillings " i guinea, • g.

Notes. — i. The legal value of a pound Sterling, or sovereign, Is $4.8665; the value of an English shilling is 24^ cents; and that of a penny about 2 cents.

2. Farthings are commonly expressed as fractions of a penny. Thus, 1 far.=^d.; 2 far.=£d. ; 3 far.=$d.

1. How many farthings in 5 pence ?

Analysis. — Since there are 4 farthings in a penny, there must be 4 times as many farthings as pence, and 4 times 5 are 20 pence. Therefore, etc.

2. How many pence in 3 shillings ? In 5 s. ? In 7 s. ?

3. How many shillings in £3? In £5 ? In £10 ?

4. In 15 farthings how many pence ?

Analysis. — Since in 4 farthings there is 1 penny, in 15 far. there are as many pence as 4 far. are contained times in 15 far ;l and 4 is in 15, 3 times and 3 over. Therefore, in 15 far. there are 3d. and 3 far. over, or 3^d.

5. How many shillings in 18 pence ? . In 24d. ? .

6. How many pounds in 21 shillings? In 30 s. ? In 40 s. ? In 85 s.? In 100 s. ?

\* If the teacher desires further practice upon the Tables, as they are recited, he will find corresponding Examples in the Slate Exercises, pp. 173-178.

COMPOUND NUMBERS. 161

TROY WEIGHT.

1 4. For what is Troy Weight used ? For weighing gold, silver, and jewels.

15. What are the denominations ?

Pounds, ounces, pennyweights, and grains.

TABLE.

24 grains (gr.) are 1 pennyweight, pwt. 20 pennyweights a 1 ounce, oz.

12 ounces " 1 pound, lb.

Note. — The best method of imparting to children a correct idea of Weights and Measures, is to let them see and handle the actual standards, or some material objects which are equal to the several units of length, surface, capacity, and weight. In this way, the Compound Tables afford a wide field for object teaching.

1. How many grains in 2 pennyweights?

2. How many pennyweights in 3 ounces ? In 5 oz. ?

3. How many ounces in 3 pounds ? In 5 pounds ?

4. How many ounces in 40 pwt. ? In 45 pwt. ?

AVOIRDUPOIS WEIGHT.

16. For what is Avoirdupois weight used?

For weighing all coarse articles; as, hay, cotton groceries, etc., and all metals except gold and silver.

17. What are the denominations ? Tons, hundreds, pounds, and ounces.

TABLE.

16 ounces {oz) are 1 pound, lb.

100 pounds " 1 hundred weight, ciot.

20 cwt., or 2000 lbs., " 1 ton, T.

8 oz. = J lb. ; 4 oz. = J pound.

162 COMPOUND NUMBEES.

Notes. — i. The ounce is often divided into halves, quarters, etc.

2. In business transactions, the dram, the quarter of 25 lbs., and the firkin of 56 lbs., are not used as units of Avoirdupois Weight.

3. Net weight is the weight of goods, without the bag, cask, etc.

4. Gross weight is the weight of goods with the bag, cask, or box in which they are contained. It calls 28 lbs. a quarter, 112 pounds a hundred weight, and 2240 pounds a long ton.

1. How many ounces in 3 pounds ? In 4 lbs.?

2. How many pounds in 3 quarters ? In 5 qrs. ?

3. How many hundreds in 3 tons ? In 5 tons ?

4. In 40 ounces, how many pounds ? In 48 oz. ?

5. In 30 hundreds, how many tons ? In 60 cwt. ?

6. In 3500 pounds, how many tons? In 5000 lbs. ?

APOTHECARIES' WEIGHT.

1 8. For what is Apothecaries' Weight used ? For mixing medicines.

19. What are the denominations ?

Pounds, ounces, drams, scruples, and grains.

TABLE.

20 grains (gr.) are 1 scruple, sc, or 3.

3 scruples " 1 dram, dr., or 3 .

8 drams " 1 ounce, oz., or f .

12 ounces " 1 pound, g>.

Note. — The only difference between Troy and Apotliecaries3 weight, is in the division of the ounce. The pound, ounce, and grain are the same in each.

1. How many grains in 2 scruples ? In 3 scruples ?

2. How many scruples in 4 drams ? In 8 drams ?

3. How many drams in 5 ounces ? In 100 ounces ?

4. How many ounces in 6 pounds ? In 1 2 pounds J

COMPOUND NUMBERS. 163

LINEAR MEASURE.

20. For what is Linear Measure used ?

For measuring that which has length, without breadth; as, lines, distances, etc. It is often called Long Measure.

21. What are the denominations ?

Leagues, miles, furlongs, rods, yards, feet, and inches,

TABLE.

ix inches {in.) are i foot, • ft.

3 feet " i yard, yd.

5 \ )rds., or 1 6 \ ft. " i rod, perch, or pole, r., or p. 40 rods, " 1 furlong, fur.

8 fur., or 3 20 rods " i mile m.

3 miles " 1 league, I.

Note. — The inch is commonly divided into halves, fourths, eighths, or tenths ; sometimes into twelfths, called lines.

1. Draw a straight line 2 inches long on your slate or blackboard.

2. Draw one 4 in. long. 6 in. long. 9 in. long. A foot long. A yard long.

3. How long is your pencil ? This pen-knife ? This pen-holder? This paper-folder ? This ruler?

4. How long is this table ? How wide ? How long is the school-room ? How wide ? How high ? How long is the play-ground ? *

5. How many inches in 3 feet ? In 5 feet ? In 8 feet ?

6. How many inches in 2 ft. and 5 in. ? 4 ft. and 6 in. ?

7. How many feet in 4 yards ? In 7 yds. ? In 9 yds. ?

8. How many feet in 5 yards and 2 feet ? In 6 yds. and 4 ft. ?

* This exercise should be varied, and continued till the class obtain a clear idea of the ordinary measures of length.

164 COMPOUND NUMBBBB.

9. How many miles in 5 leagues ? In 8 leagues ?

10. How many feet in 2 rods? In 3 rods? • In 37 inches, how many feet? In 60 in.? In

? In 100 in.?

In 1 8 feet, h ow many yards ? In 2 8 ft. ? In 40 ft. ?

In 32 furlongs, how many miles? In 41 fur.?

fur.?

11.

75 in. ? In 100 in.: 12

In 50 fur,

CLOTH MEASURE.

22. For what is Cloth Measure used?

For measuring those articles of commerce whose length only is considered; as, cloths, laces, ribbons, etc.

23. What are the denominations ?

The Linear Yard is the principal unit. This is divided into quarters, eighths, and sixteenths.

TABLE.

3 ft. or 36 in-> are 1 yard, - - - yd. 18 in., " 1 half yard, - - i yd.

9 in., " 1 quarter yard, - J yd.

4-J in., . " 1 eighth " - I yd.

2\ in., " 1 sixteenth, " - TV yd.

Note. — Ells Flemish, English, and French, are no longer used iu the United States ; and the nail, as a measure, is practically obsolete.

1. How many quarters in 14 in. ? In 26 in. ? 2i How many fourths in 3J yards ? In 5 J yards ?

3. How many eighths in 25 in. ? In 37 in. ?

4. How many eighths in 2| yards ? In 3I yards ?

5. How many yards in 14 half yards ? In 30 halves ? In 35 halves ?

6. How many yards in 25 fourths of a yard ? In 32 fourths ? In 48 sixteenths ?

COMPOUND NUMBERS,

165

SQUARE MEASURE.

24. For what is Square Measure used ?

For measuring surfaces, or that which has length and breadth, without thickness; as, land, flooring, etc. It is often called Land Measure.

25. What are the denominations?

Acres, square rods, square yards, square feet, and square inches.

TABLE.
144 square in. (sq. in.)	are 1 square foot,	sq.ft.
9 square feet	" 1 square yard,	sq. yd.
30 J square yards, or ) 2 72 J square feet, )	u j 1 sq. rod, perch, 1 or pole,	sq. r.
160 sq. rods	" 1 acre,	A.
640 acres	" 1 square mile,	M.

Note. — The acre was formerly divided into 4 roods ; but in practice the rood is no longer used as a unit of measure.

26. What is a Square ?

A Square is a rectilinear figure which has/owr equal sides and four right angles. Thus,

A Square Inch is a square, each side of which is 1 inch in length.

A Square Yard is a square, each side of which is 1 yard in length.

Note. — The corners of any square figure, also of a table, a room, etc., are right angles.

1. Make a right angle upon your slate, or the black- board.

3eq. ft.x3=9pq. ft.

9 eq. ft. = i eq. yd.

166 COMPOUND NUMBERS.

2. Make a square inch.

3. Make a square whose side is 3 inches. 6 inches.

4. Make a square foot.

5. Make a square yard.

6. Divide a square yard into square feet.

7. Divide a square foot into square inches.

8. How many square inches in 2 sq. ft. ? In 3 sq. ft. ?

9. How many square feet in 3 sq. yds. ? In 5 sq. yds. ?

10. In 27 sq. feet, how ma*hy sq. yards ? In $6 sq. feet ?

11. What is the difference between 3 feet square, and 3 square feet ?

CUBIC MEASURE.

27. For what is Cubic Measure used ? For measuring solids ; as, timber, boxes of goods, the capacity of rooms, ships, etc. It is often called Solid

2§. What are the denominations ?

Cords, cubic yards, cubic feet, and cubic inches.

TABLE.

1738 cubic inches (cu. in.) are 1 cubic foot, cu.ft.

27 cubic feet " 1 cubic yard, cu. yd.

128 cubic feet " 1 cord, C. 1

28, a. Describe a cord ? A foot of wood ?

A Cord of wood is a pile 8 ft. long, 4 ft. wide, and 4 ft. high : for, 8 x 4 x 4= 128.

A Cord Foot is one foot in length of such a pile ; hence, 8 cord feet make one cord.

Note. — Timber is now measured by cubic feet and inches.

The old cubic ton of 40 feet of round timber, and 50 feet of hewn timber, has fallen into disuse in the United States.

COMPOUND NUMBERS.

16?

27 cu. ft. = 1 cu. yd.

29. What is a Cube?

A Cube is a regular solid bounded by six equal squares, called its faces. Thus,

A Cubic Inch is a cube, each side of which is a square inch.

A Cubic Yard is a cube, each side of which is a square yard.

1. Draw a cubic inch.

2. Draw a cube whose sides are 3 inches square.

3. Draw a cubic foot.

4. How long and wide must a block of marble whose height is 3 feet, to form a cubic yard ?

5. How many cubic feet in 2 cubic yards ?

6. How many cubic yards m 54 cubic feet ?

7. In 2 cords, how many cubic feet ?

be,

DRY MEASURE.

30. For what is Dry Measure used ?

For measuring grain, fruit, salt, etc.

31. What are the denominations ?

Chaldrons, bushels, pecks, quarts, and pints.

TABLE.

2 pints (pt.) are 1 quart, qt.

8 quarts " 1 peck, ph.

4 pecks, or 32 qts., " 1 bushel, bu.

36 bushels " 1 chaldron, ch.

Notes. — 1. The dry quart is equal to i\ liquid quart nearly. 2. The chaldron is used for measuring coke and bituminous coal.

1

168 COMPOUND NUMBERS.

i. In 8 pints, how many quarts ? In 16 pints ?

2. In 32 quarts, how many pecks ? In 40 quarts ?

3. How many pecks in 5 bushels ? In 7 bushels ?

4. How many quarts in 5 pecks ? In 9 pecks

5. How many bushels in 12 pecks? In 17 pecks

6. How many quarts in 2 bushels and 3 pecks ?

7. How many quarts in 3 pecks and 4 quarts ?

8. How many bushels in 40 quarts ? In 64 quarts ?

LIQUID MEASURE.

32. For what is Liquid Measure used ?

For measuring milk, wine, vinegar, molasses, etc.

33. What are the denominations ?

Hogsheads, barrels, gallons, quarts, pints, and gills.

TABLE.

4 gills (gi.) are 1 pint, pt.

2 pints " 1 quart, qt.

4 quarts " 1 gallon, gal.

31^ gallons " 1 barrel, bar., or bbl.

63 gallons ■ 1 hogshead, hlid.

Notes. — 1. Liquid Measure is often called Wine Measure. 2. The old Beer Measure is practically obsolete in this country

1. How many gills in 4 pints ? In 10 pints ?

2. How many pints in 7 quarts ? In 9 quarts ?

3. How many quarts in 5 gallons? In 10 gallons?

4. In 20 quarts, how many gallons ?

5. In 24 pints, how many quarts?

6. In 16 gills, how many pints?

7. In 24 gills, how many pints ? How many quarts ?

8. In 32 pints, hoW many quarts ? How many gallons.?

9. How many gallons in 2 barrels ? 10. How many gallons in 2 hogsheads ?

COMPOUND NUMBEKS.

169

CIRCULAR MEASURE.

34. For what is Circular Measure used ?

For measuring angles, land, latitude and longitude, the motion of the heavenly bodies-, etc.

35. What are the denominations ? Signs, degrees, minutes, and seconds.

TABLE

6o seconds (") are i minute,

6o*minutes " i

30 degrees

12 signs,. or 3600

Note. — Signs as a measure are used only in Astronomy

1 sign, s.

1 circumference, cir.

36. What is a Circle ? A Circle is a plane figure

bounded by a curve line, every part of which is equally distant from a point within called the center.

37. What is the Circumference of a Circle ?

The Circumference of a Circle is the curve line by which it is bounded. It is divided into 3600.

38. What is the Diameter?

The Diameter is a straight line drawn through the centre, terminating at each end in the circumference.

39. What is the Radius ?

The Radius is a straight line drawn from the center to the circumference, and is equal to half the diameter.

8

170 COMPOUND NUMBERS.

40. What is an Arc of a Circle ?

An Arc of a Circle is any part of the circum- ference.

In the preceding figure, ADEBFis the circum- ference; A B the diameter; C A, C D, C E, etc., are radii ; A D, D E, etc., are arcs.

Draw a circle. Draw a diameter. Draw another 1 diameter perpendicular to the first.

Note. — These two diameters divide the circumference into four equal parts, called quadrants.

41. How many degrees in a quadrant ?

Ninety.

42. How many and what angles do these two diameters form ?

Four right angles.

43. How many degrees in a right angle ?

Ninety.

MEASUREMENT OF TIME.

44. What are the denominations of Time ?

Centuries, years, months, tveeks, days, hours, minutes, and seconds.

TABLE.

6o seconds (sec.) 6o minutes 24 hours 7 days

365 days, 52 w.

366 days 12 calendar months (mo.)

100 years

Note. — In most business transactions, 30 days are considered a month. Four weeks are sometimes called a lunar month.

ays, or ) r. and 1 d., )

are	i minute,	mm.
a	1 hour,	h.
«	1 day,	d.
u	1 week,	w.
a	1 common year,	c.y.
tt	1 leap year,	E*
t(	1 civil year,	y-
a	1 century,	ce*„

COMPOUND NUMBERS.

171

45. What is a common year ?

A common year is one which contains 365 days.

46. What is a solar year ?

A solar year is the time in which the earth revolves around the sun, and equals 365 d. 5I1. 48 min. and 49.7 sea

Note.— The excess of the solar above the common year is about 6 hours, or \ of a day, nearly ; hence, in 4 years, it amounts to about 1 day.

47. What is a leap year?

A leap year is one which contains 366 days.

48. How caused, and why so called ?

It is caused by the excess of a solar above a common year ; and is so called because it leaps over the limit, or runs on one day more than a common year.

This day is given to February, because it is the short- est month; hence, in leap years, February has 29 days.

49. What is a civil year ?

A civil year is the year adopted by government for computing time, and includes both common and leap years as they occur.

50. How is the civil year divided ?

It is divided into 1 2 calendar months, viz. : January {Jan.), the first month, has 3 1 days.

February	(mi	" second	a	28	a
March	{Mar.),	" third	a	31	a
April	{Apr.),	" fourth	a	30	a
May	{May),	" fifth	it	31	it
June	{June),	" sixth	it	30	a
July	{July),	" seventh	a	31	a
August	{Aug.),	" eighth	a	31	a
September	{Sept.),	" ninth	a	30	a
October	{Oct.),	" tenth	it	31	a
November	{Nov),	" eleventh	a	30	a
December	{Dec),	« twelfth	u	31	t(

172

COMPOUND NUMBERS,

Note. — The following couplet will aid the leaner In remem- bering the months that have 30 days each : " Thirty days hath September, April, June, and November."

Each of the other months has 31 days, except February, which in common years has 28 days, but in leap years, 29.

51. Into how many seasons is the year divided?

Four, viz.: Spring, Summer, Autumn, and Winter.

52. Name the months of each season ? Spring consists of March, April, and May. Summer " June, July, and August. Autumn * September, October, and November. Winter " December, January, and February.

MISCELLANEOUS

12 things are 1 dozen. 12 doz. " 1 gross.

24 sheets are 1 quire of paper. 2c quires " 1 ream "

2 leaves are

TABLES.

1 2 gross are 1 great gross. 20 things " 1 score.

2 reams are 1 bundle. 5 bundles " 1 bale.

1 folio.

1 quarto, or 4to. 1 octavo, or 8vo. 1 duodecimo, or 12 mo. 1 eighteen mo. 1 twenty-four mo. Note. — The terms folio, quarto, octavo, etc., denote the nurrfim- of leaves into which a sheet of paper is folded in making books.

Aliquot Parts of a Dollar, or 100 Cents.

4 leaves

8 leaves

12 leaves

18 leaves

24 leaves

50 cents as $}. 33I cents ss %\. 25 cents ss $J. 20 cents = %\. i6f cents = %\.

12J cents == $|.

10 cents as %^. 8-J cents = %JV 6J cents = $tV 5 cents = $-2*ff.

REDUCTION

1. What is Reduction ?

Reduction is changing a number from one denom- ination to another, without altering its value. It is of two kinds, Ascending and Descending.

2. What is Reduction Descending?

Reduction Descending is changing higher de- nominations to lower j as, feet to inches, etc.

3. What is Reduction Ascending ?

Reduction Ascending is changing lower de- nominations to higher; as, inches to feet, etc.

To Reduce Higher Denominations to Lower.

i. How many farthings are there in £16, 5 s. 4d. 2 far. ?

Analysis. — Since there are 20s. in a Operation.

pound, there must be 20 times as many £16, 5s. 4d. 2 far. shillings as pounds, plus the given shil- 2o

lings. Now 20 times 16 are 320s., and

3203. + 5s.=325s. Again, since there are **3

I2d. in a shilling, there must be 12 times

as many pence as shillings, plus the given 3 9046L

pence. Now 12 times 325 are 3900, and 4

390od. + 4d.=3go4d. Finally, since there — ~z „ .

are 4 far. in a penny, there are 4 times as 5 Iar« A-ns.

many farthings as pence, plus the given

farthings. Now 4 times 3904 are 15616 far., and 15616 far. +

2 far.— 15618 far. Therefore, etc.

4. How reduce higher denominations to lower ?

Multiply the highest denomination hy the number re- quired of the next lower to make one of the higher, and to the product add the lower denomination.

Proceed in this manner with the successive denomina- tions, till the one required is reached.

.174 REDUCTION".

2. Reduce £5, 6s. 9^d. to farthings ? Suggestion.— £5, 6s. g£d. = £5, 6s. gd. 2 far. Arts. 5126 far.

3. Eeduce £9, is. 6Jd. to farthings ?

4. In 17s. 4d. 3 far., how many farthings ?

5. In £43, 4s. how many pence ?

6. In £115, how many farthings ?

To Reduce Lower Denominations to Higher.

7. Eeduce 156 18 farthings to pounds ?.

Analysis. — Since in 4 farthings there is Operation.

1 penny, in 15618 far. there are as many a\ 15618 far.

pence as 4 is contained times in 15618 ; and

15618 divided by 4=3904^ and 2 far. over. I2 ) 39°4d. 2 far„ Again, since in I2d. there is is., in 3go4d. 2Q\ -g ^

there are as many shillings as 12 is con- —1

tained times in 3904; and 3904-7-12=3255. £16, 5s.

and 4d. over. Finally, in 325s. there &re Am. £16, 5S.4d. 2f.

as many pounds as 20 is contained times in

325 ; and 325-f-20=£i6 and 5s. over. Therefore, etc.

5. How reduce lower denominations to higher ?

Divide the given denomination by the number required of this denomination to make a unit of the next higher.

Proceed in this manner with the successive denomina- tions, till the one required is reached. The last quotient, with the several remainders annexed, ivill be the answer.

Note. — The remainders, it should be observed, are the same denomination as the respective dividends from which they arise (P. 62, Q. 5, Rem.)

Pkoof. — Reduction Ascending and Descending prove each other ; for one is the reverse of the other.

8. Reduce 1231 pence to pounds? Ans. £5, 2s. 7d. Proof. — Reversing the operation we have 20 £5, 2S. 7<L

times 5 = ioos., and ioos. + 2s.=iG2s. Again, 20

12 times io2 = i224d., and I224d. + 7d.=i23id., j~g,

the same as the given number of pence. There- j 2

fore, the work is right. ,

1231&

REDUCTION. 175

9. In 1 46 1 pence, how many pounds, shillings, and pence ?

10. In 27035 farthings, how many pounds, etc. ?

11. What will 25 pen-knives cost, at 2s. 6d. apiece?

Solution — 2s. 6d.=3od. Now 30CL x 25 = 7500!., and 75od.= £3, 23. 6d.

1 2. What will 75 slates cost, at 1 1 pence each ?

13. How many pennyweights in 7 lb. 5 oz. troy ?

14. How many grains in 10 lb. 7 oz. 6 pwt. 9 gr. ?

15. Eeduce 156 1 pwt. to pounds and ounces ?

16. Eeduce 6575 grains to pounds, etc.

17. A goldsmith made 12 gold rings, each weigh- ing 3 pwt. 4 gr. : how many ounces of gold did he use?

18. A lady bought a gold chain weighing 2 oz. 12 pwt., at $1.50 a pennyweight: how much did she pay for her chain ?

19. Eeduce 2265 ounces Avoirdupois to hundreds.

20. In 15 T. 2 cwt. 31 lb. 8 oz., how many ounces ?

21. Eeduce 100 tons, 75 lb. 4 oz. to ounces.

22. What will 5 lbs. 4 oz. of candy come to, at 6 cts. an ounce ?

23. A farmer sold 2 tons, 375 lbs. of maple sugar, at 15 cts. a pound : how much did he receive for it ?

24. In 5 lb., apothecaries' weight, how many drams ?

25. In 7 lb. 4 oz., how many scruples ?

26. In 469 scruples, how many apothecaries' pounds?

27. In 1578 grains, how many apothecaries' ounces?

28. How many feet in 45 rods ?

Note. — For multiplying by 5^ or i6£, the number of yards or feet in a rod, see Note, p. 113.

29. How many feet in 12 miles, 10 rods, and 7 feet ?

30. How many yards in 26 miles, 3 fur. 2 yards ?

170 REDUCTION.

31. In 456 yards, how many rods ?

Remake. — To divide by 5 A- or i6£ (the Operation.

number of feet or yards in a rod) we 5^) 45 6 yd.

reduce both the divisor and dividend to 2 2

halves ; then divide in the usual way. —

Thus, 5i=n halves, and 456=912 halves; n ) 912

now 11 is contained in 912, 82 times and * " 7T j

10 remainder. But the dividend is half

yards ; therefore the 10 remainder is half yards, and is equal to

5 yards. (P. 120, Q. 45.)

32. In 1560 feet, how many rods ?

^2,. How many miles, rods, etc., in 11278 feet?

34. In 5 1. 17 m. 3 fur. 6 r. 10 ft., how many feet ?

35. When the fare is 5 cts. a mile, what will it cost you to ride 1 5 leagues ?

36. How many rods of fence are required on both sides of a road 2 miles long ?

37. Allowing a military step to be 2-J ft, how many steps will a soldier take in marching 5 miles ?

38. How many quarters of a yard in 45 inches ?

39. How many eighths of a yard in 27 inches ?

40. Keduee 151 yards to sixteenths.

41. In 951 eighths of a yard, how many yards?

42. If you pay 8 cts. for \ yard of muslin, how much would you have to pay for 20 yards ?

43. A lady paid $3 for J yard of lace; what would a piece of 35 yards come to at that rate ?

44. How many square feet in 160 sq. rods ?

45. How many sq. feet in 5 A. 61 sq. rods?

46. How many sq. yd. in 21 A. 36 sq. rods?

47. How many sq. rods in a sq. mile ?

48. In 85 1 sq. rods, how many acres ?

49. In 75625 sq. yards, how many acres?

50. In 46273 sq. inches, how many sq. yards?

REDUCTION. 177

51. What will be the cost of a village-lot containing 20 sq. rods of land, at 25 cts. per sq. foot ?

5 2 What will it cost to sod a park of 2 acres, at 1 2 cts. a sq. yard ?

53. How many cu. in. in 41 cu. yards, 16 cu. feet?

54. How many cu. yards in 96365 cu. inches ?

55. Eeduce 4250 cu. feet to cords ?

56. Eeduce 75^ cords to cu. feet ?

57- Eeduce 15 cords and 2^ cu. feet to cu. feet?

58. At 4 cts. a cu. foot, what will it cost to excavate a cellar containing 450 cu. yards ?

59. What will 12 cords of wood come to, at 5 cts. a cubic foot ?

60. What is the worth of 168 cord feet of wood, at $4 a cord ?

61. Eeduce 6 bu. 3 pk. 5 qt. to pints ?

62. How many bushels in 1647 quarts ?

6$. What cost 5 oushels of chestnuts, at 12 cts. a quart ?

64. A lad bought 2 bushels of apples for $2.50, and sold them at 40 cts. a half peck : what was his profit ?

65. How many quart boxes are required to hold 4 bu. 1 pk. of blackberries ?

66. At 6 cts. a quart, how many bushels of peanuts' can be bought for $6.42 ?

67. How many gills in 6 gal. 1 qt. 1 pt. 3 gills?

68. How many quarts in 3 hhd. 3 gal. 2 qt. ?

69. In 1832 gills, how many gallons ?

70. In 2560 quarts, how many hogsheads?

71. A milkman having a 15 gallon can full of milk, sold 15 quarts, and spilt the rest : how many quarts did he lose ?

72. What cost a hogshead of maple syrup, at 25 cents a quart ?

178 REDUCTION.

73. A druggist paid $126 for a cask of alcohol con- taining 42 gal., and sold it at 20 cts. a gill : how much did he make ?

74. How many seconds in 3 days, 5 hr. 17 minutes?

75. How many days in 4565 minutes ?

76. How many minutes in 7 weeks, 5 days ?

77. How many minutes in a common year?

78. How many common years in 48256 hours?

79. If a clock ticks seconds, how many times does it tick in a week ?

80. At $3.50 per day, what will it cost me to board 12 weeks?

81. If a man's pulse beats 73 times a minute, how many times will it beat in 3 1 days ?

82. If a steamer sails 11 miles an hour, how long will it take her to sail 3000 miles ?

83. Eeduce 45 ° 13' to seconds.

84. How many degrees in 10000"?

85. How many signs in 8275'?

86. The earth revolves 3600 on its axis in 24 hours : how many degrees does it revolve in 1 hour ? How far in 4 minutes ?

87. How many sheets of paper in 5 reams, 10 quires?

88. How many reams in 12258 sheets ?

89. If you pay $2.50 a ream for paper, what is that a sheet ?

90. How many crayons are there in 40 boxes, each containing a gross ?

91. What will 25 gross of lead pencils cost, at 42 cts. a dozen ?

92. Pens are packed in boxes containing a gross: how many pens are there in 6 boxes ?

93. A man having 100 dozen eggs, sent them to mar- ket in 16 boxes : how many eggs did he put in a box ?

RECTANGULAR SURFACES. 179

MEASUREMENT OF RECTANGULAR SURFACES.

6. What is a Rectangular Figure ?

A Rectangular Figure is one which has four sides and four right angles. (See next Fig.)

When all the sides are equal, it is called a square , when the opposite sides only are equal, it is called an oblong, or parallelogram.

7. What is the area of a figure ?

The Area of a Figure is the quantity of surface it contains. It is often called the superficial contents.

Note. — The term rectangular signifies right angled.

i. How many square feet of canvas in a rectangular painting 3 feet long and 2 feet wide ?

Illustration. — Let the painting be rep- resented by the figure in the margin ; its length being divided into three equal parts, and its breadth into two ; each denoting a linear foot. It will be seen that there are 2 rows of squares in the figure, and 3 3 feet'

squares in a row. Therefore, the painting must contain 2 times 3, or 6 square feet.

8. How find the area of a rectangular surface ? Multiply the length and breadth together.

2. How many square feet in a blackboard 8 ft. long and i>i ft. wide ?

3. How many square inches in a pane of glass 32 in. long and 24 in. wide ?

4. How many square rods in a garden 15 rods long and 6 rods wide ?

5. How many yards of carpeting 1 yard wide are required to cover a room 18 feet long and 15 feet wide ?

180

RECTANGULAR SOLIDS.

6. How many sq. feet in a board 16 ft. long and 1} ft wide ?

7. How many acres in a farm 100 rods long and 80 rods wide ?

8. How many acres in a township 6 miles square ?

9. A flower garden is 30 yards long and 18 yards wide- what are its contents ?

10. How many brick 8 in. long and 4 in. wide, will it iake to pave a sidewalk 40 ft. long and 5 ft. wide ?

11. What is the cost of a pine board 18 ft. long and 2 \ ft. wide, at 8 cts. a square foot ?

MEASUREMENT OF RECTANGULAR SOLIDS.

4 feet.

~^

P

9. What is a Rectangular Body ?

A Rectangular Body

is one bounded by six rectan- gular sides, each opposite pair being equal and parallel; as, boxes of goods, blocks of hewn stone, etc. When all the sides are equal, it is called a cube.

10. What are the Contents of a body?

The Contents or Solidity of a body is the quai* tity of matter or space it contains.

1. How many cu. feet are there in a box of books 4 ft. long, 3 ft. wide, and 2 ft. deep ?

Illustration. — Let the box be represented by the preceding figure ; its length being divided into four equal parts, its breadth into three, and its depth into tiw ; each part denoting a linear foot. In the upper surface of the box there are 3 time* 4, or 12

RECTANGULAR SOLIDS. 181

pq. feet. Now, if the box were i foot deep, it would contain i time as many cubic feet as there are square feet in its upper face, and i time 4 x 3=12 cu. ft. But the box is 2 fe^t deep ; therefore it must contain 2 times 4 x 3=24 cu. feet.

11. How find the contents of a rectangular body. Multiply the length, breadth, and thickness together.

2. How many cu. inches in a Lrick 8 in. long, 4 in. wide, and 2 in. thick ?

3. How many cubic feet in a box of sugar 5 ft. long,

3 ft. wide, and 3 ft. deep ?

4. Henry made a pile of cubic letter blocks, the length of which was 8 blocks, the width 6 blocks, and the height 5 blocks : how many blocks were in the pile ?

5. How many cubic feet in a pile of brick 13 ft. long, 7 ft. wide, and 5 ft. high ?

6. How many cubic feet in a load of wood 7 ft. long,

4 ft. wide, and 3 \ ffc. high ?

7. How many cu. feet in a bin 12 ffc. long, 6 ft. wide, and 5 \ ft. deep ?

8. How many cu. yards of earth must be removed to dig a cellar 36 ft. long, 20 ft. wide, and 6 ft. deep ?

9. How many cu. feet in a stick of timber 36 ft. long, 1 J ft. wide, and i| ft. thick ?

10. What will it cost to build a wall 120 ft. long, 1} ft. thick, and 9 ft. high, at 15 cts. a cubic foot?

11. What will it cost to dig a trench 100 ft. long, 9 ft wide, and 4^ ft. deep, at 27 cts. a cu. yard ?

12. What is the worth of a pile of wood 48 ft. long, 6 ft. high, and 4 ft. wide, at $4} a cord ?

13. A rectangular bin is 10 ft. long, 6 ft. wide, and 4 feet deep : what are its contents ?

14. A load of wood is 7-| feet long, 4 ft. wide, and 3 ft. high : what are its contents ?

182 DENOMINATE FEACTIONS.

REDUCTION OF DENOMINATE FRACTION&

12. What is a Denominate Fraction?

A Denominate Fraction is one or more of the equal parts into which a Compound or Denominate number may be divided.

13. How are they expressed ?

Denominate Fractions are expressed either as common fractions, or as decimals; as, \ pound, .8 yard.

To reduce Denominate Fractions to Units of Lower Denominations.

i. Reduce f gallon to quarts and pints.

Analysis. — Since there Operation.

are 4 qts. in a gal., there J g. x 4=f, or 1 qt. and f qt. rem. must be 4 times as many 3 qt. X 2 = 6 or I1 t)t.

quarts as gallons; and 4 ■ Am, \ 1 xi *t '

times £ gal. are § , equal x ■ r

to 1 qt. and £ qt. rem. Again, since there are 2 pt. in a quart, there must be 2 times f or f pt., equal to i£ pt. Therefore, eic.

11. How reduce denominate fractions to units of a lower denomination ?

1. Multiply the given numerator by the number required to reduce the fraction to the next lower denomination, and divide the product by the denominator. (P. 173, Q. 4.)

II. Multiply and divide the successive remainders in the same manner till the lowest denomination is reached. The several quotients will be the answer required.

2. Reduce | of a yard to feet and inches.

3. Reduce f of a pound sterling to shillings and pence.

4. Reduce fa of a week to days and hours.

5. What part of a pint is fa of a gallon ?

Solution. — This example is the same in principle as the pre- ceding. Reducing the numerator to the required denomination, place it over the givon denominator : /* gal. x 4 x 2— J$, or | pt.

DENOMINATE FRACTIONS. 183

6. What part of a quart is ^ of a bushel ?

7. What part of a pennyweight is ^-J^ pound Troy ?

8. Eeduce .6 yard to feet and inches.

Analysis. — Reducing .6 yard to feet, we have -6 jd.

.6 yd. x 3 = 1 ft. and .8 ft. over. Again, reducing _^

.8 ft. to inches, we have .8 ft. x 12=9.6 in. There- 1.8 ft. fore, .6 yard equals 1 ft. 9.6 in., which is the answer I2

required. Ans. 1 ft. 9.6 in. 9.6 in.

15. How reduce a denominate decimal to units of lower denominations ?

I. Multiply the denominate decimal by the number re- quired of the next lower denomination to make one of the given denomination, and point off the product as in multiplication of decimals.

II. Proceed in this manner with the decimal part of the successive products, as far as required. The integral part of the several products will he the answer.

Note. — The preceding operations in Denominate Fractions are the same in principle as those in Reduction Descending.

9. Reduce .84 gal. to quarts and pints.

10. Reduce .625 week to days, etc.

11. Reduce .875 bushel to pecks, etc.

To reduce a Compound Number from a Lower to a Denonv inate Fraction of a Higher Denomination.

12. What part of a gallon is 1 pint and 2 gills ? Analysis. — 1 pt. 2 gi. — 6 gills ; 1 pt. 2 gi. = 6 gi.

and 1 gallon=i x 4 x 2 x 4=32 gills. r gal. x 4 X 2 X 4 = 32 gi. But 6 gills are & of 32 gi, equal to , 6 3 <«

-h gal. Therefore, etc. MS' * °r T * gaL

16. How reduce a compound number to a denominate fraction of a higher denomination ?

I. Reduce the given number to its lowest denomination for the numerator.

II. Reduce to the same denomination, a unit of the required fraction, for the denominator.

i.5 pt.

184 DENOMINATE FRACTIONS.

Note. — If the lowest denomination of the given number con- tains a fraction, the number must be reduced to the parts indi- cated by the denominator of the fraction. (Ex. 16.)

13. Reduce 2 ft. 5 in. to the fraction of a yard.

14. Reduce 3 qt. 1 pt. to the fraction of a bushel.

15. What part of a pound sterling is 12s. 6d. ?

1 6. What part of a pound Troy is f pennyweight ? Solution — The lowest denomination is sths of a pwt. Now

1 lb. Troy = 1 x 12 x 20 x 5 = 1200 fifths pwt. Ana. t^ott* or Tou lb.

17. What part of a mile is j of a rod ?

18. Reduce f of a quart to the fraction of a bushel ?

19. What decimal part of a gallon is 3 qt. 1 pt. 2 gi.? Analysis. — Since 4 gi. are 1 pt., there Operation.

must be 1 fourth as many pints as gills, and 4 2 gi.

i of 2 gi.=f, or .5 pc. Place the .5 pt. on

the right of the given pints. Again, 2 pt.

are 1 qt. ; hence, there is 1 half as many

quarts as pints; and £ of 1.5 pt.=.75 qt.,

which we place on the right of the given Ans. O.g-IJK °"al.

quarts. Finally, 4 qt. are 1 gal. ; hence,

there is 1 fourth as many gallons as quarts; and J- of 3.75 qt.

=0.9375 gal. Therefore, etc.

17. How reduce a compound number to a denominate decimal of a higher denomination.

I. Write the numbers in a column, placing the lowest denomination at the top.

II. Beginning with the lowest, divide it by the number required of this denomination to make a unit of the nex-t higher, and annex the quotient to the next higher.

Proceed in this manner with the successive denomina- tions, till the one required is reached.

20. Reduce 3 fur. 20 rods to the decimal of a mile.

21. What decimal of a pound is 6s. 8d. ?

22. Reduce 5 gal. 3 qt to the decimal of a hogshead.

3-75 qt.

COMPOUND ADDITION

	Operation.
Gal.	qt.	pt.	gi.
5	3	I	3
8	2	I	I
4	3	I	3

i. What is the sum of 5 gal. 3 qt. 1 pt. 3 gi.; 8 gal. 2 qt, 1 pt. 1 gi.; 4 gal. 3 qt. 1 pt. 3 gi.?

Analysis. — Write the numbers so that the same denominations shall stand in the same column, and begin- ning at the right, add the columns sep- arately. Thus, 3 gi. and 1 gi. are 4 gills, and 3 are 7 gills, equal to 1 pt. and 3 gi. Set the 3 gi. under the column of gills, Ans. 19 2 o 3 and carrying the 1 pt. to the column of

pints, the sum is 4 pints, equal to 2 qts. and no remainder. Write a cipher under the pints, and carrying the 2 qts. to the column of quarts, the sum is 10 qts., equal to 2 gal. and 2 quarts. Set the 2 qts. under the quarts, and carrying the 2 gal. to the column of gallons, the sum is 19 gals. Hence, 19 gal. 2 qt. o pt. 3 gi. is the sum required.

1. How add Compound Numbers ?

I. Write the same denominations one under another, and beginning at the right, add each column separately.

II. If the sum of a column is less than a unit of the next higher denomination, write it under the column added.

If equal to one or more units of the next higher denomination, carry these units to that denomination, wd write the excess under the column as in Simple Ad- dition. (P. 28, Q. 13.)

Remake. — Addition, Subtraction, etc., of Compound Numbers are the same in principle as the corresponding operations in Simple Numbers. The only difference between them arises from their scales of increase. The orders of the latter increase by the constant scale of 10. The denominations of the former increase by a variable scale. In both we carry for the number which it

186 COMPOUND ADDITION.

takes of a lower order or denomination to make one in the next higher. In the former, this number is always 10 ; in the latter, it is variable.

(4 (3.) (4.)

£. s. d. far. Lb. oz. pwt. Yd. ft. in.

10 13 4 2 13 8 9 825

7 5 3 3 8 6 8 7 1 &

8352 583 527

(5-)			(6.)			(7.)
Bu. pk. qt.	pt.	T.	cwt.	lb.	M.	fur. r.
15 3 7	1	5	18	35	35	3 28
30 2 3	1	8	3	83	84	5 is
8 3 5	0	3	17	35	38	3 8
53 1 6	1	7	5	70	27	4 13

8. In one bin there are 35 bu. 3 pk. and 7 qts. of oats; in another, 27 bu. 2 pk. 5 qts.; and in another, 28 bu. 3 pk: how many bushels are there in all?

9. A merchant sold 3 pieces of muslin : one contain- ing 35 yds. and 1 fourth ; another, 43 yds. and 5 eighths ; and the other, 38 yds. and 3 eighths: how many yards did he sell ?

10. In one garden there are 13 sq. r. 5 sq. yd. 3 sq.ft.; in another, 18 sq. r. 8 sq. yd. 5 sq. ft.; and in another, 23 sq. r. 5 sq. yd. and 8 sq. ft: how much land in all?

1 1. A farmer has 4 fields : one containing 18 A. 35 sq. rods; another 30 A. 78 sq. r. ; another 45 A. 30 sq. r., the other, 23 A. 65 sq. r. : how many acres has he?

12. How much wood is there in 3 loads, one of which contains 1 0. 45 cu. ft.; another, 1 C. and 58 cu. ft; and the other 1 0. 85 cu. ft. ?

COMPOUND SUBTRACTION.

i. From 27 yd. i ft. 8 in.," take 18 yd. 2 ft. 5 in.

Analysis. — Write the less number under Operation.

the greater, placing the same denominations Yd. ft. in.

in the same column. Beginning at the right, 27 I 8

we proceed thus : 5 in. from 8 in. leave 3 in. ; 182 S

set the 3 under the column of inches. Next,

since 2 ft. cannot be taken from 1 ft., we bor- Ans. 823 row a unit of the next higher denomination, . which is yards. Now 1 yd. or 3 ft. added to 1 ft. make 4 ft., and 2 ft. from 4 ft. leave 2 ft. Finally, 1 to carry to 18 makes 19, and 19 yds. from 27 yds. leave 8 yds. Hence, the difference is 8 yds. 2 ft. 3 in.

2. How subtract Compound Numbers ?

I. Write the several denominations of the subtrahend under those of the same name in the minuend.

II. Beginning at the right, subtract each denomination of the subtrahend from that above it, and set the remain- der under the term subtracted.

III. If the number in any denomination of the subtra- hend is larger than that above it, add to the upper num- ber as many as are required to make a unit of the next higher ; then subtract and carry 1 to the next denomina- tion in the subtrahend, as in Simple Subtraction.

(*•) (3-)

From £25, 7s. 6d. 2 far. 13 lb. 7 oz. 18 pwt. 23 gr. Take $23, 5s. 3d. 3 far. 7 lb. 8 oz. 13 pwt. 18 gr.

4. From 2 bu. take 3 pk. 5 qt.

5. From 8 m. 130 r. take 250 r. 3 yd. 2 ft.

6. From a hogshead of molasses 35 gal. 3 qt. were drawn : how many gallons were left ?

188 COMPOUND SUBTRACTION.

7. If one farm contains 165 A. 118 sq. r., and another 100 A. 135 sq. r., what is the difference between them?

8. What is the difference between two loads of wood, one of which contains 1 0. 38 cu. ft., the other 125 cu. ft.?

9. What is the difference in the weight of two stacks of hay, one of which contains 5 T. 135 lb., the other 7 T. 387 lb.?

10. The longitude of New York is 740 o' 3* W. ; that of Chicago, 870 35' W.: what is the difference in their longitude ?

11. The latitude of New Orleans is 290 57' 30" N«; that of Montreal 45° 31' K: what is the difference in their latitude ?

12. What is the difference of time between Dec. 25th, 1865, and April 20th, 1872 ?

Solution. — Place the earlier date under the later, the years on the left, the months next, and the days on the right, and proceed as in subtracting other Compound Numbers.

Remark. — In finding the difference between two dates, and in most business transactions, 30 days are considered a month, and 12 months a year.

13. If a man was born Jan. 1st, 1850, how old will he be July 4th, 1876 ?

14. A note dated March 13th, 1870, was paid Feb. 25th, 1872 : how long did it run ?

15. Charles was born July 30th, 1865, and his brother Oct. 24-th, 1869 : what is the difference in their ages ?

16. A whale-ship started on a voyage Aug. 25th, 1867, and returned July 18th, 187 1, how long was she gone?

	Y.	m.	d.
	1872	4	20
	1865	12	25
Ans	. 6	3	25

COMPOUND MULTIPLICATION.

i. If a family uses 2 lbs. 12 oz. of butter a day, how much will they use in 3 days ?

Analysis. — They will use 3 times 2 lb. and 12 oz. Now 3 times 2 lb. are 6 lb., and 3 times 12 oz. are 36 oz., equal to 2 lb. 4 oz. ; which, added to 6 lb., make 8 lb. 4 oz. Therefore, etc.

2. If it takes 4 gal. 3 qt. of water to fill a demijohn, how much will it take to fill 2 of the same size ?

3. A farmer gave a bag of corn, containing 2 bu. 3 pk.. to each of 4 beggars : how much did he give to all ?

4. If it takes 3 yd. 1 qr. of cloth to make a boy's suit, how many yards will it take to make 5 suits ?

5. How long will it take a man to chop 3 cords of wood, if he chops at the rate of a cord in 4 hr. 30 min. ?

6. If you pick 2 qt. 1 pt. of blackberries an hour, how many can you pick in 6 hours ?

7. If 1 book costs 2 shillings and 6 pence, what will 5 books cost ?

SLATE EXERCISES.

1. A miller ground 5 grists, each containing 2 bush- els, 3 pecks, 5 quarts of wheat : how much wheat did he grind?

ANALYSIS. — 5 grists contain 5 times as much Operation as 1 grist. Now 5 times 5 qt. are 25 qt., equal ' Bu. pk. qt. to 3 pk. and 1 qt. Set the 1 qt. under the 23c quarts, and carry the 3 pk. to the product of

pecks. Next, 5 times 3 pk. are 15 pk., and 3 are 1

18 pk., equal to 4 bu. and 2 pk. Set the 2 under 14 2 1 the pecks, and carrying the 4 bu. to the product of bu. we have 5 times 2 are 10 bu., and 4 are 14 bu. Therefore, he ground 14 bu. 2 pk. 1 qt.

190 COMPOUND MULTIPLICATION.

3. How multiply Compound Numbers?

I. Write the multiplier under the lowest denomination of the multiplicand, and beginning at the right, multiply each term in succession.

II. If the product of any term is less than a unit of the next higher denomination, set it under the term mul- tiplied.

III. If equal to one or more units of the next higher denomination, carry these units to that denomination, and write the excess under the term multiplied.

		(2.)		(30
	B.	yd. ft.	M.	fur. r.	yd.
Mult.	i3	2 2	30	3 18	4
By		5			8
Ans.	67	2 I	243	3 29	4}

4. Bought 5 casks of vinegar, each containing 36 gal. 3 qt. 1 pt. : how much did they all contain ?

5. Sold 6 pieces of cloth, each containing 42 yards and 3 quarters : how much did all contain ?

6. A farmer has 4 pastures, of 15 A. 63 sq. r. each : how much land in all ?

7. A man bought 10 loads of wood, each containing 1 C. 35 cu. ft. : how much wood did lie buy?

8. If you read 5 h. 35 min. per day, how many hours will you read in 1 2 days ?

9. Bought 7 loads of hay, averaging 1 T. 375 lbs.: how much did all contain ?

10. If the price of one cow is £8, 15s. 6-Jd., what will 8 cows cost, at the same rate ?

11. If you have 11 apple-trees, and they yield 7 bu. 3 p"k. apiece, how many apples will you have ?

COMPOUND DIVISION.

i. If 48 lb. 12 oz. of rice are divided equally among

15 persons, what part, and how much, will each receive ?

i

! Analysis. — 1 is fV of 15 ; therefore, each per- Operation.

son will receive 1 fifteenth part. lb. oz.

Again, 1 fifteenth of 48 lb. is 3 lb., and 3 15)48 12

remainder. Reducing the remainder 3 lb. to .

ounces, they become 48 oz., and adding the 12 *

oz. we have 60 oz. Now 1 fifteenth of 00 oz. is 4 oz. Therefore, each received -fa part, which is 3 lb. 4 oz.

Note. — 1. The object in this example is to divide a compound number into equal parts, in order to find the value of one part.

2. A farmer sent 29 bu. 1 pk. of wheat to mill, in bags of 3 bu. 1 pk. each : how many bags did he use ?

Analysis. — Reducing the whole 29 bu. 1 pk.= H7 pk.

quantity to pecks, it becomes 117 pk. , l,u j r)k.— j -y pk.

The quantity in each bag, 3 bu. 1 pk., \tf*

equal 13 pk. We now divide as in i_„JL

simple numbers. AllS, 9 bags.

Note. — 2. The object of this example is to find how many times one compound number is contained in another.

4. How divide Compound Numbers?

I. When the divisor is an abstract number, Beginning at the left, divide each denomination in suc- cession, and set the quotient under the term divided.

If there is a remainder, reduce it to the n*x{ lower denomination, and, adding it to the given up-tts of that denomination, divide as before.

II. When the divisor is a compound number, Reduce the divisor and dividend to the lowest denom- ination contained in either, and divide as in simple numbers.

192 COMPOUND DIVISION.

Eemakk. — It will be observed from the preceding examples, that the object of Compound as well as Simple Division is twofold :

ist, To divide a compound number into equal parts, the divisor being abstract. In this case the quotient is the same denomination as the dividend.

2d, To find now many times one compound number is contained in another. In this case the quotient is times, or an abstract num- ber. (P. 63, Q. 10.)

Perform the following divisions :

(3-) (5-)

5 ) 16 A. 75 sq. r. 35 sq. ft. 2 lb. 4 oz. ) 17 lb. 6 oz.

(4.) (6.)

6)505 cu. ft. 154 cu. in. £2, i2S.)£23, 8s.

7. If I sell 46 bu. 1 pk. of plums in equal quantities to 7 market-men, how many will each receive ?

8. Charles having a kite-line 72 ft. 4 in. long, cut it into 7 equal parts : what was the length of each part ?

9. If a man travels 48 m. 3 fur. in 9 hours, how far will he go in 1 hour ?

10. A goldsmith made 5 lb. 3 oz. of silver into 24 spoons : what was the weight of each ?

11. How many iron rails 18 ft. long are required to lay both sides of a track 7 m. 160 r. in length ?

12. A man gathered 57 bu. 3 pk. of oranges from 9 trees : what was the average yield ?

13. If 8 men mow 17 A. 32 sq. r. in a day, how much can 1 man mow ?

14. How many times does a car-wheel 16 ft. 6 in. in circumference turn around in going 2 miles ?

15. How many bags, holding 2 bu. 3 pk. each, can be filled from a bin which contains 19 bu. 1 pk. of corn ?

16. How many bundles of hay, each weighing 465 pounds, can be made from a scaffold which contains 5 tons, 125 pounds?

PERCENTAGE

1. What is meant by per cent ? Per cent denotes hundredths.

Note. — The term is from the Latin per and centum, by or in a hundred.

2. What is the rate ?

The Mate is the number which shows how many hundredths are taken. Thus i per cent of a number is i hundredth part of that number ; 2 per cent, 2 hun- dredths ; 3 per cent, 3 hundredths, &c.

3. With what do the terms rate per cent, correspond ?

The terms Mate per cent correspond with the terms of a fraction, the denominator of which is always 100, and the numerator the given rate.

4. How then may per cent be expressed ?

By a Common or a Decimal Fraction.

5. How is per cent expressed by decimals ?

Write the figures denoting the per cent in the first two places on the right of the decimal point, and the parts of 1 per cent in the succeeding places toward the right.

1 per cent is	written	.OI	\ per cent is written	.005
6 per cent	a	.06	\ per cent "	.0025
10 per cent	a	.IO	2\ per cent "	.025
100 per cent	tt	1.00	6 J per cent *	.0625
106 per cent	<(	1.06	33t Per cent "	•33l
125 per cent	a	1.25	I07i Per cen^ "	i-°75

€-. How many figures are required to express per cent? Per cent denotes himdredths ; therefore every per cent requires at least two figures, to express it decimally.

194 PEKCENTAGE.

7. If the given per cent is less than 10, what must be done ? A cipher must be prefixed to the figure denoting it. Thus, i per cent is written .01 ; 3 per cent. .03, etc.

Note. — When a given part of 1 per cent cannot he exactly expressed by one or two decimal figures, it is written as a common fraction, and annexed to the figures expressing hundredths, or the per cent. Thus, &% is written .04^, instead of .043333 + .

§. To what is 100 per cent of a number equal ?

A hundred per cent of a number is equal to the number itself; for $%% is equal to 1.

9. When the rate is 100 per cent or over, how is it expressed ?

By a mixed number, or by an improper fraction. Thus 125% is written 1.25, or -ffj.

10. What is the sign of per cent ?

The Sign of per cent is an oblique line between two ciphers (%). Thus, 2% is read 2 per cent, etc. "Write the following per cents decimally :

1. 4%; 7r/c; 10%; 45%; 103%; 110%; 205%.

2. 2|%; 6-V/o; 7-|%; i8f%; io6J%; iii{£

1 1 . How read a given per cent expressed decimally ?

Bead the first Uvo decimal figures as per cent, and those on the right as decimal parts of 1 per cent. Copy and read the following as rates per cent :

1. .03; .06; .045; .11J; .625; 1.25; 1.50; 2.00.

2. 1.06; 1.07; 1.08; 1.10^ ; 1.62-I; 1.005; 2.00.

12. What are the elements or parts in calculating percentage ?

The base, the rate per cent, the percentage, and the amount.

13. Explain each.

The Base is the number on which the percentage is calculated.

The Hate per cent is the number which shows how many hundredths of the base are to be taken.

PERCENTAGE. 195

The Percentage is the number obtained by taking that portion of the base indicated by the rate per cent.

The Amount is the base, increased or diminished by the percentage.

Note. — The conditions of the question show whether the percentage is to be added to, or subtracted from the base to form the amount.

CASE I To find the Percentage, the Base and Rate being given.

MENTAL EXERCISES. i. How many are f of 40 ? (P. 90.) Analysis. — 1 fifth of 40 is 8, and 3 fifths are 3 times 8, or 24.

2. How many are 40 multiplied by | ? Ana. 24.

3. To how many hundredths is f equal ?

Analysis. — 1 = \%% ; hence \ equals \ of |g&, or -fife ; and 3 fifths = 3 times -£&, or -&%-. (P. 95, Q. 20.)

4. What is T6o°{j- of 40 ? Ans. 24. (P. 90.)

5. To what per cent is ■££$ equal ? Ans. 60 per cent.

6. What is 60 per cent of 40 ?

Analysis. — 60% is 60 times 1 % . Now, 1 % of 40 is -fa of 40* or YVo, and 60 times -,*„% = *&&t or 24> Ans.

7. To what per cent is -j-jfo equal ? yj^? A0t> ? r&?

8. Eeduce J to hundredths, f to hundredths. ^ to hundredths.

9. To what per cent of a number is -J- of it equal ? Analysis. — £ equals -ftftj ; therefore, £ of a number is 20%.

(P. 95, Q- 20.)

10. To what per cent is i equal ? J ? -^ ? ^ ?

11. What is 5 per cent of 200 yards?

Analysis. — $% is the same as juu- Now, ffa of 200 yards is 2 yards, and 5 hundredths are 5 times 2, or 10 yards. Therefore, $% of 200 yards is 10 yards.

12. What is 7 % of $300 ? 8% of 500 barrels ?

196 PERCENTAGE.

13. Which is the greatest, $ of 200; or 200 x •$; or «oo x .60; or 60 per cent of 200 ?

SLATE EXERCISES.

Remakk. — From the preceding illustrations, it will be seen that finding a fractional part of a number, multiplying it by a wmmon or a decimal fraction, and finding a per cent of it, are identical in principle. With the first two the learner is supposed to be familiar ; if not, he should carefully review them before going further. (P. 90, 113, 139.)

1. What is4# of $315 ?

Analysis.— 4 per cent is the same as rfo* operation. and t^u expressed decimally is .04 ; therefore, * g

4 per cent of $315 is .04 times $315. Multiply- -. '

ing the base by the rate, expressed decimally, *

we have $315 x .04= $12.60, the percentage re- p

quired.

14. How find the Percentage, when the base and rate are given? Etjle. — Multiply the base by the rate, expressed

decimally.

Notes. — 1. When the rate is an even part of 100, the per- centage may be found by taking a like part of the base. Thus, for 20%, take £ ; for 25^, take \, etc. (Ex. 3.)

2. The amount is found by adding the percentage to, or sub- tracting it from the base, as the case may be. (Ex. 10.)

2. What is 6% of $415.50? Ans. $24.93.

3. What is 25 % of 460 pounds ? Solution. — 460 pounds x \ = 115 pounds, Ans.

4. s?o of 640 yards. 7. 8% of 1000 rods.

5. 6% of $765.60. 8. 12 # of 1 1 10 barrels.

6. yfo of 600 bushels. 9. 20% of 2040 men.

10. A farmer having 163 acres of land, bought 12 fc more, how many acres did he then own ?

Solution.— 163 A. x .12 = 19.5C A. bought; 163 A. + 19.56 A. = 182.56 A. owned.

PERCENTAGE. 197

n. A man having 560 sheep, lost 2\% of them by sickness : how many did he have left ?

Ans.— 5<5o s. x .02^ = 14 s. lost ; 560 s. — 14 s. = 546 s. left.

12. A teacher's salary is 12% more this year than last; it was then $1500 : what is it now ?

13. From a school of 750 pupils, 20 per cent were absent : how many were present ?

14. What is 62} % of $25000 ?

CASE II. To find the Hate, the Base and Percentage being given. 1. What part of 4 is 3 ?

Analysis. — 1 is \ of 4, and 3 must be 3 times 1 fourth, or 3 fourths of 4.

?. What part of 5 is 3 ? What part is 4 ?

3. To how many hundredths is f equal ?

4. What part of 100 is 4 ? What part is 5 ? 7 ? 9 ?

5. What per cent of a number is y^ ? T£g- ? -f^ ?

6. What per cent of $5 are $2 ?

Analysis.— $2 are } of $5 ; and £ equals ^0% ; therefore, $2 are 40% of $5. (P. 95, Q. 20.)

7. What per cent of 10 yards are 3 yards? 5 yds.?

',yds.?

8. What part of a number is 20%, expressed in the lowest terms of a common fraction ?

Analysis.— 20 per cent = -ftfr ; and -flfo =a \ Ans. (P. 95.)

9. What part of a number is 25%, expressed by deci- mals?

10. What part of a number is 5 per cent? 10 per cent ? 40 per cent ?

198 PERCENTAGE.

SLATE EXERCISES.

Remark. — From the preceding illustrations, it will be seen that finding the rate, when the base and percentage are given, is the same in principle as finding what part one number is of another ; then changing the common fraction to hundredths.

i. A man paid $28 for a cow, and sold it so as to make $7 : what per cent did he make ?

Analysis. — In this example the base $28, operation. and the percentage $7, are given, to find the 28 ) 7.00

rate per cent. Now, $7 are -^8- of $28 ; and

■fa = 7 -*- 28 = .25 or 25%. Ans. 25%

Or fa = i = flfo or 25 % . (P. 95, Q- 20.)

15. How find the Rate, when the Base and Percentage are given?

Divide the percentage by the base ; the first tivo deci- mal figures will be hundredths, or the rate per cent j the others, parts of one per cent

Note. — The number denoting the base is always preceded by the word of, which distinguishes it from the percentage.

2. A lady having $40, spent $7 for a collar : what per cent of her money did she spend ? Ans. 174%.

3. What % of 18 is 6 ? 8. What % of 135 is 4-5 ?

4. What % of £1 are £3 ? 9. What % of 150 is 30?

5. What % of $35 are $7 ? 10. What % of J is J ?

6. What % of 54 is 6 ? 11. What % of .8 is .2 ?

7. What% of 150 is 75? 12. What % of f is A?

13. A man bought a farm of 200 acres, and sold 50 acres of it : what per cent of his farm did he sell ?

14. A farmer having 500 sheep, sold 125 of them: what per cent of his flock did he sell ?

15. If a man earns $450 a year, and lays up $225 of it, what per cent of his earnings does he save ?

16. From a school of 750 pupils, 250 were absent: what per cent were absent?

COMMISSION. 199

APPLICATIONS OF PERCENTAGE.

J. To what classes of problems is percentage applied ?

First, To those in which time is one of the elements uf calculation ; as Interest, etc.

Second, To those which are independent of time ; a? Commission, Profit and Loss, etc.

COMMISSION.

2. What is Commission, and how computed ?

Commission is an allowance made to Agents, Collec- tors, etc., for the transaction of business, and is com- puted like percentage.

Notes. — i. An Agent is one who transacts business for an- other, and is often called a Commission Merchant.

2. A Collector is one who collects debts, taxes, duties, etc.

3. Goods sent to an agent to sell, are called a consignment ; the person to whom they are sent, the Consignee ; and the person sending them, the Consignor.

3. What answers to the base, the rate, etc. ? The amount of sales, etc., is the base. The per cent for services, the rate. The commission, the percentage. The amount of sales, etc., plus or minus the commis- sion, the amount.

To find the Commission, the Amount of Sales and the Rate being given.

1. A merchant sold goods to the amount of $250, at 3 % commission : what was his commission ?

Analysis. — In this example the amount of Operation. sales $250, is the base, and 3% the rate. The $250 B.

question then is, what is 3% of $250. Now, #0, J^

3% equals .03 expressed decimally ; therefore,

3% of $250 is .03 times $250, and $250 x .03 Ans. $7.50 C. 3= $7 50, the commission required.

200 COMMISSION.

2. A broker sold 3 shares of bank stock for $300: what was his commission at -J per cent ?

Solution. — ^% — .005, and $300 x .005 = $1.50, Ans.

4. How find the Commission, when the amount of sales and rate are ^iven ?

Multiply the amount of sales by the rate, expressed decimally. (P. 196, Q. 14.)

Remakes. — 1. The net proceeds of a business transaction, are the gross amount Of sales, etc., minus the commission and other charges.

2. When the amount of sales, etc., and the commission are known, the net proceeds are found by subtracting the commission from the amount of sales. (Ex. 5.) Conversely,

3. When the net proceeds and commission are known, the amount of sales is found by adding the commission to the net proceeds. (Ex. 6.)

3. A man collected a school tax of $1250, at 5^ com- mission : how much did he receive ?

4. If you sell a consignment of goods for $1175.60, what will be your commission at 4% ?

5. My agent sold a quantity of flour for $1585, and charged me 4$ : what was Irs commission, and what the net proceeds ?

Analysis. — $1585 x .04 = $6340 com. ; $1585 — $63.40 = $1521.60, net proceeds.

6. Received $115.20 for selling a consignment of goods, and sent the consignor $2444.80 : what was the amount of sales, and what % was my commission ?

Analysis.— $2444.804- $115.20 — $2560 sulos ; ti^.co -*- 2560 ■ ,o$\% commission.

7. My agent bought 28 shares of !N". Y. Central R. R. for $2800, and charged me ij per cerA ivouuivaioii: wfrat was his commission ?

PROFIT AND LOSS, £01

PROFIT AND LOSS.

5. "What are Profit and Loss, and how computed ?

Profit and Loss are sums gained or lost in busi- ness transactions, and are computed like Percentage.

6. What answers to the base, the rate, etc. ? The Cost or sum invested is the Base ; The Per cent profit or loss, the Rate ; The Profit or Loss, the Percentage ;

The Selling Price, that is, the cost plus or minus the profit or loss, the Amount.

To find the Profit or Loss, the Cost and the Rate per cent Profit or Loss being given.

i. Bought a horse for $150, and sold it for 12 per cent profit : how much did I gain by the transaction ?

Analysis. — In this case the cost $150 is the $150 B.

base, and 12% the rate. Now, 12% is the 12 R.

same as .12, and .12 times $150 ss $150 x .12 = .

$18.00. Therefore I gained $18. Am. $18.00 €L

2. Bought a carriage for $250, and sold it at a loss of 8?c : how much did I lose ?

Analysis. — Here the cost $250 is the base, $2^0 B.

and 8% the rate. Now, 8% is the same as 08 E

.08, and .08 times $250 = $250 x .08 = $20.00. .

Therefore, my loss was $20. Ans. $20.00 L.

7. How find the Profit or Loss, when the cost and rate are given ?

Multiply the cost ly the rate expressed decimally, as in 'percentage. (P. 196, Q. 14.)

Remarks.— 1. When the 'per cent is an even part of 100 it is generally shorter, and therefore preferable to nse the fraction.

2. The selling price is found by adding the profit to or subtracts ing the loss from the cost, as the case may be. (Ex. 9.)

202 PROFIT AND LOSS.

3. A man bought a cow for $35, and sold it at 20% profit : how much did he gain ?

4. Henry bought a pair of skates for $3.75, and sold them for 20% more than he gave : what was his gain ?

5. A grocer bought flour at $8.50 a barrel, and sold it at 25 % loss : what did he lose on a barrel ?

6. A merchant bought a piece of silk for $585, and, sold it for 18% advance : what was his gain ?

7. Bought a house lot for $230, and sold it at 8% less than cost : what was my loss ?

8. A man paid $260 for a buggy, and sold it at 25% profit : how much did he make by the operation ?

9. If a man pays $150 for a watch, for what must he sell it to gain 12%?

Analysis. — To gain 12% he must sell it for the cost, plus 11%. Now, 12% of $150 = $150 x .12 == $18.00; and $150 + $18 = $168, Ana.

10. A grocer paid $200 for a lot of peaches, and finding them damaged, sold them at a loss of 25% : for what did he sell them ?

Analysis. — To lose 25% he must sell them for the cost, minus 25%. Now, 25% of $200 is $50; and $200 — $50 = $150, the selling price.

11. Bought a case of 12 hats at $6.50 apiece: for what must I sell the whole to make 20% ?

12. A dealer bought 50 buffalo robes, at $8 apiece, and sold them at 16% loss : what did he get for them ?

13. A merchant bought 240 barrels of flour, at $7.50 a barrel, and sold it at a profit of 1 2 J % : what did the flour come to ?

14. Bought a farm for $4200,. and sold it for 23$ % more than cost : for how much was it sold ?

PROFIT AND LOSS. 203

To find the per cent Profit or Loss, the cost and the amount of Profit or Loss being given.

i. I bought a sleigh for $50, and sold it for $20 more than the cost : what per cent was the profit ?

Analysis. — The gain $20 is H of $50, the cost ; now § % = tW or 40 per cent. (P. 198, Q. 15.) Therefore the profit was 40 per cent.

8. How find the rate per cent, the cost and the amount of profit or loss being given ?

Divide the amount of profit or loss by the cost, as in percentage. (P. 198, Q. 15.)

Remark. — When the cost and selling price are given, the Profit is found by subtracting the cost from the selling price. The Loss is found by subtracting the selling price from the cost.

2. Bought a horse for $200, and sold it for $50 less than cost : what per cent was the loss ?

3. A fruit dealer bought oranges at 4 cents, and sold them so as to make 2 cents on each : what per cent was his profit ?

4. If you buy a slate for 10 cents, and sell it for 5 cents more than cost : what per cent is your gain ?

5. Paid $35 for a cow, and sold her for $10 less than cost : what per cent was the loss ?

6. If a man buys flour at $7.50 a barrel, and sells it for $8.50, what per cent is his profit ?

7. If you buy tea at 60 cents a pound, and sell it at 80 cents, what per cent is your profit ?

8. If a grocer buys sugar at 8 cents a pound, and sells it at 6 cents, what per cent is his loss ?

9. A fruit dealer bought bananas at $3.50 a hundred, and gpld them at $5 a hundred : what per cent did he male?

INTEREST.

1. What is interest?

Interest is a compensation for the use of money.

, 2. What are the elements or parts to be considered ?

The Principal, the Rate, the Interest, the Time, and the Amount. 3. Explain each.

The Principal is the money lent. The Mate is the per centner annum. The Interest is the percentage. The Time is the period for which the principal draws interest. The Amount is the sum of the principal and interest.

Note. — The term per annum, from the Latin per and annus, signifies by the year.

To find the interest of $1, at 6 pep cent for months.

Remark. — The learner should observe that Interest differs from the preceding applications of Percentage by introducing time as an element in connection with the rate per cent. The terms rate and rate per cent always mean a certain number of hundredths yearly, and pro rata for longer or shorter periods.

i. If I charge 6% yearly for the use of $i, how much 'shall I receive ?

Analysis. — 6 per cent is Tott, and $i is ioo cents. Now, Tfj j i>f ioo cents is 6 cents. Therefore, I shall receive 6 cents.

2. If the interest of $i for i year is 6 cents, how much is it for i month ? For 2 months ? For 3 months ?

Analysis.— Since the interest of $1 at 6 per cent for 12 months (1 year) is 6 cents, for 1 month it is 1 twelfth of 6 cents, and f§ of 6 cents is -ft or \ cent. Again, since the interest of $1 at 6% Is \ cent for 1 month ; for 2 months it is 2 halves or 1 cent ; for 3 months, 3 halves or li cent ; for 6 mos., 6 halves or 3 cents, etc.

INTEREST. 205

4. What then is the interest of $i at 6 per cent, for any number of months ?

The interest of SI, at 6 per cent, for any number of months, is half as many cents as months.

4. What is the interest of $1, at 6% for 5 months ?

5. What is the interest of $1, at 6fo for 7 months?

To find the interest of $1, at 6% for days.

6. What is the interest of Si, at 6% for 1 d. ? For 2 d. ? 3d.? 4 d. ? 5 d. ? 6 d. ? etc.

Analysis. — Since the interest of $1 for 30 days (1 month), is \ cent, or 5 mills, for 1 day it is 1 thirtieth of 5 mills, and -3Vr of 5 is -nr, or £ mill. Again, since the interest of $1 for 1 day is £ mill, for 2 days, it is § mill ; for 3 days, % ; for 6 days, f , or 1 mill.

5. What then is the interest of $i,at 6 per cent for any num- ber of days ?

The interest of SI, at 6 per cent, for any number of days, is 1 sixth as many mills as days.

7. What is the interest of $1, at 6% for 19 days?

8. What is the interest of $1, at 6 ft for 15 days ? For 23 d.? 25 d. ?

To find the interest of $f, at 6 per cent for months and day9.

9. What is the interest of $1, at 6fo for 4 m. 21 d. ? Analysis. — The interest of $1 for 4 m. is £ of 4 or 2 cents ;

and the interest of $1 for 21 d. is £ of 21 or ?>\ mills, which, added to 2 cts., make 2 cts. + 3^ mills, or $.0235.

6. How find the interest of $1, at 6fc for months and days. Take half the number of months for cents, and one

sixth the number of days for mills. TJie sum will be the interest.

Note. — In finding 1-sixth of the days, it is commonly sufficient to carry the quotient to tenths or hundredths of a mill.

10. What is the interest of $1, at 6% for 6 m. 24 d. ?

11. What is the interest of $1, at 6% for 8 m. 27 d. ?

206 INTEREST.

General Method of Computing Interest, the Principal, the Time, and the Rate being given.

12. What is the interest of $115.20 for 1 y.im. and 12 d.,at6#?

Analysis. — 1 year equals 12 m., Operation.

and 12 m. + 1 m. - - =13111. $1 15.20

The int. of $1 for 13 months - = $.065 0fij

" " 12 days - . - ss .002

The int. of $1 for iy.im. and 12 d. = $.067 $80640

Since the int. of $1 for the given time is 69120

$.067, or .067 times the principal, the int. of a o $115.20 must be .067 times that sum, and '*' $115.20 x .067 = $7.7184, Afl8.

13. What is the interest of $150 for 10 months, at 7 per cent ?

Analysis. — At 6% the int. of $150 for the time is $150 x .05 =$7.50. But 7% is 1%, or £ more than b% : and $ of $7.50 is $1.25. Now. $7.50 + $1.25 = $8.75, the interest at 7 per cent.

Ans. $8.75

9*. How find the interest on a given principal, for any given time and rate ?

I. When the rate is 6 per cent,

Multiply the principal by the interest of $1, at 6 per wit for the time, expressed decimally.

II. When the rate is greater or less than 6 per cent, Add to or subtract from the interest at 6 yer cent such

a part of itself as the given rate exceeds or falls short of 6 per cent.

§. How find the amount ?

Add the interest to the principal.

Notes. — 1. In finding the time, first determine the number of entire calendar months ; then the number of days left.

1KTEKEST. 207

2. In computing interest, if the mills are 5 or more, it is customary to add 1 to the cents ; if less than 5, they are dis- regarded.

Only three decimal figures are retained in the following answers :

14. Find the amount of $75.60 for 1 y. 3 m. 9 d., at

6%.

Solution. — Int. of.$i for the given time and rate, is $.0765. Now, $75.60 x .0765 = $5.7834, the int. The prin. $.75.60 -V $57834 == $8l-383> Amt.

15. Find the int. of $45.50 for 1 y. 7 m. 15 d., at y%.

16. Find the amt. of $58.75 for 1 y. 10 m. 21 d., at 6#>*

17. The int. of $85 for 8 m. 9 d., at 5%.

18. The int. of $113 for 7 m. 18 d., at 6%.

19. The int. of $150 for 1 y. 3 m., at 6%.

20. The int. of $265 for 1 y. 7 m., at 6%.

21. The amt. of $500 for 2 y., at 8%.

22. The amt. of $763.25 for 1 y. 9 m. 27 d., at 5$.

23. The int. of $1500 for 3 years, at 2>%.

24. The int. of $2678 for 1 y. 7 m. 19 d., at 7%.

25. The int. of $2750 for ^ days, at 6%.

26. The int. of $3700 for 6^ days, at 7%.

27. The int. of $2500.73 for 93 days, at 5%.

28. What is the int. on a note of $500, from March 10th, 1872, to July 25th, 1872, at 6% ?

Analysis. — The time from March 10th to July 10th, is 4 m. ; from July 10th to July 25th, it is 15 d. Now, $500 x .0225 = $11.25, Ans.

29. What is the amt. of $1250, from July 20th, 1873, to Dec. 29th, 1873, at 6% ?

30. What is the amt. of a note of $2000, bearing int. from March 1st, 1872, to Jan. 25th, 1873, at 7 % ?

31. What is the interest of $4500 for five years, at 6% ? The amount ?

ANSWERS

ADDITION.

Ex.

Ans.

Page 21.

2. 24

3. 25 *

4- 32

5- 27

6. 26

7. 27

8. 34 9- 37

1. 45

2- 45

3- 52 4 59

5. 60

6. 61 7.58 8. 66

Page 23.

1. 47

2. 41

3- 44

4- 55

5- 53

6. 62

7. 64

8. 70

9. 22 pounds 10. 35 yards

Page 24.

2. 67

3- 88

4- 79 5. 98

Ex.

Ans.

6. 97

7. 887

8. 888

9. 999

10. 67

11. 77 89 95

12.

13- 14.

i5-

99

rage 28.

13142 16424 16189 17 140

17374 1714 yds.

2453 lbs. 2359 rods. $2263

2454 A.

Page 29.

6. 3280

7- 1936

8. 1232

9. 2093

10. 2377

11. 861 pages

12. 2764 days

13. $369

14. $119

15. 939 sheep

16. 1682 sheep

Ex.

AN8.

17. 27514

18. 84OI8

19. 1799 A.D.

20. $7835

21. 1886 A.D.

Page 30.

22. 2387 yds.

23. $2123 318 years 208 mar. $2526 45913 men

28. $58020

29. 1492 days

30. 705 miles

31. 998 acres

32. 11252 years

33. 2413 oz.

34. 2969 lbs.

35. $1700

36. 1003 A.

24.

25- 26.

27.

Page 31.

37. #5<>3

$593 $6674

$53.68

$48.57 $62.55

#539-°3 $829.03 $648.62

I7M-57

$6366.10

38.

39- 40.

41. 42.

43-

44.

45- 46.

47-

ANSWERS,

209

SUBTRACTION.

Ex. Ans.	Ex. Ans.	Ex. Ans.
Page 38.	4. 2223	17. 3240230
2. 213	5. I409	18. 1765509
3- 324	6. 1804	19. 3929992
4. 320		20. 6706495
5. 2232	I- 235	21. 2235IO9
6. 4063	2. 3108	22. IOOOOO4
	3- 3^	23. 582 sh.
Page 39.	4. 144	24. $422
1. 223 lbs.	5- JI74	25. 83 years
2. 263 yds.	6. 5218	26. $588
3. 4316 hats	7. 3101	27. 64 years
4. $1111	8. 2228	28. 1722
7. 3562 in.	9. 2167 qts.	29. $930
8. 5101 oz.	10. 1447 1 7 bar.	30. 156 years
9. 3000 weeks
10. 32 1 1	Pagre 45.	Page 44.
11. 4000	11. 738 bu.	1. 504
	12. $723	2. 100
Page 42.	13. $405	3. 1409
i. Given	14. 990	4. 2640
2. 2191	15. 990	5- !5i64
3- 2313	16. 7272	6. 4325

MULTIPLICATION

Page 50.

2. 84828

3. 66963

4. 48848

5- 55555

6. 606606

7. 996092

8. 770707

9. 888888

Page 52.

2. 1825 days

3. $4500

4. 7000 A.

5. 4272

6. 22035

7. 114144

8. 325215

9. 4207380

10. 4450096

11. 6643240

12. 8757720

Page 54.

2. 93100 1 3- I39502

4. 247388 5- 344109 6. 681252

2/574476

3. 1086912

4. 1830048

5. 3299541

6. 99764360

7. 126257940

8. 296989550 9- M37399648

210	ANSWEBS.
Ex. Ans.	Ex. Ans.	Ex. Ans.
io. 8760 hrs.	31. $655500	2°- 33995°°
11. 48000 rods	32. 1 1 6300 p.	21. 85442OOO
12. $23055	33- #59985	22. 24139500
13. $24250	34. $188040	23. 195500000
14- 5393°58		24. 21375000000
l5- 18305988	Page 57*	25. 4186100000
16. 143225262	1-3. Given	26. 480480120000
17. 168465500	4. 36100	27. $14000
18. 213882848	5- 453°°	28. 1260000 cts.,
19. 229152462	6. 2045000	or $12600
20. 186691875	7. 46208000	29. 36000 miles
21. 411290946	8. 58241000	30. 150000 bu.
	9. 3260720000	31. 62500000
Page 56.	10. 400728900000	32. 3920000000
22. $55250	11. $510	33. 1412019000
23. 22770 miles	12. $26500	34. 28860000000
24. 16884 yards	13. $20500
25. $160000	14. 63000 cts., or	Page 59.
26. $19250	$630	1. 5130 mo.
27. $40250		2. 56250 d.
28. $123375	Page 58.	3- $576
29. $282750	15-18. Given	4. $460
30. $200000 SF	19. 1 147000 [ O R T D I VI S I 0	5. $1464 N.
i*age 65.	5. I201lf	11. 460 hats
~ 2131		12. 587 barrels
3. 2031	Fage 68.	13. 52-f weeks
4. 2101	2. 1 1708	14. 125 hrs.
5. IOIOI	3- 11389!	15. 52 boats
6. 2021	4. 12004	16. 24 weeks
7. IIOI	5. 18062	17. 70 barrels
8. 1010		18. 170 boxes
£. IOIOI	Page 60.	19. $6253
	6. 112363-	20. 2808 boxes
?age 67.	7. 10715}	21. 2203 A.
2. 3H2	8. II202|	22. $10603
j. 20143	9. 1 0338 £	23. 2500 hours
4 3121?	10. 282 barrels	24. 125 stages

		ANSWERS.	211
	LONG DIVISION.
Ex	Ans.	Ex. Ans.	Ex. Ans.
	Page 71.	24. 60 days	2. 90000 Cts.,
I.	Given	25. $68Jf£	or %oo
2.	Given	26. $28	3. $1605
3- 4-	3476J- 7275t	Page 74,	4. $5 5. 105^ tons
		1. Given	6. $945
	Page 72. QTI9	2. 85, 64 rem. 3- 46, 53 T rem.	-P«*/e 78.
i.	4. 48	7. 685 barrels;
2.	2 242f|	5. 437, 5681 rem.	$7 a bar.
3-	2I75A	6. 3, 9467 rem.	8. $2733
4-	I756HI	7. 2, 72364 rem.	9. $2310
5-	I2 724H	8. 10	10. $1305
6.	l6lo44	9- 85> 325764r.	«• 25435
	53	10. Given	12. 29379
7-	I333Iff	11. 426, 14 rem.	13. 41884
8.	13379	12. 411, 15 rem.	14. 30 days
9-	10328IJ	13- 85> J545 rem.	15. 4 years
IO.	99289H	14. 78, 281 rem.	16. 164 cts., 2d
ii.	204 cows	15. 31, 342 rem.	311 cts., 3d
12.	200-JI A.	16. 8, 16 168 rem.	17. 25 days
13- I4.	36 months 180 stoves	Page 75.	Page 81.
15- 16.	i57ff years 83JJ months	1. 244 2. 775	1. Given 2. 2x3x7 3. 2x2x2x2x3
*7- 18.	99^ hhd. 75 yoke	Page 76. 3. 1089	4. 2x2x3x5 5. 2x2x2x3x3 6. 2x2x5x5
	Page 73.	4. 2510 5. 606 6. 1747 7. 6500	7- 5X5X5
19. 20. 21.	Given i329*£ff 10266^	8. 2x2x3x11 9- 5x5x7 10. 2x2x2x5x5 II.2X2X2X2X?
22.	3oio|IM	jPagre 77.	X 2 X 2 X 2
23-	3°2 2y-6-2T	1. 5 days	12. 5X5X13

212	ANSWERS.
Ex. Ans.	Ex. Ans.	Ex. Ans.
13- 2x5x5x3x3	IO. II	14. 32
14. 5X5X5X5	11. -*/, or i2| yd.	15. 12 ft.
15. 2X2X2x5x5	12. 36 barrels	16. 20 yds.
x5	13. 44 days.
16. 2x2x2x2x2
X2X3X3X3 fj, 2x2x2x2x3	Page 85.	Page 87. 1. Given
X3X13	1. Given	2. 24
	2. 9	3- 48
Page 82.	3. 16 4- J5	4. 90 5. 1008
1. Given	5-24 6. 15 7. 27	6. 1800
2. Given	7. 660
3. Given	8. 156
4. Given	8. 4	9. 648
5- h or 2 J	9. 256	10. 240
6. J£, or if	10. 12	11. 168
7. V> or 3?	11. 15	12. 12960
8. 4	12. 15	13- 864
9. 14	13. 48
REDUCH	:iON OF FRACTIONS.
J'aflre 95.	13. i	2. Givem
1. Given	14- i	3-1
2. «	15. f	4- I
~ 25 3' ^HT 4- «	16. } 17. i	5- i 6.|
5 t#	18.21	7-1
6. 41	9	8. 1
7.H	19. f	9. A
8- tV\	20. -J	10. i
9. *&	•i.2i	11. V
"• TOlfe	12	12. -B
i'. Given	Page 96.	13. *
' .*	1. Given	14- A

ANSWERS.

213

Ex. Ans.	Ex. ANS.	Ex.	Ans.
i5- U	19. $3i	2.	Given
ik&t		3-	a
17- i	Pagre 99.
is. a	1. Given	JPttgre J 00.
19- i	2. Given	4-	Given
20. J 21. H	• 3- ¥ 4- *¥■	5- 6.	Given *
22. tV*	5- H*	7-	Tl^f
23- A	6. ±fi	8.	*
24. i	7. ¥	9-	j
25.* 26.il	8. ^ 9- -v-	10. 11.	i
	10. ip-	12.	i A H A
rage 97.	11. 4*	13-
1. Given	12. ^ it. •s-47 14.5 i5- W 16 xt§7	14. *5-
2. 19 3- »4l	16. 17-
4- 16 J	18.	5 T
5- i6|	xu* 15 T- 2003	19.	3
6. 12	*7« —20- 18. a$|* 19. ^P 20. V 21. ^F 22. 7 ° ° 3 „„ 85 19	20.	ft
7- My 8. 12A 9. 15	1. Given
10. 8	2.	1*
". 5ft	3-	if
12. si 13. H 14. 7f	3* — ?r — 24. a^ul 25. 15233	4. 5- 6.	if 51
i5- 7fi	26. i^o-U.	7-	81 T77
16. 8	27. 103 beg.	8.	87 TTT
17. 12		9-	225
18. 2|sh.	1. Given	10.	500 TWO"

214	ANSWEES	1.
Ex. Ans.	Ex.	AN8.		Ex. Ans.
Page 102.	- 8 3* T2>	9			fi 36 40 45 °- o"T7> 6"T3T> ^0"
i. Given	/i 6 5 4' TT? TT			7- 18, ft, IS
- 4 0 4 5 12	5* "2~8">	12
2' ZH> 3T» ^TT	1^
8 JUL 144 X85. °- TT5> ^T6> 2T#			4-	8 5 6 ~2Ut T0~> TO"
« 315 616 297 9* 3~9~T> ^9~3> ^9T			5-	21 10 4 T4> T4> ?4~
T~ 31080 47730 16340 IO* S~TSWUt 6 3 640? 6 364tf		6.	30 2 8 4 9 TO? T0> TU
XI« J^OOO? IfOOO? TTTTTT		7-	Ti'5> T5> T6~
12. Given			8.	TUT? TOT? TOT
13- n, a, f?			9-	TT> TT> IT
i4.a%%1			IO. n.	4T> if> Jf ? A 1320 1617 525 2695 ~3~4~6T> T4~6T> 3 46 5> TT6T
Page 103	•		12.	Given
i, 2. Given			13-	16 2 10 2 5 4T(5> ~4TT? 4~tf
3* 3~0~> 3~0~> 317			14.	t > ¥> 1. ¥

ADDITION OF FRACTIONS

Page 105,

i. Given 2. ^ = 2| A. 3-*!= * 4- M = ^ft

5- «=»*

6. «=i«

Page 106.

4- H=H

6-ft=i*

7- W=«*

«•« = '« 9- W = 'ft

II. «f = 2T«&

13. ioif-lbs.

14. 52-^ yd. i5- 77W m-

16. $24J

17. l7i«

18. Given

19. 5«

20. 28J

■I. 23I

22. 46f}

ANSWERS

315

SUBTRACTION OF FRACTIONS.

Ex. Ans.	Ex. Ans.	Ex. Ans.
Jt-tef/d 107.	3-T5-	Pa#e J 0,9.
i, 2. Given	4- A	15. Given
*Hf&$	5- A	16. 2 if bu.
4. U bu.	6-Ni	17. Given
e 60	7- T%	18. 35lgal.
6-tWr	8. A	19. 18J
7- jib	9. ?£=$	20. 35 A
8-T§*T	i°- u	21. 20f
	TT 44	22. 29J
JPacre 20S.	12. A	24. A=*
i. Given	13- A*	25-A=J
2-H	*4-Ab

MULTIPLICATION OF FRACTIONS.

Page 111
1.	Given
2.	*6f
3-	$toA
4-	$i5°A
5-	»*A
6.	i5ii
7-	8H
8.	7t3t
9-	*H
10.	"A
11.	26J.

12. 36 If

13- 62J

14. 42f J

15. Given

16. $2.87}

17. $843 J

18. 781J

19- 1575

20. 2478I

21. 3562I

22. 5000

23. 8775

24. $654

25. $75°

26. $2224j

1. Given

2. 45 cts. 3- $62^

4- 33i

5-48f

6. 22f

7- 3*1

8.30

9-741

10. 257A

11. 135

12. 292 days

Page 114.

13. Given

14. $7

15. 486 miles

16. 325 i7-44i|

216

ANSWERS,

Ex. Ans.	Ex. Ans.	Ex. Ans.
i 8. 663	8.|	4- Hfj
19. 1090J	9-1%	5- 5°f
20. 4161^	10. A	6. 14J
21. 6250	"•A
	I2- A	Jtogre J 16.
Page 115.	13- A	7- Ii.6si
1. Giyen	14. A	8. 240I
2.f&	15. A	9- 159ft
3-$A	16. Given	10. 585«
4-1		11.625
5-1	i-t4t	12. 1431J
M	2. J	13- 2503H
7-H	3-f	H. 6737M

DIVISION OF FRACTIONS.

Page 117.	17. 3H	Page 120.
1, 2. Given	TQ ,,293	2.6
3- A	Tn ?293 19- 23^<J	3- 4j
4-H	20. 4fli	4.8f
5- A		5-4
6. A	rage 119.	Mtt
7-t**	1, 2. Given	7- 4i
S-tH*	3-53J
9- A	4.88	JPwflre Jf£l.
i°- dflft	5- H3i	2. i| lb.
11. A^	6. 192^	3. 2^ pines
12. Aft	7- 5A Mtt	4. 6} lb.
Paf/e iiS.	9-6f	Page 122.
14. $7i	10. 8f	5-3}
15-51	11. $4	6.1*
16. 3A	12. 4 miles	7- iA

ANSWERS,

217

Ex. Ans.	Ex. Ans.	Ex. Ans.
8. 2f	18. I»	Page 123.
9. Iff	19- 2^	4- &
10.5	20. 2||	5- ^T
"*irtfr	21- 3if	6-i
12. 2f|	2 2. 2f|	7. A
13- 2|	23.2^	8. A
14- 2^6T	25-5	9- *i*
15- 2j	26. A	10. ^
*"• X 228 25	27. 60	11.*

QUESTIONS FOR REVIEW.

Page 124.

1. to!

2. $I2|

5- *38«

6. 37 J lb.

7. 96 p. k.

8. 1 46 J lb.

9. 4| lb.

10. 8 cords

11. 5 bar. is. -It* sum;

TO?

dif.; A> prod.; 3i quot.

13. 2 if days

14. 8 bales 15- Mi

16. $9 J i7-4ilb.

18. 3f

19. $i7f

20. 3500 lb.

21.

fjsold; U left

22. 45 weeks

23. hsi

24. $9A

25.fi yd.

26. A

27. 35

28. i«

29. 6f

30. $8, flour

31. 14 oranges

32. 5 lb.

FRACTIONAL rage 126.

28 — 4

1 27 — I . 3« "ST — U 6 3 — 7 .

RELATION Page 127.

4-y?

5. A

bu.;

fbu.

6.i

OF NUMBERS.

»7 10

7" T3

8. Given

9. 24 cts.

10. $1093^

11. $264

218

ANSWERS.

Ex. Ans.	Ex. Ans.	Ex. Ans.
12. Given	18. io|£ bu.	5. 180
I3-T&T		6. 261
14- T&g	Page 128.	7. 46of J
15- A	2. 42I	8. $40
l&irff	3-56J	9. $i6of
17- $5	4. 86f	10. 84 yrs.

DECIMAL

I'asre 132.

1. .12

2. .25

3- -o5 .49

FRACTIONS

,119

,027 ,009 ,013 •1345

10. .0236

11. .0039

12. .OOOO7

13. .06 ; .041 ; .007

14. .0201 ; .00752

REDUCTION OF DECIMALS.

Page 133.	I3-7WU	5- -3333 +
1. Given	T/1 1881	6..5
3-Tto = *	*5* 20000	7- -375 8.4 9. .4166-h 10. .9 "••75
4* tuvu—^vu - 404—101 5- -nnnr— if5Tr	J6- tAW I7.tJ!S* 18. iWft
7- A		12. .3125
8- A	P«flre 134.	13. .2
9. AV	1. Given	14. .0625
t t 2 5 2	2. -5	15. .025 16. . 1
XI- Z7ZJ	3. .2	T f 2
12. 2 oJ0 0	4.-75	I f. .6
ADDI1	ION OF DEC!	MALS.
IVrflre J36.	4. 274.251	8. $74-375
1. Given	5.6.6516	9. 72.946 A.
2. 33.079	6. 31.465	10. 109.841 g.
3. 16.027	7. 45.66 yd.	11. 176.15 r.

ANSWERS.

219

SUBTRACTION OF DECIMALS.

Ex. Ans.	Ex. Ans.	Ex. Ans.
Page 137*	7. 3.782	13. O.045
2. 3.262	8. 99.162	14. O.O054
3. 6.1682	9. 214.25	15. O.OOOO9
4- 27.3797	10. 7.3992	16. 6.25 yds.
5. O.76442	11. 14.993	17. 0.45 ship
6. 49.525	12. 0.6306	18. 62.3 A.

MULTIPLICATION OF DECIMALS.

Page 139.

1. Given

2. Given

3. Given

4. Given

5. 0.00381

6. 0.0363

7. 0.001058

8. 760.2128

9. 25.664

10. 0.3159

11. 117.351

12. 18.25

13. 0.114015

14. 8.09792

15. 50.06223

16. 0.0060024

17. 53-7758

18. 70

19-3

20. 0.000804

21. 111.375

22. 24.375

23- 393-75

Page 140.

24. 10.125

25. 0.00804

26. 0.00007

27. 0.00035

DIVISION OF DECIMALS

Page 142.

1. Given 2 Given

3. Given

4. Given

5. 11 lbs.

6. 51 lots

7- 4-312

8. 0.0^312

9. 0.0002806

10. 0.0734201

11. 142.5

12. 76

13. 2.0454 +

14. 0.4885 +

15. 0.015

16. 3.65

17. 0.0385

18. 0.5

19. 0.39104

20. 2.9029 +

21. 1000

22. 100

23. 0.01

24. O.OOOI

25- o-75

26. $2.5

27. 4.8 d.

28. $0.5

29. $10.5

30. 561.7 + r.

31. 181.05 -1 A.

32. 74 times

33. T.066+ t.

34. 4.2857 + t.

220

ANSWERS,

ADDITION OF U. S. MONEY.

Ex.

Page 148.

i, 2. Given

3. $1026.692

4. $1631.03

5. $2274.52

6. $284.37

Ex.

7. $769.73

8. $21.00

9. $284,375

10. $557.43

11. $165,846

12. $265,525

Ex.

Ans.

13. $1390.758

14. $1967.06

15. $3071.58

16. $156.13

17. $73.50

SUBTRACTION OF U. S. MONEY

Page 149.

2. $19,585

3. $15,085

4. $48,918

5. $99,125

rage 150.

6. $0,875

7. $20,625

8. $1.25

9. $187,375

10. $83.58

11. $990.00

12. $160,065

13. $94.86

14. $296,967

15. $19,705

MULTIPLICATION OF U. S. MONEY.

Page 151.

3- '$432.85 $9.01125 $650,052 $61987.50

$22.50

DIVISI Page 153.

7. 1 1.308 + t.

8. 12 times

9. 63 melons

10. 85.5 lb.

11. $2,125

12. 1 2 -J cts.

13. 14 y earl's

14. 80 A.

15. #5-375

16. $35.63

17. $64.03

8. $494.00

9. $28.1875

10. $753-75

11. $13500

12. $1038

13. $31262.50

ON OF U. S.

18. $2.35

19. 25 cts.

20. $9,625

Page 154.

1. $61,895

2. $1.75 pro.

3. $9.i25dif.

4. $38.31 dif.

5. $25.84

6. $534.60

7. 17 cts.

Page 152.

14. $756.00

15. $19,875

16. $78.80

17. $186.20 s.; $3.80 dif.

MONEY.

8. $7.64$

Page 155.

9. $10,788 +

10. $46.50

11. $67.85

12. 71 hoi^es I3.$i5i3.i.5 s.;

$377,875 a.

13. 12.5 tubs

15. $6

16. 50O \YdlV

ANSWEKS.

221

Ex.

APPLICATIONS OF U. S. MONEY.

Ans.

Page 156.

i. $9.59 ami

2. $35.25 +$3.1 9 + $5.04 + $18 +$2 1 = $82.48, amt.

3. $19.44 + $16.50 + $8.80 + $7.20 =$5 1. 94, debits; $9.00 + $8.40 + $8.25 +$18.70 =$44.35, credits.

Page 157.

4. $i.56 + $28.5o + $4.o8 + $4.32 + $io.5o=:$48.96j amt.

5. $16.70 + $15 + $9 + $10.52 =$51.22, debits; $24.00 + ^36 + $62 +$26.25 —$148.25, credits ; Balance due J. Barker, =$97.03

6. $47.34 + $24.50 + $28.50 + $9.75 +$n.io + $7.5o + $10.80 =$139.44 amt.

Page 174.

3. 8714 far.

4. 835 far.

5. io368d.

6. 1 1 0400 far.

Page 175.

9. £6, is. 9d.

10. £28, 3s. 2d. 3 f.

11. Given

1 2. £3, 8s. 9d.

13. 1780 pwt.

14. 61 1 13 gr.

15. 61b. 6 oz. 1 pwt,

16. 1 lb. 1 oz. 13 p. 23 gr.

17. 1 oz. 18 pwt.

18. $78

19. 1 cwt. 41 lb. 9 oz.

20. 483704 oz.

REDUCTION.

21. 3201204 OZ.

22. $5.04

23. $656.25

24. 480 dr.

25. 21 12 sc.

26. 1 lb. 7 oz. 4 dr. 1 sc.

27. 30Z. 2 dr. 18 g.

28. 742 J ft

29- 63532 ft-

30. 46422 yd.

Page 176.

31. Given

32. 94 r. 9 ft.

33. 2 m. 43 r. 8J ft.

34. 1 7 1049 ft.

35. $2.25

36. 1280 rods

37. 10560 st.

38. 5 quarters

39. 6 eighths

40. 2416 i6ths

41. 118J yd.

42. $3.20

43. $420

44. 43560 sq. ft.

45. 234407 J sq.ft.

46. 102729 sq. yd.

47. 102400 sq. r.

48. 5 A. 51 sq. r.

49. 15 A. 100 sq. r.

50. 35 s.y. 6 sq.ft. 49 sq. in.

Page 177.

51. $1361.25

52. $10454.40

53. 1940544 cu. in.

54. 2 cu. y. 1 cu. ft 1325 cu. in.

55. 33 C. 26 cu. ft

222

ANSWERS.

Ex.

Ans.

56. 9664 cu. ft.

57. 1943 cu. ft.

58. $486

59. $76.80

60. $84

61. 442 pts.

62. 51 bu. 1 p. 7

63. $19.20

64. $3.90 prof.

65. 136 boxes

66. 3 bu. 1 p. 3 (

67. 207 gills

68. 770 qt.

69. 57 gaL 1 q.

Ex.

Ans.

70. 10 hhd. 10 g.

71. 45 qt.

72. $63

Page 178.

73. $142.80

74. 278220 sec.

75. 3d.4h.5m.

76. 77760 m.

77. 525600 m.

78. 5y.185d.16h.

79. 604800 t.

80. $294

81. 3258720 t.

Ex.

Ans.

82. 11 d. S^/yh.

83. 16*780"

84. 20 46' 40"

85.4 s. 17° 55'

86. 15" in 1 h.; i° in 4 m.

87. 2640 sh.

88. 25 r. 10 qu. 1 8 sh.

89. $0,005^

90. 5760 cra)rons

91. $126

92. 864 pens 93- 75 eggs

SURFACES AND SOLIDS. Page 179.

1. Given

2. 28 sq. ft.

3. 768 sq. in.

4. 90 sq. r. 5- 3° yds.

Page 180.

6. 24 sq. ft.

7. 50 A.

8. 23040 A.

9. 540 sq. yd.

10. 900 brick

11. $3.60

Page 181.

1. Given

2. 64 cu. in.

3. 45 cu. ft.

4. 240 blocks

5. 455 cu. ft

6. 98 cu. ft.

7. 378 cu. ft.

8. 160 cu. yd.

9. 81 cu. ft.

10. $243

11. $40.50

12. $40.50

13. 240 cu. ft.

14. 90 cu. ft.

REDUCTION

Page 182.

1. Given

2. 1 ft. io| in.

3. 78. 6d.

4. 5 d. 6 hr.

Page 183.

6. § qt.

OF DENOMINATE FRACTIONS

15. £f

16. Given

18. .^bu.

19. Given

20. 0.38125 m.

21. £0.33$

22. 0.09-^ hhd.

7. f pwt.

9. 3 qt. .72 pt.

10. 4 d. 9 hr.

11. 3 pk. 4qt.

Page 184.

13- tt yd.

14. & bu.

ANSWERS. COMPOUND ADDITION

Ex. Ans.		Ex. Ans.
Page 186.		7. 186 m. 24 r.
2. £26, 28. id. 3 far.		8. 92 bu. 1 pk. 4 qt.
3. 27 lb. 11 oz.		9. H7iyd.
4. 22 yd. 8 in.		10. 54 sq. r. 19 sq. yd. 7 sq. ft
5. 108 bu. 3 pk. 6 qt. 6. 25 T. 5 cwt. 23 lb.	1 pt.	11. 117 A. 48 sq. r.
	1 2. 4 C. 60 cu. ft.

COMPOUND SUBTRACTION.

Page 187.

1. Griven

2. £2, 2s. 2d. 3 far.

3. 5 lb. 11 oz. 5 pwt.5 gr.

4. 1 bu. o p. 3 qt.

5. 7 m. 199 r. i}y. 1 ft, or 7 m. 199 r. 1 y. 2 ft. 6 in.

6. 27 gal. 1 qt.

Page 188.

7. 64 A. 143 sq. r.

8. 41 cu. ft.

9. 2 T. 252 lb.

10. 13° 34' 57"

11. 15° 33' 30"

13. 26 yr. 6 mo. 3 d.

14. 1 yr. 11 mo. 12 d.

15. 4 yr. 2 mo. 24 d.

16. 3 y. 10 m. 23 d.

COMPOUND MULTIPLICATION.

Page 190.

4. 184 g. 1 q. 1 p.

5. 256 yd. 2 qr.

6. 61 A. 92 sq. r.

7. 12 C. 94 c. ft.

8. 67 hr.

9. 9 T. 625 lb.

10. £70, 4s. 4d.

11. 85 bu. 1 pk.

COMPOUND DIVISION.

rage 192.

3. 3 A. 47 sq. r. 7 sq. ft.

4. 84 cu. ft. 31 3§ cu. in.

5- 7i I2 g*

6.9

7. 6 bu. 2\ pk.

8. 10 ft. 4 in.

9. 5 m. 3 fur. 10. 2 oz. 12 pwt.

11. 4400 rails

12. 6 bu. 1 J pk.

13. 2 A. 24 sq. r

.. 24 sq. r.

14. 640 times

15. 7 bags

16. 2 iff bundles

224

ANSWERS,

	PERCENTAGE.
Ex. Ans.	Ex. Ans.	Ex. Ans.
Page 1 96.	13. 600 pupils.	II. 25%.
i, 3. Given.	14. $15625.	12. 25%.
4. 32 yds.		13. 25%.
5. $45-936-	rage 198.	14. 25%.
6. 42 bu.	1, 2. Given.	15. 50%.
7. 80 rods.	3- 33i#.	16. 33i?o-
8. 133.2 bar.	4- 3°°^-
9. 408 men.	5. 20%.	Page 200,
	6. 1 1 \%.	1, 2. Given.
Page 197.	7. 50%.	3. $62.50.
10, 11. Given.	8. 33i%-	4. $47,024.
12. 81680.	9. 20$.	5, 6. Given.
	10. i6f#. •	7- $35-

PROFIT AN D LOSS.
Page 202.	9, 10. Given.	2. 25%.
1, 2. Given.	11. $93.60.	3- 5°%-
3. $7 gain.	12. $336.	4. 50 %
4. $0.75.	13. $2025.	5. 28}%.
5. $2,125.	14. $5600.	6. isifo
6. $105.30.		7- 33i7°-
7. $18.40.	Page 203.	8. 25%.
8. $65.	1. Given. INTEREST.	9. 427#-
JPaflN? 207.	21. $80.00 int. ;	28. Given.
1- 14. Given.	$580 amt.	29. 5 m. 9 d. time;
15. $5,176.	22. $69,646 int. ;	$33,125 int.;
16. $6,668 int.;	$832,897 amt.	$1283. 1 25 am.
$65,418 amt	23. $360.	30. 10 m. 24 d. t.;
17. $2.94.	24. $306,683.	$126.00 int.;
18. $4,294.	25. $15,125.	$2126 amt.
19. $11.25.	26. $45-325-	31. $1350 int.;
20. £25.175.	27. $32,301.	$5850 amt.

YB 35847

■ ~ T~ — : : ~ ;

M249550

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