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COMPLETE

ARITHMETIC,

ORAL AND WRITTElSr.

PART SECOND.

BY

MALCOLM MacVICAR, Ph.D., LL.D.,

FBINCIFAIi STATE NOBMAL SCHOOL, POTSDAM, N. Y.

PUBLISHED BY

TAINTOR BROTHERS, MERRILL & CO.,

NEW YORK.

\

Copyright by

TAINTOR BROTHERS, MERRILL & CO.,

1878.

Electrotyped by Smith & MoDougal, 83 Beekinan St., N. Y.

^^ (ft ^ ^

QA 10:

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â€¢-*g^| PREFACE, l^^^**

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THE aim of the author in the preparation of this work may

be stated as follows :

1. To present each subject in arithmetic in such a manner

as to lead the pupil by means of preparatory steps and proposi-

tions which he is required to examine for himself, to gain clear

perceptions of the elements necessary to enable him to grasp

as a reality the more complex and complete processes.

2. To present, wherever it can be done, each process object-

ively, so that the truth under discussion is exhibited to the eye

and thus sharply defined in the mind.

3. To give such a systematic drill on oral and written exer-

cises and review and test questions as will fix permanently in

the mind the principles and processes of numbers with their

applications in practical business.

4. To arrange the pupil's work in arithmetic in such a manr

ner that he will not fail to acquire such a knowledge of princi-

ples and facts, and to receive such mental discipline, as will fit

him properly for the study of the higher mathematics.

The intelligent and experienced teacher can readily deter-

mine by an examination of the work how well the author has

succeeded in accomplishing his aim.

M57?0.'f7

PREPACK.

Special attention is invited to the method of presentation

given in the teacher's edition. This is arranged at the begin-

ning of each subject, just where it is required, and contains

definite and full instructions regarding the order in which the

subject should be presented, the points that require special

attention and illustration, the kind of illustrations that should

be used, a method for drill exercise, additional oral exercises

where required for the teacher's use, and such other instructions

as are necessary to form a complete guide to the teacher in the

discussion and presentation of each subject.

The plan adopted of having a separate teacher's edition

avoids entirely the injurious course usually pursued of cum-

bering the pupil's book with hints and suggestions which are

intended strictly for the teacher.

Attention is also invited to the Properties of Numbers, Great-

est Common Divisor, Fractions, Decimals, Compound Num-

bers, Business Arithmetic, Ratio and Proportion, Alligation,

and Square and Cube Root, with the belief that the treatment

will be found new and an improvement upon form.er methods.

The author acknowledges with pleasure his indebtedness to

Prof. D. H. Mac Vicar, LL.D., Montreal, for valuable aid

rendered in the preparation of the work, and to Charles D.

McLean, A. M., Principal of the State Normal and Training

School, at Brockport, N. Y., for valuable suggestions on

several subjects.

M. MacVICAR.

Potsdam, September^ 1877.

.^

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C..

BEVIMW OF PART FIRST.

PAGE

Notation and Numeration. 1

Addition 4

Subtraction 5

Multiplication 7

Division 11

Properties of Numbers. ... 16

Exact Division 16

Prime Numbers 19

Factoring 20

Cancellation 31

Greatest Common Divisor.. 22

Least Common Multiple. ... 25

Fractions 29

Complex Fractions 40

Decimal Fractions 46

Denominate Numbers 59

Metric System 75

FAllT SFCOND,

Business Arithmetic 79

Aliquot Parts 82

Business Problems 85

Applications 100

Profit and Loss 101

Commission 103

Insurance 105

Stocks 107

Taxes Ill

Duties or Customs 113

Review and Test Questions. 114

Interest 115

PAGB

Method by Aliquot Parts. 117

Method by Six Per Cent.. 119

Method by Decimals 121

Exact Interest 122

Compound Interest 127

Interest Tables 120

Annual Interest 131

Partial Payments 1 32

Discount 136

Bank Discount 138

Exchange 141

Domestic Exchange 142

Foreign Exchange 146

Equation of Payments 151

Review and Test Questions. 160

Ratio 161

Proportion 169

Simple Proportion 170

Compound Proportion 174

Partnership 177

Alligation Medial 181

Alligation Alternate 182

Involution 187

Evolution 189

Progressions 205

Arithmetical Progression. 206

Geometrical Progression.. 208

Annuities 210

Mensuration 213

Review and Test Examples. 227

Answers 243

â™¦>â– â™¦â– â™¦Â»â–

REVIEW OF PART FIRST.*

.(Â»H^^> Y^85^^V^^

NOTATION AND NUMERATION,

DEFINITIONS.

11. A Unit is a single thing, or group of single things,

regarded as one ; as, one ox, one yard, one ten, one hundred.

12. Units are of two kinds â€” Mathematical and

Common. A mathematical unit is a single thing which has a

fixed value ; as, one yard, one quart, one hour, one ten, A

common unit is a single thing which has no fixed value ; as,

one house, one tree, one garden, one farm.

* Note. â€” The first 78 pages of this part contains so much of the matter

In Part First as is necessary for a thorough review of each subject, in-

cluding all the tables of Compound Numbers. For convenience in

making references, the Articles retained are numbered the same as in

Part First. Hence the numbers of the Articles are not consecutive.

2 NOTATION AND NUMERATION.

13. A Number is a unit, or collection of units ; as, one

man, three houses, /(92<r, six hundred.

Observe, the nwnher is "the how many" and is represented by what-

ever answers the question, How many ? Thus in the expression seven

j&vda, seven represents the number.

14. The Unit of a Number is one of the things

numbered.

Thus, the unit of eight bushels is one bushel, of five boys is one hoy^

of nine is one.

15. A Concrete Number is a nnmhor which is applied

to objects that are named ; as four chairs, ten lells,

16. An Abstract Number is a number which is not

applied to any named objects ; as nine, five, thirteen.

17. Like Numbers are such as have the same unit.

Thus, four windows and eleven windows are like numbers, eight and

ten, three hundred and seven hundred.

18. Unlike Numbers are such as have different units.

Thus, twelve yards and five days are unlike numbers, also six cents

and nine minutes.

19. Figures are characters used to express numbers.

20. The Value of a figure is the number whidi it

represents.

21. The Simple or Absolute Value of a figure is the

number it represents when standing alone, as 8.

22. The Local or Representative Value of a figure

is the number it represents in consequence of the place it

occupies.

Thus, in 66 the 6 in the second place from the right represents a num-

ber ten times as great as the 6 in the first place.

2.3. Notation is the method of writing numbers by

means of figures or letters.

NOTATION AND NUMERATION. 3

24. Numeration is the method of reading numbers

which are expressed by figures or letters.

35. A Scale in Arithmetic is a succession of mathematical

units which increase or decrease in value according to a fixed

order.

26. A Decimal Scale is one in which the fixed order

of increase or decrease is uniformly ten.

This is the scale used in expressing numbers by figures.

â€¢ 27. Arithmetic is the Science of Numbers and the Art

of Computation.

REVIEW AND TEST QUESTIONS.

31. Study carefully and answer each of the following

questions :

1. Define a scale. A decimal scale.

2. How many figures are required to express numbers in the decimal

scale, and why ?

3. Explain the use of the cipher, and illustrate by examples.

4. State reasons why a scale is necessary in expressing numbers.

5. Explain the use of each of the three elementsâ€” /^wres, plare, and

comma â€” in expressing numbers.

6. What is meant by the simple or absolute value of figures ? What

by the local or representative value?

7. How is the local value of a fig'ure affected by changing it from the

first to the third place in a number ?

8. How by changing a figure from the second to the fourth ? From

the fourth to the ninth ?

9. Explain how the names of numbers from twelve to twenty are

formed. From twenty to nine hundred ninety.

10. What is meant by a period of figures ?

11. Explain how the name for each order in any period is formed.

12. State the name of the right-hand order in each of the first six

periods, commencing with units.

13. State the two things mentioned in (9) which must be observed

when writing large numbers.

14. Give a rule for reading numbers ; also for writing numbers.

ADDITION.

ADDITIOIT,

50. Addition is the process of uniting two or more

numbers into one number.

51. Addends are the numbers added.

53. The Sum or Amount is the number found by

addition.

53. The Process of Addition consists in forming

units of the same order into groups of ten, so as to express

their amount in terms of a higher order.

54. The Sign of Addition is +, and is redAplus,

When placed between two or more numbers, thus, 8 + 3 + 6 + 2 + 9, it

means that they are to be added.

55. The Sign of Equality is =, and is read equals,

or equal to ; thus, 9 + 4 = 13 is read, nine plus four equals

thirteen,

5Q. Principles. â€” I. Only numbers of the same denom-

ination and units of the same order can he added.

II. The sum is of the same denomination as the

addends.

III. The whole is equal to the sum of all the parts.

REVIEW AND TEST QUESTIONS.

57. 1. Define Addition, Addends, and Sum or Amount.

2. Name each step in the process of Addition.

3. Why place the numbers, preparatory to adding, units under units,

tens under tens, etc.?

4. Why commence adding with the units' column ?

5. What objections to adding the columns in an irregular order?

Illustrate by an example.

6. Construct, and explain the use of the addition table.

7. How many combinations in the table, and how found?

8. Explain carrying in addition. What objection to the use of the

word ?

SUBTRACTION. S

9. Define counting and illustrate by an example.

10. Write five examples illustrating the general problem of addition,

" Given all the parts to find the whole,"

11. State the difference between the addition of objects and the addi-

tion of numbers.

12. Show how addition is performed by using the addition table.

13. What is meant by the denomination of a number? What by

units of the same order ?

14. Show by analysis that in adding numbers of two or more places,

the orders are treated as independent of each other.

SUBTEACTION.

â€¢70, Subtraction is the process of finding the difference

between two numbers.

71# The Minuend is the greater of two numbers whose

difference is to be found.

72. The Subtrahend is the smaller of two numbers

whose difference is to be found.

73. The Difference or Memainder is the result

obtained by subtraction.

74. The JPi^ocess of Subtraction consists in com-

paring two numbers, and resolving the greater into two parts,

one of which is equal to the less and the other to the differ-

ence of the numbers.

75. The Sign of Subtraction is â€” , and is called

minus.

When placed between two numbers it indicates that their difference

is to be found ; thus, 14 â€” 6 is read, 14 minus 6, and means that the dif-

ference between 14 and 6 is to be found.

76. JParentheses ( ) denote that the numbers inclosed

between them are to be considered as one number.

77. A Vinculum affects numbers in the same

manner as parentheses.

Thus, 19 + (13â€” 5), or 19 + 13â€”5 signifies that the difference between

13 and 5 is to b*e added to 19.

6 SUBTRACTION.

78. Pri:n'Ciples. â€” /. Only like mtmbers and units of

the same order can he subtracted.

II. The minuend is the sum of the subtrahend and>

difference, or the minuend is the whole of which the

subtrahend and difference are the parts.

III. An equal increase or decrease of the minuend

and subtrahend does not change the difference.

REVIEW AND TEST QUESTIONS.

79. 1. Define the process of subtraction. Illustrate each step by

an example.

2. Explain how subtraction should be performed when an order in the

subtrahend is greater than the corresponding order in the minuend.

Illustrate by an example.

3. Indicate the difference between the subtraction of numbers and the

subtraction of objects.

4. When is the result in subtraction a remainder, and when a di {Ter-

ence?

5. Show that so far as the process with numbers is concerned, the

result is always a difference.

6. Prepare four original examples under each of the following prob-

lems and explain the method of solution :

Prob. I.â€” Given the wliole and one of the parts to find the other part.

Prob. II. â€” Given the sum of four numbers and three of them to find

the fourth.

7. Construct a Subtraction Table.

8. Define counting by subtraction.

9. Show that counting by addition, when we add a number larger

than one, necessarily involves counting by subtraction.

10. What is the difference between the meaning of denomination and

orders of units ?

11. State Principle III and illustrate its meaning by an example.

12. Show that the difference between 63 and 9 is the same as the

difference between (63 + 10) and (9 + 10).

13. Show that 28 can be subtracted from 92, without analyzing the

minuend as in (64), by adding 10 to each number.

14. What must be added to each number, to subtract 275 from 829

without analyzing the minuend as in (C>4:) ?

15. What is meant by borrowing and canning in subtraction ?

MULTIFLICATION. 7

MULTIPLIOATIOIT.

ILIjITSTMATION of l*JtOCESS.

92. Step II. â€” To multiply hy using the parts of the

midtiplier.

1. The multiplier may be made into any desired parts, and tlie mul-

tiplicand taken separately the number of times expressed by each part.

The sum of the products thus found is the required product.

Thus, to find 9 times 12 we may take 4 times 12 which are 48, then 5

times 12 which are 60. 4 times 12 plus 5 times 12 are 9 times 12 ;

hence, 48 plus 60, or 108, are 9 times 12.

2. When we multiply by one of the equal parts of the multiplier, we

find one of the equal parts of the required product. Hence, by multi-

plying the part thus found by the number of such parts, we find the

required product.

For example, to find 12 times 64 we may proceed thus :

(1.) ANALYSIS. (2.)

64x4 = 256^ 64

64x 4 = 256>=3 times 256 4

\

64x4 = 256 J 256

64x12 = 768 3

768

(1.) Observe, that 12 = 4 + 4 + 4 ; hence, 4 is one of the 3 equal

parts of 12.

(2.) That 64 is taken 12 times by taking it 4 times + 4 times -t- 4 times,

as shown in the analysis.

(3.) That 4 times 64, or 256, is one of the 3 equal parts of 12 times 64.

Hence, multiplying 256 by 3 gives 12 times 64, or 768.

3. In multiplying by 20, 30, and so on up to 90, we invariably multi-

ply by 10, one of the equal parts of these numbers, and then by the

number of such parts.

For example, to multiply 43 by 30, we take 10 times 43, or 430, and

multiply this product by 3 ; 430 x 3 = 1290, which is 30 times 43. We

multiply in the same manner by 200, 300, etc., 2000, 3000, etc.

f MULTIPLICATION.

93. Prob. II. â€” To multiply by a number containing

only one order of units.

1. Multiply 347 by 500.

(1.) ANALYSIS. (2.)

Firststep, 347x100= 34700 347

Second Btep, 34700 X 5 = 173500 500

173500

Explanation. â€” 500 is equal to 5 times 100 ; hence, by taking 347,

as in first step, 100 times, 5 times this result, or 5 times 34700, as shown

in second step, will make 500 times 347. Hence 173500 is 500

times 347.

In practice we multiply first by the significant figure, and annex to

the product as many ciphers as there are ciphers in the multiplier, as

shown in (2).

96. Prob. III. â€” To multiply by a number containing

two or more orders of units.

1. Multiply 539 by 374.

(1.) analysis.

Multiplicand.

Multiplier.

539X 4= 2156 1st partial product.

539X374 = ^539X 70= 37 7 30 2d partial product

539x300 = 161700 3d partial product.

2 015 8 6 Whole product.

=1

Explanation. â€” 1. The multiplier, 374, is analyzed into the parts 4,

70, and 300, according to (92).

2. The multiplicand, 539, is taken first 4 times = 2156 (8G) ; then

70 times = 37730 (93) ; then 300 times = 161700 (93).

3. 4 times + 70 times + 300 times are equal to 374 times ; lience the

sum of the partial products, 2156, 37730, and 161700, is equal to 374

times 539 = 201586.

4. Observe, that in practice we arrange the partial products as shown

in (2), omitting the ciphers at the right, and placing the first significant

figure of each product under the order to which it belongs.

MULTIP LIGATION. SF

DEFINITIONS.

100. Multiplication is the process of taking one

number as many times as there are units in another.

101. The Multiplicand is the number taken, or mul-

tiplied.

103. The Multiplier is the number which denotes how

many times the multiplicand is taken.

103. The JProduct is the result obtained by multipli-

cation.

104. A Partial Product is the result obtained by

multiplying by one order of units in the multiplier, or by any

part of the multiplier.

105. The Total or Whole Product is the sura of all

the partial products.

106. The Process of Multiplication consists, /rs^,

in finding partial products by using the memorized results of

the Multiplication Table; second, in uniting these partial

products by addition into a total product.

107. A Factor is one of the equal parts of a number.

Thus, 12 is composed of six 2's, four 3's, three 4's, or two 6's ; hence,

2, 3, 4, and 6 are factors of 12.

The multiplicand and multiplier are factors of the product. Thus,

37 X 25 = 925. The product 925 is composed of twenty-five 37's, or

tMrtyse'cen 25's. Hence, both 37 and 25 are equal parts or factors

of 925.

108. The Sign of Multiplication is x, and is read

times, or mitltipUed hy.

When placed between two numbers, it denotes that either is to be

multiplied by the other. Thus, 8x6 shows that 8 is to be taken 6 times,

or that 6 is to be taken 8 times ; hence it may be read either 8 times 6 or

6 times 8.

10 MULTIPLICATION.

109. Principles. â€” /. Tlve muUiplicand may he either

an abstract or concrete number.

II. Tlve multiplier is always an abstract number.

III. Tlxe product is of the same denomination as the

multiplicand.

KEVIEW AND TEST QUESTIONS.

110. 1. Define Multiplication, Multiplicand, Multiplier, and

Product.

2. What is meant by Partial Product ? Illustrate by an example.

3. Define Factor, and illustrate by examples.

4. What are the factors of 6 ? 14? 15? 9? 20? 24? 25? 27?

32? 10? 30? 50? and 70?

5. Show that the multiplicand and multiplier are factors of the

product.

6. What must the denomination of the product always be, and why ?

7. Explain the process in each of the following cases and illustrate

by examples :

I. To multiply by numbers less than 10.

II. To multiply by 10, 100, 1000, and so on.

III. To multiply by one order of units.

XV. To multiply by two or more orders of units.

V. To multiply by the factors of a number (92 â€” 2). -

8. Give a rule for the third, fourth, and fifth cases.

9. Give a rule for the shortest method of- working examples where

both the multiplicand and multiplier have one or more ciphers on

the right.

10. Show how multiplication may be performed by addition.

11. Explain the construction of the Multiplication Table, and illus-

trate its use in multiplying.

12. Why may the ciphers be omitted at the right of partial

products ?

13. Why commence multiplying the units' order in the multi-

plicand first, then the tens', and so on? Illustrate your answer by

an example.

14. Multiply 8795 by 629, multiplying first by the tens, then by the

hundreds, and last by the units.

DIVISION. 11

15. Multiply 3573 by 483, commencing with the thousands of the

multiplicand and hundreds of the multiplier.

16. Show that hundreds multiplied by hundreds will give ten thou-

sands in the product.

17. Multiplying thousands by thousands, what order will the pro-

duct be?

18. Name at sight the lowest order which each of the following

examples will give in the product :

(1.) 8000 X 8000 ; 2000000 x 3000 ; 5000000000 x 7000.

(2.) 40000 X 20000 ; 7000000 x 4000000.

19. What orders in 3928 can be multiplied by each order in 473, and

not have any order in the product less than thousands ?

DIVISION.

ILLUSTRATION OF PROCESS.

119. Prob. I. â€” To divide any number by any divisor

not greater tlian 12.

1. Divide 986 by 4.

Explanation. â€” Follow the analysis and notice each step in the

process; thus,

1. We commence by dividing the higher order of units. We know

that 9, the figure expressing hundreds, contains twice the divisor 4.

and 1 remaining. Hence 900 contains, according to (117 â€” 2), 200

times the divisor 4, and 100 remaining. We multiply the divisor 4 by

200, and subtract the product 800 from 986, leaving 186 of the dividend

yet to be divided.

ANALYSIS. 2. We know that 18, the number

4)986(200 expressed by the two left-hand

4x200 = 800 40 figures of the undivided dividend,

contains 4 times 4, and 2 remaining,

â– ^ ^ " Hence 18 tens, or 180, contains,

4x40 =160 6 according to (117â€”2), 40 times 4,

4x6 =24

n n n An 2 ^^^ 20 remaining. We multiply

^ the divisor 4 by 40, and subtract

the product 160 from 186, leaving

2 26 yet to be divided.

12

DIVISION,

3. We know that 26 contains 6 times 4, and 2 remaining, wliicli is

less than the divisor ; hence the division is completed.

4. We have now found that there are 200 fours in 800, 40 fours in

160^, and 6 fours in 26, and 2 remaining ; and we know that 800 + 160

+ 26 = 986. Hence 986 contains (200 + 40 + 6) or 246 fours, and 2

remaining. The remainder is placed over the divisor and written after

the quotient ; thus, 246f .

SHORT AND Tj O N G DIVISION C O M P JL H E D .

121. Compare carefully the following forms of writing the

work in division :

(1-

FORM USED FOR

Two Bteps in the

4

)

EXPLANATION.

process written.

) 986 ( 200

(2.)

liONG DIVISION.

One step written.

4 ) 986 ( 246

(3.)

SHORT DIVISION.

Entirely mental.

4)986

4x200=

800

40

8

""2461

186

6

18

4x 40 =

160

246

16

26

26

4x 6 =

24

24

To divide any number by any given

Explanation. â€” 1. We find how many

times the divisor is contained in the

fewest of the left-hand figures of the

dividend which will contain it.

59 is contained 3 times in 215, with a

remainder 38, hence, according to

(115â€”1), it is contained 300 times in

21500, with a remainder 3800.

2. We annex the figure in the next

lower order of the dividend to the

remainder of the previous division, and divide the number thus found

by the divisor. 2 tens annexed to 380 tens make 382 tens.

129. Prob. II.â€” To

ivisor.

1. Divide 21524 by 59,

59)21524(364

177

382

354

284

236

48

DIVISION. 13

59 is contained 6 times in 382, with a remainder 28 ; hence, according

to (1 15 â€” 1), it is contained 60 times in 3820, with a remainder 280.

3. We annex the next lower figure and proceed as before.

137. Prob. III. â€” To divide by using tlie factors of

the divisor.

Ex. 1. Divide 315 by 35.

5)315 Explanation.â€” 1. The divisor 35 = 7 fives.

7 nveVy^ fives ^- ^^^^^^"^ *^^^ ^^^ ^^ ^' ^^^ ^"^ *^'*

â€” q 315 = 63>eÂ«. (138.)

3. The 63 fives contain 9 times 7 fives;

hence 315 contains 9 times 7 fives or 9 times 35.

Ex. 2. Divide 359 by 24.

2 |^5_9

3 tivos I 1 7 9 twos and 1 remaining = 1

4 (3 twos) I 5 9 (3 twos) and 2 twos remaining = 4

Quotient, 1 4 and 3 (3 tivos) remaining = 18

True remainder, 2 3

Explanation.â€” 1. The divisor 24 = 4x3x2 = 4 (3

2. Dividing 359 by 2, we find that 359=179 twos and 1 unit remaining.

3. Dividing 179 twos by 3 twos, we find that 179 twos â€”. 59 (3 twos) and

2 twos remaining.

4. Dividing 59 (3 twos) by 4 (3 twos), we find that 59 (3 twos) contain

4 (3 twos) 14 times and 3 (3 twos) remaining.

Hence 359, which is equal to 59 (3 twos) and 2 twos + 1, contains

4 (3 twos), or 24, 14 times, and 3 (3 twos) + 2 twos + 1, or 23, remaining.

143. Prob. IV. â€” To divide v^hen the divisor con-

sists of only one order of units.

1. Divide 8736 by 500. Explanation.â€” 1. We divide

~ \ ft 7 I ^ fi ^^^^ ^^ *^^^ factor 100. This is

^ LlÂ±Â±râ€” done by cutting off 36, the units

1 7 and 236 remaining. and tens at the right of the divi-

dend.

2. We divide the quotient, 87 hundreds, by the factor 5, which gives

a quotient of 17 and 2 hundred remaining, which added to 36 gives 236,

-^Jf

^<^/>-^^.^

COMPLETE

ARITHMETIC,

ORAL AND WRITTElSr.

PART SECOND.

BY

MALCOLM MacVICAR, Ph.D., LL.D.,

FBINCIFAIi STATE NOBMAL SCHOOL, POTSDAM, N. Y.

PUBLISHED BY

TAINTOR BROTHERS, MERRILL & CO.,

NEW YORK.

\

Copyright by

TAINTOR BROTHERS, MERRILL & CO.,

1878.

Electrotyped by Smith & MoDougal, 83 Beekinan St., N. Y.

^^ (ft ^ ^

QA 10:

Eaac ,

â€¢-*g^| PREFACE, l^^^**

"^ "^^^^

THE aim of the author in the preparation of this work may

be stated as follows :

1. To present each subject in arithmetic in such a manner

as to lead the pupil by means of preparatory steps and proposi-

tions which he is required to examine for himself, to gain clear

perceptions of the elements necessary to enable him to grasp

as a reality the more complex and complete processes.

2. To present, wherever it can be done, each process object-

ively, so that the truth under discussion is exhibited to the eye

and thus sharply defined in the mind.

3. To give such a systematic drill on oral and written exer-

cises and review and test questions as will fix permanently in

the mind the principles and processes of numbers with their

applications in practical business.

4. To arrange the pupil's work in arithmetic in such a manr

ner that he will not fail to acquire such a knowledge of princi-

ples and facts, and to receive such mental discipline, as will fit

him properly for the study of the higher mathematics.

The intelligent and experienced teacher can readily deter-

mine by an examination of the work how well the author has

succeeded in accomplishing his aim.

M57?0.'f7

PREPACK.

Special attention is invited to the method of presentation

given in the teacher's edition. This is arranged at the begin-

ning of each subject, just where it is required, and contains

definite and full instructions regarding the order in which the

subject should be presented, the points that require special

attention and illustration, the kind of illustrations that should

be used, a method for drill exercise, additional oral exercises

where required for the teacher's use, and such other instructions

as are necessary to form a complete guide to the teacher in the

discussion and presentation of each subject.

The plan adopted of having a separate teacher's edition

avoids entirely the injurious course usually pursued of cum-

bering the pupil's book with hints and suggestions which are

intended strictly for the teacher.

Attention is also invited to the Properties of Numbers, Great-

est Common Divisor, Fractions, Decimals, Compound Num-

bers, Business Arithmetic, Ratio and Proportion, Alligation,

and Square and Cube Root, with the belief that the treatment

will be found new and an improvement upon form.er methods.

The author acknowledges with pleasure his indebtedness to

Prof. D. H. Mac Vicar, LL.D., Montreal, for valuable aid

rendered in the preparation of the work, and to Charles D.

McLean, A. M., Principal of the State Normal and Training

School, at Brockport, N. Y., for valuable suggestions on

several subjects.

M. MacVICAR.

Potsdam, September^ 1877.

.^

^i^^

EZL

C..

BEVIMW OF PART FIRST.

PAGE

Notation and Numeration. 1

Addition 4

Subtraction 5

Multiplication 7

Division 11

Properties of Numbers. ... 16

Exact Division 16

Prime Numbers 19

Factoring 20

Cancellation 31

Greatest Common Divisor.. 22

Least Common Multiple. ... 25

Fractions 29

Complex Fractions 40

Decimal Fractions 46

Denominate Numbers 59

Metric System 75

FAllT SFCOND,

Business Arithmetic 79

Aliquot Parts 82

Business Problems 85

Applications 100

Profit and Loss 101

Commission 103

Insurance 105

Stocks 107

Taxes Ill

Duties or Customs 113

Review and Test Questions. 114

Interest 115

PAGB

Method by Aliquot Parts. 117

Method by Six Per Cent.. 119

Method by Decimals 121

Exact Interest 122

Compound Interest 127

Interest Tables 120

Annual Interest 131

Partial Payments 1 32

Discount 136

Bank Discount 138

Exchange 141

Domestic Exchange 142

Foreign Exchange 146

Equation of Payments 151

Review and Test Questions. 160

Ratio 161

Proportion 169

Simple Proportion 170

Compound Proportion 174

Partnership 177

Alligation Medial 181

Alligation Alternate 182

Involution 187

Evolution 189

Progressions 205

Arithmetical Progression. 206

Geometrical Progression.. 208

Annuities 210

Mensuration 213

Review and Test Examples. 227

Answers 243

â™¦>â– â™¦â– â™¦Â»â–

REVIEW OF PART FIRST.*

.(Â»H^^> Y^85^^V^^

NOTATION AND NUMERATION,

DEFINITIONS.

11. A Unit is a single thing, or group of single things,

regarded as one ; as, one ox, one yard, one ten, one hundred.

12. Units are of two kinds â€” Mathematical and

Common. A mathematical unit is a single thing which has a

fixed value ; as, one yard, one quart, one hour, one ten, A

common unit is a single thing which has no fixed value ; as,

one house, one tree, one garden, one farm.

* Note. â€” The first 78 pages of this part contains so much of the matter

In Part First as is necessary for a thorough review of each subject, in-

cluding all the tables of Compound Numbers. For convenience in

making references, the Articles retained are numbered the same as in

Part First. Hence the numbers of the Articles are not consecutive.

2 NOTATION AND NUMERATION.

13. A Number is a unit, or collection of units ; as, one

man, three houses, /(92<r, six hundred.

Observe, the nwnher is "the how many" and is represented by what-

ever answers the question, How many ? Thus in the expression seven

j&vda, seven represents the number.

14. The Unit of a Number is one of the things

numbered.

Thus, the unit of eight bushels is one bushel, of five boys is one hoy^

of nine is one.

15. A Concrete Number is a nnmhor which is applied

to objects that are named ; as four chairs, ten lells,

16. An Abstract Number is a number which is not

applied to any named objects ; as nine, five, thirteen.

17. Like Numbers are such as have the same unit.

Thus, four windows and eleven windows are like numbers, eight and

ten, three hundred and seven hundred.

18. Unlike Numbers are such as have different units.

Thus, twelve yards and five days are unlike numbers, also six cents

and nine minutes.

19. Figures are characters used to express numbers.

20. The Value of a figure is the number whidi it

represents.

21. The Simple or Absolute Value of a figure is the

number it represents when standing alone, as 8.

22. The Local or Representative Value of a figure

is the number it represents in consequence of the place it

occupies.

Thus, in 66 the 6 in the second place from the right represents a num-

ber ten times as great as the 6 in the first place.

2.3. Notation is the method of writing numbers by

means of figures or letters.

NOTATION AND NUMERATION. 3

24. Numeration is the method of reading numbers

which are expressed by figures or letters.

35. A Scale in Arithmetic is a succession of mathematical

units which increase or decrease in value according to a fixed

order.

26. A Decimal Scale is one in which the fixed order

of increase or decrease is uniformly ten.

This is the scale used in expressing numbers by figures.

â€¢ 27. Arithmetic is the Science of Numbers and the Art

of Computation.

REVIEW AND TEST QUESTIONS.

31. Study carefully and answer each of the following

questions :

1. Define a scale. A decimal scale.

2. How many figures are required to express numbers in the decimal

scale, and why ?

3. Explain the use of the cipher, and illustrate by examples.

4. State reasons why a scale is necessary in expressing numbers.

5. Explain the use of each of the three elementsâ€” /^wres, plare, and

comma â€” in expressing numbers.

6. What is meant by the simple or absolute value of figures ? What

by the local or representative value?

7. How is the local value of a fig'ure affected by changing it from the

first to the third place in a number ?

8. How by changing a figure from the second to the fourth ? From

the fourth to the ninth ?

9. Explain how the names of numbers from twelve to twenty are

formed. From twenty to nine hundred ninety.

10. What is meant by a period of figures ?

11. Explain how the name for each order in any period is formed.

12. State the name of the right-hand order in each of the first six

periods, commencing with units.

13. State the two things mentioned in (9) which must be observed

when writing large numbers.

14. Give a rule for reading numbers ; also for writing numbers.

ADDITION.

ADDITIOIT,

50. Addition is the process of uniting two or more

numbers into one number.

51. Addends are the numbers added.

53. The Sum or Amount is the number found by

addition.

53. The Process of Addition consists in forming

units of the same order into groups of ten, so as to express

their amount in terms of a higher order.

54. The Sign of Addition is +, and is redAplus,

When placed between two or more numbers, thus, 8 + 3 + 6 + 2 + 9, it

means that they are to be added.

55. The Sign of Equality is =, and is read equals,

or equal to ; thus, 9 + 4 = 13 is read, nine plus four equals

thirteen,

5Q. Principles. â€” I. Only numbers of the same denom-

ination and units of the same order can he added.

II. The sum is of the same denomination as the

addends.

III. The whole is equal to the sum of all the parts.

REVIEW AND TEST QUESTIONS.

57. 1. Define Addition, Addends, and Sum or Amount.

2. Name each step in the process of Addition.

3. Why place the numbers, preparatory to adding, units under units,

tens under tens, etc.?

4. Why commence adding with the units' column ?

5. What objections to adding the columns in an irregular order?

Illustrate by an example.

6. Construct, and explain the use of the addition table.

7. How many combinations in the table, and how found?

8. Explain carrying in addition. What objection to the use of the

word ?

SUBTRACTION. S

9. Define counting and illustrate by an example.

10. Write five examples illustrating the general problem of addition,

" Given all the parts to find the whole,"

11. State the difference between the addition of objects and the addi-

tion of numbers.

12. Show how addition is performed by using the addition table.

13. What is meant by the denomination of a number? What by

units of the same order ?

14. Show by analysis that in adding numbers of two or more places,

the orders are treated as independent of each other.

SUBTEACTION.

â€¢70, Subtraction is the process of finding the difference

between two numbers.

71# The Minuend is the greater of two numbers whose

difference is to be found.

72. The Subtrahend is the smaller of two numbers

whose difference is to be found.

73. The Difference or Memainder is the result

obtained by subtraction.

74. The JPi^ocess of Subtraction consists in com-

paring two numbers, and resolving the greater into two parts,

one of which is equal to the less and the other to the differ-

ence of the numbers.

75. The Sign of Subtraction is â€” , and is called

minus.

When placed between two numbers it indicates that their difference

is to be found ; thus, 14 â€” 6 is read, 14 minus 6, and means that the dif-

ference between 14 and 6 is to be found.

76. JParentheses ( ) denote that the numbers inclosed

between them are to be considered as one number.

77. A Vinculum affects numbers in the same

manner as parentheses.

Thus, 19 + (13â€” 5), or 19 + 13â€”5 signifies that the difference between

13 and 5 is to b*e added to 19.

6 SUBTRACTION.

78. Pri:n'Ciples. â€” /. Only like mtmbers and units of

the same order can he subtracted.

II. The minuend is the sum of the subtrahend and>

difference, or the minuend is the whole of which the

subtrahend and difference are the parts.

III. An equal increase or decrease of the minuend

and subtrahend does not change the difference.

REVIEW AND TEST QUESTIONS.

79. 1. Define the process of subtraction. Illustrate each step by

an example.

2. Explain how subtraction should be performed when an order in the

subtrahend is greater than the corresponding order in the minuend.

Illustrate by an example.

3. Indicate the difference between the subtraction of numbers and the

subtraction of objects.

4. When is the result in subtraction a remainder, and when a di {Ter-

ence?

5. Show that so far as the process with numbers is concerned, the

result is always a difference.

6. Prepare four original examples under each of the following prob-

lems and explain the method of solution :

Prob. I.â€” Given the wliole and one of the parts to find the other part.

Prob. II. â€” Given the sum of four numbers and three of them to find

the fourth.

7. Construct a Subtraction Table.

8. Define counting by subtraction.

9. Show that counting by addition, when we add a number larger

than one, necessarily involves counting by subtraction.

10. What is the difference between the meaning of denomination and

orders of units ?

11. State Principle III and illustrate its meaning by an example.

12. Show that the difference between 63 and 9 is the same as the

difference between (63 + 10) and (9 + 10).

13. Show that 28 can be subtracted from 92, without analyzing the

minuend as in (64), by adding 10 to each number.

14. What must be added to each number, to subtract 275 from 829

without analyzing the minuend as in (C>4:) ?

15. What is meant by borrowing and canning in subtraction ?

MULTIFLICATION. 7

MULTIPLIOATIOIT.

ILIjITSTMATION of l*JtOCESS.

92. Step II. â€” To multiply hy using the parts of the

midtiplier.

1. The multiplier may be made into any desired parts, and tlie mul-

tiplicand taken separately the number of times expressed by each part.

The sum of the products thus found is the required product.

Thus, to find 9 times 12 we may take 4 times 12 which are 48, then 5

times 12 which are 60. 4 times 12 plus 5 times 12 are 9 times 12 ;

hence, 48 plus 60, or 108, are 9 times 12.

2. When we multiply by one of the equal parts of the multiplier, we

find one of the equal parts of the required product. Hence, by multi-

plying the part thus found by the number of such parts, we find the

required product.

For example, to find 12 times 64 we may proceed thus :

(1.) ANALYSIS. (2.)

64x4 = 256^ 64

64x 4 = 256>=3 times 256 4

\

64x4 = 256 J 256

64x12 = 768 3

768

(1.) Observe, that 12 = 4 + 4 + 4 ; hence, 4 is one of the 3 equal

parts of 12.

(2.) That 64 is taken 12 times by taking it 4 times + 4 times -t- 4 times,

as shown in the analysis.

(3.) That 4 times 64, or 256, is one of the 3 equal parts of 12 times 64.

Hence, multiplying 256 by 3 gives 12 times 64, or 768.

3. In multiplying by 20, 30, and so on up to 90, we invariably multi-

ply by 10, one of the equal parts of these numbers, and then by the

number of such parts.

For example, to multiply 43 by 30, we take 10 times 43, or 430, and

multiply this product by 3 ; 430 x 3 = 1290, which is 30 times 43. We

multiply in the same manner by 200, 300, etc., 2000, 3000, etc.

f MULTIPLICATION.

93. Prob. II. â€” To multiply by a number containing

only one order of units.

1. Multiply 347 by 500.

(1.) ANALYSIS. (2.)

Firststep, 347x100= 34700 347

Second Btep, 34700 X 5 = 173500 500

173500

Explanation. â€” 500 is equal to 5 times 100 ; hence, by taking 347,

as in first step, 100 times, 5 times this result, or 5 times 34700, as shown

in second step, will make 500 times 347. Hence 173500 is 500

times 347.

In practice we multiply first by the significant figure, and annex to

the product as many ciphers as there are ciphers in the multiplier, as

shown in (2).

96. Prob. III. â€” To multiply by a number containing

two or more orders of units.

1. Multiply 539 by 374.

(1.) analysis.

Multiplicand.

Multiplier.

539X 4= 2156 1st partial product.

539X374 = ^539X 70= 37 7 30 2d partial product

539x300 = 161700 3d partial product.

2 015 8 6 Whole product.

=1

Explanation. â€” 1. The multiplier, 374, is analyzed into the parts 4,

70, and 300, according to (92).

2. The multiplicand, 539, is taken first 4 times = 2156 (8G) ; then

70 times = 37730 (93) ; then 300 times = 161700 (93).

3. 4 times + 70 times + 300 times are equal to 374 times ; lience the

sum of the partial products, 2156, 37730, and 161700, is equal to 374

times 539 = 201586.

4. Observe, that in practice we arrange the partial products as shown

in (2), omitting the ciphers at the right, and placing the first significant

figure of each product under the order to which it belongs.

MULTIP LIGATION. SF

DEFINITIONS.

100. Multiplication is the process of taking one

number as many times as there are units in another.

101. The Multiplicand is the number taken, or mul-

tiplied.

103. The Multiplier is the number which denotes how

many times the multiplicand is taken.

103. The JProduct is the result obtained by multipli-

cation.

104. A Partial Product is the result obtained by

multiplying by one order of units in the multiplier, or by any

part of the multiplier.

105. The Total or Whole Product is the sura of all

the partial products.

106. The Process of Multiplication consists, /rs^,

in finding partial products by using the memorized results of

the Multiplication Table; second, in uniting these partial

products by addition into a total product.

107. A Factor is one of the equal parts of a number.

Thus, 12 is composed of six 2's, four 3's, three 4's, or two 6's ; hence,

2, 3, 4, and 6 are factors of 12.

The multiplicand and multiplier are factors of the product. Thus,

37 X 25 = 925. The product 925 is composed of twenty-five 37's, or

tMrtyse'cen 25's. Hence, both 37 and 25 are equal parts or factors

of 925.

108. The Sign of Multiplication is x, and is read

times, or mitltipUed hy.

When placed between two numbers, it denotes that either is to be

multiplied by the other. Thus, 8x6 shows that 8 is to be taken 6 times,

or that 6 is to be taken 8 times ; hence it may be read either 8 times 6 or

6 times 8.

10 MULTIPLICATION.

109. Principles. â€” /. Tlve muUiplicand may he either

an abstract or concrete number.

II. Tlve multiplier is always an abstract number.

III. Tlxe product is of the same denomination as the

multiplicand.

KEVIEW AND TEST QUESTIONS.

110. 1. Define Multiplication, Multiplicand, Multiplier, and

Product.

2. What is meant by Partial Product ? Illustrate by an example.

3. Define Factor, and illustrate by examples.

4. What are the factors of 6 ? 14? 15? 9? 20? 24? 25? 27?

32? 10? 30? 50? and 70?

5. Show that the multiplicand and multiplier are factors of the

product.

6. What must the denomination of the product always be, and why ?

7. Explain the process in each of the following cases and illustrate

by examples :

I. To multiply by numbers less than 10.

II. To multiply by 10, 100, 1000, and so on.

III. To multiply by one order of units.

XV. To multiply by two or more orders of units.

V. To multiply by the factors of a number (92 â€” 2). -

8. Give a rule for the third, fourth, and fifth cases.

9. Give a rule for the shortest method of- working examples where

both the multiplicand and multiplier have one or more ciphers on

the right.

10. Show how multiplication may be performed by addition.

11. Explain the construction of the Multiplication Table, and illus-

trate its use in multiplying.

12. Why may the ciphers be omitted at the right of partial

products ?

13. Why commence multiplying the units' order in the multi-

plicand first, then the tens', and so on? Illustrate your answer by

an example.

14. Multiply 8795 by 629, multiplying first by the tens, then by the

hundreds, and last by the units.

DIVISION. 11

15. Multiply 3573 by 483, commencing with the thousands of the

multiplicand and hundreds of the multiplier.

16. Show that hundreds multiplied by hundreds will give ten thou-

sands in the product.

17. Multiplying thousands by thousands, what order will the pro-

duct be?

18. Name at sight the lowest order which each of the following

examples will give in the product :

(1.) 8000 X 8000 ; 2000000 x 3000 ; 5000000000 x 7000.

(2.) 40000 X 20000 ; 7000000 x 4000000.

19. What orders in 3928 can be multiplied by each order in 473, and

not have any order in the product less than thousands ?

DIVISION.

ILLUSTRATION OF PROCESS.

119. Prob. I. â€” To divide any number by any divisor

not greater tlian 12.

1. Divide 986 by 4.

Explanation. â€” Follow the analysis and notice each step in the

process; thus,

1. We commence by dividing the higher order of units. We know

that 9, the figure expressing hundreds, contains twice the divisor 4.

and 1 remaining. Hence 900 contains, according to (117 â€” 2), 200

times the divisor 4, and 100 remaining. We multiply the divisor 4 by

200, and subtract the product 800 from 986, leaving 186 of the dividend

yet to be divided.

ANALYSIS. 2. We know that 18, the number

4)986(200 expressed by the two left-hand

4x200 = 800 40 figures of the undivided dividend,

contains 4 times 4, and 2 remaining,

â– ^ ^ " Hence 18 tens, or 180, contains,

4x40 =160 6 according to (117â€”2), 40 times 4,

4x6 =24

n n n An 2 ^^^ 20 remaining. We multiply

^ the divisor 4 by 40, and subtract

the product 160 from 186, leaving

2 26 yet to be divided.

12

DIVISION,

3. We know that 26 contains 6 times 4, and 2 remaining, wliicli is

less than the divisor ; hence the division is completed.

4. We have now found that there are 200 fours in 800, 40 fours in

160^, and 6 fours in 26, and 2 remaining ; and we know that 800 + 160

+ 26 = 986. Hence 986 contains (200 + 40 + 6) or 246 fours, and 2

remaining. The remainder is placed over the divisor and written after

the quotient ; thus, 246f .

SHORT AND Tj O N G DIVISION C O M P JL H E D .

121. Compare carefully the following forms of writing the

work in division :

(1-

FORM USED FOR

Two Bteps in the

4

)

EXPLANATION.

process written.

) 986 ( 200

(2.)

liONG DIVISION.

One step written.

4 ) 986 ( 246

(3.)

SHORT DIVISION.

Entirely mental.

4)986

4x200=

800

40

8

""2461

186

6

18

4x 40 =

160

246

16

26

26

4x 6 =

24

24

To divide any number by any given

Explanation. â€” 1. We find how many

times the divisor is contained in the

fewest of the left-hand figures of the

dividend which will contain it.

59 is contained 3 times in 215, with a

remainder 38, hence, according to

(115â€”1), it is contained 300 times in

21500, with a remainder 3800.

2. We annex the figure in the next

lower order of the dividend to the

remainder of the previous division, and divide the number thus found

by the divisor. 2 tens annexed to 380 tens make 382 tens.

129. Prob. II.â€” To

ivisor.

1. Divide 21524 by 59,

59)21524(364

177

382

354

284

236

48

DIVISION. 13

59 is contained 6 times in 382, with a remainder 28 ; hence, according

to (1 15 â€” 1), it is contained 60 times in 3820, with a remainder 280.

3. We annex the next lower figure and proceed as before.

137. Prob. III. â€” To divide by using tlie factors of

the divisor.

Ex. 1. Divide 315 by 35.

5)315 Explanation.â€” 1. The divisor 35 = 7 fives.

7 nveVy^ fives ^- ^^^^^^"^ *^^^ ^^^ ^^ ^' ^^^ ^"^ *^'*

â€” q 315 = 63>eÂ«. (138.)

3. The 63 fives contain 9 times 7 fives;

hence 315 contains 9 times 7 fives or 9 times 35.

Ex. 2. Divide 359 by 24.

2 |^5_9

3 tivos I 1 7 9 twos and 1 remaining = 1

4 (3 twos) I 5 9 (3 twos) and 2 twos remaining = 4

Quotient, 1 4 and 3 (3 tivos) remaining = 18

True remainder, 2 3

Explanation.â€” 1. The divisor 24 = 4x3x2 = 4 (3

2. Dividing 359 by 2, we find that 359=179 twos and 1 unit remaining.

3. Dividing 179 twos by 3 twos, we find that 179 twos â€”. 59 (3 twos) and

2 twos remaining.

4. Dividing 59 (3 twos) by 4 (3 twos), we find that 59 (3 twos) contain

4 (3 twos) 14 times and 3 (3 twos) remaining.

Hence 359, which is equal to 59 (3 twos) and 2 twos + 1, contains

4 (3 twos), or 24, 14 times, and 3 (3 twos) + 2 twos + 1, or 23, remaining.

143. Prob. IV. â€” To divide v^hen the divisor con-

sists of only one order of units.

1. Divide 8736 by 500. Explanation.â€” 1. We divide

~ \ ft 7 I ^ fi ^^^^ ^^ *^^^ factor 100. This is

^ LlÂ±Â±râ€” done by cutting off 36, the units

1 7 and 236 remaining. and tens at the right of the divi-

dend.

2. We divide the quotient, 87 hundreds, by the factor 5, which gives

a quotient of 17 and 2 hundred remaining, which added to 36 gives 236,

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