Malcolm MacVicar.

A complete arithmetic : oral and written online

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Online LibraryMalcolm MacVicarA complete arithmetic : oral and written → online text (page 1 of 18)
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•-*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.



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 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.

Potsdam, September^ 1877.







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


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


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



.(»H^^> Y^85^^V^^



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 — 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. — 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.


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

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

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

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.


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

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.

• 27. Arithmetic is the Science of Numbers and the Art
of Computation.


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 elements— /^wres, plare, and
comma — 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.



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

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

5Q. Principles. — 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

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


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 ?


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.


•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 — , and is called

When placed between two numbers it indicates that their difference
is to be found ; thus, 14 — 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 + (13— 5), or 19 + 13—5 signifies that the difference between
13 and 5 is to b*e added to 19.


78. Pri:n'Ciples. — /. 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.


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-

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.— Given the wliole and one of the parts to find the other part.
Prob. II. — 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 ?




92. Step II. — To multiply hy using the parts of the

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


(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.


93. Prob. II. — 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


Explanation. — 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. — To multiply by a number containing
two or more orders of units.

1. Multiply 539 by 374.

(1.) analysis.


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.


Explanation. — 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.



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-

103. The Multiplier is the number which denotes how
many times the multiplicand is taken.

103. The JProduct is the result obtained by multipli-

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.


109. Principles. — /. 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


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

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

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 — 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.


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 ?



119. Prob. I. — To divide any number by any divisor
not greater tlian 12.

1. Divide 986 by 4.

Explanation. — 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 — 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,

■^ ^ " Hence 18 tens, or 180, contains,

4x40 =160 6 according to (117—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.



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 .


121. Compare carefully the following forms of writing the
work in division :



Two Bteps in the




process written.

) 986 ( 200



One step written.

4 ) 986 ( 246



Entirely mental.










4x 40 =






4x 6 =



To divide any number by any given

Explanation. — 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
(115—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.— To

1. Divide 21524 by 59,









59 is contained 6 times in 382, with a remainder 28 ; hence, according
to (1 15 — 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. — To divide by using tlie factors of
the divisor.

Ex. 1. Divide 315 by 35.

5)315 Explanation.— 1. The divisor 35 = 7 fives.

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

— q 315 = 63>e«. (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.— 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 —. 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. — To divide v^hen the divisor con-
sists of only one order of units.

1. Divide 8736 by 500. Explanation.— 1. We divide
~ \ ft 7 I ^ fi ^^^^ ^^ *^^^ factor 100. This is
^ Ll±±r— done by cutting off 36, the units

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

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,

1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Online LibraryMalcolm MacVicarA complete arithmetic : oral and written → online text (page 1 of 18)