1

6

8

98. If the numerator of the fraction be multiplied by 2, without altering the denominator, the product is, which fraction is twice as great as , for g

contains twice as many equal parts as does. If the denominator be divided by 2, keeping the numerator as it is, the quotient is,

and as the unit is now divided into half

as many equal parts, each of these is twice as great; therefore, twice 3.

It follows that, to multiply a fraction by any whole number, we may either multiply its numerator or divide its denominator by that number.

8

99. Let us take the fraction : if its numerator be divided by 4, the quotient is, which is 4 times as small, for it contains 1-fourth of this number of equal parts. If the de

nominator 9 be multiplied by 4 the result is, a fraction 4 times as small as §, for it contains the same number of parts,

each of which is 4 times less. Therefore, to divide a fraction by any whole number, we may either divide its numerator or multiply its denominator by that number.

100. EXERCISES UPON ARTICLE 98.

1. Multiplication of a fraction by a whole number, when operating upon its numerator.

8

Multiply by 2: 3, 4, 7, 7

5 8'

6

2

Multiply by 3: 4, 4, 75, 75, 1.

2. Multiplication of a fraction by any whole number operating on its denominator.

3. Multiplication of a fraction by a whole number, operating on its numerator, or on its denominator, if possible.

Multiply by 3: 8, fr, 12, 14, 3, 18, 19, 27.

19

Multiply by 8: 75, 14, 18, 13, 11, 15, 25, 17.

101. EXERCISES UPON ARTICLE 99.

1. Division of a fraction by a whole number, operating on its

2. Division of a fraction by any whole number, operating on its denominator.

3. Division of a fraction by any whole number, operating on the numerator, if possible, or on the denominator.

Divide by 4 17, 8, 11, 7, 14, 29, 37. 18.

67

9 36

5 25

Divide by 6:17, 11, 34, 13, 8, 38, 11, 11.

ין

102. Since a fraction is multiplied by an integer, when we multiply its numerator, and is divided by an integer when its denominator is multiplied, what will result if we multiply both the numerator and denominator by the same quantity?

instance,

==

3×2 6
4X2 8

For

Now, evidently, on one hand, we have made

the fraction twice as great as, and on the other twice as small; therefore, the value of the fraction is not altered. The accompanying diagrams illustrate the truth of this deduction :—

It is easily shown that the same result is obtained if the numerator and denominator be both divided by the same number; 6÷2 3

for instance,

8÷2 1

Therefore, if the terms of a fraction

be multiplied or divided by the same number, the value of the fraction will be unaltered.

103. EXERCISES.

1. Multiplication of the terms of a fraction by the same number.

Multiply by 4: 3, 3, 3, 73, 18.

Multiply by 7: 1, § 3, 11, 14.

2. Division of the terms of a fraction by the same number.

104. By article 102, we see that a fraction can be expressed by different terms, for instance :

But as we form a clearer idea of a fraction when expressed by small numbers than large ones, and as the numbers being small shorten the operations, it will be useful to simplify fractions, or to divide the terms by any common factor, or common measure ; the fraction will be expressed in its lowest terms when the factor is the greatest common measure.

Thus, in 33, it is found that 8 is contained 4 times in the numerator and 5 times in the denominator; then, 3 is equivalent to, or 3, and 8 is called the greatest common measure of 32 and 40. It must be observed that the same result can be obtained by dividing the terms by 2, three times in succession, 3 3 = 18 = 8% = 4; 2 is only a common measure of 32 and 40.

105. A number which contains another number exactly, is said to be a multiple of it (see Art. 69). 32 and 40 are multiples. of 8.

106. Beginners find it easier to reduce fractions by dividing the terms successively by common measures, as follows: =12 ==. The terms are divided by 2 as often as possible, and then by 3.

107. The following observations will be found useful:1. If the terms of the fraction are even, or if the figures in the units' place be divisible by 2, the numbers are divisible by 2.

2. If 4 is a measure of the figures of the last two places, the terms are divisible by 4. Ex. 13. Since 24 and 56 are 1244 31 divisible by 4 the fraction is reducible by 4:

=

156 4 39

3. If 8 is a measure of the figures in the three last places, then the terms are divisible by 8. Ex. 88. Since 8 is a measure both of 168 and 760, therefore the fraction is reducible

271
345

fraction end in 0, cut the same number of Ex. 33%

31000

=

6. If the sum of the figures of each term be divisible by 3 or 9, the fraction is reducible by 3 or 9. Ex. 7. Now, 4+7+1 12; and 5+7+3=15. Both 12 and 15 are divisible by 3, 471÷3 157 therefore the terms are reducible by 3, and

=

573÷3 191

7. If the sum of the figures of the even places be equal to the sum of the figures of the uneven places, or if one sum exceeds the other by 11, or by any multiple of eleven, in each term, then the fraction is reducible by 11. Ex. 45,

19162

Sum of figures of even places of numerator=4+9=13) dif. 0. =8+5=13)

وو

odd

Sum of figures of even places of denominator=9+6=152 dif. 11.

=1+1+2=45

8. Every number consisting of more than 2 figures is divisible. by 7, if after having doubled the units' figure and subtracted that product from the figure on the left hand of the units' place the remainder is 7 or a multiple of 7. If the remainder contains more than 2 figures, the units' figure of the remainder must be doubled, and the product subtracted from the figures on the left of the units' place, &c. The only exceptions to this law are 98 and 119. Here the figures on the left of the units must be subtracted from the units

doubled. Ex. 252. Multiply the units' figure of the numerator by 2: 2×2=4, subtract 4 from 25 leaves 21, double the units' figure of the denominator, 2×5=10, subtract 10 from 101 leaves 91; then, because 21 and 91 are multiples 252÷7 of 7, the fraction admits of being reduced by 7:

36 145

1015÷7

=

108. It would save time, and very often useless labour, if we had a method to ascertain directly the greatest common divisor, factor, or measure of two numbers. In order to make this inquiry, let us establish a few principles.

109. Since the common measure of two numbers is a number which divides them both exactly, we shall find that the measures of 36 and 60 are,

For 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.

For 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.

The measures common to both are: 1, 2, 3, 4, 6, 12. We see, then, that 12 is the greatest common measure of 36 and 60.

110. If we examine two numbers, as 36 and 60, and find their sum and difference, 36+60=96, 60—36=24, we notice that the common measure of the numbers 1, 2, 3, 4, 6, 12 measure also their sum and difference. Therefore, if one quantity measures two others, it measures likewise their sum and difference.

111. But as the multiple of any number is the sum of several numbers equal to the one expressed, it follows that if any number measures another number, it measures any multiple of the latter. Ex. 4 measures 8; and, therefore, measures 2 × 8, 3×8, 4 x 8, &c.

112. We may observe that the remainder of the division of two numbers is nothing but the difference between the dividend and the product of the divisor by the quotient, viz., a multiple of the latter. Hence, every number which measures both the dividend and the divisor, measures, likewise, the remainder. Ex. 13 measures both 234 and 65. Divide 234 by 65: the quotient is 3, and the remainder 39, which 13 measures.

113. These principles being understood, let it be required to find the greatest common measure of 1749 and 477.