19. The "Greatest Common Measure" of two numbers is found by dividing the greater by the less, and the preceding divisor by the remainder, continually, till nothing is left: the last divisor is the greatest common measure required.

DEF. The Greatest Common Measure of two or more numbers is the greatest number which will divide each of them without remainder.

To find the greatest common measure of 189 and 224.

189) 224 (1
189

35) 189 (5
175

14) 35 (2

28

7) 14 (2

14

0

By proceeding according to the rule, it appears that 7 is the last divisor, or the Greatest Common Measure sought. The proof of this rule will be given hereafter. See Art. 103.

20. A fraction is reduced to its lowest terms by dividing its numerator and denominator by their greatest Common Measure.

By the Rule given in the last Art. the greatest Common Measure of the numerator and denominator is found to be 11;

and therefore

35

36

is the fraction in its lowest terms.

COR. If unity be the greatest Common Measure of the numerator and denominator, the fraction is already in its lowest terms.

21. To reduce any number of fractions to a common denominator.

Having reduced, if there be any, compound fractions to simple ones, and mixed numbers to improper fractions, multiply each numerator by all the denominators except its own, for the new numerator, and all the denominators together for a common denominator.

are the fractions required. These fractions are respectively equal to the former, the numerator and denominator in each case having been multiplied by the same numbers, namely, the product of the denominators of the rest (Art. 12).

22. If the denominator of one of two fractions contain the denominator of the other a certain number of times exactly, multiply the numerator and denominator of the latter by that number, and it will be reduced to the same denominator with the former.

Since 12 contains 3 four times exactly, multiply both the

2
3

numerator and denominator of by 4, and it becomes

fraction having the same denominator with

5

In the reduction of fractions to a common denominator the following rule is frequently required, in order that the reduced fractions may be in their lowest terms:

23. To find the "Least Common Multiple" of any numbers.

DEF. The "Least Common Multiple" of any numbers is the least number which is divisible by each of them without remainder.

To find it, write down in one line the numbers of which the least common multiple is required, separating them by some mark, as a comma. Divide all those which have a common measure by that common measure, and bring down the other numbers placed in a line with the quotients, separated as before; and repeat this process as long as any common measure exists between two or more of them. The Least Common Multiple required will be the continued product of the divisors and of the final quotients.

Ex. Required the Least Common Multiple of 8, 12, and 18. 2 8, 12, 18

Least Com. Mult. is 2×2×3×2×1×3 or 72.

The proof of this rule will be given hereafter. See Art. 115.

24. To reduce fractions to a common denominator, in their lowest terms, find the "Least Common Multiple" of all the denominators, and make that the common denominator by multiplying both the numerator and denominator of each fraction by the quotient of Least Common Multiple divided by the denominator.

Ex. Reduce to a common denominator

1 2 3 12
3'7' 14' 21

3

and

4

The Least Common Multiple of the denominators by Art. 23 is found to be 84; and therefore the required fractions are

28×1 12×2 6×3 4x12 21×3
28×3' 12×7' 6×14' 4×21' 21×4'

25. COR. By reducing fractions to a common denominator their values may be compared.

48

49

are and that is, the fractions have the same relative

84

84

values that 48 and 49 have.

26. By reducing fractions to a common numerator also their values may be compared.

12

52

Thus is

12

and

4 17'

when reduced to a common numerator, are

and and since the former of these fractions signifies that the 51 unit is divided into 52 equal parts of which 12 are taken, and the latter signifies that the unit is divided into 51 equal parts of which 12 are taken, it is obvious that the latter fraction is the greater of the two, or that which has the smaller denominator.

27. To find the value of a fraction of a proposed denomination in terms of a lower denomination.

Multiply the fraction by the number of integers of the lower denomination contained in one integer of the higher, and the product is the value required. The value of any fractional part of the lower denomination may be obtained in the same manner, till we come to the lowest.

The operation is usually performed in the following manner:

28. To reduce a quantity to a fraction of any denomination.

Make the given quantity the numerator, and the number of integers of its denomination in one of the proposed deno