.5

7

3

Or more generally thus: let of be the compound fraction. Then

proceed with any number of fractions, first reducing two of them to a simple fraction, and then taking that and a third, and so on.

Hence it appears that the word of in a compound fraction signifies multiplication.

CASE 5. To reduce fractions of different denominators to equivalent fractions having a common denominator.

45. THE general rule for this purpose may be derived thus. Let the

And the terms of the fraction multiplied by the denominator 3 gives

2

common denominator 21, are respectively equal to the fractions and

Next, taking and, and multiplying the terms of the former frac

15 21

tion by 11, and those of the latter by 21, we get

having the common de

having the common denoininator 3 x 7 x 11, are equal to

spectively. And the same method may be extended to any number of fractions,

Hence it appears that the new numerators are found by multiplying each numerator into all the denominators except its own, and that the common denominator is the continued product of all the denominators.

Ex. 2. Reduce, , and to equivalent fractions having a common denominator.

When any factors in the new numerators and common denominator have a common measure or divisor, resolve them into other factors, then (39) reject the like number of like factors in the numerators and denominator, and the fractions will be reduced to the lowest terms which admit of a common de nominator.

Ex. 3. Let 4, 1, and, be reduced to a common denominator.

6 X 9

4 X 9

The fractions with a common denominator are

4 X 6 X 9'

4X6X 9,

6 and 9; therefore if 6 in the first and third fractions, and 6, 4 and 9 in

now 2, and 3, are the respective divisors of 4 and 6, and

the second, are resolved into the factors 2 and 3, the fractions will be

4

9 6

or

4

and ; where the common denominator 36 is the least 4 x 9' 30' 36' 36

common multiple or number divisible by 4, 6, and 9. And in the same manner the least common multiple of other proposed numbers may be found, first making them the denominators of fractions having 1 for each

numerator.

46. But the least common multiple is readily found by the following rule. (See art. 212. vol. 2.)

Write down the proposed numbers in a line, and divide by the prime number 2 as long as it will divide two or more of them without a remainder, and set down the quotients together with the undivided numbers in a line below.--Divide this second line by 2, and also the third line, &c. in the same manner, if they will divide. This done, proceed with 3 the next prime number, and so on to 5, or 7, &c, till there are no two numbers that can be thus divided: Then the continued product of the divisors, the last quotients, and the undivided numbers, is the multiple sought.

Examp. 1. To find the least common multiple of 7, 24, 40, 45, and 72,

2520 is the multiple required;

Then 2 x 2 x 2 x 3 x 3 x 5 x 7

or the least number divisible by 7, 24, 40, 45, and 72.

Exump. 2. Required the least common multiple of 27, 66, 135, 275,

and 675.

47. When the least denominator of two fractions exactly divides the greatest, multiply the terms of that fraction which hath the least denominator by the quotient.

Thus and are brought to a common denominator by multiplying the numerator and denominator of 3 by 2 (the quotient of 8 divided by 4).

And,, are brought to a common denominator by multiplying the terms of by 4; and those of % by 2; the three required fractions being, 12, 11.

10

Or thus:

48. HAVING reduced the given fractions to their lowest terms, find the least common multiple of the denominators, which divide by the denominators, and multiply the numerators by the corresponding quotients; then the products placed over the said multiple give the fractions in their lowest terms.

3 5 14' 22' valent fractions having the least common denominator.

Thus, let it be required to reduce the fractions

The least common multiple of 14, 22, and 121 is 1694:

49. REDUCE Compound fractions to simple ones; and all the fractions to a common denominator. Then add the numerators together and place the sum over the common denominator for the answer.

When the fractions are large, or numerous, it will be best to reduce them to the least common denominator.

The fractions when brought to a common denominator will be 29,

and brought to a common denominator are 7 and 12:

then 7+12=19.

Ans. 12=14.

་

50. When mixed numbers, or mixed numbers and fractions are to be added together, bring the fractions to a common denominator, then set down the integers as in common addition, and the fractions on the right hand :

Add the fractions together, and carry the integers (if any) from