EXAMPLES.

1. What is the least common multiple of 4, 5, 6 and 10% Operation,

×5)4 5 6 10

2. What is the common multiple of 6 and 8?

Ans. 24.

3. What is the least number that 3, 5, 8 and 12 will measure?

Ans. 120.

4. What is the least number that can be divided by the 9 digits separately, without a remainder?

Ans. 2520.

REDUCTION OF VULGAR FRACTIONS,

IS the bringing them out of one form into another, in order to prepare them for the operation of Addition, Subtraction, &c.

CASE I.

To abbreviate or reduce fractions to their lowest terms.

RULE.-1. Find a common measure, by dividing the greater term by the less, and this divisor by the remainder, and go on, always di viding the last divisor by the last remainder, till nothing remains; the last divisor is the common measure.*

2. Divide both of the terms of the fraction by the common measure, and the quotients will make the fraction required.

*To find the greatest common measure of more than two numbers, you must find the greatest common measure of two of them as per rule above; then, of that common measure and one of the other numbers, and so on through all the numbers to the last; then will the greatest common mea. sure last found be the answer.

Or, if you choose, you may take that easy method in Problem I. page 69.)

To reduce a mixed number to its equivalent improper fraction.

RULE.-Multiply the whole number by the denominator of the given fraction, and to the product add the numerator, this sum written bove the denominator will form the fraction required.

EXAMPLES.

1. Reduce 45% to its equivalent improper fraction.

4581-7=387 Ans. 2. Reduce 192 to its equivalent improper fraction.

18

3. Reduce 16, to an improper fraction.

Ans. 354

18

To find the value of an improper fraction. RULE.-Divide the numerator by the denominator, and the quo

tient will be the value sought.

CASE IV.

To reduce a whole number to an equivalent fraction, hav ing a given denominator.

RULE.-Multiply the whole number by the given denominator, place the product over the said denominator, and it will form the fraction required.

EXAMPLES.

1. Reduce 7 to a fraction whose denominator will be 9. Thus, 7×9=-63, and 3 the Ans. 2. Reduce 18 to a fraction whose denominator shall be 12. Ans. 216

3. Reduce 100 to its equivalent fraction, having 90 for a denominator. 00 Ans. 88° 8° = 1 8 °

CASE V.

To reduce a compound fraction to a simple one of equal value.

RULE.-1. Reduce all whole and mixed numbers to their equiva lent fractions.

2. Multiply all the numerators together for a new numerator, and all the denominators for a new denominator; and they will form the fraction required.

EXAMPLES.

1. Reduce of 3 of 3 of to a simple fraction.

2. Reduce of

3. Reduce of 1

10

4 Reduce 3 of 3 of 8 to a simple fraction.

Ans. 836

1500

Ans. 120-31

5. Reduce of 12 of 42 to a simple fraction.

36

Ans. 1880-21

NOTE. If the denominator of any member of a com pound fraction be equal to the numerator of another men

ber thereof, they may both be expunged, and the other members continually multiplied (as by the rule) will produce the fraction required in lower terms. 6. Reduce of 3 of 4 to a simple fraction.

7. Reduce of of 1⁄2 of 11⁄2 to a simple fraction.

CASE VI.

Ans.

To reduce fractions of different denominations to equiva lent fractions having a common denominator.

RULE I.

1. Reduce all fractions to simple terms.

2. Multiply each numerator into all the denominators except its own, for a new numerator; and all the denominators into each other continually for a common denominator; this written under the several new numerators will give the fractions required.

EXAMPLES.

1. Reduce,,, to equivalent fractions, having a common denominator.

1⁄2 + 1⁄2 + ?=24 common denominator.

2. Reduce, %, and 11, to a common denominator.

Ans. 848, 58, and $38,

3. Reduce,, §, and 7, to a common denominator.

192

Ans. 11, 17, 118, and 153

2899

6

4. Reduce, 2%, and, to a common denominator 800 300 400

5. Reduce 3, 3, and 12ļ, to a common denominator.

6. Reduce 3, 3, and § of 11, to a common denominatorAns. -768 2592 1980

3456 3456 3456

The foregoing is a general rule for reducing fractions to a common denominator; but as it will save much labour to keep the fractions in the lowest terms possible, the following Rule is much preferable.

RULE II.

For reducing fractions to the least common denominator. (By Rule, page 143) find the least common multiple of all the denominators of the given fractions, and it wili be the common denominator required, in which divide each particular denominator, and multiply the quotient by its own numerator, for a new numerator, and the new numerators being placed over the common denominator, will express the fractions required in their lowest terms,

EXAMPLES.

1. Reduce, 2, and §, to their least common denominator 4)2 4 8

2)2 1 2

1 1

1 4×2-8 the least com. denominator.

8÷2×1-4 the 1st numerator.

8-4×3-6 the 2d numerator.

8-8×5-5 the 3d numerator.

5

These numbers placed over the denominator, give the answer,, equal in value, and in much lower term than the general Rule would produce 2. Reduce, f, and, to their least

tor.

common denomin Ans. 4, 1, 45

48

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