EXAMPLES.

1. Reduce of of of to a simple fraction.

Or, by expunging the equal numerators and denominators, it will give as before.

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

Or, by expunging the equal numerators and denominators, it will

3. Reduce of of of 12 to a simple fraction.

Ans. 71=111.

CASE V.

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

RULE.-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.

158.

What is the rule for reducing fractions of different denominators to equivaalent fractions, having a common denominator ?

EXAMPLES.

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

Therefore, the new equivalent fractions are %, 8%, and 188, the

answer.

2. Reduce,, †, §, and 7, to fractions, having a common denominator.

Ans. 576 763 864 960 1008

1152 1152 11529 11529 1152°

NOTE. When there are whole numbers, mixed numbers, or compound fractions, given in the question, they must first be reduced to their simple forms.

CASE VI.

To reduce any given fractions to others, which shall have the least common denominator.

RULE. Find the least common multiple (by problem II, page 72,) of all the denominators of the given fractions, and it will be the common denominator required; then divide the common denominator by the denominator of each fraction, and multiply the quotient by the numerator for a new numerator; the new numerators written over the common denominator will form the fractions required in their lowest terms.

159. What is the rule for reducing any given fractions to others which shall have the least common denominator?

NOTE. The common denominator is a multiple of all the denominators, and consequently will divide by any of them; therefore, proper parts may be taken for all the numerators as required by

the rule.

EXAMPLES.

1. Reduce,, and to fractions having the least common denominator possible.

4)3 4 8

3 1 2

4×3×2-24 least common
denominator.

24-3X18 the first numerator; 24-4X3-18 the second numera

tor; 24-8X7=21 the third numerator.

Whence the required fractions are

8 18

2. Reduce 1, 3, 2, and to fractions having the least common denominator.

To reduce a fraction of one denomination to an equivalent fraction of a higher denomination.

RULE. Reduce the given fraction to a compound one by comparing it with all the denominations between it and that denomination you would reduce it to; lastly, reduce this compound fraction to a single one, by Case V, and you will have a fraction of the required denomination, equal in value to the given fraction.

NOTE. The reason of the rule may be shown in the following manner-As there are 12 pence in a shilling, four-fifths of one penny can be only a twelfth part as much of 12 pence or a shilling, as it is of one penny. Hence, to reduce four-fifths of a penny to the fraction of a shilling, the given fraction must be diminished 12 times, or one-twelfth of it will be the equivalent fraction of a shilling. And in general, the fraction of one denomination must be as much diminished to be an equivalent fraction of a higher denomination, as is indi

160. How do you reduce a fraction of one denomination to an equivalent fraction of a higher?

cated by the number of parts of the given denomination which make one of the higher denomination.

EXAMPLES.

1. Reduce of a cent to the fraction of a dollar.

By comparing it, it becomes 4 of of, which, reduced by Case V. will be 4X1 X1 = 4

and 7X10X10= 700

1 doll. Ans.

2. Reduce of a penny to the fraction of a pound.

Ans. £t

3. Reduce of an ounce to the fraction of a lb. Avoirdupois.

Ans. lb.

28

4. Reduce of a pennyweight to the fraction of a lb. Troy. Ans. 1lb.

CASE VIII.

To reduce a fraction of one denomination to the fraction of another, but less, retaining the same value.

RULE.-Multiply the given numerator by the parts of the denominations between it and that denomination you would reduce it to, for a new numerator, which place over the given denominator: Or, only invert the parts contained in the integer, and make of them a compound fraction as before; then reduce it to a simple one.

NOTE. This rule is the reverse of the preceding, and the reason of it may be shown in a similar manner. The fraction of a higher denomination is obviously less than the equivalent fraction of a lower denomination; for example, of a pound are shillings, or 12 shillings. Whence the value of the fraction must be increased, to render it an equivalent fraction of a lower denomination, so many times as there are parts of the less denomination in the higher.

EXAMPLES.

1. Reduce of a dollar to the fraction of a cent.

161. What is the rule for reducing a fraction of one denomination to another less, retaining the same value?

Ꮮ

By comparing the fraction it will be of of; then

2. Reduce of a pound to the fraction of a penny.

Ans. 3 d.

3. Reduce of a lb. Avoirdupois to the fraction of an ounce.

4. Reduce of a lb. Troy to the fraction of a pwt.

Ans. 4 oz.

Ans. pwt.

CASE IX.

To find the value of a fraction in the known parts of the integer, as of coin, weight, measure, &c.

RULE.-Multiply the numerator by the parts in the next inferior denomination, and divide the product by the denominator; and if any thing remain, multiply it by the next inferior denomination, and divide by the denominator as before, and so on, as far as necessary; and the quotients placed after one another, in their order, will be the answer required. Or, reduce the numerator, as if it were a whole number, to the lowest denomination, and divide the result by the denominator; the quotient will be the number of the lowest denomination, (which must be brought into higher denominations as far as it will go,) and the remainder will be a numerator, to be placed over the given denominator for a fraction of the lowest denomination.

NOTE. From this rule, in connexion with what has been said of Reduction of Federal Money, it appears, that, annexing to the given numerator as many ciphers, as will fill all the places to the lowest denomination, and dividing the number so formed by the denominator, the quotient will be the answer in the several denominations, and the remainder a numerator to be placed over the given denominator, forming a fraction of the lowest denomination.

162. How do you find the value of a fraction in the known parts of the integer ?