CASE 2d. To reduce a mixed number to its equivalent improper fraction.

Multiply the integer by the denominator of the fraction, and to the product add the numerator, and the denominator being placed under this sum, will give the fraction required.

Example.

Let 54 be reduced to its equivalent improper fraction.

Reduce the following mixed numbers to their equivalent improper fractions.

CASE 3d. To reduce an improper fraction to its equivalent whole or mixed number.

Divide the numerator by the denominator for the integral part, and place the remainder, if any, over the numerator for the fractional part, and it will form the whole or mixed number required.

Example.

Let 397 be reduced to its equivalent whole or mixed

number.

7)39
54

Reduce the following improper fractions to their equivalent whole or mixed numbers.

CASE 4th. To reduce a whole number to an equivalent fraction, having a given denominator.

Multiply the whole number by the given denominator, and under the product thereof, place the same denominator, and it will form the fraction required.

If there is no given denominator, place an unit as a denominator.

Example.

Reduce the following whole numbers to the form of fractions, with the denominators opposite said numbers, respec

CASE 5th. To reduce a compound fraction, to an equivalent simple one.

Multiply all the numerators together for a new numerator, and all the denominators together for a new denominator, and the produets will form the fraction required,

If a part of the compound fraction or quantity be a whole or mixed number, it must be reduced to an improper fraction; and when it can be done, any numerator and denominator of the given fractions may be divided by the same number, and the quotients used instead of themselves; also when any two contrary terms are the same, they may be wholly expunged.

Let of 1 of of 8 be reduced to a simple fraction. *X÷×+×÷=4=6 the Answer.

Let the following compound fractions be reduced to their equivalent simple fractions or value, each set respectively.

CASE 6th. To reduce fractions of different denominators, to equivalent fractions, having one common denominator.

Multiply each numerator continually into all the denominators, but its own, for the numerators, and all the denominators together, for a common denominator.

But if the least common denominator is required, find the least common multiple of all the given denominators, which will be the least common denominator required; divide this common denominator severally, by each denominator of the given fractions, and the several quotients multiplied by the numerators of the given fractions, will be the respective

numerators.

Examples.

Let, and be reduced Let be reduced tofracto equivalent fractions, having tions equivalent thereto hava common denominator. ing one common denominator.

When in any set of fractions, one of the denominators is such, that each of the others may be evenly divided thereinto, it is evident that this denominator is the least required.

Let&,, and be reduced to fractions equivalent thereto, having the least common mon denominator.

Let 4, 4, 4 and, be reduced to fractions equivalent thereto having the least common denominator.

Let the following fractions be reduced to other fractions equivalent thereto, each set having one common denominator.

Let the following fractions be reduced to other fractions

equivalent thereto, having the least common denominator.

CASE 7th.-To find the value of a fraction in the inferior denominations of the integer of which it is a part

Multiply the numerator by the number of parts in the next inferior denomination, and divide the product by the denominator, as before directed in Simple Division, in that part of the rule which referred to the treatment of the remainder, the quotient or several quotients thus found will be the value required.

is no more of an cwt. to applied to any

Here it is evident that to find the value of & of than to divide 51 by 8, or to find the value of divide 5 cwt. by 16, and the same may be fraction of any integer.

Reduce the following fractions to their respective values.

CASE 8th. To bring any compound quantity to the frac tion of the integer of which it is a part.

Bring the compound quantity (by multiplying by the parts of its inferior denominations) to the lowest name mentioned in it, which make the numerator of the fraction to be found, and the integer expressed in parts of the same name or value, will be the denominator.