To reduce a mixed number to its equivalent improper fraction.

Bule.*

Multiply the whole number by the denominator of the fraction, and add the numerator to the product; this sum written over the denominator will be the fraction required.

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

Rule.†

Divide the numerator by the denominator and the quotient will be the whole number, and the remainder, if any, will be the numerator to the given denominator.

*All fractions represent a division of the numerator by the denominator, which taken together are proper and adequate expressions for the quotient. Thus 2 divided by 3, is 2-3; whence the reason of the rule is manifest; for if a quantity be multiplied and divided by the same number, it evidently remains the same. A whole number may be changed into an equivalent fraction with a given denominator, by multiplying the whole number by the denominator and writing the product over said denominator.

†This rule is evidently the reverse of the preceding, and is the same as Simple Division.

To reduce a compound fraction to an equivalent single one.

RULE.* Multiply all the numerators together for a new numerator, and all the denominators together for a new denominator; then reduce this new fraction to its lowest terms.

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 together for a common denominator.

* If part of the compound fraction be a whole or mixed number, it must be reduced to an improper fraction. If any denominator of a compound fraction be equal to a numerator of the same, both may be expunged, and the other numbers, multiplied as by the rule, will produce the fraction required in lower

terms.

+ By examining the operation it will be seen that the numerator and denominator of every fraction are multiplied by the very same numbers, and conse quently their values are not altered.

Case VIII.

To reduce fractions of different denominators to equivalent fractions having

the least common denominator.

RULE 1.*-Find the least common multiple of all the denominators of the given fractions, and it will be the common denominator required.

2. Divide the common denominator by the denominator of each fraction, and multiply the quotient by the numerator, the products will be the numerators required.

To find the value of a fraction in known parts of an integer. RULE.-Multiply the numerator by the parts of the next inferior denomination, and divide the product by the denominator; 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; the quotients will be the answer required.

*The common denominator is a multiple of all the denominators, and consequently will divide by any of them: therefore, proper parts may betaken for all the numerators as required.

Case X.

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

RULE.-Make a compound fraction of it, and reduce it to a single

RULE. Reduce compound fractions to single ones; mixed numbers to improper fractions, fractions of different integers to those of the same, and all of them to a common denominator; then the sum of the numerators, written over the common denominator, will be the sum of the fractions required.

* By reducing fractions to a common denominator, they are made to express similar parts of the same unit, and as each numerator shows how many of those parts are signified by the fraction, the sum or difference of the numerators written over the common denominator, is evidently the sum or difference of the fractions.

3. SUBTRACTION OF VULGAR FRACTIONS.

RULE.-Prepare the fractions as for addition, and the difference of the numerators written over the common denominator will be the difference of the fractions required.

RULE.--Reduce compound fractions to single ones, and mixed numbers to improper fractions; then multiply the numerators together for the numerator, and the denominators together for the denominator of the fraction required.

Examples.

1. Multiply 4, of, and 184 | 4. Multiply of 3 by of 33. continually together.

Ans.

RULE-Prepare the fractions as for Multiplication, then invert the divisor, and proceed exactly as in Multiplication.

* Fractions are sometimes most conveniently brought to a common denominator by Multiplication or Division. In the first example

common denominator with by multiplying both its terms by 7.

is brought to a