Fractions

Thus, for example in the fraction 3/4 the denominator 4 tells us how many parts the whole is divided into, and the numerator 3 tells us how many of those parts we're dealing with.

When the numerator is less than the denominator, the fraction is proper, i.e. less than unity.

When the numerator is greater than or equal to denominator, the fraction is improper, i.e. greater than unity:

A fraction greater than 1 can also be written as a mixed number, i.e. as the sum of a positive integer and a

proper fraction:

Example:

Converting mixed numbers to improper fractions

To convert a mixed number to an improper fraction, multiply the whole number by the denominator of the

fraction, add the numerator of the fraction to the product and write the sum over the given denominator.

Converting improper fractions to mixed numbers

To convert an improper fraction to a mixed number divide the fraction's numerator by its denominator. The

integer part of the division is the integer part of the mixed number. The remainder of the division is the

numerator of the resulting fraction, as is shown in the above example.

Converting and reducing fractions

Fractions that represent the same number are called equivalent fractions.

For any fraction, multiplying the numerator and denominator by the same nonzero number gives an equivalent fraction.

A fraction is in lowest terms when the greatest common factor of its numerator and denominator is 1.

To reduce a fraction to lowest terms, divide the numerator and denominator by their greatest common

factor.

Example:

equivalent fractions:

and

Comparing fractions

To compare fractions with the same denominator, look at their numerators. The larger fraction is the one

with the larger numerator.

If the numerators of two fractions are the same, the fraction with the smaller denominator is the larger

fraction.

Example: To compare fractions with different denominators, e.g. 4/5 and 7/9, multiply the numerator

and the denominator of both fractions, the first fraction by the denominator of the second fraction and the

second fraction by the denominator of the first fraction. We get equivalent fractions with the same

denominators, thus

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Decimal fractions

Decimal fractions are vulgar fractions whose denominator is a power of ten. Or, it is a fraction written in the

place-value notation with base 10 preceded by dot, after which each successive digit indicates a multiple of

the successive negative powers of 10.

Example:

Adding and subtracting like fractions

To add or subtract two fractions with the same denominator, add or subtract the numerators and write the

sum over the common denominator.

Examples:

Adding and subtracting fractions

To add or subtract fractions with different denominators: First find the least common denominator (the

smallest number that can be divided by each denominator). Write equivalent fractions using this denominator.

Then add or subtract the fractions.

Examples:

Multiplying fractions

To multiply a whole number by a fraction means to multiply it by a numerator and to divide a product by a denominator.

When two fractions are multiplied, the result is a fraction with a numerator that is the product of the fractions' numerators and a denominator that is the product of the fractions' denominators.

Examples:

Reciprocal fractions

The fractions a/b and b/a are reciprocals. Reciprocal fractions are two fractions whose product is 1.

Dividing fractions

To divide a number by a fraction, multiply the number by the reciprocal of the fraction.

To divide mixed numbers, you should always convert them to improper fractions, then multiply the first

number by the reciprocal of the second.

To divide a fraction by a whole number, write the whole number as an improper fraction with a denominator of

1, then multiply as fractions.

Examples:

Simplifying complex fractions

Complex fractions or compound fractions are fractions in which the numerator and/or denominator contain

fractions.

To simplify complex fractions, change the complex fraction into a division problem, i.e., divide the numerator

by the denominator.

Examples:

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