fraction (mathematics)

In mathematics, a number that indicates one or more equal parts of a whole. Usually, the number of equal parts into which the unit is divided (denominator) is written below a horizontal or diagonal line, and the number of parts comprising the fraction (numerator) is written above; for example, 2/3 has numerator 2 and denominator 3. Such fractions are called vulgar fractions or simple fractions. The denominator can never be zero.

Proper and improper fractions
A proper fraction is one in which the numerator is less than the denominator. For example, 2/5, 3/5, and 7/8 are all proper fractions.

An improper (or top heavy) fraction has a numerator that is larger than the denominator. For example:

14/5 means 14 ÷ 5



14/5 = 24/5

This is called a mixed number.

Denominator of zero
A combination such as 5/0 is not regarded as a fraction (an object cannot be divided into zero equal parts). Zero divided by any number is zero, and any number divided by zero is infinity.

Decimal fractions
A decimal fraction has as its denominator a power of 10. Exact decimals can be changed into fractions using place values. For example:

0.37 means 3 tenths and 7 hundredths



0.37 is the equivalent to 37/100 as a fraction and 37% as a percentage.

Most fractions can be expressed exactly as decimal fractions (1/3 = 0.333...). Fractions are also known as rational numbers; that is, numbers formed by a ratio. Integers may be expressed as fractions with a denominator of 1, so 6 is 6/1, for example.

Addition and subtraction
To add or subtract with fractions a common denominator (a number divisible by both the bottom numbers) needs to be identified. For example:

3/4 + 5/6

First both denominators should be the same. 12 is the lowest number of which both 4 and 6 are factors – it is the lowest common denominator. To change 3/4 into twelfths the denominator is multiplied by 3. The numerator must also be multiplied by 3:



To change 5/6 into twelfths the denominator is multiplied by 2. The numerator must also be multiplied by 2:



3/4 + 5/6 = 9/12 + 10/12 = 19/12 = 17/12

If whole numbers appear in the calculation they can be added/subtracted separately first.

Multiplication and division
All whole numbers in a division or multiplication calculation must first be converted into improper fractions. For multiplication, the numerators are then multiplied together and the denominators are then multiplied to provide the solution. For example:

72/3 × 41/2 = 23/3 × 9/2 = 207/6 = 341/2

In division, the procedure is similar, but the second fraction must be inverted before multiplication occurs. For example,

55/12 ÷ 11/8 = 65/12 ÷ 9/8 = 65/12 × 8/9 = 520/108 = 422/27

Fraction of an amount
For example:

to find 1/2 of £300 divide by 2 to get £150 to find 1/5 of 250 m divide by 5 to get 50 m

The fraction wall
This is a useful visual tool when working with fractions.

© RM 2010. Helicon Publishing is division of RM.

 

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