# Selecting the 'right' denominator

Selecting the ‘right’ denominator is key to any questions where we are asked to compare, to add, or to subtract fractions. This is a good opportunity to apply some basic number skills to a fraction topic.

## Case One: When the denominators are coprime

When adding \(\frac{3}{7} + \frac{3}{5}\) what denominator should we use?

7 and 5 are both prime numbers and therefore 7 and 5 have no common factors above the number 1. 7 and 5 are said to be coprime. In this case the lowest common denominator is found by multiplying the two denominators. Here the denominator we use is 35 (7 x 5):

\[\frac{3}{7} + \frac{3}{5} = \frac{15}{35} + \frac{21}{35} = \frac{36}{35} = 1\frac{1}{35}\]## Case Two: When one denominator is a factor of the other

When adding \(\frac{2}{7} + \frac{1}{21}\) what denominator should we use?

7 is a factor of 21. If one number is a factor of the other then use the larger of the two denominators. In this case we use 21.

\[\frac{2}{7} + \frac{1}{21} = \frac{6}{21} + \frac{1}{21} = \frac{7}{21} = \frac{1}{3}\]## Case Three: When both denominators share one or more factors

When adding \(\frac{1}{6} + \frac{5}{9}\) what denominator should we use?

Both 6 and 9 are in the three times table.

When this occurs, consider the first few terms of the times tables of each denominator.

- 6, 12,
**18**, 24 - 9,
**18**, 27, 36

18 is the smallest number which is a multiple of both 6 and 9. We use 18 as the denominator.

\[\frac{1}{6} + \frac{5}{9} = \frac{3}{18} + \frac{10}{18} = \frac{13}{18}\]## Prime Factors

In each of the above three cases, expressing the denominators as the product of their prime factors will enable the ‘best’ denominator to be found. This is explored further in the Understanding Numbers course.

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