Adding and subtracting fractions

Adding/ subracting fractions with common denominators

For two fractions \(\frac{a}{c}\) and \(\frac{b}{c}\) with a common denominator, their sum is defined as:

$$ \frac{a}{c} + \frac{b}{c} = \frac{a + b}{c} $$

For example:

$$ \frac{2}{8} + \frac{3}{8} = \frac{5}{8} $$

The same applies to subtraction:

$$ \frac{a}{c} - \frac{b}{c} = \frac{a - b}{c} $$

Adding/ subracting fractions without common denominators

  • Find the lowest common denominator for the two fractions
  • Use this to create two equivalent fractions
  • Add/subtract
  • Reduce

Lowest common denominator and lowest common multiple

Given the symmetry between factors and divisors these properties are related. Note however that the LCM is more generic: it applies to any set of numbers not just fractions. Whereas the LCD is explicitly to do with fractions (hence ‘denominator’).

  • For two fractions \(a, b\) (or a set), the LCD is the smallest number divisble by both the denominator of \(a\) and the denominator of \(b\) (or each member of the set).

  • For two fractions \(a, b\) (or a set), the LCM is the smallest number that is a multiple of the denominator of \(a\) and the denominator of \(b\) (or each member of the set).

In order to find the LCM of the set \(\{12, 16\}\) we list the multiples of both:

$$ 12, 24, 36, 48 \\ 16, 32, 48 $$

Until we identify the smallest number common to both lists. In this case it is 48. Thus the LCM of 12 and 16 is 48.

The relationship between LCM and LCD is that the least common denominator is the least common multiple of the fractions’ denomintors.

Demonstration: addition

We can now use this to calculate the addition of two fractions without common denominators: \(\frac{4}{9} + \frac{1}{6}\).

First identify the common multiples of 9 and 6:

$$ 9, 18, ... \\ 6, 12, 18, ... $$

The least common multiple is 18. We then think: what do we need to multiply each denominator by to get 18?

In the case of the first fraction (\(\frac{4}{9}\)) it is 2:

$$ \frac{4}{9 \cdot 2} = \frac{4}{18} $$

But what we do to the denominator, we must also do to the numerator, hence:

$$ \frac{4 \cdot 2}{9 \cdot 2} = \frac{8}{18} $$

We then do the same to the second fraction (\(\frac{1}{6}\)). We need to multiply its denominator by 3 to get 18 and we apply this also to the numerator.

$$ \frac{1 \cdot 3}{6 \cdot 3} = \frac{3}{18} $$

We now have two fractions that share a common denominator so we can sum:

$$ \frac{8}{18} + \frac{3}{18} = \frac{11}{18} $$

Demonstration: subtraction

Calculate:

$$ \frac{3}{5} - \frac{2}{3} $$

Once again we need to find the least common denominator for the two fractions. We start by listing the common multiples for the two denominators 5 and 3:

$$ 5, 10, 15, ... \\ 3, 6, 9, 12, 15,... $$

The lowest common multiple is 15. From the first fraction we get 15 by multiplying by 3. With the second fraction we get 15 by multiplying by 5. Thus:

$$ \frac{3 \cdot 3}{5 \cdot 3} = \frac{9}{15} $$

$$ \frac{2 \cdot 5}{3 \cdot 5} = \frac{10}{15} $$

We can now carry out the subtraction:

$$ \frac{9}{15} - \frac{10}{15} = -\frac{1}{15} $$

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