Fraction arithmetic
Adding and subtracting fractions
Fractions with the same denominatorThe bottom part of a fraction. For â…ť, the denominator is 8, which represents 'eighths'. can be added (or subtracted) by adding (or subtracting) the numerators.
For instance, \(\frac{2}{9} + \frac{3}{9} = \frac{5}{9}\) or \(\frac{6}{11} - \frac{4}{11} = \frac{2}{11}\).
If two fractions do not have the same denominator, then find a common denominator by making equivalent fractionsTwo fractions are equivalent when they are worth the same amount but written differently. 3/5 is equivalent to 6/10 as the numerator and denominator have both been multiplied by 2..
Example
Work out \(\frac{4}{7} + \frac{1}{3}\).
Work out the common denominator by looking for the lowest common multiple of 7 and 3. This is 21 (\(7 \times 3 = 21\)).
Write the equivalent fractions using 21 as the new common denominator.
\(\frac{4}{7} = \frac{12}{21}\)
\(\frac{1}{3} = \frac{7}{21}\)
So: \(\frac{4}{7} + \frac{1}{3} = \frac{12}{21} + \frac{7}{21} = \frac{19}{21}\)
This is the final answer as the fraction cannot be simplifyA fraction is simplified when there are no more common factors shared by the numerator and denominator. For example, the fraction 8/10 will simplify to 4/5 by removing a common factor of 2..
Question
Work out \(2 \frac{2}{5} - \frac{3}{4}\).
This sum contains a mixed numberA mix of a whole number and a fraction. (\(2 \frac{2}{5}\)), which must be converted to an improper fractionsIn improper fractions (or top-heavy fractions) the numerator is larger than the denominator, eg 7/5 , 9/4 ..
- Change the whole number part to a fraction using the denominator of the fraction part. So \(5 = \frac{10}{5}\)
- Add on the fraction part: \(\frac{10}{5} + \frac{2}{5} = \frac{12}{5}\)
So \(2\frac{2}{5} = \frac{12}{5}\)
This gives: \(2 \frac{2}{5} - \frac{3}{4} = \frac{12}{5} - \frac{3}{4}\)
Now look for a common denominatorA common denominator exists when the denominator (the number at the bottom of a fraction) of two or more fractions is the same. Common denominators help to compare or add/subtract two or more fractions. Common denominators are made by using equivalent fractions, eg a common denominator for 1/4 and 1/3 would be twelfths. by looking for the lowest common multiple of 5 and 4 which is 20.
Write equivalent fractions using 20 as the common denominator.
\(\frac{12}{5} = \frac{48}{20}\)
\(\frac{3}{4} = \frac{15}{20}\)
So: \(\frac{12}{5} - \frac{3}{4} = \frac{48}{20} - \frac{15}{20} = \frac{33}{20}\)
The question was asked in mixed number format, so the answer should be given as a mixed number. \(\frac{33}{20}\) is an improper fraction so it can be written as a mixed number.
Divide the numeratorThe top part of a fraction. For â…ť , the numerator is 5. by the denominator:
\(\frac{33}{20} = 1 \frac{13}{20}\)
This fraction cannot be simplified further, so this is the final answer.