A household reviews its monthly spending against its total income. Rent takes \(\frac{2}{5}\) of the income, while food takes \(\frac{1}{4}\) of the same in...
Question 1 Report
A household reviews its monthly spending against its total income. Rent takes \(\frac{2}{5}\) of the income, while food takes \(\frac{1}{4}\) of the same income.
- Show that rent takes a greater fraction of income than food. (2)
- Find the fraction of income by which rent exceeds food, in its simplest form. (1)
Answer Details
This question compares two fractions with different denominators by converting them to a common denominator, and then finds the difference between them.
Writing both fractions over the common denominator \(20\) (the lowest common multiple of \(5\) and \(4\)):
\[\dfrac{2}{5} = \dfrac{8}{20} \quad \text{and} \quad \dfrac{1}{4} = \dfrac{5}{20}\]Since \(\dfrac{8}{20} \gt \dfrac{5}{20}\), rent takes the greater fraction of income than food [2 marks].
The fraction by which rent exceeds food is:
\[\dfrac{8}{20} - \dfrac{5}{20} = \dfrac{3}{20}\][1 mark]
Two fractions can only be compared or subtracted directly once they share the same denominator; \(20\) is chosen here because it is the smallest number that both \(5\) and \(4\) divide into exactly.
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