A swimming club's annual membership fee went up twice in successive years: first by 12%, then by 4% of the new amount. After both increases, the fee is £582...
Question 1 Report
A swimming club's annual membership fee went up twice in successive years: first by 12%, then by 4% of the new amount. After both increases, the fee is £582.40.
- Show that these two increases combined are equivalent to a single percentage rise of 16.48%. (3)
- Hence find the original fee, before either increase, correct to the nearest penny. (3)
- The committee claims this is the same overall effect as a single 17% rise. Determine, with a calculation, whether this claim is correct. (2)
Answer Details
This question chains two percentage rises into a single multiplier, reverses that multiplier to recover the starting value, and then checks a claim about an alternative single rise.
A \(12\%\) rise gives a multiplier of \(1.12\), and a further \(4\%\) rise (of the new amount) gives a multiplier of \(1.04\). Combining these:
\[1.12 \times 1.04 = 1.1648\]which is \(1 + 0.1648\), equivalent to a single rise of \(16.48\%\), as required [3 marks].
The original fee is the final fee divided by the combined multiplier:
\[£582.40 \div 1.1648 = £500.00\][3 marks]
A single \(17\%\) rise on the original fee would give:
\[£500 \times 1.17 = £585.00\]Since \(£585.00 \neq £582.40\), the committee's claim is not correct [2 marks].
The two increases combine to \(16.48\%\), which is close to but not the same as \(17\%\); testing a claim like this always means calculating the true equivalent rate rather than estimating it.
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