The resale value of a festival wristband depreciates by the same percentage, \(r\%\), each day once the festival sells out. A wristband bought for \(\pounds...
Question 1 Report
The resale value of a festival wristband depreciates by the same percentage, \(r\%\), each day once the festival sells out. A wristband bought for \(\pounds120\) on Day 0 is worth \(\pounds108.30\) on Day 2, as shown in the diagram.
- Set up an equation in \(r\) for the value on Day 2. (2)
- Solve your equation to find the value of \(r\). (3)
Answer Details
(a) Depreciating by the same percentage \(r\%\) each day means multiplying by \(\left(1-\dfrac{r}{100}\right)\) once per day. After 2 days, this multiplier is applied twice, so the value on Day 2 is \(120\left(1-\dfrac{r}{100}\right)^{2}=108.30\), matching the bar shown on the diagram. [2 marks]
(b) Dividing both sides by \(120\): \(\left(1-\dfrac{r}{100}\right)^{2}=\dfrac{108.30}{120}=0.9025\). Taking the square root of both sides: \(1-\dfrac{r}{100}=\sqrt{0.9025}=0.95\) (taking the positive root, since a resale value multiplier must be positive). Rearranging, \(\dfrac{r}{100}=1-0.95=0.05\), so \(r=5\). [3 marks]
Squaring the daily multiplier to model 2 days' depreciation, rather than doubling the percentage \(r\), reflects that the second day's fall is calculated on the already-reduced Day 1 value, not on the original \(\pounds120\).
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