An allotment holder's crop weight, \(x\) kg, increases by 25% one year, then decreases by 20% the following year. Show that the weight after these two chang...
Question 1 Report
An allotment holder's crop weight, \(x\) kg, increases by 25% one year, then decreases by 20% the following year.
- Show that the weight after these two changes can be written as \(1.25 \times 0.8x\), and simplify this to a single multiplier. (2)
- Hence determine the overall percentage change in the crop weight over the two years, stating whether it is an increase or a decrease. (3)
Answer Details
(a) A 25% increase corresponds to a multiplier of \(1.25\), and the following 20% decrease corresponds to a multiplier of \(1-0.20=0.8\), applied to the already-increased weight. So the weight after both changes is \(1.25\times0.8x\), which simplifies to \(1.0x\), since \(1.25\times0.8=1.0\). [2 marks]
(b) Since the combined multiplier is exactly \(1.0\), the final weight equals the starting weight exactly, so there is no overall change in the crop weight over the two years, it is neither an increase nor a decrease. [3 marks]
It is a common misconception that a 25% increase followed by a 20% decrease would cancel to leave a net 5% increase; because each percentage is calculated on a different amount (the second on the already-larger weight), the two multipliers instead combine to exactly \(1.0\) here.
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