Priya places a stationery order for the office where she works. Pens are sold only in boxes of 24 and printer cartridges are sold only in boxes of 36. Priya...

Three different pieces of number work appear here: a lowest common multiple, reversing a percentage increase, and combining two successive percentage falls. The percentage parts are both handled with multipliers, which is safer than working out each change separately.

1. Smallest number of pens [2 marks]
   The order must be a whole number of boxes of each kind and end with equal numbers, so the total must be a common multiple of 24 and 36. The smallest such number is the lowest common multiple. Using prime factors, \[24 = 2^3 \times 3, \qquad 36 = 2^2 \times 3^2\] Take the highest power of each prime: \[\text{LCM} = 2^3 \times 3^2 = 8 \times 9 = 72 \tag{1 mark}\] So the smallest number of pens is 72, which is 3 boxes of pens and 2 boxes of cartridges.
2. Price before tax [2 marks]
   Adding 20 per cent tax multiplies the pre-tax price by 1.2, so to reverse it divide: \[\frac{276.00}{1.2} \tag{1 mark}\] \[= 230.00\] The price before tax is £230. Taking 20 per cent off £276 gives £220.80 and is wrong, because the tax was 20 per cent of the smaller pre-tax figure, not of the total. Check: \(230 \times 1.2 = 276\).
3. Overall percentage fall [3 marks]
   A fall of 16 per cent has multiplier \(1 - 0.16 = 0.84\), and a fall of 25 per cent has multiplier \(1 - 0.25 = 0.75\). Applying them one after the other: \[0.84 \times 0.75 \tag{1 mark}\] \[= 0.63 \tag{1 mark}\] The overall multiplier is 0.63, so the price ends at 63 per cent of where it started: \[1 - 0.63 = 0.37 \quad \Rightarrow \quad \text{an overall fall of } \mathbf{37 \ \text{per cent}} \tag{1 mark}\] Adding the percentages to give 41 per cent is wrong, because the second fall is applied to the already reduced price, not to the original.

Examination reminder: successive percentage changes multiply, they never add, and reversing a percentage change means dividing by the multiplier rather than applying the percentage in the other direction.