A stationery order's shipping cost, \(S\) pounds, is directly proportional to both the number of boxes, \(b\), and the distance, \(m\) miles, delivered. Whe...
Question 1 Report
A stationery order's shipping cost, \(S\) pounds, is directly proportional to both the number of boxes, \(b\), and the distance, \(m\) miles, delivered. When \(b = 3\) and \(m = 20\), \(S = 24\).
- Find the constant of proportionality, \(k\), in \(S = kbm\). (2)
- Work out \(S\) when \(b = 5\) and \(m = 12\). (1)
Answer Details
Being directly proportional to two quantities at once means shipping cost equals a constant, \(k\), multiplied by both the number of boxes and the distance together; \(k\) is found from the one known combination of values before being used to price any other combination.
- \[ 24=k\times3\times20=60k \] so \[ k=0.4 \] [2 marks]
- \[ S=0.4\times5\times12=\pounds24 \] [1 mark]
It is a coincidence that both orders cost the same, \(\pounds24\); the number of boxes rose from \(3\) to \(5\) (a factor of \(\frac{5}{3}\)) while the distance fell from \(20\) to \(12\) miles (a factor of \(\frac{3}{5}\)), and since these two factors are exact reciprocals of each other, their combined effect on the cost exactly cancels out.
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