A phone company uses the formula \(C = \dfrac{k}{n} + f\) to set the monthly cost per person of a shared family data plan, where \(C\) is the cost per perso...
Question 1 Report
A phone company uses the formula \(C = \dfrac{k}{n} + f\) to set the monthly cost per person of a shared family data plan, where \(C\) is the cost per person in pounds, \(k\) is the total data cost, \(f\) is a fixed line fee per person, and \(n\) is the number of people sharing the plan.
- Work out \(C\) when \(k = 60\), \(f = 4\) and \(n = 5\). (2)
- Make \(k\) the subject of the formula. (2)
- Make \(n\) the subject of the formula. (2)
- Show that if \(n\) doubles while \(k\) and \(f\) stay the same, \(C\) does not halve. (1)
Answer Details
Substituting values evaluates the shared cost per person directly; making \(k\) or \(n\) the subject instead reverses that calculation, always by first isolating the fraction (subtracting \(f\)) before undoing the division by \(n\). Testing what happens algebraically when \(n\) doubles shows that a fixed cost like \(f\) does not shrink even when the shared part does.
- \(C=\dfrac{60}{5}+4=12+4=\pounds16\) [2 marks]
- Subtracting \(f\) from both sides: \(C-f=\dfrac{k}{n}\); multiplying by \(n\): \(k=n(C-f)\) [2 marks]
- From \(C-f=\dfrac{k}{n}\), rearranging for \(n\): \(n=\dfrac{k}{C-f}\) [2 marks]
- Doubling \(n\) gives a new cost \(\dfrac{k}{2n}+f\); half of the original cost is \(\dfrac12\left(\dfrac{k}{n}+f\right)=\dfrac{k}{2n}+\dfrac{f}{2}\). Since the fixed fee \(f\) is not equal to \(\dfrac{f}{2}\) (unless \(f=0\)), the new cost is not half the original, so \(C\) does not halve when \(n\) doubles. [1 mark]
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