A household's annual electricity bill is \(\pounds C\), found from \(C=4.5\times10^2 \times n\), where \(n\) is the number of people in the household. A fam...
Question 1 Report
A household's annual electricity bill is \(\pounds C\), found from \(C=4.5\times10^2 \times n\), where \(n\) is the number of people in the household. A family is billed \(\pounds2.7\times10^3\).
- Find \(n\), the number of people in the family. (2)
- A smaller family of 4 people uses the same supplier. Work out their bill, giving your answer in standard form. (1)
- A rival supplier charges \(\pounds1.6\times10^3\) plus \(\pounds1\times10^2\) per person. Work out the rival's cost for a family of 6 people, and decide which supplier is cheaper for that family. (2)
Answer Details
A formula linking a bill to the number of people can be rearranged to find an unknown quantity, then reused for other household sizes, and comparing two suppliers means evaluating each one's total charge for the same scenario.
(a) Rearranging \(C=4.5\times10^{2}\times n\) for \(n\) by dividing both sides by \(4.5\times10^{2}\):
\[n=2.7\times10^{3}\div4.5\times10^{2}=2700\div450=6\] [2 marks](b) Substituting \(n=4\) into the original formula:
\[C=4\times4.5\times10^{2}=1800=1.8\times10^{3}\] [1 mark](c) The rival supplier's cost for 6 people is a fixed charge plus a per-person charge:
\[1.6\times10^{3}+6\times1\times10^{2}=1600+600=2200=2.2\times10^{3}\]Comparing this to the original supplier's charge for 6 people, \(\pounds2.7\times10^{3}=\pounds2700\): since \(\pounds2200 \lt \pounds2700\), the rival supplier is cheaper, by \(2700-2200=\pounds500=\pounds5\times10^{2}\). [2 marks]
The original supplier charges purely per person, while the rival charges a fixed amount plus a smaller per-person rate; comparing the two total costs for the same household size, rather than comparing the per-person rates alone, is what correctly identifies which is cheaper.
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