A bill in two parts
Recovering a two-part linear model from two bills, then using it to explain a comparison.
Notes and data
An electricity tariff charges a fixed standing charge for every day of the billing period, plus a rate for each kilowatt-hour used.
A household received these two bills, both for 30-day months:
210 kWh used: £71.40 330 kWh used: £96.60
Answers in order, explanations withheld, and a report on where the chain broke.
- 13.1Systems of equationsFoundation
What is the rate charged per kilowatt-hour?
- A£0.21
- B£0.34
- C£0.29
- D£0.12
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£0.21
Both months have the same standing charge, so subtracting one bill from the other removes it entirely: £96.60 - £71.40 = £25.20 for the extra 330 - 210 = 120 kWh, giving £25.20 / 120 = £0.21 per kWh. The choice £0.34 is the first bill divided by its units, which mixes the two parts together.
- 13.2Systems of equationsStandard
Builds on: The rate from question 1. Without it there is no way to separate the two parts of a single bill. The choice £27.30 stops at the monthly figure.
What is the standing charge per day?
- A£0.91
- B£27.30
- C£2.38
- D£1.36
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£0.91
The 210 kWh cost 210 x £0.21 = £44.10, so the standing charge for the month is £71.40 - £44.10 = £27.30. That covers 30 days, so the daily charge is £27.30 / 30 = £0.91.
- 13.3Linear functionsStandard
Builds on: Both earlier answers, and the reason the second choice is wrong is precisely the step between them — question 2 divided by 30, and this question has to multiply back.
Which function gives the cost in pounds of a 30-day month in which x kilowatt-hours are used?
- AC(x) = 27.30 + 0.21x
- BC(x) = 0.91 + 0.21x
- CC(x) = 27.30x + 0.21
- DC(x) = 0.21 + 30x
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C(x) = 27.30 + 0.21x
The constant term is the standing charge for the whole period, not for one day, so it is 30 x £0.91 = £27.30. The second choice is the commonest slip: a daily rate used as if it were a monthly one.
- 13.4Interpreting linear modelsChallenge
Builds on: The model from question 3, which is what confirms the neighbour is on the same tariff rather than a worse one. Without it the first and second choices are indistinguishable.
A neighbour on the same tariff used 160 kWh in a 30-day month, was billed £60.90, and works this out as 38p per unit against this household's 34p. What explains the difference?
- AThe standing charge is spread over fewer units, so the average cost per unit is higher.
- BThe neighbour is being charged a higher rate per kilowatt-hour.
- CThe neighbour was billed for more days than the household.
- DAverage cost per unit does not depend on how much is used.
Show the answer and reasoningHide the reasoning
The standing charge is spread over fewer units, so the average cost per unit is higher.
Check the bill against the model: 27.30 + 0.21 x 160 = £60.90, so the neighbour really is on this tariff. Both pay 21p a unit; the same £27.30 divided among 160 units rather than 210 is what lifts the average. Average cost per unit falls as usage rises on any tariff with a fixed part.