Two firms price a taxi ride. Ace charges \(y = 2x + 6\) pounds for \(x\) miles. Bell charges \(y = 3x + 2\) pounds. The grid shows both lines. 0123456789100...
Question 1 Report
Two firms price a taxi ride. Ace charges \(y = 2x + 6\) pounds for \(x\) miles. Bell charges \(y = 3x + 2\) pounds. The grid shows both lines.
- Write down Ace's fixed charge. (1)
- Use the graph to write down where the fares are equal. (1)
- Solve \(2x + 6 = 3x + 2\) to confirm your answer. (2)
- Find the fare both firms charge for that journey. (1)
- For a 9 mile ride, find the difference in fares. (1)
- Advise the passenger which firm to choose, and why. (1)
Answer Details
Each fare is a fixed charge plus a rate per mile, so on the graph the intercept is the fixed charge and the gradient is the charge per mile.
(a) The Ace line meets the vertical axis at 6, which is the fare for zero miles, so the fixed charge is £6. This matches the \(+6\) in \(y = 2x + 6\). [1]
(b) The two lines cross at a distance of 4 miles, so that is where the fares are equal. [1]
(c)
- \(2x + 6 = 3x + 2\). Subtract \(2x\) from both sides and subtract 2: \(6 - 2 = 3x - 2x\), that is \(4 = x\). [1]
- So \(x = 4\), confirming the reading taken from the graph. [1]
(d) Substituting into either formula: \(2\times 4 + 6 = 14\), and \(3\times 4 + 2 = 14\), so both firms charge £14 for that journey. [1]
(e) For 9 miles, Ace charges \(2\times 9 + 6 = 24\) pounds and Bell charges \(3\times 9 + 2 = 29\) pounds, a difference of £5. [1]
(f) For a journey such as the 9 mile ride, choose Ace: its gradient of 2 pounds per mile is smaller than Bell's 3 pounds per mile, so beyond the 4 mile crossing point Ace is always cheaper, and the gap widens by £1 for every further mile. [1]
The advice depends on the distance. Below 4 miles Bell is the cheaper firm, because its smaller fixed charge of £2 outweighs its higher rate; the crossing point is exactly where the two effects balance.
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