A community library charges £18 a year, including 5 free book loans; each further loan costs £1.20. Fee £18 Free loans 5 Extra loan cost £1.20 A member borr...
Question 1 Report
A community library charges £18 a year, including 5 free book loans; each further loan costs £1.20.
| Fee | £18 |
|---|---|
| Free loans | 5 |
| Extra loan cost | £1.20 |
- A member borrows 9 books in a year. Work out the total amount, \(T\) pounds, paid. (2)
- Write down an expression, in terms of the number of books borrowed \(b \ge 5\), for \(T\). (1)
- A different member pays £32.40 for the year. Work out how many books were borrowed. (2)
- A third member claims that borrowing 20 books costs less than £40. Determine, with working, if this is correct. (1)
Answer Details
This question builds a piecewise linear cost model from a fee-plus-per-unit description, evaluates it, forms and solves an equation from a known total, and checks a numerical claim.
Borrowing \(9\) books means \(9-5=4\) loans beyond the \(5\) free ones, each costing \(£1.20\):
\[T = £18 + (9-5) \times £1.20 = £18 + £4.80 = £22.80\][2 marks]
For a general number of books \(b \ge 5\):
\[T = 18 + 1.2(b-5)\][1 mark]
Setting \(T=£32.40\):
\[18+1.2(b-5) = 32.40\] \[1.2(b-5) = 14.40\] \[b-5 = 12\] \[b = 17 \text{ books}\][2 marks]
For \(b=20\):
\[T = 18+1.2(20-5) = 18+18 = £36\]Since \(£36 \lt £40\), the third member's claim is correct [1 mark].
The formula in part (b) only applies once at least \(5\) books have been borrowed (since \((b-5)\) must not be negative), which matches the "\(b \ge 5\)" condition stated in the question.
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