A community library receives a delivery split between large-print books and standard-print books in the ratio \(2:9\). The library then receives an addition...
Question 1 Report
A community library receives a delivery split between large-print books and standard-print books in the ratio \(2:9\). The library then receives an additional delivery of standard-print books only, with no large-print books added, changing the overall ratio of large-print to standard-print books to \(2:15\).
- Given that the library held \(14\) large-print books throughout, find the number of standard-print books held before the additional delivery. (2)
- Work out how many standard-print books were in the additional delivery. (3)
Answer Details
Knowing one actual quantity that corresponds to a stated ratio lets every other quantity in that ratio be recovered by finding the value of "one part"; a later one-sided addition then gives an equation for the unknown addition.
(a) The ratio large-print:standard-print is \(2:9\), and large-print \(=14\) corresponds to the "2" part, so one part is worth \( 14\div2=7 \). The standard-print count, corresponding to the "9" part, is \( 9\times7=63 \) [2 marks].
(b) Let \(p\) be the number of standard-print books added. Since large-print stays at 14 while standard-print becomes \(63+p\), and the new ratio is \(2:15\): \( \dfrac{14}{63+p}=\dfrac{2}{15} \). Cross-multiplying gives \( 2(63+p)=14\times15=210 \), so \( 63+p=105 \), giving \( p=42 \) standard-print books [3 marks].
Only the standard-print count changes between the two ratios (large-print stays at 14 throughout), which is exactly why the large-print number appears unchanged as the numerator on both sides of the working in parts (a) and (b).
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