A community library receives a donation of \(\pounds T\), split between the children's section and the adult section in the ratio \(5:7\). The children's se...
Question 1 Report
A community library receives a donation of \(\pounds T\), split between the children's section and the adult section in the ratio \(5:7\). The children's section then receives a further \(\pounds 48\) from a literacy grant, changing the ratio of children's section funding to adult section funding to \(3:4\).
- Write down an expression, in terms of \(T\), for the new amount held by the children's section. (1)
- Form an equation in \(T\) and solve it to find the value of \(T\). (4)
- Hence find the original amount allocated to the adult section. (2)
Answer Details
A total split in a given ratio gives each section's original amount as a fraction of the total, and adding a fixed sum to only one section then creates a new ratio equation that can be solved for the original total.
(a) The children's section originally holds \( \dfrac{5}{12}T = \dfrac{5T}{12} \). After the £48 grant, its new amount is \( \dfrac{5T}{12}+48 \) [1 mark].
(b) The adult section is unaffected, staying at \( \dfrac{7T}{12} \). Since the new ratio of children's to adult funding is \(3:4\): \( \dfrac{\frac{5T}{12}+48}{\frac{7T}{12}}=\dfrac{3}{4} \) [1 mark]. Cross-multiplying gives \( 4\left(\dfrac{5T}{12}+48\right)=3\left(\dfrac{7T}{12}\right) \), i.e. \( \dfrac{20T}{12}+192=\dfrac{21T}{12} \) [1 mark]. Subtracting \( \dfrac{20T}{12} \) from both sides gives \( 192=\dfrac{T}{12} \) [1 mark], so \( T=192\times12=2304 \) [1 mark].
(c) The original amount allocated to the adult section is \( \dfrac{7(2304)}{12}=7\times192=£1344 \) [2 marks].
Only the children's section receives the £48 grant, which is why the adult section's expression, \(\dfrac{7T}{12}\), appears unchanged in the new ratio equation while the children's expression gains the extra "+48" term.
Download The App On Google Playstore
Everything you need to excel in your exams
