(a) A boy had M Dalasis (D). He spent D15 and shared the remainder equally with his sister. If the sister's share was equal to \(\frac{1}{3}\) of M, find th...

Answer Details

(a) The boy had \(M\) dalasis, spent D15, leaving \(M - 15\). This remainder is shared equally between him and his sister, so the sister's share is \(\dfrac{M - 15}{2}\). We are told this equals \(\dfrac13 M\):

\[\frac{M - 15}{2} = \frac{M}{3}\]

\[3(M - 15) = 2M \;\Rightarrow\; 3M - 45 = 2M \;\Rightarrow\; M = 45\]

So \(M = \text{D}45\).

(b) Let the total number of tourists be \(N\). Travelled by road \(= \tfrac23 N\), by air \(= \tfrac{13}{30}N\), by both \(= \tfrac{4}{15}N\).

(i) Venn diagram: two intersecting circles (Road, Air). Both region \(= \tfrac{4}{15}N\); road only \(= \tfrac23 N - \tfrac{4}{15}N = \tfrac{6}{15}N\); air only \(= \tfrac{13}{30}N - \tfrac{4}{15}N = \tfrac{5}{30}N\); outside both \(= 20\).

(ii)(1) Those using road or air:

\[\frac23 + \frac{13}{30} - \frac{4}{15} = \frac{20 + 13 - 8}{30} = \frac{25}{30} = \frac56\]

So the fraction using neither is \(1 - \tfrac56 = \tfrac16\), and \(\tfrac16 N = 20 \Rightarrow N = 120\). 120 tourists were interviewed.

(2) Air only \(= \tfrac{13}{30}N - \tfrac{4}{15}N = \tfrac{5}{30}\times 120 = 20\) tourists.