(a) If \(\log 5 = 0.6990, \log 7 = 0.8451\) and \(\log 8 = 0.9031\), evaluate \(\log (\frac{35 \times 49}{40 \div 56})\). (b) For a musical show, x children...
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(a) If \(\log 5 = 0.6990, \log 7 = 0.8451\) and \(\log 8 = 0.9031\), evaluate \(\log (\frac{35 \times 49}{40 \div 56})\).
(b) For a musical show, x children were present. There were 60 more adults than children. An adult paid D5 and a child D2. If a total of D1280 was collected, calculate the
(i) value of x ; (ii) ratio of the number of children to the number of adults ; (iii) average amount paid per person ; (iv) percentage gain if the organisers spent D720 on the show.
Answer Details
(a) Simplify inside first. \(40 \div 56 = \dfrac{40}{56} = \dfrac{5}{7}\), so \[\frac{35\times 49}{40\div 56} = \frac{35\times 49}{5/7} = 35\times 49\times\frac{7}{5} = 7\times 49\times 7 = 7^4 = 2401.\] \[\log 2401 = \log 7^4 = 4\log 7 = 4(0.8451) = \mathbf{3.3804}.\]
(b) Let there be \(x\) children; adults \(= x + 60\). Adult pays \(D5\), child \(D2\), total \(D1280\): \[5(x + 60) + 2x = 1280 \Rightarrow 7x + 300 = 1280 \Rightarrow 7x = 980 \Rightarrow x = 140.\]
(i) \(x = \mathbf{140}\) children (adults \(= 200\)).
(ii) children : adults \(= 140 : 200 = \mathbf{7 : 10}.\)
(iii) Total people \(= 340\); average \(= \dfrac{1280}{340} = \mathbf{D3.76}\) per person.
(iv) Gain \(= 1280 - 720 = D560\); percentage gain \(= \dfrac{560}{720}\times 100 = \mathbf{77.8\%}.\)
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