(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.
Explanation
(a) \(\log (\frac{35 \times 49}{40 \div 56})\) Given \(\log 5 = 0.6990 , \log 7 = 0.8451 , \log 8 = 0.9031\) \(\log (\frac{35 \times 49}{40 \div 56} = \log (\frac{(7 \times 5) \times 7^{2}}{(8 \times 5) \div (8 \times 7)}\) = \(\log (\frac{7^{3} \times 5}{5 \div 7}\) = \(\log 7^{3} + \log 5 - (\log 5 - \log 7)\) = \(3 \log 7 + \log 5 - \log 5 + \log 7\) = \(4 \log 7\) = \(4 \times 0.8471\) = \(3.3804\) (b) (i) Since there are 60 more adults than children, then the number of adults = x + 60 \(\therefore D 5(x + 60) + D x(2) = D 1280\) \(5x + 300 + 2x = 1280\) \(7x + 300 = 1280 \implies 7x = 1280 - 300 = 980\) \(x = \frac{980}{7} = 140\) There were 140 children. (ii) Ratio of children to adults = \(x : x + 60\) = \(140 : (140 + 60)\) = \(140 : 200\) = \(7 : 10\) (iii) Total number of persons at the show : 140 + 200 = 340 persons. Total amount gotten = D 1280 Average paid per person = \( D \frac{1280}{340}\) = \(D \frac{64}{17}\) = D 3.765 (iv) Percentage profit Profit : D (1280 - 720) = D 560 % profit : \(\frac{560}{720} \times 100% = \frac{700}{9} %\) = \(77.78%\)
