(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%)