(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.

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(a) If $\log 5 = 0.6990, \log 7 = 0.8451$ and $\log 8 = 0.9031$ , evaluate \(\log ...

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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.

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

∴ D 5 (x + 60) + D x (2) = D 1280

5 x + 300 + 2 x = 1280

7 x + 300 = 1280 ⟹ 7 x = 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

77.78

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