Work and Wages Questions in English

(B) The man's $1$ day work is $\frac{1}{20}$.
The man and the boy's $1$ day work together is $\frac{1}{12}$.
The boy's $1$ day work is $\frac{1}{12} - \frac{1}{20} = \frac{5-3}{60} = \frac{2}{60} = \frac{1}{30}$.
This means the boy can complete the work alone in $30 \text{ days}$.
The ratio of work done by the man and the boy is proportional to their efficiency,which is the inverse of the time taken.
Ratio of work (Man : Boy) $= \frac{1}{20} : \frac{1}{30} = 3 : 2$.
Total share $= Rs. 500$.
Boy's share $= \frac{2}{3+2} \times 500 = \frac{2}{5} \times 500 = Rs. 200$.

(D) The $1^{st}$ man's $3$ days' work $= \frac{3}{6} = \frac{1}{2}$.
The $2^{nd}$ man's $3$ days' work $= \frac{3}{8}$.
The boy's $3$ days' work $= 1 - (\frac{1}{2} + \frac{3}{8}) = 1 - (\frac{4+3}{8}) = 1 - \frac{7}{8} = \frac{1}{8}$.
The ratio of their work done is $\frac{1}{2} : \frac{3}{8} : \frac{1}{8}$.
Multiplying by $8$,we get the ratio $4 : 3 : 1$.
Thus,the money should be divided in the ratio $4 : 3 : 1$.

(B) First,calculate the daily wage of one woman:
Daily wage of $12$ women $= \frac{30240}{24} = Rs. 1260$.
Daily wage of $1$ woman $= \frac{1260}{12} = Rs. 105$.
Since the daily wage of a man is equal to that of a woman,the daily wage of $1$ man $= Rs. 105$.
Let the number of men required be $x$.
Total wages for $x$ men working for $16$ days $= x \times 16 \times 105 = 10080$.
$1680x = 10080$.
$x = \frac{10080}{1680} = 6$.
Therefore,$6$ men are required.

(C) Let the daily wage of a man be $M$ and a boy be $B$.
From the given data:
$3M + 4B = \frac{4200}{7} = 600$ ....$(1)$
$11M + 13B = \frac{16600}{8} = 2075$ ....$(2)$
Multiply equation $(1)$ by $13$ and equation $(2)$ by $4$:
$39M + 52B = 7800$ ....$(3)$
$44M + 52B = 8300$ ....$(4)$
Subtracting $(3)$ from $(4)$:
$5M = 500 \Rightarrow M = 100$.
Substitute $M = 100$ in $(1)$:
$3(100) + 4B = 600 \Rightarrow 4B = 300 \Rightarrow B = 75$.
Now,for $7$ men and $9$ boys,the daily earning is:
$7(100) + 9(75) = 700 + 675 = 1375$.
Time taken to earn $Rs. 19250$ is:
$\frac{19250}{1375} = 14 \text{ days}$.

(D) The efficiency of a person is inversely proportional to the time taken to complete a work.
Let the time taken by $P$ be $T_P = 12$ days and by $Q$ be $T_Q = 18$ days.
The ratio of their efficiencies is $\frac{1}{12} : \frac{1}{18} = 3 : 2$.
Wages are distributed in the ratio of their efficiencies.
Total ratio sum $= 3 + 2 = 5$.
Share of $P = \frac{3}{5} \times 1200 = Rs. 720$.
Share of $Q = \frac{2}{5} \times 1200 = Rs. 480$.