Question

# $12$ men and $16$ boys can do a piece of work in $5$ days. $13$ men and $24$ boys can do it in $4$ days. The ratio of the daily work done by a man to that of a boy is

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Solution

## Step $1:$ Let us assumeBoy finish the work in $x$ days and man finish the work in $y$ daysOne day work of a boy $=\frac{1}{x}$and one day work of a man $=\frac{1}{y}$One day work of $16$ boys $=\frac{16}{x}$and one day work of $12$ men $=\frac{12}{y}$Step $2:$ Making the equation on the given dataGiven that,One day work of $12$ men and $16$ boys $=\frac{1}{5}$$\frac{16}{x}+\frac{12}{y}=\frac{1}{5}...\left(1\right)$Similarly,One day work of $13$ men and $24$ boys $=\frac{1}{4}$$\frac{24}{x}+\frac{13}{y}=\frac{1}{4}...\left(2\right)$Step $3:$ Multiplying by $3$ to equation $\left(1\right)$ and $2$ to equation $\left(2\right)$$\frac{48}{x}+\frac{36}{y}=\frac{3}{5}...\left(3\right)$$\frac{48}{x}+\frac{26}{y}=\frac{2}{4}...\left(4\right)$Subtract equation $\left(4\right)$ from equation $\left(3\right)$ we get,$⇒\frac{10}{y}=\frac{3}{5}-\frac{2}{4}$$⇒\frac{10}{y}=\frac{12-10}{20}$$⇒\frac{10}{y}=\frac{1}{10}$$⇒y=100$Now, substitute $y=100$ in equation $\left(1\right)$$\frac{16}{x}+\frac{12}{100}=\frac{1}{5}$$\frac{16}{x}=\frac{1}{5}-\frac{12}{100}$$\frac{16}{x}=\frac{8}{100}$$x=\frac{1600}{8}$$x=200$Required Ratio $=\frac{\frac{1}{y}}{\frac{1}{x}}$ $=\frac{x}{y}$ $=\frac{200}{100}$ $=\frac{2}{1}$Hence, the required ratio $=2:1$.

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