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

Answer 1

Step $1:$ Let us assume

Boy finish the work in $x$ days and man finish the work in $y$ days

One 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 data

Given 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,

$\Rightarrow \frac{10}{y}=\frac{3}{5}-\frac{2}{4}$

$\Rightarrow \frac{10}{y}=\frac{12-10}{20}$

$\Rightarrow \frac{10}{y}=\frac{1}{10}$

$\Rightarrow y=100$

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 $=\frac{{\displaystyle \frac{1}{y}}}{{\displaystyle \frac{1}{x}}}$

$=\frac{x}{y}$

$=\frac{200}{100}$

$=\frac{2}{1}$

Hence, the required ratio $=2:1$.