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$(a){\text{ 2:1}} \\ (b){\text{ 3:1}} \\ (c){\text{ 3:2}} \\ (d){\text{ 5:4}} \\$

Hint: In this question let one boy take x days to finish a work and one man takes y days to finish work, thus using the unitary method one day work for boys and men can be taken out. Then use this to find the work done by 12 men and 16 boys and then 13 men and 24 boys to form two different equations in two variables to get the answer.

Let one boy finish the work in x days.
And one man finished the work in y days.
So the one day work of one boy is $\dfrac{1}{x}$ and one day's work of one man is $\dfrac{1}{y}$.
So the one day work of 16 boys is $\dfrac{{16}}{x}$ and one day work of 12 men is $\dfrac{{12}}{y}$.
Now it is given that 12 men and 16 boys can do a piece of work together in 5 days.
So 12 men and 16 boys one day work together is $\dfrac{1}{5}$.
So one day work of 16 boys + one day work of 12 men = one day work of 12 men and 16 boys together.
$\Rightarrow \dfrac{{16}}{x} + \dfrac{{12}}{y} = \dfrac{1}{5}$............................ (1)
Similarly
One day work of 24 boys + one day work of 13 men = one day work of 13 men and 24 boys together.
Together they will finish in four days.
$\Rightarrow \dfrac{{24}}{x} + \dfrac{{13}}{y} = \dfrac{1}{4}$................................... (2)
Now multiply by 3 and 2 in equation (1) and (2) respectively we have,
$\Rightarrow \dfrac{{48}}{x} + \dfrac{{36}}{y} = \dfrac{3}{5}$................................... (3)
And
$\Rightarrow \dfrac{{48}}{x} + \dfrac{{26}}{y} = \dfrac{2}{4}$....................................... (4)
Now subtract equation (4) from equation (3) we have,
$\Rightarrow \dfrac{{48}}{x} + \dfrac{{36}}{y} - \dfrac{{48}}{x} - \dfrac{{26}}{y} = \dfrac{3}{5} - \dfrac{2}{4}$
$\Rightarrow \dfrac{{10}}{y} = \dfrac{{12 - 10}}{{20}} = \dfrac{2}{{20}} = \dfrac{1}{{10}}$
$\Rightarrow y = 100$
Now substitute this value in equation (1) we have,
$\Rightarrow \dfrac{{16}}{x} + \dfrac{{12}}{{100}} = \dfrac{1}{5}$
$\Rightarrow \dfrac{{16}}{x} = \dfrac{1}{5} - \dfrac{{12}}{{100}} = \dfrac{8}{{100}}$
$\Rightarrow x = \dfrac{{1600}}{8} = 200$
So the ratio(R) of daily work done by a man to that of a boy is
$\Rightarrow R = \dfrac{{\dfrac{1}{y}}}{{\dfrac{1}{x}}} = \dfrac{x}{y} = \dfrac{{200}}{{100}} = \dfrac{2}{1}$
So the ratio is (2 : 1).
Hence option (A) is correct.

Note: There are always two ways to solve any two linear equations involving two variables. The method of elimination is being used above,in that the coefficients of any variable are made the same for both the equations and then added or subtracted to eliminate that variable to get the value of another variable. In substitution one variable is taken out in terms of another variable and then substituted into another equation to take out the variable.

Alternate Solution:

Let a man do ‘m’ work in a day and a boy do ‘b’ work in a day. Let the total work be ‘W’.

W = 5 (12m + 16b)  -> Equation (1)

Also, W = 4 (13m + 24b)  -> Equation (2)

Combining equations (1) and (2)

60m + 80b = 52m + 96b

8m = 16b

m/b = 2/1

Therefore, the ratio m:b = 2:1