Question

4 men and 6 women get Rs. 1600 by doing a piece of work in 5 days. 3 men and 7 women get Rs. 1740 by doing the same work in 6 days. In how many days, 7 men and 6 women can complete the same work getting Rs. 3760?
(a) 6 days
(b) 8 days
(c) 10 days
(d) 12 days

Answer 1

Hint: Assume that each man is getting ‘$x$’ rupees and each woman is getting ‘$y$’ rupees for their work in one day. Form two equations in two variables and solve them to find $x\text{ and }y$. Assume that the required number of days 7 men and 6 women have to work is ‘$n$’. Form another equation and use the value of $x\text{ and }y$ to find $n$.

Complete step-by-step answer:
Let us assume that 1 man gets Rs. ‘$x$’ for his work in one day.

And, 1 woman gets Rs. ‘$y$’ for her work in one day.

Therefore, total earnings of 4men and 6 women in 1 day $=\dfrac{1600}{5}$.

$\begin{align}
  & \therefore 4x+6y=\dfrac{1600}{5} \\
 & 4x+6y=320 \\
 & 2x+3y=160..................(i) \\
\end{align}$

Also, total earnings of 3men and 7 women in 1 day $=\dfrac{1740}{6}$.

$\begin{align}
  & \therefore 3x+7y=\dfrac{1740}{6} \\
 & 3x+7y=290.................(ii) \\
\end{align}$

Now, multiplying equation (i) by 3 and equation (ii) by 2, we get,

$\begin{align}
  & 6x+9y=480.................(iii) \\
 & 6x+14y=580................(iv) \\
\end{align}$

Subtracting equation (iii) from equation (iv), we get,

$\begin{align}
  & 5y=100 \\
 & \therefore y=20 \\
\end{align}$

Now, substituting the value of $y$ in equation (i), we have,

$\begin{align}
  & 2x=160-60 \\
 & 2x=100 \\
 & \therefore x=50 \\
\end{align}$

Hence, 1 man gets Rs. 50 and 1 woman gets Rs. 20 each day.

Let 7 men and 6 women require ‘$n$’ days to complete the same work to get Rs. 3760.

Therefore, in 1 day they will get a total of $\dfrac{3760}{n}$ rupees.

$\therefore 7x+6y=\dfrac{3760}{n}$

Substituting the value of $x\text{ and }y$ in the above equation, we get,

$\begin{align}
  & 350+120=\dfrac{3760}{n} \\
 & 470=\dfrac{3760}{n} \\
 & n=\dfrac{3760}{470} \\
 & \therefore n=8 \\
\end{align}$

Hence, option (b) is the correct answer.

Note: We have to form all equations for a single day otherwise it would be difficult to calculate the money and hence the number of days required to complete the work. If we will calculate the money for 5 days or 6 days then we have to multiply the equations by these numbers and hence the calculations will be difficult.