Question

If 5 men or 9 women can do a job in 19 days, how many days will it take 3 men and 6 women to do the same job?
\[\begin{align}
  & \text{A}.\text{12 days} \\
 & \text{B}.\text{15 days} \\
 & \text{C}.\text{18 days} \\
 & \text{D}.\text{21 days} \\
\end{align}\]

Answer 1

Hint: In this question, we are given that 5 men or 9 women can do a job in 19 days and have to calculate the time taken by 3 men and 6 women to do the same job. Here, in the first line 'or' is used so either 5 men will take 19 days or 9 women will take 19 days. Using this, we can calculate the work power of one man in terms of women. Then, we will use it to calculate women’s power for 3 men and 6 women. After that, we will use the concept of inverse proportion to evaluate the required time.

Complete step by step solution:
Here, we are given 5 men or 9 women can do a job in 19 days which means if 5 men work alone they can complete work in 19 days, or if 9 women work alone they can complete work in 19 days.
So, the work power of 5 men will be equal to the work power of 9 women.
We can say, 5 men = 9 women. So,
Work power of 1 man will be $1\text{ men}=\dfrac{9}{5}\text{women}$.
Hence, the work power of 1 man is equal to the work power of $\dfrac{9}{5}$ women.
We need the work power of 3 men and 6 women. So, the work power of 3 men will be $3\times \dfrac{9}{5}\text{ women}=\dfrac{27}{5}$ work power of women.
Total work power of 3 men and 6 women becomes $\dfrac{27}{5}+6$.
Taking LCM of 5, we get: $\Rightarrow \dfrac{27+30}{5}=\dfrac{57}{5}$.
Hence, our problem can be reduced to:
If a woman can do work in 19 days then how much time will $\dfrac{57}{5}$ women take to complete the same work.
As we know, the more the person does a job, the less time is required. So we are dealing with the inverse proportion here.
Let us suppose it takes x days to complete the task for $\dfrac{57}{5}$ women. So, our problem of inverse proportion becomes,
$9\times 19=\dfrac{57}{5}\times x$
From this, we will calculate the value of x, taking $\dfrac{57}{5}$ on other side, we get:
\[\begin{align}
  & \Rightarrow x=9\times 19\times \dfrac{5}{57} \\
 & \Rightarrow x=9\times \dfrac{5}{3} \\
 & \Rightarrow x=3\times 5 \\
 & \Rightarrow x=15 \\
\end{align}\]
Hence it takes 15 days to complete the work done by $\dfrac{57}{5}$ women or we can say it takes 15 days to complete the work done by 3 men and 6 women.
Hence, option B is the correct answer.

Note: Students should take care of the word 'or' and 'and' in the statement. Here, students should not get confused with the term $\dfrac{57}{5}$ women. $\dfrac{57}{5}$ represents the work power of women. Similarly, we can solve the entire question in terms of the work power of men. Take care while determining if we have to use direct proportion or inverse proportion.