Question

12 men can complete a piece of work in 16 days. How many days will 4 men take to complete the task? \[\]
A.60 days\[\]
B.45 days\[\]
C.54 days\[\]
D.48 days\[\]

Answer 1

Hint: We see that the problem is indirect variation and so we use a unitary method to first find the number of days 1 man will take to complete the work by multiplying $a\times b$ where $a$ the given number of men is and $b$ is the given number of days. We divide $ab$ by 4 to find the answer. \[\]

Complete step-by-step solution
We know that the unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value. There are two types of unitary method one is direct variation and the other is indirect variation. \[\]
If the one quantity $a$ decreases with another increase of quantity $b$ and also $a$ increases with a decrease in quantity $b$ then we say the quantities $a$ and $b$ are in indirect variation and the product $ab$ remains constant and represents the value of a single unit. Then we divide by the given value to get the corresponding value. \[\]
We are given the question that 12 men can complete a piece of work in 16 days. We see that if we decrease the number of men more days will be required to complete the task and if we increase the number of men fewer days will be required for completing the task. So the problem is indirect variation where $a=16,b=12$. \[\]
The single unit is the number of days one man will take to complete the task which is,
\[ab=a\times b=16\times 12\]
We are asked to find the number of days 4 men will take to complete the task. So the given value is 4. So the number of days corresponding to the given value is
\[\dfrac{ab}{4}=\dfrac{16\times 12}{4}=48\]
So the correct option is D.

Note: We must be careful of the difference between indirect and direct variation where quantity $a$ increases with another quantity $b$ and also $a$ decreases with $b$ with $\dfrac{a}{b}$ remains constant and represents the value of a single unit. Men and work, speed, and time are indirect variation problems while weights and objects are direct variation problems.