Question

If certain number of workmen can do a piece of work in \[25\] hours, in how many hours another set of an equal number of men, do a piece of work, twice as great supposing that \[2\] men of first set can do as much work in an hour, as \[3\] men of the second set do in an hour?
A) \[60\]
B) \[75\]
C) \[90\]
D) \[105\]

Answer 1

Hint: The given question is based on time and work topic. We have to find the speed of the worker from the given data and then form an equation using the formula \[\dfrac{{M_1 \times D_1 \times T_1}}{{W_1}} = \dfrac{{M_2 \times D_2 \times T_2}}{{W_2}}\]. We can find the required hours $y$ substituting the values in the formula.

Complete step by step solution:
Time and work deals with the time taken by an individual or a group of individuals to complete a piece of work and the efficiency of the work done by each of them.
If \[x\] number of people can do \[W_1\] work, in \[D_1\] days, working \[T_1\] hours each day and the number of people can do \[W_2\] work, in \[D_2\] days, working \[T_2\] hours each day, then the relation between them will be:
\[\dfrac{{M_1 \times D_1 \times T_1}}{{W_1}} = \dfrac{{M_2 \times D_2 \times T_2}}{{W_2}}\]
Moreover \[Rate\,of\,work = \dfrac{1}{{Time\,taken}}\].
Now we can solve the sum as follows:
Let the number of required hours be \[x\].
We are given that \[2\] men of the first set can do as much work in an hour, as \[3\] men of the second set do in an hour.
Using \[Rate\,of\,work = \dfrac{1}{{Time\,taken}}\], the speed of first set of workers will be \[Speed = \dfrac{1}{2}\]
For the second set of workers, it will be \[Speed = \dfrac{1}{3}\].
Now we apply the formula \[\dfrac{{M_1 \times D_1 \times T_1}}{{W_1}} = \dfrac{{M_2 \times D_2 \times T_2}}{{W_2}}\] as follows-
Since we are given a certain number of workers, let us assume \[1\] workers. So \[M_1 = 1\], speed will be \[D_1 = \dfrac{1}{3}\] and hours are the missing required figure \[T_1 = x\]. We are given that \[M_2 = 1\]having speed \[D_2 = \dfrac{1}{2}\] can complete the same work \[W_1 = W_2\] in \[T_2 = 25\]hours. Moreover, the speed of the second set of workers is twice as great.
We can form the equation as follows by substituting the above information:
\[M_1 \times D_1 \times T_1 = M_2 \times D_2 \times T_2\] since \[W_1 = W_2\]
\[1 \times \dfrac{1}{3} \times x = 2(1 \times \dfrac{1}{2} \times 25)\]
Simplifying the equation, we get,
\[\dfrac{1}{3}x = 25\]
Cross-multiplying, we get,
\[x = 25 \times 3\]
\[x = 75\]

Thus, the required number of hours is \[75\]. Hence Option (B) is correct.

Note:
Knowing the formulas allows you to jump right to an answer after reading the question. As a result, understanding the formula for every numerical skill subject simplifies the solution and calculations.
Other important points are as follows:
Efficiency and Time are inversely proportional to each other.
If a piece of work is done in \[x\] number of days, then the work done in one day = \[\dfrac{1}{x}\].
If \[\dfrac{x}{y}\] is the ratio of the number of men which are required to complete a piece of work, then the ratio of the time taken by them to complete the work will be \[\dfrac{y}{x}\].