Question

If 10 men working 7 hours a day dig a trench 147m long, how many men working 8 hours a day will dig a trench 168m long (of the same breadth and depth as the first in the same number of days)?

Answer 1

Hint: Assume the rate at which a single man can dig trench to be x meter per hour.
This rate will be the same for the other group of men too.
Find this value of x using the information that 10 men working 7 hours a day dig a trench 147m long.
 Use this value of x to find the required no, of men working 8 hours a day to dig a trench 168m long.

Complete step by step answer:
 Here, in the given question it is not given that in how many days the 147m trench was dug by the first group of 10 men. So, let us assume they dug it in 1 day.
Hence, the other group of men will also have to dig 168m long trench in 1 day. (given in question that both the groups complete their work in same no. of days)
Now, let us assume each man can dig $x$ meter long trench per hour in a day.
$\therefore $10 men can dig $10 \times x$ meter long trench in 1 hour.
$\therefore $In 7 hours, they will dig = $10 \times x \times 7 = 70x$ meter long trench. -------(1)
As we know, 10 men working 7 hours a day dug a 147m long trench.
$\begin{gathered}
\Rightarrow 70x = 147m{\text{ }}\left( {{\text{from (1)}}} \right) \\
\Rightarrow x = \dfrac{{147}}{{70}} = 2.1m \\
\end{gathered}$

Therefore, work output of each men is:
P = digging 2.1-meter-long trench per hour in a day. -------------(2)
This work output will be the same for the other group of men too.
Now, let us assume the required no. of men to dig a trench 168m long in 8hours a day be y.
We know, 1 man can dig a 2.1-meter-long trench in 1 hour.
$\Rightarrow \therefore y$ men can dig $2.1 \times y$ meter long trench in 1 hour.
$\Rightarrow \therefore $In 8 hours, they will dig = $2.1 \times y \times 8 = 16.8y$ meter long trench.
This value must be equal to 168.
$\begin{gathered}
\Rightarrow 16.8y = 168 \\
\Rightarrow y = \dfrac{{168}}{{16.8}} = 10{\text{ men}} \\
\end{gathered}$

Therefore, 10 men are required to dig a trench 168m long in 8hours a day.
So, it means if the same no. of men works 8 hours a day, they will be able to dig a 168m long trench.

Note: In above question we assumed that both the groups complete their task in 1 day, but you can assume any no. of days because until each group completes their task in the same no. of days it will not have any effect at the solution. During simplification it will be cancelled out.