8 men and 3 boys together can complete a piece of work in 5 days. If 5 boys can complete the same piece of work in 15 days, in how many days do 5 men complete the same piece of work?

10

The correct answer is Option (4) → 10

1. Find the efficiency of a Boy

We are told that 5 boys can complete the work in 15 days.

The total work can be expressed as "Boy-days":

$\text{Total Work} = 5\text{ boys} \times 15\text{ days} = 75\text{ boy-days}$

This means 1 boy would take 75 days to finish the work alone.

Efficiency of 1 boy ($B$) = $\frac{1}{75}$ of the work per day.

2. Find the efficiency of a Man

We are told that 8 men and 3 boys can complete the work in 5 days.

In 1 day, they complete $\frac{1}{5}$ of the work.

We can set up the equation (where $M$ is the efficiency of 1 man):

$8M + 3B = \frac{1}{5}$

Substitute the value of $B$ ($\frac{1}{75}$):

$8M + 3\left(\frac{1}{75}\right) = \frac{1}{5}$

$8M + \frac{1}{25} = \frac{1}{5}$

$8M = \frac{1}{5} - \frac{1}{25}$

Find a common denominator:

$8M = \frac{5 - 1}{25} = \frac{4}{25}$

$M = \frac{4}{25 \times 8} = \frac{1}{25 \times 2} = \frac{1}{50}$

Efficiency of 1 man ($M$) = $\frac{1}{50}$ of the work per day.

3. Calculate Time for 5 Men

We need to find how many days ($D$) it takes for 5 men to finish the work:

$\text{Work rate of 5 men} = 5 \times M = 5 \times \frac{1}{50} = \frac{1}{10}$

Since they do $\frac{1}{10}$ of the work per day, the total time required is:

$\text{Time} = \frac{1}{1/10} = 10\text{ days}$

Final Answer:

5 men will complete the work in 10 days.