The problem describes the progress of digging a drain over several days. We are given that $\frac{7}{15}$ of the total length was dug on the first day, and $\frac{1}{4}$ of the remaining length was dug on the second day. We need to find: (i) The fraction of the total length remaining after the first day. (ii) The fraction of the total length dug on the second day. (iii) Given that 600 meters remained to be dug after the first two days, find the total length of the drain. (iv) If 4 men can dig the remaining 600 meters in 3 days, how many more men are needed to dig the same length in 2 days?

1. Problem Description

The problem describes the progress of digging a drain over several days. We are given that $\frac{7}{15}$ of the total length was dug on the first day, and $\frac{1}{4}$ of the remaining length was dug on the second day. We need to find:

(i) The fraction of the total length remaining after the first day.

(ii) The fraction of the total length dug on the second day.

(iii) Given that 600 meters remained to be dug after the first two days, find the total length of the drain.

(iv) If 4 men can dig the remaining 600 meters in 3 days, how many more men are needed to dig the same length in 2 days?

2. Solution Steps

(i) Fraction remaining after the first day:

If $\frac{7}{15}$ of the drain was dug on the first day, then the remaining fraction is $1 - \frac{7}{15}$.

$1 - \frac{7}{15} = \frac{15}{15} - \frac{7}{15} = \frac{8}{15}$

(ii) Fraction dug on the second day:

On the second day, $\frac{1}{4}$ of the remaining length was dug. The remaining length after the first day is $\frac{8}{15}$ of the total length. So, the fraction dug on the second day is $\frac{1}{4} \times \frac{8}{15}$.

$\frac{1}{4} \times \frac{8}{15} = \frac{8}{60} = \frac{2}{15}$

(iii) Total length of the drain:

After the first two days, the fraction of the drain that has been dug is $\frac{7}{15} + \frac{2}{15} = \frac{9}{15}$.

The fraction of the drain remaining is $1 - \frac{9}{15} = \frac{15}{15} - \frac{9}{15} = \frac{6}{15} = \frac{2}{5}$.

We are given that this remaining fraction, $\frac{2}{5}$, corresponds to 600 meters. Let $L$ be the total length of the drain. Then, $\frac{2}{5}L = 600$. To solve for $L$, we multiply both sides by $\frac{5}{2}$.

$L = 600 \times \frac{5}{2} = 300 \times 5 = 1500$ meters.

(iv) Number of men needed:

4 men can dig 600 meters in 3 days. This means that the total work done is $4 \times 3 = 12$ man-days.

We want to dig the same 600 meters in 2 days. Let $x$ be the number of men required. Then $x \times 2 = 12$.

$x = \frac{12}{2} = 6$ men.

Since we already have 4 men, we need $6 - 4 = 2$ more men.

3. Final Answer

(i) $\frac{8}{15}$

(ii) $\frac{2}{15}$

(iii) 1500 meters

(iv) 2 more men