Vasant can do a piece of work in 24 days. He works at it alone for 4 days and his friend Ritu alone finishes the remaining work in 25 days. Both of them together can complete the work in:

Let’s solve this step by step.

First, let’s find out how much work Vasant can complete in one day. Since Vasant can do the entire piece of work in 24 days, his work rate is:

\[
\text{Vasant’s work rate} = \frac{1}{24} \text{ (work per day)}
\]

Vasant works alone for 4 days, so the fraction of the work he completes is:

\[
\text{Work done by Vasant} = 4 \times \frac{1}{24} = \frac{1}{6}
\]

This means that after Vasant works for 4 days, \(\frac{1}{6}\) of the work is done and \(\frac{5}{6}\) of the work is remaining.

Ritu completes the remaining \(\frac{5}{6}\) of the work in 25 days. So, Ritu’s work rate is:

\[
\text{Ritu’s work rate} = \frac{\frac{5}{6}}{25} = \frac{1}{30} \text{ (work per day)}
\]

Now, we want to find out how long it would take for Vasant and Ritu to complete the work together. The combined work rate of Vasant and Ritu is:

\[
\text{Combined work rate} = \text{Vasant’s work rate} + \text{Ritu’s work rate} = \frac{1}{24} + \frac{1}{30}
\]

To find the time taken for them to complete the work together, we take the reciprocal of their combined work rate:

\[
\text{Time taken together} = \frac{1}{\text{Combined work rate}} = \frac{1}{\frac{1}{24} + \frac{1}{30}}
\]

Let’s calculate the exact value of the time taken together using this formula.

The time taken for Vasant and Ritu to complete the work together is:

\[
\text{Time taken together} = \frac{1}{\frac{1}{24} + \frac{1}{30}} = \frac{1}{\frac{30 + 24}{24 \times 30}} = \frac{24 \times 30}{54} = \frac{720}{54} = \frac{40}{3} = 13\frac{1}{3} \text{ days}
\]

So, Vasant and Ritu together can complete the work in \(13\frac{1}{3}\) days.