If Ramesh can complete 2/5th of a work in 24 days and Satish can complete 1/3rd of the work in 30 days, then in how many days can they complete the work if they work together?

Answer

To solve the problem, we will follow these steps:

### Step 1: Find Ramesh's work rate
Ramesh can complete \( \frac{2}{5} \) of the work in 24 days. To find his work rate (work done per day), we can use the formula:

\[ \text{Work Rate of Ramesh} = \frac{\text{Work Completed}}{\text{Time Taken}} = \frac{\frac{2}{5}}{24} \]

Calculating this:

\[ \text{Work Rate of Ramesh} = \frac{2}{5 \times 24} = \frac{2}{120} = \frac{1}{60} \]

### Step 2: Find Satish's work rate
Satish can complete \( \frac{1}{3} \) of the work in 30 days. Similarly, we calculate his work rate:

\[ \text{Work Rate of Satish} = \frac{\text{Work Completed}}{\text{Time Taken}} = \frac{\frac{1}{3}}{30} \]

Calculating this:

\[ \text{Work Rate of Satish} = \frac{1}{3 \times 30} = \frac{1}{90} \]

### Step 3: Combine their work rates
Now, we can find the combined work rate of Ramesh and Satish when they work together:

\[ \text{Combined Work Rate} = \text{Work Rate of Ramesh} + \text{Work Rate of Satish} \]

Substituting the values we found:

\[ \text{Combined Work Rate} = \frac{1}{60} + \frac{1}{90} \]

### Step 4: Find a common denominator
To add these fractions, we need a common denominator. The least common multiple (LCM) of 60 and 90 is 180.

Now, we convert both fractions:

\[ \frac{1}{60} = \frac{3}{180} \]
\[ \frac{1}{90} = \frac{2}{180} \]

Adding these together:

\[ \text{Combined Work Rate} = \frac{3}{180} + \frac{2}{180} = \frac{5}{180} = \frac{1}{36} \]

### Step 5: Calculate the total time to complete the work together
To find out how many days they will take to complete the entire work together, we take the reciprocal of their combined work rate:

\[ \text{Time Taken} = \frac{1}{\text{Combined Work Rate}} = \frac{1}{\frac{1}{36}} = 36 \text{ days} \]

### Final Answer
Ramesh and Satish can complete the work together in **36 days**.

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