P can do a piece of work in 25 days and Q can do the same work in 15 days. In how many days they can complete the work by working together?

Answer

To solve the problem of how many days P and Q can complete the work together, we can follow these steps:

### Step 1: Determine the work done by P and Q in one day.
- P can complete the work in 25 days, so the work done by P in one day is:
\[ \text{Work done by P in one day} = \frac{1}{25} \]
- Q can complete the work in 15 days, so the work done by Q in one day is:
\[ \text{Work done by Q in one day} = \frac{1}{15} \]

### Step 2: Calculate the total work done by P and Q together in one day.
- To find the total work done by both P and Q in one day, we add their individual work rates:
\[ \text{Total work in one day} = \frac{1}{25} + \frac{1}{15} \]

### Step 3: Find the LCM of the denominators.
- The denominators are 25 and 15. The least common multiple (LCM) of 25 and 15 can be calculated as follows:
- Prime factorization of 25: \(5^2\)
- Prime factorization of 15: \(3 \times 5\)
- LCM = \(3 \times 5^2 = 75\)

### Step 4: Convert the fractions to have a common denominator.
- Now we convert \(\frac{1}{25}\) and \(\frac{1}{15}\) to have a denominator of 75:
\[ \frac{1}{25} = \frac{3}{75} \quad (\text{since } 1 \times 3 = 3 \text{ and } 25 \times 3 = 75) \]
\[ \frac{1}{15} = \frac{5}{75} \quad (\text{since } 1 \times 5 = 5 \text{ and } 15 \times 5 = 75) \]

### Step 5: Add the fractions.
- Now we can add the two fractions:
\[ \text{Total work in one day} = \frac{3}{75} + \frac{5}{75} = \frac{8}{75} \]

### Step 6: Calculate the total days to complete the work together.
- If P and Q can do \(\frac{8}{75}\) of the work in one day, the total number of days required to complete the work is the reciprocal of this value:
\[ \text{Total days} = \frac{75}{8} \]

### Step 7: Convert to mixed fraction.
- To convert \(\frac{75}{8}\) into a mixed fraction:
- Divide 75 by 8:
- \(8 \times 9 = 72\) (which is the largest multiple of 8 less than 75)
- Remainder: \(75 - 72 = 3\)
- Therefore, \(\frac{75}{8} = 9 \frac{3}{8}\)

### Final Answer:
- P and Q together can complete the work in \(9 \frac{3}{8}\) days.

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