A can do a piece of work in 9 days and B can do the same work in 15 days. If they work together, in what ratio A and B will receive their wages?

Options

A. `3:5`
B. `5:3`
C. `2:5`
D. `5:2`

Answer

To solve the problem of how A and B will receive their wages in ratio when working together, we will follow these steps:

### Step 1: Determine the work done by A and B in one day.
- A can complete the work in 9 days. Therefore, A's work done in one day is:
\[ \text{Work done by A in one day} = \frac{1}{9} \]
- B can complete the work in 15 days. Therefore, B's work done in one day is:
\[ \text{Work done by B in one day} = \frac{1}{15} \]

### Step 2: Find the combined work done by A and B in one day.
- To find the total work done by A and B together in one day, we add their individual work rates:
\[ \text{Combined work in one day} = \frac{1}{9} + \frac{1}{15} \]
- To add these fractions, we need a common denominator. The least common multiple (LCM) of 9 and 15 is 45. Thus, we convert the fractions:
\[ \frac{1}{9} = \frac{5}{45}, \quad \frac{1}{15} = \frac{3}{45} \]
- Now, we can add them:
\[ \text{Combined work in one day} = \frac{5}{45} + \frac{3}{45} = \frac{8}{45} \]

### Step 3: Determine the ratio of work done by A and B.
- The ratio of work done by A and B can be found by comparing their individual work rates:
- A's work rate: \(\frac{1}{9}\)
- B's work rate: \(\frac{1}{15}\)
- To find the ratio, we can express it as:
\[ \text{Ratio of A's work to B's work} = \frac{\frac{1}{9}}{\frac{1}{15}} = \frac{15}{9} \]
- Simplifying this ratio:
\[ \frac{15}{9} = \frac{5}{3} \]

### Step 4: Conclusion
- Therefore, the ratio of wages received by A and B will be:
\[ \text{Ratio of wages} = 5:3 \]