To solve the problem step by step, we will calculate the work done by X, Y, and Z, and then determine how much Z should be paid based on their contributions.
### Step 1: Determine the work done by X and Y in one day.
- X can complete the work in 15 days.
- Therefore, work done by X in one day = \( \frac{1}{15} \) of the work.
- Y can complete the work in 10 days.
- Therefore, work done by Y in one day = \( \frac{1}{10} \) of the work.
### Step 2: Calculate the combined work done by X and Y in one day.
- Combined work done by X and Y in one day = Work done by X + Work done by Y
\[
= \frac{1}{15} + \frac{1}{10}
\]
- To add these fractions, we need a common denominator. The least common multiple (LCM) of 15 and 10 is 30.
\[
= \frac{2}{30} + \frac{3}{30} = \frac{5}{30} = \frac{1}{6}
\]
### Step 3: Determine the total work done by X, Y, and Z together in one day.
- Let the work done by Z in one day be represented as \( z \).
- Together, X, Y, and Z complete the work in 5 days.
- Therefore, the total work done in one day by X, Y, and Z together = \( \frac{1}{5} \).
### Step 4: Set up the equation for the work done by Z.
- From the previous steps, we know:
\[
\frac{1}{6} + z = \frac{1}{5}
\]
- To find \( z \), we need to solve this equation.
- First, find a common denominator for \( \frac{1}{6} \) and \( \frac{1}{5} \), which is 30.
\[
\frac{5}{30} + z = \frac{6}{30}
\]
- Rearranging gives:
\[
z = \frac{6}{30} - \frac{5}{30} = \frac{1}{30}
\]
### Step 5: Calculate the total efficiency of X, Y, and Z.
- The efficiencies are:
- X: \( \frac{1}{15} \)
- Y: \( \frac{1}{10} \)
- Z: \( \frac{1}{30} \)
- Total efficiency = \( \frac{1}{15} + \frac{1}{10} + \frac{1}{30} \)
- Using a common denominator of 30:
\[
= \frac{2}{30} + \frac{3}{30} + \frac{1}{30} = \frac{6}{30} = \frac{1}{5}
\]
### Step 6: Determine the payment for Z.
- The total payment for the work is Rs. 720.
- The payment is distributed based on the efficiency of each worker.
- Z's share of the total work is \( \frac{1}{30} \) out of the total efficiency \( \frac{1}{5} \).
### Step 7: Calculate Z's payment.
- The ratio of Z's work to total work:
\[
\text{Z's payment} = \frac{Z's \text{ efficiency}}{\text{Total efficiency}} \times \text{Total payment}
\]
\[
= \frac{\frac{1}{30}}{\frac{1}{5}} \times 720 = \frac{1}{30} \times 5 \times 720
\]
\[
= \frac{720}{6} = 120
\]
### Final Answer:
Z is paid Rs. 120.
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