To solve the problem step by step, we will first determine the work rates of X, Y, and Z, and then find out how long Y and Z will take to complete \( \frac{7}{8} \) of the work together.
### Step 1: Determine the work rates of X and Y
- X can complete the work in 16 days, so the work done by X in one day is:
\[
\text{Work rate of X} = \frac{1}{16} \text{ (work/day)}
\]
- Y can complete the work in 24 days, so the work done by Y in one day is:
\[
\text{Work rate of Y} = \frac{1}{24} \text{ (work/day)}
\]
### Step 2: Calculate the combined work rate of X and Y
- The combined work rate of X and Y when they work together is:
\[
\text{Combined work rate} = \frac{1}{16} + \frac{1}{24}
\]
- To add these fractions, we find a common denominator, which is 48:
\[
\frac{1}{16} = \frac{3}{48}, \quad \frac{1}{24} = \frac{2}{48}
\]
- Therefore,
\[
\text{Combined work rate} = \frac{3}{48} + \frac{2}{48} = \frac{5}{48} \text{ (work/day)}
\]
### Step 3: Calculate the work done by X and Y in 4 days
- In 4 days, the amount of work done by X and Y together is:
\[
\text{Work done in 4 days} = 4 \times \frac{5}{48} = \frac{20}{48} = \frac{5}{12}
\]
### Step 4: Determine the remaining work
- The total work is considered as 1 unit. Therefore, the remaining work after X and Y have worked for 4 days is:
\[
\text{Remaining work} = 1 - \frac{5}{12} = \frac{7}{12}
\]
### Step 5: Determine Z's work rate
- Z completes the remaining work in \( 10 \frac{1}{2} \) days, which is \( \frac{21}{2} \) days. Thus, Z's work rate is:
\[
\text{Work rate of Z} = \frac{1}{\frac{21}{2}} = \frac{2}{21} \text{ (work/day)}
\]
### Step 6: Calculate the work rate of Y and Z together
- The combined work rate of Y and Z is:
\[
\text{Work rate of Y and Z} = \frac{1}{24} + \frac{2}{21}
\]
- To add these fractions, we find a common denominator, which is 168:
\[
\frac{1}{24} = \frac{7}{168}, \quad \frac{2}{21} = \frac{16}{168}
\]
- Therefore,
\[
\text{Combined work rate of Y and Z} = \frac{7}{168} + \frac{16}{168} = \frac{23}{168} \text{ (work/day)}
\]
### Step 7: Calculate the time taken by Y and Z to complete \( \frac{7}{8} \) of the work
- To find out how long it takes Y and Z to complete \( \frac{7}{8} \) of the work, we set up the equation:
\[
\text{Time} = \frac{\text{Work}}{\text{Work rate}} = \frac{\frac{7}{8}}{\frac{23}{168}} = \frac{7}{8} \times \frac{168}{23}
\]
- Simplifying this gives:
\[
= \frac{7 \times 21}{23} = \frac{147}{23} \text{ days}
\]
### Conclusion
Thus, Y and Z together can complete \( \frac{7}{8} \) of the work in \( \frac{147}{23} \) days.
