To solve the problem, we need to determine how many days it will take for X, assisted by Y and Z on every second day, to complete the work.
**Step 1: Calculate the work done by X, Y, and Z in one day.**
- X can complete the work in 30 days, so the work done by X in one day is:
\[
\text{Work by X in one day} = \frac{1}{30}
\]
- Y can complete the work in 45 days, so the work done by Y in one day is:
\[
\text{Work by Y in one day} = \frac{1}{45}
\]
- Z can complete the work in 90 days, so the work done by Z in one day is:
\[
\text{Work by Z in one day} = \frac{1}{90}
\]
**Step 2: Find the total work done by X, Y, and Z together in one day.**
On the second day, X is assisted by Y and Z. Therefore, the total work done on the second day is:
\[
\text{Total work on second day} = \text{Work by X} + \text{Work by Y} + \text{Work by Z} = \frac{1}{30} + \frac{1}{45} + \frac{1}{90}
\]
To add these fractions, we need a common denominator. The least common multiple (LCM) of 30, 45, and 90 is 90. Thus, we convert each fraction:
\[
\frac{1}{30} = \frac{3}{90}, \quad \frac{1}{45} = \frac{2}{90}, \quad \frac{1}{90} = \frac{1}{90}
\]
Now adding them:
\[
\text{Total work on second day} = \frac{3}{90} + \frac{2}{90} + \frac{1}{90} = \frac{6}{90} = \frac{1}{15}
\]
**Step 3: Calculate the total work done in two days.**
In two days:
- On the first day, X works alone:
\[
\text{Work done on first day} = \frac{1}{30}
\]
- On the second day, X, Y, and Z work together:
\[
\text{Work done on second day} = \frac{1}{15}
\]
Thus, the total work done in two days is:
\[
\text{Total work in 2 days} = \frac{1}{30} + \frac{1}{15}
\]
Convert \(\frac{1}{15}\) to a common denominator of 30:
\[
\frac{1}{15} = \frac{2}{30}
\]
Now add:
\[
\text{Total work in 2 days} = \frac{1}{30} + \frac{2}{30} = \frac{3}{30} = \frac{1}{10}
\]
**Step 4: Calculate the total time to complete the work.**
The total work is 1 (the whole job). Since \(\frac{1}{10}\) of the work is completed in 2 days, we can find out how many such sets of 2 days are needed to complete the work:
\[
\text{Number of sets} = \frac{1}{\frac{1}{10}} = 10
\]
Therefore, the total time taken is:
\[
\text{Total time} = 10 \text{ sets} \times 2 \text{ days/set} = 20 \text{ days}
\]
Thus, the work will be completed in **20 days**.
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