To solve this problem, we need to determine how much work each person (X, Y, and Z) completes in a day and then track their contributions over the given days.

Step 1: Calculate the Work Rates of X, Y, and Z

• X can complete the work in 36 days.
  So, X’s work rate is $\frac{1}{36}$ of the work per day.

• Y can complete the work in 90 days.
  So, Y’s work rate is $\frac{1}{90}$ of the work per day.

• Z can complete the work in 54 days.
  So, Z’s work rate is $\frac{1}{54}$ of the work per day.

Step 2: Calculate the Work Done by X, Y, and Z in the First 10 Days

For the first 10 days, all three (X, Y, and Z) work together.

The combined work rate for X, Y, and Z is:

$\text{Total work rate} = \frac{1}{36} + \frac{1}{90} + \frac{1}{54}$

To add these fractions, first find the least common denominator (LCD), which is 540.

$\frac{1}{36} = \frac{15}{540}, \quad \frac{1}{90} = \frac{6}{540}, \quad \frac{1}{54} = \frac{10}{540}$

Adding these:

$\text{Total work rate} = \frac{15}{540} + \frac{6}{540} + \frac{10}{540} = \frac{31}{540}$

This is the amount of work done per day by all three together.

So, the work done in the first 10 days is:

$\text{Work done in 10 days} = 10 \times \frac{31}{540} = \frac{310}{540} = \frac{31}{54}$

Step 3: Calculate the Work Done After Y Leaves

After 10 days, Y leaves the work. Now, only X and Z continue working together.

Their combined work rate is:

$\text{X + Z’s work rate} = \frac{1}{36} + \frac{1}{54}$

The LCD for 36 and 54 is 108.

$\frac{1}{36} = \frac{3}{108}, \quad \frac{1}{54} = \frac{2}{108}$

Adding these:

$\text{X + Z’s work rate} = \frac{3}{108} + \frac{2}{108} = \frac{5}{108}$

They work together for 2 days before X leaves. The work done in these 2 days is:

$\text{Work done in 2 days} = 2 \times \frac{5}{108} = \frac{10}{108} = \frac{5}{54}$

Step 4: Calculate the Remaining Work and Z’s Contribution

Total work done so far is:

$\text{Total work} = \frac{31}{54} + \frac{5}{54} = \frac{36}{54} = \frac{2}{3}$

So, the remaining work is:

$1 - \frac{2}{3} = \frac{1}{3}$

Now, Z alone works to complete the remaining $\frac{1}{3}$ of the work. Z’s work rate is $\frac{1}{54}$ per day.

The time taken by Z to complete $\frac{1}{3}$ of the work is:

$\text{Time} = \frac{\frac{1}{3}}{\frac{1}{54}} = \frac{1}{3} \times 54 = 18 \text{ days}$

Step 5: Calculate the Total Time Taken

The total time taken to complete the work is:

• 10 days (when X, Y, and Z worked together)
• 2 days (when X and Z worked together)
• 18 days (when Z worked alone)

$\text{Total time} = 10 + 2 + 18 = 30 \text{ days}$

Final Answer

The entire work was completed in 30 days.

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Would you like further details or have any questions?

Here are 5 related questions:

1. How would the solution change if Y left after 5 days instead of 10?
2. What if Z’s efficiency increased, and he could finish the work in 45 days?
3. How would the work distribution change if X worked alone for the first 5 days?
4. Can you derive a general formula for the time taken if workers leave at different stages?
5. What if one more worker joined Z after X left?

Tip: Breaking down complex word problems into smaller steps helps prevent errors in calculations.