A can complete a work in $x$ days, and B can complete the same work in $2x$ days. A starts the work and works for $\frac{22}{9}$ days, then B replaces A and finishes the remaining work in the same time that C and D together would take to complete the entire work. The efficiency ratio of C to D is $4:5$. If C and D work on alternate days starting with C, they complete the work in $\frac{89}{2}$ days. E and F together work for 24 days and then are replaced by A and B respectively, who complete the remaining work in 20 days. The efficiency ratio of E to F is $5:4$. If E and F together complete the entire work alone, what is the difference between the work done by E alone and the work done by F alone?

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**Solution:**

1. Let the number of days E takes to complete the work alone be $4x$ and F takes $5x$ days (since efficiency ratio E:F = 5:4, time ratio is inverse).

2. Work done by E in 24 days = $\frac{24}{4x} = \frac{6}{x}$.

3. Work done by F in 24 days = $\frac{24}{5x} = \frac{24}{5x}$.

4. After 24 days, A and B work for 20 days to finish the remaining work. The combined work done by A and B in 20 days is given as:

$$\frac{20}{	ext{time taken by A and B together}}$$

But from the problem, this combined work plus the work done by E and F in 24 days equals the whole work (1).

5. Using the given data and calculations (from the problem statement), the equation is:

$$\frac{24}{5x} + \frac{24}{4x} + \frac{20}{100/3} + \frac{20}{200/3} = 1$$

Simplifying:

$$24 	imes \frac{9}{20x} + \frac{60}{100} + \frac{60}{200} = 1$$

$$\frac{216}{20x} + 0.6 + 0.3 = 1$$

$$\frac{216}{20x} = 0.1$$

$$x = \frac{216}{20 	imes 0.1} = 108$$

6. Total time for E and F together to complete the work:

$$	ext{LCM of } 4x 	ext{ and } 5x = 20x = 20 	imes 108 = 2160 	ext{ days}$$

But since they work together, their combined rate is:

$$\frac{1}{4x} + \frac{1}{5x} = \frac{5 + 4}{20x} = \frac{9}{20x}$$

So, total time to complete work together:

$$\frac{1}{\frac{9}{20x}} = \frac{20x}{9} = \frac{20 	imes 108}{9} = 240 	ext{ days}$$

7. Work done by E alone in 240 days:

$$\frac{240}{4x} = \frac{240}{4 	imes 108} = \frac{240}{432} = \frac{5}{9}$$

8. Work done by F alone in 240 days:

$$\frac{240}{5x} = \frac{240}{5 	imes 108} = \frac{240}{540} = \frac{4}{9}$$

9. Difference between work done by E and F alone:

$$\frac{5}{9} - \frac{4}{9} = \frac{1}{9}$$

**Answer:** $\frac{1}{9}$

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