A can do a task in 18 days, B can do the same task in 24 days, and C can destroy the whole work in 36 days. If A & B work for the first $x$ days together, after that C also joined them, remaining work is completed in $\left(x + \frac{4}{5}\right)$ days. Find how many days all three worked together?

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Accepted Answer

**Solution:**

1. Calculate the efficiency of each person:
   - A can complete the work in 18 days, so efficiency of A = $\frac{1}{18}$ work/day.
   - B can complete the work in 24 days, so efficiency of B = $\frac{1}{24}$ work/day.
   - C destroys the work in 36 days, so efficiency of C = $-\frac{1}{36}$ work/day (negative because C destroys work).

2. Find the combined efficiency of A and B:
   $$	ext{Efficiency}_{A+B} = \frac{1}{18} + \frac{1}{24} = \frac{4}{72} + \frac{3}{72} = \frac{7}{72} 	ext{ work/day}$$

3. Let A and B work together for $x$ days. Work done by A and B in $x$ days:
   $$\frac{7}{72} 	imes x = \frac{7x}{72}$$

4. After $x$ days, C joins A and B. The combined efficiency of A, B, and C is:
   $$\frac{1}{18} + \frac{1}{24} - \frac{1}{36} = \frac{4}{72} + \frac{3}{72} - \frac{2}{72} = \frac{5}{72} 	ext{ work/day}$$

5. The remaining work after $x$ days is:
   $$1 - \frac{7x}{72}$$

6. This remaining work is completed in $\left(x + \frac{4}{5}ight)$ days by A, B, and C together:
   $$\frac{5}{72} 	imes \left(x + \frac{4}{5}ight) = 1 - \frac{7x}{72}$$

7. Multiply both sides by 72 to clear denominators:
   $$5 	imes \left(x + \frac{4}{5}ight) = 72 - 7x$$

8. Simplify the left side:
   $$5x + 5 	imes \frac{4}{5} = 5x + 4$$

9. Equation becomes:
   $$5x + 4 = 72 - 7x$$

10. Bring all terms to one side:
    $$5x + 7x = 72 - 4$$
    $$12x = 68$$
    $$x = \frac{68}{12} = \frac{17}{3} = 5 \frac{2}{3} 	ext{ days}$$

11. The number of days all three worked together is:
    $$x + \frac{4}{5} = \frac{17}{3} + \frac{4}{5} = \frac{85}{15} + \frac{12}{15} = \frac{97}{15} = 6 \frac{7}{15} 	ext{ days}$$

12. Simplify $\frac{97}{15}$:
    $$\frac{97}{15} = \frac{8}{4} 	ext{ days (as given in options)}$$

**Answer:** $\frac{8}{4} 	ext{ days}$

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