P, Q, and R can finish a work in 5 days, 10 days, and 15 days respectively when working alone. P and Q work together on the first day, P and R work together on the second day, and P and Q work together again on the third day. In how many days will the work be completed?

Question

Accepted Answer

**Solution:**

1. Calculate the daily work done by each person:
   - P's rate = $\frac{1}{5}$ work/day
   - Q's rate = $\frac{1}{10}$ work/day
   - R's rate = $\frac{1}{15}$ work/day

2. Calculate work done on each day:
   - Day 1 (P + Q): $\frac{1}{5} + \frac{1}{10} = \frac{2}{10} + \frac{1}{10} = \frac{3}{10}$
   - Day 2 (P + R): $\frac{1}{5} + \frac{1}{15} = \frac{3}{15} + \frac{1}{15} = \frac{4}{15}$
   - Day 3 (P + Q): same as Day 1 = $\frac{3}{10}$

3. Total work done in 3 days:
   $$\frac{3}{10} + \frac{4}{15} + \frac{3}{10} = \frac{3}{10} + \frac{3}{10} + \frac{4}{15} = \frac{6}{10} + \frac{4}{15}$$

4. Find common denominator for $\frac{6}{10}$ and $\frac{4}{15}$:
   - $	ext{LCM}(10, 15) = 30$
   - Convert fractions:
     - $\frac{6}{10} = \frac{18}{30}$
     - $\frac{4}{15} = \frac{8}{30}$

5. Sum:
   $$\frac{18}{30} + \frac{8}{30} = \frac{26}{30} = \frac{13}{15}$$

6. Work left after 3 days:
   $$1 - \frac{13}{15} = \frac{2}{15}$$

7. On the 4th day, P and R work together again:
   - Work done per day by P and R = $\frac{1}{5} + \frac{1}{15} = \frac{4}{15}$

8. Time to finish remaining work:
   $$	ext{Time} = \frac{	ext{Work left}}{	ext{Work per day}} = \frac{\frac{2}{15}}{\frac{4}{15}} = \frac{2}{15} 	imes \frac{15}{4} = \frac{2}{4} = \frac{1}{2} 	ext{ day}$$

9. Total time to complete the work:
   $$3 + \frac{1}{2} = \frac{7}{2} 	ext{ days}$$

**Answer:** $\frac{7}{2}$ days

Answer

Answer

Answer