A and B together can complete a piece of work in 24 days. C's efficiency is half that of A's. If A, B, and C all work together, they finish the work in 20 days. How many days will B and C take to complete the work if they work together?

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Accepted Answer

**Solution:**

1. Let the work done by A in one day be *a* units.
2. Since C's efficiency is half of A's, C's one day work = $ $\frac{a}{2} $$ units.
3. Let B's one day work be *b* units.

4. Given A and B together complete the work in 24 days, so:
   $$
   $a + b = \frac{1}{24}$
   $$
   (Because total work = 1 unit, so daily work rate = $ $\frac{1}{\text{days}} $$)

5. When A, B, and C work together, they finish in 20 days:
   $$
   $a + b + \frac{a}{2} = \frac{1}{20}$
   $$

6. Simplify the second equation:
   $$
   $a + b + \frac{a}{2} = \frac{1}{20} \implies (a + b) + \frac{a}{2} = \frac{1}{20}$
   $$

7. Substitute $ $a + b = \frac{1}{24}$ $ from step 4:
   $$
   $\frac{1}{24} + \frac{a}{2} = \frac{1}{20}$
   $$

8. Rearranging to find *a*:
   $$
   $\frac{a}{2} = \frac{1}{20} - \frac{1}{24} = \frac{6 - 5}{120} = \frac{1}{120}$
   $$

9. Therefore,
   $$
  $ a = \frac{2}{120} = \frac{1}{60}$
   $$

10. From step 4, find *b*:
    $$
   $ a + b = \frac{1}{24} \implies b = \frac{1}{24} - a = \frac{1}{24} - \frac{1}{60} = \frac{5 - 2}{120} = \frac{3}{120} = \frac{1}{40}$
    $$

11. Now, B and C working together daily work:
    $$
   $ b + \frac{a}{2} = \frac{1}{40} + \frac{1}{2} \times \frac{1}{60} = \frac{1}{40} + \frac{1}{120} = \frac{3 + 1}{120} = \frac{4}{120} = \frac{1}{30}$
    $$

12. Hence, B and C together will complete the work in 30 days.

**Answer:** 30 days

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