A and B together can complete a work in 20 days. B starts working alone and works for 15 days, then A takes over and finishes the remaining work in 24 days.

Quantity I: Number of days taken by B alone to finish the entire work.

Quantity II: Number of days taken by A alone to finish the entire work.

Compare Quantity I and Quantity II and choose the correct option.

Question

A and B together can complete a work in 20 days. B starts working alone and works for 15 days, then A takes over and finishes the remaining work in 24 days.

Quantity I: Number of days taken by B alone to finish the entire work.

Quantity II: Number of days taken by A alone to finish the entire work.

Compare Quantity I and Quantity II and choose the correct option.

Accepted Answer

**Solution:**

1. Let the number of days taken by A alone to finish the work be *A* days, and by B alone be *B* days.

2. Since A and B together finish the work in 20 days, their combined work rate is:
   $$\frac{1}{A} + \frac{1}{B} = \frac{1}{20} \quad 	ext{...(i)}$$

3. B works alone for 15 days, so work done by B in 15 days is:
   $$\frac{15}{B}$$

4. Remaining work after B's 15 days is:
   $$1 - \frac{15}{B}$$

5. A finishes the remaining work in 24 days, so:
   $$\frac{24}{A} = 1 - \frac{15}{B}$$

6. Rearranging the above:
   $$\frac{24}{A} + \frac{15}{B} = 1 \quad 	ext{...(ii)}$$

7. Multiply equation (i) by 24:
   $$24 	imes \left( \frac{1}{A} + \frac{1}{B} ight) = 24 	imes \frac{1}{20}$$
   $$\frac{24}{A} + \frac{24}{B} = \frac{24}{20} = 1.2$$

8. Subtract equation (ii) from this:
   $$\left( \frac{24}{A} + \frac{24}{B} ight) - \left( \frac{24}{A} + \frac{15}{B} ight) = 1.2 - 1$$
   $$\frac{24}{B} - \frac{15}{B} = 0.2$$
   $$\frac{9}{B} = 0.2$$
   $$B = \frac{9}{0.2} = 45 	ext{ days}$$

9. Substitute *B = 45* into equation (i):
   $$\frac{1}{A} + \frac{1}{45} = \frac{1}{20}$$
   $$\frac{1}{A} = \frac{1}{20} - \frac{1}{45} = \frac{9 - 4}{180} = \frac{5}{180} = \frac{1}{36}$$
   $$A = 36 	ext{ days}$$

10. Comparing the two:
    - Quantity I = *B* = 45 days
    - Quantity II = *A* = 36 days

Since 45 > 36, Quantity I > Quantity II.

**Answer:** Quantity I > Quantity II

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