24 men can complete a work in *X* days, and 32 women can complete the same work in *(X + 8)* days. The ratio of work done by 15 men to 12 women in the same time period is 3 : 1.
Find by how many days the time taken by 1 man to complete the work is less than the time taken by 1 woman to complete the same work.

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Accepted Answer

**Solution:**

1. Let the total work be *W* units.

2. Work done by 24 men in *X* days = *W* units.
   So, work done by 1 man in 1 day = $ \frac{W}{24X} $.

3. Work done by 32 women in *(X + 8)* days = *W* units.
   So, work done by 1 woman in 1 day = $ \frac{W}{32(X+8)} $.

4. According to the problem, the ratio of work done by 15 men to 12 women in the same time is 3 : 1.
   Let the time be *t* days.

Work done by 15 men in *t* days = $ 15 \times t \times \frac{W}{24X} = \frac{15tW}{24X} $.

Work done by 12 women in *t* days = $ 12 \times t \times \frac{W}{32(X+8)} = \frac{12tW}{32(X+8)} $.

Given ratio:
   $$
   \frac{15tW}{24X} : \frac{12tW}{32(X+8)} = 3 : 1
   $$

5. Simplify the ratio:
   $$
   \frac{15tW}{24X} \div \frac{12tW}{32(X+8)} = 3
   $$

$$
   \frac{15tW}{24X} \times \frac{32(X+8)}{12tW} = 3
   $$

$$
   \frac{15 \times 32 (X+8)}{24 \times 12 X} = 3
   $$

6. Simplify numerator and denominator:
   $$
   \frac{480 (X+8)}{288 X} = 3
   $$

7. Cross multiply:
   $$
   480 (X+8) = 3 \times 288 X
   $$

$$
   480X + 3840 = 864X
   $$

8. Rearranging:
   $$
   864X - 480X = 3840
   $$

$$
   384X = 3840
   $$

$$
   X = 10
   $$

9. Calculate work rates:
   - Work done by 1 man in 1 day = $ \frac{W}{24 \times 10} = \frac{W}{240} $
   - Work done by 1 woman in 1 day = $ \frac{W}{32 \times 18} = \frac{W}{576} $

10. Calculate days taken by 1 man and 1 woman to complete the work:
    - Days by 1 man = $ 240 $
    - Days by 1 woman = $ 576 $

11. Difference in days:
    $$
    576 - 240 = 336
    $$

**Answer:** None of these (Difference is 336 days)

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