A certain number of men can do a work in 60 days. If there were 8 men less it could be finished in 10 days more. Number of men are :

Question

Accepted Answer

To determine why Option 3 (56 men) is the correct answer, let's analyze the problem using the given information and logical reasoning.

### Explanation for Option 3 (56 men):

1. **Initial Scenario:**
   - Let the number of men be $ x $.
   - These $ x $ men can complete the work in 60 days.
   - Therefore, the total work can be expressed as $ 60x $ man-days.

2. **Modified Scenario:**
   - If there are 8 men less, the number of men becomes $ x - 8 $.
   - The work is completed in 70 days (10 days more than the initial scenario).
   - Therefore, the total work can also be expressed as $ 70(x - 8) $ man-days.

3. **Equating the Work:**
   - Since the total work remains the same in both scenarios, we equate the two expressions:
     $$
     60x = 70(x - 8)
     $$

4. **Solving the Equation:**
   - Expanding the equation:
     $$
     60x = 70x - 560
     $$
   - Rearranging the terms:
     $$
     560 = 70x - 60x
     $$
     $$
     560 = 10x
     $$
   - Solving for $ x $:
     $$
     x = \frac{560}{10} = 56
     $$

Thus, the number of men required to complete the work in 60 days is 56, which confirms that Option 3 is correct.

### Explanation for Why Other Options are Incorrect:

1. **Option 1 (14 men):**
   - If there were only 14 men, the total work would be $ 60 \times 14 = 840 $ man-days.
   - With 6 men (14 - 8), it would take $ \frac{840}{6} = 140 $ days, not 70 days. Hence, this option is incorrect.

2. **Option 2 (28 men):**
   - If there were 28 men, the total work would be $ 60 \times 28 = 1680 $ man-days.
   - With 20 men (28 - 8), it would take $ \frac{1680}{20} = 84 $ days, not 70 days. Hence, this option is incorrect.

3. **Option 4 (112 men):**
   - If there were 112 men, the total work would be $ 60 \times 112 = 6720 $ man-days.
   - With 104 men (112 - 8), it would take $ \frac{6720}{104} \approx 64.62 $ days, not 70 days. Hence, this option is incorrect.

In conclusion, the calculations and logical reasoning clearly support that Option 3 (56 men) is the correct answer.