A certain number of men can complete a task **2 days earlier** than a certain number of boys. The work done by one man in one hour is equal to the work done by three boys in one hour.

Which of the following ratios of the number of men to the number of boys can satisfy the above condition?

Question

A certain number of men can complete a task **2 days earlier** than a certain number of boys. The work done by one man in one hour is equal to the work done by three boys in one hour.

Which of the following ratios of the number of men to the number of boys can satisfy the above condition?

Accepted Answer

**Solution:**

1. Let the number of men be *M* and the number of boys be *B*.

2. Given that 1 man’s 1 hour work = 3 boys’ 1 hour work.
   So, 1 man = 3 boys in terms of work rate.

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

4. Work done by *M* men in 1 hour = *M* units of man work.
   Work done by *B* boys in 1 hour = *B* units of boy work.

5. Since 1 man = 3 boys, work done by *M* men in 1 hour = work done by *3M* boys in 1 hour.

6. Let the time taken by men to complete the work be *T* hours.
   Then, time taken by boys to complete the work = *T + 2 	imes 24* hours (since 2 days earlier means 48 hours difference).

7. Using work formula:
   $$	ext{Work} = 	ext{Rate} 	imes 	ext{Time}$$

For men:
   $$W = M 	imes 1 	ext{ hour work rate} 	imes T$$

For boys:
   $$W = B 	imes 1 	ext{ hour work rate} 	imes (T + 48)$$

8. Converting boys’ work rate in terms of men:
   Since 1 man = 3 boys,
   $$	ext{Work rate of boys} = \frac{M}{3}$$

So,
   $$W = \frac{M}{3} 	imes B 	imes (T + 48)$$

9. Equate the two expressions for *W*:
   $$M 	imes T = \frac{M}{3} 	imes B 	imes (T + 48)$$

10. Simplify:
    $$T = \frac{B}{3} (T + 48)$$

11. Rearranged:
    $$T = \frac{B}{3} T + 16 B$$

12. Bring terms involving *T* to one side:
    $$T - \frac{B}{3} T = 16 B$$

$$T \left(1 - \frac{B}{3}ight) = 16 B$$

13. Simplify the bracket:
    $$T \left(\frac{3 - B}{3}ight) = 16 B$$

14. Multiply both sides by 3:
    $$T (3 - B) = 48 B$$

15. Rearranged:
    $$3T - B T = 48 B$$

16. Express *B* in terms of *T*:
    $$B T + 48 B = 3 T$$

$$B (T + 48) = 3 T$$

$$B = \frac{3 T}{T + 48}$$

17. Since *M* men take *T* hours, and *B* boys take *T + 48* hours, the ratio of men to boys is:
    $$\frac{M}{B} = \frac{M}{\frac{3 T}{T + 48}} = \frac{M (T + 48)}{3 T}$$

18. But we know the ratio *M : B* is given in options. To check which ratios satisfy the condition, test each option:

- For (i) 1 : 2
  - Men = 1, Boys = 2
  - Check if men finish 2 days earlier:
    Work rate men = 1 man
    Work rate boys = 2 boys
    Since 1 man = 3 boys, 1 man = 3 boys
    So, men’s rate = 3 boys’ rate
    Boys’ rate = 2 boys
    Men work faster, so men finish earlier.
    Calculate time difference:
    Men time = W / 3
    Boys time = W / 2
    Difference = W/2 - W/3 = W/6
    For difference to be 48 hours, W/6 = 48 => W = 288 hours
    This is consistent, so ratio (1:2) can satisfy.

- For (ii) 2 : 5
  - Men = 2, Boys = 5
  - Men’s rate = 2 men = 6 boys
  - Boys’ rate = 5 boys
  - Men finish earlier
  - Time difference:
    $$\frac{W}{5} - \frac{W}{6} = \frac{W}{30}$$
  - For difference = 48 hours, $$\frac{W}{30} = 48 \Rightarrow W = 1440$$
  - Valid, so ratio (2:5) satisfies.

- For (iii) 2 : 3
  - Men = 2, Boys = 3
  - Men’s rate = 6 boys
  - Boys’ rate = 3 boys
  - Time difference:
    $$\frac{W}{3} - \frac{W}{6} = \frac{W}{6}$$
  - For difference = 48 hours, $$\frac{W}{6} = 48 \Rightarrow W = 288$$
  - Valid, so ratio (2:3) satisfies.

- For (iv) 16 : 35
  - Men = 16, Boys = 35
  - Men’s rate = 48 boys
  - Boys’ rate = 35 boys
  - Time difference:
    $$\frac{W}{35} - \frac{W}{48} = \frac{13W}{1680}$$
  - For difference = 48 hours, $$\frac{13W}{1680} = 48 \Rightarrow W = \frac{48 	imes 1680}{13} \approx 6203.08$$
  - Valid, so ratio (16:35) also satisfies.

19. However, the question states men finish 2 days earlier than boys, so the difference must be exactly 48 hours.

20. Checking the options, only (ii) and (iii) satisfy the condition exactly for reasonable values.

**Answer:** Only (ii) and (iii)

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