If 2 men and 4 boys can finish a job in 8 days and 3 men and 2 boys can finish it in 6 days, how many days will 12 boys take to finish the same ?

Given:

2 men and 4 boys can finish a job in 8 days.

3 men and 2 boys can finish the job in 6 days.

Formula used:

Let the work done by 1 man in 1 day = M units, and the work done by 1 boy in 1 day = B units.

Work = Total units of work

Calculations:

From the first equation, the total work done by 2 men and 4 boys in 1 day is:

(2M + 4B) × 8 = Total work (W)

⇒ 2M + 4B = W/8 (Equation 1)

From the second equation, the total work done by 3 men and 2 boys in 1 day is:

(3M + 2B) × 6 = Total work (W)

⇒ 3M + 2B = W/6 (Equation 2)

Now solve these two equations simultaneously:

From Equation 1: 2M + 4B = W/8

From Equation 2: 3M + 2B = W/6

Multiply Equation 1 by 3 and Equation 2 by 2 to eliminate variables:

6M + 12B = 3W/8 (Equation 3)

6M + 4B = W/3 (Equation 4)

Now subtract Equation 4 from Equation 3:

(6M + 12B) - (6M + 4B) = (3W/8) - (W/3)

⇒ 8B = (9W - 8W) / 24

⇒ 8B = W/24

⇒ B = W/192

Now substitute B = W/192 into Equation 1:

2M + 4(W/192) = W/8

⇒ 2M + W/48 = W/8

⇒ 2M = W/8 - W/48 = 5W/48

⇒ M = 5W/96

Now, we need to find how many days 12 boys will take to complete the work. The work done by 12 boys in one day is:

12B = 12 × (W/192) = W/16

The total time taken by 12 boys is:

Time = Total work / Work done by 12 boys in 1 day = W / (W/16) = 16 days

∴ 12 boys will take 16 days to finish the job.