4 men and 3 women finish a job in 6 days, and 5 men and 7 women can do the same job in 4 days. How long will 8 men and 6 women take to do the same job?

Given:

4 men + 3 women finish a job in 6 days.

5 men + 7 women finish the same job in 4 days.

Formula Used:

M₁ × D₁ = M₂ × D₂ (when work is constant)

Let the work done by 1 man in 1 day be 'm' units.

Let the work done by 1 woman in 1 day be 'w' units.

Total work = (Number of men × work per man per day + Number of women × work per woman per day) × Number of days

Calculation:

From the first statement, total work = (4m + 3w) × 6

⇒ 24m + 18w = Total Work (Equation 1)

From the second statement, total work = (5m + 7w) × 4

⇒ 20m + 28w = Total Work (Equation 2)

Equating Equation 1 and Equation 2 (since the total work is the same):

24m + 18w = 20m + 28w

⇒ 24m - 20m = 28w - 18w

⇒ 4m = 10w

⇒ m = 10w / 4

⇒ m = 2.5w

Substitute the value of m in Equation 1 to find the total work in terms of 'w':

Total Work = 24(2.5w) + 18w

⇒ Total Work = 60w + 18w

⇒ Total Work = 78w units

Now, let the number of days taken by 8 men and 6 women be 'D' days.

Work done by 8 men and 6 women in D days = (8m + 6w) × D

Substitute m = 2.5w:

(8(2.5w) + 6w) × D = Total Work

(20w + 6w) × D = 78w

26w × D = 78w

26D = 78

⇒ D = 78 / 26

⇒ D = 3

∴ 8 men and 6 women will take 3 days to do the same job.