Question

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?

• A. 6 days
• B. 3 days
• C. 9 days
• D. 8 days

Answer 1

Answer:

B. 3 days

Let one man’s 1-day work = m and one woman’s 1-day work = w.

Form equations from given data:

(1)
4 men + 3 women finish in 6 days
$6(4m + 3w) = 1 ;\Rightarrow; 4m + 3w = \frac{1}{6}$

(2)
5 men + 7 women finish in 4 days

$4(5m + 7w) = 1 ;\Rightarrow; 5m + 7w = \frac{1}{4}$
Solve the equations:

From (1):
$4m + 3w = \frac{1}{6}$

From (2):
$5m + 7w = \frac{1}{4}$

Multiply (1) by 5 and (2) by 4:

$20m + 15w = \frac{5}{6}$
20m + 28w = 1

Subtract:

$13w = 1 - \frac{5}{6} = \frac{1}{6}$
$\Rightarrow w = \frac{1}{78}$

Substitute into (1):

$4m + 3\left(\frac{1}{78}\right) = \frac{1}{6}$
$4m + \frac{1}{26} = \frac{1}{6}$
$4m = \frac{1}{6} - \frac{1}{26} = \frac{10}{78} = \frac{5}{39}$
$m = \frac{5}{156}$

Work rate of 8 men and 6 women:

$8m + 6w = 8\left(\frac{5}{156}\right) + 6\left(\frac{1}{78}\right)$

$= \frac{40}{156} + \frac{12}{156} = \frac{52}{156} = \frac{1}{3}$

Time required:

$\text{Time} = \frac{1}{\text{rate}} = \frac{1}{1/3} = 3 \text{ days}$

Final Answer:

8 men and 6 women will complete the work in 3 days.