Question

3 men and 4 women can do a work working together in 12 days whereas 9 men and 2
women can do the same work working together in 9 days. Find in how many days can 3
men and 2 women together do the same work?

Answer 1

Given:

3 men and 4 women can do the work in 12 days.

9 men and 2 women can do the work in 9 days.

To Find:

The number of days needed if 3 men and 2 women do the work.

Solution

Define M and W:

Let the number of men be M.

Let the number of women be W.

3 men and 4 women can do the work in 12 days:

⇒ 3 men and 4 women can do 1/12 of the work in 1 day.

$3M + 4W = \dfrac{1}{12}$

9 men and 2 women can do the work in 9 days:

⇒ 9 men and 2 women can do 1/9 of the work in 1 day.

$9M + 2W = \dfrac{1}{9}$

Solve for M and W:

$3M + 4W = \dfrac{1}{12} \text{ --------------[ 1 ]}$

$9M + 2W = \dfrac{1}{9} \text{ --------------[ 2 ]}$

Equation [ 1 ] x 3:

$9M + 12W = \dfrac{1}{4} \text{ --------------[ 3 ]}$

Equation [ 3 ] - Equation [ 2 ]:

$10W = \dfrac{5}{36}$

$1W = \dfrac{1}{72} \text{ --------------[ 4 ]}$

⇒ One woman can finish 1/72 of the work in 1 day

Substitute Equation [ 4 ] into [ 1 ]:

$3M + 4\bigg( \dfrac{1}{72} \bigg) = \dfrac{1}{12}$

$3M + \dfrac{4}{72} = \dfrac{1}{12}$

$3M = \dfrac{1}{12} - \dfrac{4}{72}$

$3M = \dfrac{1}{36}$

$1M = \dfrac{1}{108}$

⇒ One man can finish 1/108 of the work in 1 day

Find the amount of work 3 men and 2 women can finish in a day:

$3M + 2W =3 \bigg( \dfrac{1}{108} \bigg) +2 \bigg( \dfrac{1}{72} \bigg)$

$3M + 2W = \dfrac{1}{36} + \dfrac{1}{36}$

$3M + 2W = \dfrac{1}{18}$

Find the number of days needed:

$\text{Number of days needed } = 1 \div \dfrac{1}{18}$

$\text{Number of days needed } = 18$

Answer: 3 men and 2 women can finish the work in 18 days.