Six Men and Fourteen Women Can Complete a Work in Five Days GMAT Problem Solving

Question: Six men and fourteen women can complete a work in five days, whereas two men and three women can complete one-fourth of the same in four days. If one man and two women take up and complete the same work, earning a total wage of $11,791 for the same, what is the total share of the two women in this amount?

1. $1,814
2. $1,876
3. $2,012
4. $2,216
5. $2,543

Correct Answer: A
Solution and Explanation:
Approach Solution 1:

The objective is to find the total share of the two women.

1. 6M + 14W = \(\frac{1}{5}\)(work done in one day) or 30M + 70W = 1
2. 2M + 3W =\(\frac{1}{16}\)(\(\frac{1}{4}\)th work takes 4 days, work done in 1 day is \(\frac{1}{16}\)) or 32M + 48W = 1

Equating and solving: 30M + 70W = 32M + 48W
2M = 22W
Therefore, 1M = 11W
Putting this in equation 2, we will get:
2 * 11W + 3W = \(\frac{1}{16}\)
Therefore 1W = \(\frac{1}{16*25} = \frac{1}{400}\)
Therefore 1W = 11W = \(11 * \frac{1}{400} = \frac{11}{400}\)

Work done by 1 Man and 2 Women are in ratio \(\frac{11}{400} : \frac{2}{400} = 11:2\)

Share of the 2 women = \(\frac{2}{3}*11791 = $1814\)

Correct Option: A

Approach Solution 2:

The objective is to find the total share of the two women.
Let \(r_m\) be the rate of Man and \(r_w\) be the rate of Women
\(r_m\) ∗ 16 + \(r_w\) ∗ 14 = .2

\(r_m\) ∗ 2 + \(r_w\) ∗ 3 = .0625 ( if 2 men and 3 women can do ¼ of the work in 4 days then they can do the whole work in 16 days hence rate together 1/16 =.0625)
Solving the above two equations we get \(r_m\)=.0275 and \(r_w\)=.0025 or 1 women can do the whole work in 1/.0025 = 400 days and 1 man can do the whole work in 1/.0275 = 400/11 days
So the rate of 1 man and 2 women is: 1* .0275 + 2*.0025= .0325 or 1 Man and 2 women can do the whole work in 1/.0325 = 400/13 days
We know 1 women can do the whole work in 400 days therefore in 400/13 days 1 women can do 1/13 of the work and 2 women can do 2/13 of the work
Now they are paid in proportion to the amount of work done, so for 2/13 of work, women are paid 2/13 of the money.
Share of the two women = 2/13 * 11 ,791 = 1814

Correct Option: A

Approach Solution 3:

Let, men = x and women = y

1. 6x + 14y =\(\frac{1}{5}\)(work done in one day) or 30x + 70y = 1
2. 2x + 3y =\(\frac{1}{16}\)(\(\frac{1}{4}\)th work takes 4 days, work done in 1 day is \(\frac{1}{16}\)) or 32x + 48y = 1

Equating and solving: 30x + 70y = 32x + 48y
2x = 22y
Therefore, 1x = 11y
Putting this in equation 2, we will get:
2 * 11y + 3y =\(\frac{1}{16}\)
Therefore 1y =\(\frac{1}{16 * 25} = \frac{1}{400}\)
Therefore 1y = 11y = \(11 * \frac{1}{400} = \frac{11}{400}\)

Work done by 1 Man and 2 Women are in ratio \(\frac{11}{400} : \frac{2}{400} = 11:2\)

Share of the 2 women = \(\frac{2}{3}*11791 = $1814\)

Correct Option: A

“Six men and fourteen women can complete a work in five days, whereas two men and three women can complete one-fourth of the same in four days. If one man and two women take up and complete the same work, earning a total wage of $11,791 for the same, what is the total share of the two women in this amount?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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