Problem Solving

swerve wrote:One woman and one man can build a wall together in two hours, but the woman would need the help of two girls in order to complete the same job in the same amount of time. If one man and one girl worked together, it would take them four hours to build the wall. Assuming that rates for men, women and girls remain constant, how many hours would it take one woman, one man, and one girl, working together, to build the wall?

A. 5/7
B. 1
C. 10/7
D. 12/7
E. 22/7

The OA is D

Source: Veritas Prep

One approach is to determine the size of the job

One woman and one man can build a wall together in two hours, but the woman would need the help of two girls in order to complete the same job in the same amount of time.

The part in blue tells us that 2 girls have the same output as 1 man.
So, let's say that 1 girl has an output of 1 unit per hour
This means that 1 man has an output of 2 units per hour
So, COMBINED, 1 man and 1 girl have an output of 3 units per hour

If one man and one girl worked together, it would take them four hours to build the wall.
Working together, 1 man and 1 girl have an output of 3 units per hour
So, after 4 hours, their combined output is 12 units.
In other words, we can say that the entire job consists of 12 units.

One woman and one man can build a wall together in two hours
Since 1 man has an output of 2 units per hour, in two hours the man's output will be 4 units.
The entire job consists of 12 units, so the woman completed the other 8 units (in 2 hours).
So, 1 woman has an output of 4 units per hour

How many hours would it take one woman, one man, and one girl, working together, to build the wall?
We have:
1 girl has an output of 1 unit per hour
1 man has an output of 2 units per hour
1 woman has an output of 4 units per hour
And the entire job consists of 12 units.

The combined rate of all 3 workers = 1 + 2 + 4 = 7 units per hour
So, the time to complete the job = 12/7 hours

Answer: D

Hi All,

Each of the other explanations explains the 'math' behind this prompt, so I won't rehash any of that here. Instead, I want to talk about the 'patterns' behind this question, and how you can answer it without doing much math at all.

To start, we have to sort through a lot of information so that we can do a comparison of the given rates:

1 man + 1 woman = 2 hours to complete the job
2 girls + 1 woman = 2 hours to complete the job
1 man + 1 girl = 4 hours to do the job

Notice how "swapping" 1 woman for 1 girl really slows down the rate. This is important to realize because the question asks how long it would take 1 man + 1 woman + 1 girl to complete the job.

We know that 1 man + 1 woman can complete the job in 2 hours, so adding a girl will speed the work up... but not by much. We know that it would take LESS than 2 hours though.

If we combined (1 man + 1 woman) and (2 girls + 1 woman), we could complete the job in 1 hour (since each of those groups can complete the job in 2 hours, combining the two groups would cut the time in half). Thus, 1 man + 2 women + 2 girls = 1 hour. We don't have 2 women and 2 girls though, we have just one of each, so the work will take MORE than 1 hour.

With those two deductions, we can eliminate Answers A, B and E. Logically, since we know that the girl works so slowly (relatively speaking), we're likely looking for an answer that's relatively close to "2".... and the remaining Answers give us an obvious choice:

Final Answer: D

GMAT assassins aren't born, they're made,
Rich

swerve wrote:One woman and one man can build a wall together in two hours, but the woman would need the help of two girls in order to complete the same job in the same amount of time. If one man and one girl worked together, it would take them four hours to build the wall. Assuming that rates for men, women and girls remain constant, how many hours would it take one woman, one man, and one girl, working together, to build the wall?

A. 5/7
B. 1
C. 10/7
D. 12/7
E. 22/7

The OA is D

Source: Veritas Prep

We can let m = the time it takes one man to build the wall, w = the time it takes one woman to build the wall, and g = the time it takes one girl to build the wall. Looking at the rates of these individuals, we see that one man's rate is 1/m, one woman's rate is 1/w, and 1 girl's rate is 1/g. Thus:

1/m + 1/w = 1/2

and

1/w + 2/g = 1/2

and

1/m + 1/g = 1/4

From the first equation, let's isolate 1/m:

1/m = 1/2 - 1/w

Let's substitute this in the equation 1/m + 1/g = 1/4:

1/2 - 1/w + 1/g = 1/4

-1/w + 1/g = -1/4

Adding the equations 1/w + 2/g = 1/2 and -1/w + 1/g = -1/4 together, we obtain:

3/g = 1/4

g = 12

Since it takes a girl 12 hours to finish the job, her rate is 1/12. We are looking for 1/m + 1/w + 1/g; therefore, we add 1/12 to the equation 1/m + 1/w = 1/2:

1/m + 1/w + 1/g = 1/2 + 1/12

1/m + 1/w + 1/g = 7/12

Thus, it will take 1/(7/12) = 12/7 hours for one man, one woman, and one girl to build the wall, working together.

Alternate Solution:

Since one woman can finish the job in the same amount of time with the help of either one man or two girls, the rate of one man is equal to the rate of two girls.

Since one man and one girl can finish the job in 4 hours, and since the rate of one man is equal to the rate of two girls, three girls can finish the job in 4 hours. Since time is inversely proportional to the number of workers, one girl can finish the job in 12 hours.

Since one man and one woman finish the job in two hours, they complete 1/2 of the job in one hour. Since one girl can finish the job in 12 hours, one girl can complete 1/12 of the job in one hour. All working together, they finish 1/2 + 1/12 = 7/12 of the job in one hour. If 7/12 of the job gets done in one hour, then the entire job will get done in 1/(7/12) = 12/7 hours.

Answer: D