If 12 men and 16 women can do a piece of work in 5 days and 13 men and 24 women can do it in 4 days, how long will 7

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BTGmoderatorDC: Moderator; Posts: 7187; Joined: Thu Sep 07, 2017 4:43 pmFollowed by: 23 members

by BTGmoderatorDC » Fri Feb 26, 2021 6:21 pm

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If 12 men and 16 women can do a piece of work in 5 days and 13 men and 24 women can do it in 4 days, how long will 7 men and 10 women take to do it?

(A) 4.2 days
(B) 6.8 days
(C) 8.3 days
(D) 9.8 days
(E) 10.2 days

OA C

Source: GMAT Prep

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Source: Beat The GMAT — Problem Solving | Fri Feb 26, 2021 6:21 pm

Vivian_GMATRockstar: Newbie | Next Rank: 10 Posts; Posts: 1; Joined: Sat Feb 27, 2021 1:53 am

Re: If 12 men and 16 women can do a piece of work in 5 days and 13 men and 24 women can do it in 4 days, how long will 7

by Vivian_GMATRockstar » Sat Feb 27, 2021 2:14 am

This is a great question to teach the value of approximation and leveraging the answer choices! Doing a problem like this with rote math is not the "GMAT" way of thinking about this problem.

We start with 12 men and 16 women, and they do it in 5 days.
But 13 men and 24 women can do it in 4 day.

So an extra 1 man and 8 women were able to shave 1 day off. From this, we can tell that the women don't contribute nearly as much to do the doing of the job as the men do.

We could assume that 11 men and 8 women would do the job in 6 days, and that 10 men and 0 women could do the job in 7 days, but we only have 7 men, not 10, so it's definitely going to take more than 7 days! Looking at the answer choices, (A) and (B) are quickly out.

(A) 4.2 days
(B) 6.8 days
(C) 8.3 days
(D) 9.8 days
(E) 10.2 days

Let's look back at the original numbers:

12 men/16 women = 5 days

Therefore:
6 men / 8 women = 10 days, but we have 7 men and 10 women now, so they will definitely get it done much faster than 10 days. Therefore, the answer is (C).

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swerve: Legendary Member; Posts: 2560; Joined: Sun Oct 29, 2017 2:04 pmFollowed by: 6 members

Re: If 12 men and 16 women can do a piece of work in 5 days and 13 men and 24 women can do it in 4 days, how long will 7

by swerve » Sun Feb 28, 2021 11:21 am

BTGmoderatorDC wrote: ↑
Fri Feb 26, 2021 6:21 pm
If 12 men and 16 women can do a piece of work in 5 days and 13 men and 24 women can do it in 4 days, how long will 7 men and 10 women take to do it?

(A) 4.2 days
(B) 6.8 days
(C) 8.3 days
(D) 9.8 days
(E) 10.2 days

OA C

Source: GMAT Prep

Here is how it is solved:

$5(12 M + 16 W) = 4(13 M + 24 W)$
$60M + 80W = 52M + 96W$

That gives, $8M = 16W,$ which is, $1M=2W$

Now, $12 M + 16W$ can do work in $5$ days.
So, $12 M + 8 M$ can do work in $5$ days.
So, $20M$ can do work in $5$ days.

Now, $7M + 10W = 7M + 5M = 12 M$

Hence, if $20M$ can do work in $5$ days, $12M$ can do work in $\dfrac{20\cdot 5}{12} = 8.33$ days.

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8175; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

Re: If 12 men and 16 women can do a piece of work in 5 days and 13 men and 24 women can do it in 4 days, how long will 7

by Scott@TargetTestPrep » Sat Mar 13, 2021 3:47 pm

BTGmoderatorDC wrote: ↑
Fri Feb 26, 2021 6:21 pm
If 12 men and 16 women can do a piece of work in 5 days and 13 men and 24 women can do it in 4 days, how long will 7 men and 10 women take to do it?

(A) 4.2 days
(B) 6.8 days
(C) 8.3 days
(D) 9.8 days
(E) 10.2 days

OA C

Solution:

We can let the time it takes 1 man to finish the work = m, and thus the rate of 1 man = 1/m. Likewise, we can let the time it takes 1 woman to finish the work = w, and thus the rate of 1 woman = 1/w.

Thus, the combined rate of 12 men and 16 women is 12/m + 16/w. Since they can finish the work in 5 days, their combined rate is also equal to 1/5. Thus, we have:

12/m + 16/w = 1/5

Multiplying both sides of the equation by 5mw, we have:

60w + 80m = mw

Similarly, the combined rate of 13 men and 24 women is 13/m + 24/w. Since they can finish the work in 4 days, their combined rate is equal to 1/4. Thus, we have:

13/m + 24/w = 1/4

Multiplying both sides of the equation by 4mw, we have:

52w + 96m = mw

So, we have 60w + 80m = 52w + 96m (since they both equal mw).

60w + 80m = 52w + 96m

8w = 16m

w = 2m

We can now substitute w = 2m into the first equation, 12/m + 16/w = 1/5, to solve for m:

12/m + 16/(2m) = 1/5

12/m + 8/m = 1/5

20/m = 1/5

m = 100

Since m = 100 days, w = 200 days. The rate of 1 man is 1/100 and the rate of 1 woman is 1/200. Thus, the rate of 7 men and 10 women is 7/100 + 10/200 = 7/100 + 5/100 = 12/100, and the time for them to finish the same work is 1/(12/100) = 100/12 = 8.3 days.

Alternate Solution:

Alternatively, we can interpret the equation w = 2m as follows:

The time required for 1 woman to finish the job is twice that of a man, or, in other words, the job done by 2 women is equivalent to the job done by 1 man. We know 12 men and 16 women finish the job in 5 days, and as per the above discussion, this is equivalent to the job done by 12 + 16/2 = 20 men.

The question is asking for the time required to finish the job with 7 men and 10 women working, which is equivalent to 7 + 10/2 = 12 men working. We can set up an inverse proportion to find the required time: If 20 men finish a job in 5 days, then 12 men finish the job in how many days? Letting this unknown quantity be x, we have

20*5 = 12*x

x = 100/12 = 8.3 days

Answer: C

Scott Woodbury-Stewart
Founder and CEO
[email protected]

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