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If 12 men and 16 women can do a piece of work in 5 days and

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If 12 men and 16 women can do a piece of work in 5 days and [#permalink]

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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
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Re: If 12 men and 16 women can do a piece of work in 5 days and [#permalink]

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New post 17 Aug 2004, 18:16
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If you solve the two equations:

12M + 16W = 5
13M + 24W = 4

and then plug M and W into

7M + 10W = X,

X should be the amount of days.

Thats too much math for me, and the numbers don't look too clean either.

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Re: If 12 men and 16 women can do a piece of work in 5 days and [#permalink]

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New post 17 Aug 2004, 18:18
Wow, this is an insane problem. I wonder if someone can come up with a quick solution but it took me a long time to figure it out. Too much calculation. I came up with 1452/175 days or about 8.3 days.

1) M/12 + W/16 = 5
2) M/13 + W/24 = 4

Multiply second line by -3/2 to eliminate variable W:
2) -3M/26 - W/16 = -6

Add up line 1 and 2:
M/12 - 3M/26 = 5 - 6
(-18 + 13)M/156 = -1
-5M = -156
M = 156/5

Now plug in second equation to get W:
156/65 + W/24 = 4
12/5 + W/24 = 20/5
W/24 = 8/5
W = 192/5

Plug in back to the question asked:
M/7 + W/10 = X
156/(5*7) + 192/(5*10) = X
156/35 + 96/25 = X
(672+780)/175 = X
X = 1452/175 = approx 8.3 days

Will certainly not be on the GMAT.
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Re: If 12 men and 16 women can do a piece of work in 5 days and [#permalink]

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New post 17 Aug 2004, 18:24
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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 (20*5)/12 = 8.33 days.

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Re: If 12 men and 16 women can do a piece of work in 5 days and [#permalink]

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New post 17 Aug 2004, 18:35
These are some the toughest problems that I have. I do have the solutions, but I am throwing them out in the hope that somebody can come up with a simpler solution as well as to challenge people.

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Re: If 12 men and 16 women can do a piece of work in 5 days and [#permalink]

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Here is how I solved it:

Create two equations based on the information given. Start from the basics. Each man can do 1/x work per day and each woman can to 1/y work per day. If we have 12 men and 16 women, then in total they can do 12/x work per day and 16/y work per day together. We also know that they can do the work together in 5 days. So in 1 day, how much of the job did they finish? 1/5 or 20%. Repeat this logic to create the second formula.
12/x + 16/y = 1/5
13/x + 24/y = 1/4

Solve for common denominators in each formula:
12y+16x=((xy)^2)/5
13y+24x=((xy)^2)/4

Multiply by denominator of fraction in each formula to get nice numbers. Since both would equal the same variable, make them equal to each other:
60y+80x=52y+96x
8y=16x
y=2x

Plug y=2x into the first equation to get:
12/x+16/2x=1/5
12/x+8/x=1/5
20/x=1/5
x=100

If x=100 and y=2x, y=200

Now in the original equation, it asks us how long 7 men and 10 women can do the work. We can create the following formula, similar to what we did in the first step. This formula tells us that in one day 7 men and 10 women can do X% of the job. We can simply take the reciprocal of the fraction to get the number of days the job will be completed in:
7/x+10/y=1/z (we want to solve for z)
7/100+10/200=1/z
7/100+5/100=1/z
12/100=1/z
z=100/12 or 8.33

Last edited by psal on 16 Jan 2014, 13:36, edited 1 time in total.

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Re: If 12 men and 16 women can do a piece of work in 5 days and [#permalink]

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New post 16 Jan 2014, 03:57
smcgrath12 wrote:
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?


Similar questions to practice:

it-takes-6-days-for-3-women-and-2-men-working-together-to-82718.html
if-w-women-can-do-a-job-in-d-days-then-how-many-days-will-83771.html
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Re: If 12 men and 16 women can do a piece of work in 5 days and [#permalink]

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smcgrath12 wrote:
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?


You will often be able to ballpark or find the exact solution by just working on the multiples of the given equations.

Say rate of work of each man is M and that of each woman is W.

Given: 12M + 16W = 1/5 (Combined rate done per day)
______3M + 4W = 1/20 (in lowest terms) ...........(I)
Given: 13M + 24W = 1/4 ......................................(II)

Reqd: 7M + 10W = ?

Using equations (I) and (II) we need to find the sum of 7M and 10W. We can get a multiple of 7M in various ways:
(3M + 4W = 1/20) * 5 gives 15M + 20W = 1/4
Adding this to equation II, we get 28M + 44W = 1/2
7M + 11W = 1/8

7 men and 11 women complete the work in 8 days. 7 men and 10 women will take a little more than 8 days to complete the work. If you do get such a question in GMAT, you will easily be able to manipulate the equations to get the desired equation. Finding the values of M and W is way too painful to interest GMAT.
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Re: If 12 men and 16 women can do a piece of work in 5 days and [#permalink]

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New post 11 Mar 2014, 18:15
My answer is 8.33 days. Here is the way I`ve done:

(1) - 12M (Men) + 16W (Woman) = 100 (The amount of work) in 5 days, or 20/day

(2) - 13M + 24W = 100 in 4 days, or 25/day

Now, to find M, You do:

(1)*3 = 36M + 48W = 60/day
(2)*-2 = -26M - 48W = -50/day

Now You subtract one from the other:

10M = 10 => 1M = 1/day

then, on (1), 12*1 + 16W = 20

16W = 8 => W = 1/2.

Finally: 7M + 10W = 100 in: 7*1 + 10*(1/2) = 12.

100/12 = 8.33

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Re: If 12 men and 16 women can do a piece of work in 5 days and [#permalink]

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Let a man do \(\frac{1}{x}\) work per day and

each woman do \(\frac{1}{y}\) work per day

\(\frac{12}{x} + \frac{16}{y} = \frac{1}{5}............\)(1)
\(\frac{13}{x} +\frac{24}{y} = \frac{1}{4}\) ............ (2)

Multiply (1) by 3 & (2) by 2

\(\frac{36}{x} + \frac{48}{y} = \frac{3}{5}\) ....... (3)

\(\frac{26}{x} + \frac{48}{y} = \frac{1}{2}\) ........ (4)

Equation (3) - (4)

\(\frac{10}{x} = \frac{1}{10}\)

x = 100

y = 200

We require to find z; substituting the values

\(\frac{7}{x}+\frac{10}{y}= \frac{1}{z}\)

\(\frac{7}{100}+ \frac{10}{200} = \frac{1}{z}\)

\(\frac{7}{100}+ \frac{5}{100} = \frac{1}{z}\)

\(\frac{12}{100} = \frac{1}{z}\)

\(z = \frac{100}{12} = 8.33 = Answer\)
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Re: If 12 men and 16 women can do a piece of work in 5 days and [#permalink]

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New post 04 Jul 2016, 22:58
An easier way to do it with simple equations :

Let rate of work for men be M and women be W

The 2 equations we get are :

12M + 16W = 1/5 ..... (1)
13M + 24 W = 1/4......(2)

Now notice 16W could be converted to 24W by multiplying it with 3/2. Hence multiply statement 1 by 3/2

18M + 24W = 3/10
13M + 24W = 1/4

Now subtract both equations and we get value of M = 1/100

Substitute value of M in equation 1 we get value of W = 1/200

Use M & W values to find value of 7 M + 10 W =>

1 = (7/100 + 20/100) * T

T = 200/24 = 8.3 days

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Re: If 12 men and 16 women can do a piece of work in 5 days and [#permalink]

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New post 17 Jul 2016, 21:31
let d=number of days needed
let m=rate of 1 man per 1 day
w=rate of 1 woman per 1 day
5(12m+16w)=4(13m+24w)
m=2w
substituting,
d(7m+5m)=5(12m+8m)
d=8.3 days

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Re: If 12 men and 16 women can do a piece of work in 5 days and [#permalink]

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New post 29 Aug 2016, 10:22
smcgrath12 wrote:
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



Answer: Option C

Please find solution as attached.
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Re: If 12 men and 16 women can do a piece of work in 5 days and [#permalink]

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New post 29 Aug 2016, 11:27
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

We can solve this using algebra but as others have noted doing this on GMAT may not be feasible.
So I thought about an intuitive explanation and so forum members let me know if you concur.
So we have 13 Men and 24 Women who together will finish the task in 4 days (per the prompt). Now if we reduce women and men by half. The task is going to take twice the time.
We see that we reduce Men to 7 (reduce by slightly less than half 6/13) and Women to 10 (reduce by slightly more than half 14/24 = 7/12). So my time would have nearly doubled. 9.8 is next nearest to 8 , but that would imply that time required has become 2.5 times. 8.8 is a good guess using this reasoning and turns out to be the correct answer.

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Re: If 12 men and 16 women can do a piece of work in 5 days and [#permalink]

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New post 30 Aug 2016, 01:22
ajdse22 wrote:
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

We can solve this using algebra but as others have noted doing this on GMAT may not be feasible.
So I thought about an intuitive explanation and so forum members let me know if you concur.
So we have 13 Men and 24 Women who together will finish the task in 4 days (per the prompt). Now if we reduce women and men by half. The task is going to take twice the time.
We see that we reduce Men to 7 (reduce by slightly less than half 6/13) and Women to 10 (reduce by slightly more than half 14/24 = 7/12). So my time would have nearly doubled. 9.8 is next nearest to 8 , but that would imply that time required has become 2.5 times. 8.8 is a good guess using this reasoning and turns out to be the correct answer.


Nothing wrong with the solution but I would be a little uncomfortable making this approximation. If I know the approximate relation between that the rate of work of men and women, then perhaps I will have an easier time making these assumptions.
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Re: If 12 men and 16 women can do a piece of work in 5 days and [#permalink]

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New post 05 Jun 2017, 18:18
smcgrath12 wrote:
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

1. Let us take the same number of days in both the cases, say 20 days
2. In the first case 3 men and 4 women can do the work in 20 days and in the second case 2.6 men and 4.8 women can do the work in 20 days.
3. 0.8 more women does the work of 0.4 less men or 2 women does the work of 1 man.
4. We can find the number of days taken by 1 man and 1 woman as 100 and 200 days
5. So the time taken by 7 men and 10 women is 1/ (7/100+10/200)=8.3 days

Do not mind the decimals if you can reach the solution faster.
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Re: If 12 men and 16 women can do a piece of work in 5 days and [#permalink]

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New post 08 Jun 2017, 17:04
smcgrath12 wrote:
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


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 also 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.

Answer: C
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Re: If 12 men and 16 women can do a piece of work in 5 days and [#permalink]

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New post 25 Jun 2017, 11:36
smcgrath12 wrote:
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


Refer to solution in the picture
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Re: If 12 men and 16 women can do a piece of work in 5 days and [#permalink]

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New post 25 Jun 2017, 11:56
12m + 16w = 5d
So, 60md + 80wd, where md = man-days, wd = woman-days

13m + 24w = 4d
So, 52md + 96wd.

60md + 80wd = 52md + 96wd
8md = 16wd
1 md = 2 wd
Efficiency of 1 man = 2 woman


7m + 10w = 14w + 10w = 24w

13m + 24w = 4d
26w + 24w = 4d
50 women can do the work in 4 days, so 24 women can do it in 50*4/24 = 8.3 days. Ans - C.
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Re: If 12 men and 16 women can do a piece of work in 5 days and [#permalink]

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smcgrath12 wrote:
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


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