One woman and one man can build a wall together in two hours, but the

Question

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

Kudos for a correct solution.

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

BrentGMATPrepNow · Accepted Answer

Another 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

anudeep133 · Answer

Solution:  Let work done by man, women and girl per hour be m,w,g respectively.
Then, m+w = 1/2-->(1) , w + 2g = 1/2-->(2) and m + g =1/4-->(3).
No. of hours it would take forone woman, one man, and one girl, working together, to build the wall,n = 1/m+w+g
From (1) and (2), m=2g and from (3) g=1/12,m=1/6 and w=1/3.
So, n = 1/(7/12) = 12/7

Option, D

Mo2men · Answer

My 2 cents

Consider: 
W = the rate for one woman 
M = the rate for one man 
G = the rate for one girl.

W + M =1/2 
W + 2G=1/2 
M + G = 1/4

Sum all equations above:

2W + 2M +3G =5/4

take 2 as common factor

2 (W+M+(3/2)G)= 5/4

W+ M+ 1.5G = 5/8.... so

Time taken= 8/5 = 1.6 BUT we need to 1G and we have more 0.5G. Henece we need number around it by decreasing 0.5G which implies that time will be little more than 1.6.

Let's scan answer Choices

Choices A & B is less than 1.6. Eliminate A &B

choice E is more than 2 hrs. Too big and should be logically less than 2 hrs when all of them (woman, man and girl) combined work together. Eliminate E

Choice C =10/7= 1 3/7 which means that it is less than 1.5, which is lower that 1.6

The only correct answer left

Answer: D

Bunuel · Answer

VERITAS PREP OFFICIAL SOLUTION:

This question is certainly quite tricky but if you understand the relation between work and rate, you can still solve this question easily. Mind you, we are using variables here only because I don’t want to write man, woman and girl again and again. Notice that there are no ‘=’ signs i.e. we are not making equations so we are not doing any algebraic manipulations.

The question is long so take one line at a time and analyze it. We will keep condensing the information we get from each sentence and figuring out the implications of new and previous information as we go along.

“One woman and one man can build a wall together in 2 hrs,”
1w + 1m -> 2 hrs ……(I)

“but the woman would need the help of 2 girls in order to complete the same job in the same amount of time.”
1w + 2g -> 2 hrs …..(II)

From (I) and (II), we can say that 1m is equivalent to 2g (i.e. 1 man does the same work as 2 girls do in the same amount of time; 1m ? 2g)

“If 1 man and 1 girl worked together, it would take them four hours to build the wall.”
1m + 1g -> 4hrs (Since 1m ? 2g, we can say that 3g will take 4 hrs to build the wall.)
or 2m + 2g -> 2 hrs …..(III) (If number of workers double, time taken to do the work becomes half)

From (II) and (III), 1w ? 2m (i.e. 1 woman does the same work as 2 men do in the same amount of time)
Hence, 1w ? 2m ? 4g

“Assuming that rates for women, men and girls remain constant, how many hours would it take 1 woman, 1 man and 1 girl working together to build the wall?”
1w + 1m + 1g ? 4g + 2g + 1g ? 7g. Since 3g take 4 hrs to build the wall, 7g will take 3*4/7 = 12/7 hrs to complete the wall.

Answer (D)

OreoShake · Answer

can someone post a solution in 1/m, 1/2g form? I am finding it difficult to understand this. Thank you.

cbh · Answer

A - man
B - woman
C - girl
Then piece of work done per hour
1. (a + b) = ½
2. (a+c) = ¼ 
3. (b+2c) = ½
b=½-a
a=¼ +c
For the third equation, (½-¼+c+2c)=½
¼+3c=½
3c=¼
c=1/12 so one girl's rate is 1/12
Then plug the value of c for others equation and we get a= ⅙ and b = ¼
Then ⅙+ ¼+1/12= 7/12 so get the entire work done then need to work 12/7 hours

JeffTargetTestPrep · Answer

We can let m = the time is takes the man to build the wall, w = the time it takes the 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 a man, a woman, and a girl to build the wall, working together.

Alternate Solution:

Since the 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

hellosanthosh2k2 · Answer

M + W = 1/2 (men and women combined rate) ---------(1)
W + 2G = 1/2 (women and 2 girls combined rate) ------- (2)
M + G = 1/4 (men and girl combined rate) ------ (3)

-----(1) multiply by 2 and add with equations (2) and (3)
we get 3(M+W+G) = 1 + 1/2 + 1/4 
 M +W + G = 7/12
so M + W +G will take 12/7 days to the job

danielarasan · Answer

1) W + M = 2h
2) W + 2G = 2h
3) M + G = 4h

From 2) - 1) we learn that M = 2G
In 3), M + G = 3G; 3G = 4h
If we double the number in 3), we know that 6G = 2h
From 1) we know that W + 2G = 2h; therefore, W = 4G

So if we transform all in G, we need 4G + 2G + G = 7 Girls!

If 3G = 4h, G = 12h. 
7G = 12h/7 to finish the job!

GMATGuruNY · Answer

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. 
Since replacing 1 man with 2 girls does not change the amount of time required to complete the job, 1 man is the equivalent of 2 girls:
M = 2G.

Let G = 1 unit per hour, implying that M = 2 units per hour.

If one man and one girl worked together, it would take them four hours to build the wall. 
Since M+G = 2+1 = 3 units per hour, the resulting wall produced in 4 hours = 3*4 = 12 units.

One woman and one man can build a wall together in two hours .
Since the 12-unit wall is built in 2 hours, we get:
W+M = 12/2 = 6 units per hour.
Since M = 2 units per hour, W = 4 units per hour.

How many hours would it take one woman, one man, and one girl, working together, to build the wall? 
Since W+M+G = 4+2+1 = 7 units per hour, the time to build the 12-unit wall = 12/7 hours.

D

fskilnik · Answer

My solution is in the image attached.

hkkat · Answer

May I know how did you come up with 12hr here?
I understood the well, but not this one.

TIA!

chloee.11 · Answer

Rate of M,W, G are: rM, rW, rG
Work = 1 (1 wall)
1. rM + rW = 1/2 
2. rM=2rG
3. rM + rG = 1/4

=> rG= 1/12
rM=1/6
rW=1/3

Time taken when W, M, G work together = work/ total rate = 1 / (1/3+1/6+1/12) = 12/7

=> Answer D

sid_the_kid · Answer

I solved this by three equations for three variables using substitution.

Usernamevisible · Answer

Bigger learning in the end

m = 2g

m + g = 4h
=> 3g = 4h

w + 2g = 2h
2m + 2g = 2h
=> w = 2m = 4g

w + m + g
= 4g + 2g + g
= 7g

3g -> 4h
7g -> 12/7 h

Doubt - I am always confused as to for hours when to use 1/h vs just h as done in this solution

This is a very important distinction.
There are two valid languages for work problems.
Language 1: Rate language (uses 1/h)
If a worker takes h hours to do 1 job, then:
Rate = 1/h
Example:
Man + Girl take 4 hours.
Then:
M + G = 1/4
jobs per hour.
This is the algebraic/rate approach.
Language 2: Time language (uses just h)
Instead of converting to rates, you compare groups that finish the same job in the same time.
Example:
Woman + Man → 2 hours
Woman + 2 Girls → 2 hours
Since both groups take exactly 2 hours, they must have equal total productivity.
So:
1m = 2g
No 1/2 appears anywhere.
You're reasoning with equivalent workers, not rates.
Why the solution never writes 1/4
When it says:
1m + 1g → 4h
it is not saying:
m + g = 4
It is simply recording a fact:
"This group needs 4 hours."
Then it replaces m by 2g:
3g → 4h
meaning:
"Three girls need 4 hours."
Still no rates.
When MUST you use 1/h?
Use 1/h when you want to:
 
 Add rates 
 Subtract rates 
 Solve equations
Example:
A takes 6 hours.
B takes 12 hours.
You cannot write:
6 + 12
Instead:
A = 1/6
B = 1/12
A + B = 1/4
So together they take 4 hours.
Whenever workers are being combined mathematically, use rates.  
 
When can you avoid 1/h?
When you're only making equivalence statements:
 
 Same work 
 Same time 
 Therefore same productivity
Example:
W + M → 2h
W + 2G → 2h
Same job. Same time.
Therefore:
M = 2G
No rates needed.  
 
Quick rule
Ask yourself:
"Am I adding/subtracting rates?"
Yes → use rates (1/h).
No → if I'm only comparing groups that finish the same work in the same time, use worker-equivalence.
In this problem, the elegant solution never needed to add rates, so it stayed entirely in the worker-equivalence world and never converted hours into 1/h.
That's why you saw 4h instead of 1/4.

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