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

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

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)

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

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

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

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

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!

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.

My solution is in the image attached.

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

TIA!

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

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