One smurf and one elf can build a treehouse together in two

Question

One smurf and one elf can build a treehouse together in two hours, but the smurf would need the help of two fairies in order to complete the same job in the same amount of time. If one elf and one fairy worked together, it would take them four hours to build the treehouse. Assuming that work rates for smurfs, elves, and fairies remain constant, how many hours would it take one smurf, one elf, and one fairy, working together, to build the treehouse?

(A) 5/7

(B) 1

(C) 10/7

(D) 12/7

(E) 22/7

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

walker · Accepted Answer

D

s,e,f - speed of smurf, elf, and fairy
t- time to build the treehouse with one smurf, one elf, and one fairy, working together

s+e=1/2
s+2f=1/2
e+f=1/4
s+e+f=1/t

fast way to solve: multiply first equation by 2 and sum three equations:

3*(s+e+f)=1+1/2+1/4
3*1/t=7/4
t=12/7

KarishmaB · Answer

If you understand the relation between work and rate, you can solve this question easily. Take one line at a time and analyze it:

One smurf and one elf can build a treehouse together in 2 hrs, 
1s + 1e -> 2 hrs ......(I)

but the smurf would need the help of 2 fairies in order to complete the same job in the same amount of time . 
1s + 2f -> 2 hrs .....(II)

which means 1e = 2f (i.e. 1 elf does the same work as 2 fairies do in the same amount of time)

If 1 elf and 1 fairy worked together,it would take them four hours to build the treehouse.

1e + 1f -> 4hrs (We can say 3f will take 4 hours)
or 2e + 2f -> 2 hrs .....(III)
(If number of workers double, time taken to do the work becomes half)
From (I) and (III), 1s = 1e + 2f = 4f
 
 Assuming that work rates for smurfs elves,and fairies remain constant how many hours would it take 1 smurf ,1 elf and 1 fairy working together to build the treehouse 
1s + 1e + 1f = 4f + 2f + 1f = 7f
Since 3f take 4 hrs, 7f will take 3*4/7 = 12/7 hrs

GMATBLACKBELT · Answer

1/s+1/e=1/2

1/s+2/f=1/2

1/e+1/f=1/4

here is where I got stuck on for a long time. I tried manipulating the equations in which we have

(s+e)/es=1/2 --> 2s+2e=es.... ya this doesnt work out too well.

Dawned on me on my way back from training... just make 1/s+1/e=1/s+2/f

1/e=2/f --> 2e=f

1/e+1/2e=1/4 --> 3/2e=1/4 --> e=6

1/s+1/6=1/2 --> 1/s= 2/6 --> s=3

1/3+2/f=1/2 --> 2f+12=3f --> f=12

Now its just

1/6+1/12+1/3 --> 1/12+2/12+4/12 --> 7/12 equals combined rate. we need t=1/7/12 --> 12/7hrs or ~1.71hrs

D.

JCLEONES · Answer

One smurf and one elf can build a treehouse together in two hours, but the smurf would need the help of two fairies in order to complete the same job in the same amount of time. If one elf and one fairy worked together, it would take them four hours to build the treehouse. Assuming that work rates for smurfs, elves, and fairies remain constant, how many hours would it take one smurf, one elf, and one fairy, working together, to build the treehouse?

(A) 5/7

(B) 1

(C) 10/7

(D) 12/7

(E) 22/7

(A) 5/7

(B) 1

(C) 10/7

(D) 12/7

(E) 22/7

srp · Answer

D
--

1 elf ~ 2 Fairys

elf completes job in x hrs
therefore (1 fairy and 1 elf) gives 1/2x + 1/x = 1/2, x = 6

time to complete job:
1 elf: 6 hrs
1 Fairy: 12 hrs
1 Smurf: 3 hrs (1/s + 1/6 = 1/2)

all 3 together (t hrs):
1/3 + 1/6 + 1/12 = 1/t
t = 12/7

maratikus · Answer

i was trying to do something similar to what walker did (which is probably the most beautiful solution of such problem) but couldn't find it right away (didn't spend more than 10 sec). It's obvious that e = 2f, then since e+f = 1/4, f = 1/12.

then s+e+f = (s+e) + f = 1/2 + 1/12 = 7/12 -> the answer is D.

then s+e+f = (s+e) + f = 1/2 + 1/12 = 7/12 -> the answer is D.

gbosl · Answer

Would you mind going through your line of thought on this. I do not understand why everything is 1/e, 1/f, 1/4, etc... why isn't it just e, f, 4, etc? Please help!!! Thanks

incognito1 · Answer

Let me try to explain based on my understanding:

e is the time that it takes an elf to build the treehouse
Hence, 1/e is the rate or speed of work for an elf (for eg. if it takes 8 hours for an elf to build the treehouse, the rate of work for the elf is 1/8th of the treehouse per hour)

Similarly, 1/f and 1/s are the RATES of work that a fairy and smurf can get done in an hour.

The equations represent the rate of work on both sides. For instance,
(1/s) + (1/e) = (1/2)
represents that in an hour, a smurf and elf together can build half the treehouse (think of 1/2 as -- if the total treehouse is 1, the RATE or speed of construction is 1/2 per hour)

Think of it as a speed, time and distance relationship. Rates can be added on both sides of the equation, as long as the distance is constant, and the time slice is fixed (per hour). Here the distance = 1 and the time we set the equations up for is 1 hour. So this would look like
  +   =  
  +   =

Hope that helps!

DoctorGMAT · Answer

It's simpler to set it up on same hours. The algebra is a mess.

a) 1 smurf and 1 elf : 4h: 2tree
b) 1 smurf and 2 fairy : 4h: 2tree
c) 1 elf and 1 fairy : 4h: 1tree

a and b mean 1 elf = 2 fairies
then c means 3 fairies make 1 tree in 4 hours, or 1 fairy makes (1/3)t in 4h
then b means 1 smurf makes (4/3)t in 4h
and a implies 1 elf makes (2/3)t in 4h

adding up parts of trees: 4h : (7/3)t and from here it's simple to get trees per hour, hours per tree, or any variation, just by scaling up or down.

- DoktorGMAT
who does tutor in the Tel Aviv Area

alimad · Answer

1/e=2/f
f=2e

Can someone explain to me how this works.

I thought that the rate of 2 faries = 1 elf , since 1/s + 1/e = 1/2 and 1/s = 1/2f = 1/2 too. 
2f = e 
f = e/2 , isn't that right.

durgesh79 · Answer

if you are stuck with this question when last few seconds are left. Try eliminating options.

if 1 elf and 1 smurf can do the job in 2 hrs and we add 1 fairy to the team the time will be less than 2 hrs. Option E is out
From second condition we can see 2 faries are equal to 1 elf in terms of work doing capability. So adding only 1 fairy will not reduce the time by half (1 hr). Option A and B are out.

you have a 50:50 chance in C and D.

fluke · Answer

Let their respective rate per hour be;

Smurf: S work/hour
Elf: E work/hour
Fairy: F work/hour

2S + 2E = 1; 2S = 1-2E --- A
2S + 4F = 1; 1-2E+4F=1; 2E-4F=0 ---- 1
4E + 4F = 1 ----------2

Solving 1 and 2
E = 1/6; F=1/12;
Substituting them in A
S = 1/3

Now;
1/6+1/12+1/3 = 1/t
t = 12/7

Ans: "D"

subhashghosh · Answer

1/S + 1/E = 1/2

1/S + 2/F = 1/2

1/E + 1/F = 1/4

2/F + 1/F = 1/4 => 1/F = 1/12; 1/E = 1/6 and 1/S = 1/2 - 1/6 = (3-1)/6 = 2/6 = 1/3

So 1/S + 1/E + 1/F = 1/3 + 1/6 + 1/12 = (4 + 2 + 1)/12 = 7/12

So the answer is D.

Spidy001 · Answer

1/S + 1/E = 1/2 -----1

1/S + 2/F = 1/2------2

1/E + 1/F = 1/4------3

1/S + 1/E + 1/F = 1/t ---4

t=?

substituting 1 in 4 ; 4=> 1/2+1/F = 1/t

solving 2 & 3 by substituting 1, 1/F = 1/12 ----5

substituting 5 in 4 => 1/2+1/12 = 1/t

=> t = 12/7

Answer is D.

jitbec · Answer

the key to reduce calculation is e=2f as done by maraticus

RFRaj · Answer

Here's how I solved it -

1/s + 1/e = 1/2

1/s + 1/f + 1/f = 1/2

Therefore, 1/e = 1/f + 1/f = 2/f

1/e + 1/f = 1/4
2/f + 1/f = 1/4
3/f = 1/4
f = 12

Thus, e = 6
and therefore , s = 3

For the final answer -- 1/s + 1/e + 1/f = 1/3 + 1/6 + 1/12 = 7/12

Therefore, the time taken would be 12/7

Ans  D

kthxbye · Answer

I'm really struggling with this question.

I understand that 1e = 2f but can't progress from there. Can anyone write out their thinking in words instead of math? Maybe that would help me better understand how they arrived at the correct answer.

Thank you!

Bunuel · Answer

Please read Karishma's post: one-smurf-and-one-elf-can-build-a-treehouse-together-in-two-58306.html#p891453 Hope it helps.

email2vm · Answer

Work done = Rate of work * Time taken

for first eq: One smurf and one elf can build a treehouse together in two hours
 
(S + E) 2 = 1 (a tree house = one tree house = 1 work done)
S+ E = 1/2

similarly : 1/S + 2/F = 1/2
1/E + 1/F = 1/4

Rest you can follow other posts

One smurf and one elf can build a treehouse together in two

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