Pastry Chef Pierre takes x hours to decorate a wedding cake. Pastry

Question

Pastry Chef Pierre takes x hours to decorate a wedding cake. Pastry chef Franco takes y hours to decorate a wedding cake. If Pierre works alone for z hours and is then joined by Franco until 20 cakes are decorated, for how long did the two pastry chef's work together.

A. \(20xz-\frac{y}{(x+y)}\)

B. \(y+\frac{z}{(20xz)}\)

C. \(\frac{20x(y-z)}{(x+y)}\)

D. \(20xy-\frac{z}{(x+y)}\)

E. \(\frac{y(20x-z)}{(x+y)}\)

A.  20xz-\frac{y}{(x+y)}

B.  y+\frac{z}{(20xz)}

C.  \frac{20x(y-z)}{(x+y)}

D.  20xy-\frac{z}{(x+y)}

E.  \frac{y(20x-z)}{(x+y)}

Hi Bunuel,
Request you to provide me your approach. I couldn't solve it.

I usually solve such questions by using below approach-
(Workdone by entity A)/(Time taken by entity a) + (Workdone by entity b)/(time taken by entity b) = (Work done together)/(Time taken together)

However, I was unable to apply the above framework on this question. Please help me out.
Thanks

Bunuel · Accepted Answer

Pierre takes x hours to decorate a cake --> the rate of Pierre is  \frac{1}{x}  cake/hour, hence in  z  hours, that he worked alone, he decorated  \frac{z}{x}  cakes;

Franco takes y hours to decorate a cake --> the rate of Franco is  \frac{1}{y}  cake/hour;

Their combined rate is  \frac{1}{x}+\frac{1}{y}=\frac{x+y}{xy}  cake/hour and if they worked for  t  hours together then they decorated  t*\frac{x+y}{xy}  cakes in that time;

Since total of 20 cakes were decorated then:  \frac{z}{x}+t*\frac{x+y}{xy}=20  -->  t*\frac{x+y}{xy}=\frac{20x-z}{x}  -->  t=\frac{y(20x-z)}{x+y} .

Answer: E.

Hope it's clear.

KarishmaB · Answer

Let me give you my approach to these problems (I am sure Bunuel will give his solution too so you needn't be disappointed

) I like to solve questions with variables by plugging in values: Say x = 1 (Pierre decorates 1 cake in 1 hr), y = 2 (Franco decorates 1 cake in 2 hrs) Together, they decorate 1 cake in 1/(1+1/2) = 2/3 hrs Say Pierre decorates 17 cakes in 17 hrs and then Franco joins in and they decorate the remaining 3 cakes in 2 hrs (For 1 cake, they take 2/3 hrs so for 3 cakes, they will take 3*(2/3) = 2 hrs) Now just plug these values x = 1, y = 2 and z = 17 in the options and whichever option gives you 2, that should be the answer. Only option (E) gives you 2 so it should be your answer. Note: While you are trying the options, you can quickly see that numerator will not be divisible by denominator or one of them is way too big so you don't have to perform the complete calculation since you are looking for 2 as your answer. e.g. you try option (A) 20xz-y/(x+y) = /3 (numerator is way too big so not possible) Option (B) y+z/(20xz) = (2+17)/20*1*17 (denominator is way too big) You can also use algebra: Remember two things: 1. Work = Rate*Time 2. We add rates. Their combined rate of work = 1/x + 1/y Pierre does some work on his own. Work done by Pierre alone = Pierre's rate * Time = (1/x)*z Leftover work = 20 - z/x Time taken to complete leftover work together = Work/Rate = (20 - z/x)/(1/x + 1/y) = y(20x - z)/(x+y)

kraizada84 · Answer

time by pierre to complete 1 cake=x
time by franco to complete 1 cake=y

in z hours pierre completes= z/x work

then the both chef guy works together; let the time taken together be t

then we have
(work done by Pierre guy in z hours)+(work done by both guy in t hours)=20

=> (z/x)+(1/x+1/y)*t=20

=> t= (20-(z/x))/(1/x+1/y)=(20x-z)*y/(x+y)

hence Option 5 is correct.

Hope this helps..!!!

cangetgmat · Answer

nice solutions, thanks Bunuel you are awesome.

I am new to gmatclub and i feel i got late coming here; it so much here to learn.

thanx guys you all rock..!!

imhimanshu · Answer

Thanks every one for your responses. @Karishma - I always enjoy reading your posts and I have learnt a lot from them. I have been a regular visitor of your blog as well. :-D

imhimanshu · Answer

Below colored part is what I was missing. I thought the statement saying - Franco and Pierre working together decorated 20 cakes rather than the complete job is to decorate 20 cakes. so as per my equation - Leftover work = 1-z/x which was making no sense. Thanks again. Appreciate your efforts

harshavmrg · Answer

Dude...ur explanation is superb...thanks for making it so simple....
dude

galiya · Answer

1/x - Pierre's rate
1/y - Franco's rate

RT=W
after z hours Pierre alone has decorated z/x cakes

then Franco appeared with his rate 1/y and  together  the cooks were working t hours

1/x*(z+t)+1/y*t=20 -->t=y(20x-z)/(y+x)

vibhav · Answer

great explanation guys! cleared a few cobwebs in my head.

jpvemula · Answer

Solution to the above problem is as followed:

z/x +t   = 20 
t= (20x - z)y/ (x+y) (ans)

Erjan_S · Answer

I can't believe that Kaplan seriously suggests to handle "this tough problem is perfect for Picking Numbers". It is easy to pick numbers like x = 2, y = 2, and z = 10 and find out that the two chefs would need to work together for 15 hours, however, after that you need to substitute x, y and x into five (five, Karl!) answer choice to check which one yields 15! How much time would it take? Solving by equation as indicated above seems to be so much easier!

lostnumber · Answer

Can someone explain how to get the highlighted line? I am able to get 1 over x + 1 over Y, but no further

Bunuel · Answer

\frac{1}{x}+\frac{1}{y}=\frac{y}{xy}+\frac{x}{xy}=\frac{y+x}{xy}

How do we add or subtract fractions?

lostnumber · Answer

It seems so obvious now. Doh! I have not touched a math problem in 10 years or so, so I am very rusty. Thank you for having patience with my silly questions

WheatyPie · Answer

I'd just like to point that if you think about the units in the problem, answers A, B, and D can be eliminated very quickly (because they don't give you a unit of time in the end). This makes plugging in values a much more reasonable choice than it would've been otherwise.

Furthermore, when choosing between C and E, notice that choosing x=y=z=1 gives you different results in C and E, only one of which makes sense.

In this way we can choose E in ~30 seconds.

Kinshook · Answer

Given: Pastry Chef Pierre takes x hours to decorate a wedding cake. Pastry chef Franco takes y hours to decorate a wedding cake.

Asked: If Pierre works alone for z hours and is then joined by Franco until 20 cakes are decorated, for how long did the two pastry chef's work together.

Work completed by Pierre in z hours alone = z/x 
Let the two work together for w hours

Work done by two for w hours together = w(1/x+1/y) = w(x+y)/xy

Total work done = z/x + w(x+y)/xy = 20
 w = \frac{(20 - z}{x)xy/(x+y)} = \frac{(20x-z)y}{(x+y)}

IMO E

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Pastry Chef Pierre takes x hours to decorate a wedding cake. Pastry

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