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

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Senior Manager
Joined: 07 Sep 2010
Posts: 289
Pastry Chef Pierre takes x hours to decorate a wedding cake. Pastry [#permalink]

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01 Apr 2012, 22:06
7
11
00:00

Difficulty:

55% (hard)

Question Stats:

69% (02:16) correct 31% (02:28) wrong based on 363 sessions

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

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)

Thanks
Math Expert
Joined: 02 Sep 2009
Posts: 47037
Pastry Chef Pierre takes x hours to decorate a wedding cake. Pastry [#permalink]

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02 Apr 2012, 00:15
10
5
imhimanshu wrote:
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. $$\frac{y(20x-z)}{(x+y)}$$

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: $$\frac{z}{x}+t*\frac{x+y}{xy}=20$$ --> $$t*\frac{x+y}{xy}=\frac{20x-z}{x}$$ --> $$t=\frac{y(20x-z)}{x+y}$$.

Hope it's clear.
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Re: Pastry Chef Pierre takes x hours to decorate a wedding cake. Pastry [#permalink]

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01 Apr 2012, 23:40
7
1
imhimanshu wrote:
Hi Bunuel,
Request you to provide me your approach. I couldn't solve it.

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.
1- 20xz-y/(x+y)
2- y+z/(20xz)
3- 20x(y-z)/(x+y)
4- 20xy-z/(x+y)
5- y(20x-z)/(x+y)

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)

Thanks

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..!!!
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##### General Discussion
GMAT Club Legend
Joined: 16 Oct 2010
Posts: 8124
Location: Pune, India
Re: Pastry Chef Pierre takes x hours to decorate a wedding cake. Pastry [#permalink]

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01 Apr 2012, 23:42
7
3
imhimanshu wrote:
Hi Bunuel,
Request you to provide me your approach. I couldn't solve it.

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.
1- 20xz-y/(x+y)
2- y+z/(20xz)
3- 20x(y-z)/(x+y)
4- 20xy-z/(x+y)
5- y(20x-z)/(x+y)

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)

Thanks

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) = [20*1*17 - 2]/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

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)
_________________

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Contact: bansal.karishma@gmail.com

Intern
Joined: 01 Apr 2012
Posts: 9
Re: Pastry Chef Pierre takes x hours to decorate a wedding cake. Pastry [#permalink]

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02 Apr 2012, 00:30
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..!!
Senior Manager
Joined: 07 Sep 2010
Posts: 289
Re: Pastry Chef Pierre takes x hours to decorate a wedding cake. Pastry [#permalink]

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02 Apr 2012, 03:11
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.
Senior Manager
Joined: 07 Sep 2010
Posts: 289
Re: Pastry Chef Pierre takes x hours to decorate a wedding cake. Pastry [#permalink]

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02 Apr 2012, 03:27
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.

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

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.
1- 20xz-y/(x+y)
2- y+z/(20xz)
3- 20x(y-z)/(x+y)
4- 20xy-z/(x+y)
5- y(20x-z)/(x+y)

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)

Thanks

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) = [20*1*17 - 2]/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

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)
Manager
Status: I will not stop until i realise my goal which is my dream too
Joined: 25 Feb 2010
Posts: 207
Schools: Johnson '15
Re: Pastry Chef Pierre takes x hours to decorate a wedding cake. Pastry [#permalink]

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03 Apr 2012, 07:14
imhimanshu wrote:
Hi Bunuel,
Request you to provide me your approach. I couldn't solve it.

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.
1- 20xz-y/(x+y)
2- y+z/(20xz)
3- 20x(y-z)/(x+y)
4- 20xy-z/(x+y)
5- y(20x-z)/(x+y)

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)

Thanks

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..!!!

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

Note: Give me kudos if my approach is right , else help me understand where i am missing.. I want to bell the GMAT Cat

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Manager
Joined: 16 Jan 2011
Posts: 102
Re: Pastry Chef Pierre takes x hours to decorate a wedding cake. Pastry [#permalink]

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13 Apr 2012, 04:49
1
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)
Senior Manager
Joined: 28 Dec 2010
Posts: 294
Location: India
Re: Pastry Chef Pierre takes x hours to decorate a wedding cake. Pastry [#permalink]

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27 May 2012, 04:17
great explanation guys! cleared a few cobwebs in my head.
Intern
Joined: 23 Dec 2015
Posts: 23
Re: Pastry Chef Pierre takes x hours to decorate a wedding cake. Pastry [#permalink]

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13 Jul 2016, 03:49
imhimanshu wrote:
Hi Bunuel,
Request you to provide me your approach. I couldn't solve it.

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-y/(x+y)
B. y+z/(20xz)
C. 20x(y-z)/(x+y)
D. 20xy-z/(x+y)
E. y(20x-z)/(x+y)

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)

Thanks

Solution to the above problem is as followed:

z/x +t [ (x+y)/xy] = 20
t= (20x - z)y/ (x+y) (ans)
Manager
Joined: 12 Nov 2016
Posts: 140
Concentration: Entrepreneurship, Finance
GMAT 1: 620 Q36 V39
GMAT 2: 650 Q47 V33
Re: Pastry Chef Pierre takes x hours to decorate a wedding cake. Pastry [#permalink]

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14 Jan 2017, 11:03
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!
Intern
Joined: 28 Mar 2018
Posts: 48
Re: Pastry Chef Pierre takes x hours to decorate a wedding cake. Pastry [#permalink]

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06 Apr 2018, 12:17
Bunuel wrote:
imhimanshu wrote:
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. $$\frac{y(20x-z)}{(x+y)}$$

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/hourand 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: $$\frac{z}{x}+t*\frac{x+y}{xy}=20$$ --> $$t*\frac{x+y}{xy}=\frac{20x-z}{x}$$ --> $$t=\frac{y(20x-z)}{x+y}$$.

Hope it's clear.

Can someone explain how to get the highlighted line? I am able to get 1 over x + 1 over Y, but no further
Math Expert
Joined: 02 Sep 2009
Posts: 47037
Re: Pastry Chef Pierre takes x hours to decorate a wedding cake. Pastry [#permalink]

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06 Apr 2018, 12:38
lostnumber wrote:
Bunuel wrote:
imhimanshu wrote:
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. $$\frac{y(20x-z)}{(x+y)}$$

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/hourand 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: $$\frac{z}{x}+t*\frac{x+y}{xy}=20$$ --> $$t*\frac{x+y}{xy}=\frac{20x-z}{x}$$ --> $$t=\frac{y(20x-z)}{x+y}$$.

Hope it's clear.

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

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

How do we add or subtract fractions?
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Joined: 28 Mar 2018
Posts: 48
Re: Pastry Chef Pierre takes x hours to decorate a wedding cake. Pastry [#permalink]

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06 Apr 2018, 13:06
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
Re: Pastry Chef Pierre takes x hours to decorate a wedding cake. Pastry   [#permalink] 06 Apr 2018, 13:06
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