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.
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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
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)
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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.

Dude...ur explanation is superb...thanks for making it so simple....
dude
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great explanation guys! cleared a few cobwebs in my head.

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!

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

How do we add or subtract fractions?
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