How to find the least number of chips to get right each color?

Question

The problem is as follows:

Claire has as porcelain jar (it is not transparent) with 141 plastic chips inside. They have the same weight and size. But they have different colors, some are red, others green and others black. If she knows that the number of green chips is to the number of black chips as 5 is to 3 and the number of red tokens is to the number of green as 3 is to 4. How many chips, at least, does she has to take out from the jar at random to have with certainty among them 11 chips of each color?

The alternatives given in my book are as follows:

$\begin{array}{ll} 1.&\textrm{136}\\ 2.&\textrm{116}\\ 3.&\textrm{120}\\ 4.&\textrm{166}\\ \end{array}$

I'm not sure exactly how should I solve this problem. Something tells me that the approach to follow is first to find the number of chips which is from each color, so with that then I can expect that to be sure the number of chips will be by considering the most unlikely case scenario.

Since it says that the green tokens it to the number of blacks as $5$ is to $3$. Then is:

$\frac{G}{B}=\frac{5}{3}$

And it also mentions that the red is to the green as $3$ is to $4$. Then this would mean:

$\frac{R}{G}=\frac{3}{4}$

Then we know that:

$R+G+B=141$

Therefore: One could tell the number of the other chips by expressing one in terms of the other as given by the earlier clues.

$R+\frac{4}{3}R+\frac{4}{3}\cdot\frac{3}{5}R=141$

Solving this yields:

$R=45$

$G=15\times 4 =60$

$B=36$

Then these are the number of chips what we have in the jar.

Since it mentions to be certain to get 11 chips of the same color each this would translate as:

$45+60+10+1=116$

Answer number 2.

Thus I believe the answer would be to draw 116 tokens out of the jar at minimum. But is this interpretation the correct one?. Can someone help me here?

Answer 1

The number of chips are correctly counted. When you have taken up 116 chips as the answer, to verify it, you can try subtracting it from the total number of chips. 141-116= 25.

The worst thing that can happen is that the 25 remaining chips remaining in the bowl are of the same color. The chip with the lowest number is black. So the worst case is that the 25 chips remaining in the bowl are all black.

So if you subtract 25 black chips in the bowl from 36 total black chips, you get 11 black chips. Which proves that there certainly are 11 chips of each color. I hope this helps.