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CAT 2023 Slot 1 — QA Question 3

Removal & ReplacementEasy

A mixture P is formed by removing a certain amount of coffee from a coffee jar and replacing the same amount with cocoa powder. The same amount is again removed from mixture P and replaced with same amount of cocoa powder to form a new mixture Q. If the ratio of coffee and cocoa in the mixture Q is 16 : 9, then the ratio of cocoa in mixture P to that in mixture Q is

Answer & solution

  • A

    4 : 9

  • B

    1 : 3

  • C

    1 : 2

  • 5 : 9

Solution

Easy

This is the classic "replace-and-repeat" dilution. After two identical replacements of fraction r100\tfrac{r}{100}, the coffee remaining is 100(1−r100)2100\left(1-\tfrac{r}{100}\right)^2. Use the final 16:916:9 ratio to pin down rr, then compare the cocoa present in mixture PP (after one replacement) with the cocoa in mixture QQ (after two).

Take the jar to start with 100100 kg of coffee and let rr kg be removed-and-replaced each time. The ratio 16:916:9 has total 2525 parts, so after the second replacement coffee =1625×100=64=\tfrac{16}{25}\times100=64 kg and cocoa =36=36 kg.

1

Coffee after two replacements equals 6464 kg:

100(1−r100)2=64 ⇒ (1−r100)2=64100100\left(1-\frac{r}{100}\right)^2=64\ \Rightarrow\ \left(1-\frac{r}{100}\right)^2=\frac{64}{100}
2

Take the square root and solve for rr:

1−r100=810=45 ⇒ r100=15 ⇒ r=201-\frac{r}{100}=\frac{8}{10}=\frac45\ \Rightarrow\ \frac{r}{100}=\frac15\ \Rightarrow\ r=20
3

Cocoa in mixture PP (just after the first replacement): we removed r=20r=20 kg of coffee and added 2020 kg cocoa.

cocoa in P=20 kg\text{cocoa in }P=20\ \text{kg}
4

Cocoa in mixture QQ (after the second replacement) is whatever is not coffee:

cocoa in Q=100−64=36 kg\text{cocoa in }Q=100-64=36\ \text{kg}
5

Required ratio cocoa(P)(P) : cocoa(Q)(Q):

20:36=5:920:36=5:9
5:9(option d)\mathbf{5:9}\quad\text{(option d)}

From (1−r100)2=1625\left(1-\tfrac{r}{100}\right)^2=\tfrac{16}{25} you instantly get 1−r100=451-\tfrac{r}{100}=\tfrac45, so cocoa fraction in PP is 15\tfrac15 and in QQ is 1−1625=9251-\tfrac{16}{25}=\tfrac{9}{25}. Ratio =15:925=5:9=\tfrac15:\tfrac{9}{25}=5:9 on the same base.

CAT 2023 Slot 1 QA Q3: A mixture P is formed by removing a certain amount of coffee from a coffee jar and replacing the same amount w — Solution | TheCATExam