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CAT 2024 Slot 1 — QA Question 15

Removal & ReplacementEasy

A glass is filled with milk. Two-thirds of its content is poured out and replaced with water. If this process of pouring out two-thirds the content and replacing with water is repeated three more times, then the  final ratio of milk to water in the glass, is

Answer & solution

  • A

    1 : 26

  • B

    1 : 27

  • 1 : 80

  • D

    1 : 81

Solution

Medium

Each "pour out two-thirds and top up with water" keeps one-third of the milk. The replacement happens 44 times in total (the first time plus three repeats), so the milk fraction is (1/3)4(1/3)^4.

1

Effect of one operation. Pouring out 23\tfrac23 leaves 13\tfrac13 of the milk; topping up with water restores the volume but adds no milk.

milk fraction after one step=13\begin{aligned} &\text{milk fraction after one step}=\tfrac13 \end{aligned}
2

Count the operations. The process is done once, then "repeated three more times" ⇒\Rightarrow 44 times total.

milk fraction=(13)4=181(from step 1, 4 times)\begin{aligned} &\text{milk fraction}=\left(\tfrac13\right)^4=\dfrac{1}{81} \quad\text{(from step 1, 4 times)} \end{aligned}
3

Milk : water. Take total volume =81=81 units; milk =1=1, so water =80=80.

milk:water=1:(81−1)=1:80\begin{aligned} &\text{milk}:\text{water}=1:(81-1)=1:80 \end{aligned}
milk:water=1:80\text{milk}:\text{water}=1:80

The trap option is 1:811:81 — that mistakenly uses milk:total instead of milk:water. Final water == total −- milk =81−1=80=81-1=80.

CAT 2024 Slot 1 QA Q15: A glass is filled with milk. Two-thirds of its content is poured out and replaced with water. If this process — Solution | TheCATExam