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CAT 2021 Slot 2 — QA Question 8

Removal & ReplacementEasy

From a container filled with milk, 9 litres of milk are drawn and replaced with water. Next, from the same container, 9 litres are drawn and again replaced with water. If the volumes of milk and water in the container are now in the ratio of 16 : 9, then the capacity of the container, in litres, is

Answer & solution

Answer: 45

Solution

Easy

This is the classic replacement formula: after each draw-and-replace of 99 litres from a tank of capacity VV, the milk fraction is multiplied by (1−9V)\left(1-\tfrac{9}{V}\right). Convert the final 16:916:9 ratio into a milk fraction, then solve.

1

Replacement formula. Each draw removes the fraction 9V\tfrac{9}{V} of whatever is in the tank, so milk left after two replacements is

milk=V(1−9V)2\begin{aligned} &\text{milk}=V\left(1-\tfrac{9}{V}\right)^{2} \end{aligned}
2

Convert the ratio to a fraction. Milk : water =16:9=16:9, so milk is 1616+9=1625\tfrac{16}{16+9}=\tfrac{16}{25} of the full tank.

V(1−9V)2=1625 V⇒ (1−9V)2=1625\begin{aligned} &V\left(1-\tfrac{9}{V}\right)^{2}=\tfrac{16}{25}\,V\\ &\Rightarrow\ \left(1-\tfrac{9}{V}\right)^{2}=\tfrac{16}{25} \end{aligned}
3

Take the root and solve. Take the positive root (1−9V>01-\tfrac{9}{V}>0).

1−9V=45⇒ 9V=15⇒ V=45\begin{aligned} &1-\tfrac{9}{V}=\tfrac{4}{5}\\ &\Rightarrow\ \tfrac{9}{V}=\tfrac{1}{5}\\ &\Rightarrow\ V=45 \end{aligned}
Capacity of the container=45 litres\text{Capacity of the container}=45\ \text{litres}
CAT 2021 Slot 2 QA Q8: From a container filled with milk, 9 litres of milk are drawn and replaced with water. Next, from the same con — Solution | TheCATExam