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CAT 2023 Slot 1 QA Question 9

Multiple choice (+3 / −1) · Arithmetic · Mixture & Alligation · Try it, then check the answer and solution below.

CAT 2023 Slot 1QAArithmetic • Mixture & AlligationEasy
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 PP and replaced with same amount of cocoa powder to form a new mixture QQ. If the ratio of coffee and cocoa in the mixture QQ is 16:9,thentheratioofcocoainmixture16: 9 , then the ratio of cocoa in mixturePtothatinmixtureto that in mixtureQ$ is
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Answer and solution

Answer: A) 5:95:9

📌 Core Concept
The problem involves repeated replacement, where each time a fraction of the mixture is replaced. The concentration of the original component after multiple replacements can be calculated using the formula: $Concentration after n replacements=(1−f)n\text{Concentration after } n \text{ replacements} = (1 - f)^n where ff is the fraction replaced each time.
🔢 Step-by-Step Solution
1
Let the total volume be TT.
2
After first replacement (mixture P):
Coffee: T−xT - x
Cocoa: xx
Fraction of coffee: T−xT\frac{T - x}{T}
Fraction of cocoa: xT\frac{x}{T}
3
After second replacement (mixture Q):
Coffee removed: T−xT×x\frac{T - x}{T} \times x
Cocoa removed: xT×x\frac{x}{T} \times x
Remaining coffee: (T−x)−T−xT×x=(T−x)×(1−xT)(T - x) - \frac{T - x}{T} \times x = (T - x) \times \left(1 - \frac{x}{T}\right)
Remaining cocoa: x−x2T+x=x×(1−xT)+xx - \frac{x^2}{T} + x = x \times \left(1 - \frac{x}{T}\right) + x
4
Given ratio in Q: Coffee:Cocoa = 16:9.
Total parts = 25
Coffee fraction = 1625\frac{16}{25}$
Cocoa fraction = 925\frac{9}{25}$
5
Using concentration formula:
Coffee concentration after two replacements: (1−xT)2=1625(1 - \frac{x}{T})^2 = \frac{16}{25}
Taking square root: 1−xT=451 - \frac{x}{T} = \frac{4}{5}
Thus, xT=15\frac{x}{T} = \frac{1}{5}
6
Cocoa in P: 15T\frac{1}{5}T
Cocoa in Q: 925T\frac{9}{25}T
Ratio \frac{15\frac{1}{5}T}{925\frac{9}{25}T} = 59\frac{5}{9}$
⚡ 30-Second Shortcut
Recognize the repeated replacement pattern. After two replacements, the concentration ratio leads directly to the fraction 15\frac{1}{5}$ for each step. The ratio of cocoa in P to Q simplifies to $5:9$ by comparing their fractions.
🎯 Final Answer
Correct Answer: Option A

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