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CAT 2024 Slot 2 — QA Question 12

PercentageEasy

A fruit seller has a stock of mangoes, bananas and apples with at least one fruit of each type. At the beginning of a day, the number of mangoes make up 40% of his stock. That day, he sells half of the mangoes, 96 bananas and 40% of the apples. At the end of the day, he ends up selling 50% of the fruits. The smallest possible total number of fruits in the stock at the beginning of the day is

Answer & solution

Answer: 340

Solution

Hard

Let total stock be TT. Mangoes =0.4T=0.4T, and bananas ++ apples =0.6T=0.6T. Total sold =50%=50\% of TT. Write the "sold" equation, express TT in terms of bananas BB, and minimise TT subject to every fruit count being a positive integer.

1

Translate the selling condition. Sold =12(0.4T)+96+0.4A=0.5T=\tfrac12(0.4T)+96+0.4A=0.5T, where AA is apples and 0.4T0.4T mangoes.

0.2T+96+0.4A=0.5T⇒ 96+0.4A=0.3T\begin{aligned} &0.2T+96+0.4A=0.5T\\ &\Rightarrow\ 96+0.4A=0.3T \end{aligned}
2

Eliminate AA using A=0.6T−BA=0.6T-B.

96+0.4(0.6T−B)=0.3T⇒ 96+0.24T−0.4B=0.3T(expand)⇒ 96−0.4B=0.06T ⇒ T=1600−203B\begin{aligned} &96+0.4(0.6T-B)=0.3T\\ &\Rightarrow\ 96+0.24T-0.4B=0.3T \quad\text{(expand)}\\ &\Rightarrow\ 96-0.4B=0.06T\ \Rightarrow\ T=1600-\tfrac{20}{3}B \end{aligned}
3

Make everything integral and minimise TT. TT decreases as BB grows, so take BB as large as possible. TT integer needs BB a multiple of 33; put B=3kB=3k, giving T=1600−20kT=1600-20k, mangoes =0.4T=0.4T, apples A=0.6T−B=960−15kA=0.6T-B=960-15k. Require A≥1A\ge1.

960−15k≥1 ⇒ k≤63.9 ⇒ kmax⁡=63⇒ T=1600−20(63)=1600−1260=340\begin{aligned} &960-15k\ge1\ \Rightarrow\ k\le 63.9\ \Rightarrow\ k_{\max}=63\\ &\Rightarrow\ T=1600-20(63)=1600-1260=340 \end{aligned}
4

Verify feasibility at T=340T=340. Mangoes =0.4(340)=136=0.4(340)=136, bananas =3(63)=189=3(63)=189, apples =960−945=15=960-945=15; all positive integers and 136+189+15=340136+189+15=340.

(M,B,A)=(136, 189, 15)all ≥1 ✓\begin{aligned} &(M,B,A)=(136,\,189,\,15)\quad\text{all }\ge1\ \checkmark \end{aligned}
Tmin⁡=340T_{\min}=340
CAT 2024 Slot 2 QA Q12: A fruit seller has a stock of mangoes, bananas and apples with at least one fruit of each type. At the beginni — Solution | TheCATExam