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CAT 2021 Slot 1 — QA Question 11

3 Variable EquationsEasy

A basket of 2 apples, 4 oranges and 6 mangoes costs the same as a basket of 1 apple, 4 oranges and 8 mangoes, or a basket of 8 oranges and 7 mangoes. Then the number of mangoes in a basket of mangoes that has the same cost as the other baskets is

Answer & solution

  • 13

  • B

    12

  • C

    11

  • D

    10

Solution

Easy

Let prices be a,o,ma,o,m. The three equal-cost baskets give two independent equations; solve them to write aa and oo in terms of mm, then convert any basket's total cost into mangoes.

1

Write the equal-cost conditions. All three baskets cost the same.

2a+4o+6m=a+4o+8m=8o+7m\begin{aligned} &2a+4o+6m=a+4o+8m=8o+7m \end{aligned}
2

First pair. Equate basket 1 and basket 3.

2a+4o+6m=8o+7m⇒ 2a=4o+m...(1)\begin{aligned} &2a+4o+6m=8o+7m\\ &\Rightarrow\ 2a=4o+m \quad\text{...(1)} \end{aligned}
3

Second pair. Equate basket 2 and basket 3.

a+4o+8m=8o+7m⇒ a=4o−m...(2)\begin{aligned} &a+4o+8m=8o+7m\\ &\Rightarrow\ a=4o-m \quad\text{...(2)} \end{aligned}
4

Solve (1) and (2). Substitute (2) into (1): 2(4o−m)=4o+m2(4o-m)=4o+m.

8o−2m=4o+m⇒4o=3m⇒o=3m4a=4o−m=3m−m=2m(from (2))\begin{aligned} &8o-2m=4o+m \Rightarrow 4o=3m \Rightarrow o=\tfrac{3m}{4}\\ &a=4o-m=3m-m=2m \quad\text{(from (2))} \end{aligned}
5

Convert a basket's cost to mangoes. Use basket 1 and substitute a=2m, o=3m4a=2m,\ o=\tfrac{3m}{4}.

2a+4o+6m=2(2m)+4⋅3m4+6m⇒ =4m+3m+6m=13m\begin{aligned} &2a+4o+6m=2(2m)+4\cdot\tfrac{3m}{4}+6m\\ &\Rightarrow\ =4m+3m+6m=13m \end{aligned}
Number of mangoes=13(option (a))\text{Number of mangoes}=13 \quad\text{(option (a))}
CAT 2021 Slot 1 QA Q11: A basket of 2 apples, 4 oranges and 6 mangoes costs the same as a basket of 1 apple, 4 oranges and 8 mangoes, — Solution | TheCATExam