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CAT 2021 Slot 2 — QA Question 1

2 Variable EquationsEasy

A box has 450 balls, each either white or black, there being as many metallic white balls as metallic black balls. If 40% of the white balls and 50% of the black balls are metallic, then the number of non-metallic balls in the box is

Answer & solution

Answer: 250

Solution

Easy

Scale the white and black counts so the percentages are clean, use the "equal number of metallic white and black balls" condition to relate the two counts, then subtract the metallic balls from the total.

1

Set up convenient variables. Let there be 100w100w white balls and 100b100b black balls. Their total is 450450.

100w+100b=450⇒ w+b=4.5(divide by 100)…(1)\begin{aligned} &100w+100b=450\\ &\Rightarrow\ w+b=4.5 \quad\text{(divide by 100)} \quad\dots(1) \end{aligned}
2

Use the equal-metallic condition. Metallic white =40%=40\% of 100w=40w100w=40w; metallic black =50%=50\% of 100b=50b100b=50b. These are equal.

40w=50b⇒ w=1.25 b(solve for w)…(2)\begin{aligned} &40w=50b\\ &\Rightarrow\ w=1.25\,b \quad\text{(solve for }w\text{)} \quad\dots(2) \end{aligned}
3

Solve (1) and (2). Substitute (2) into (1).

1.25 b+b=4.5(from (1),(2))⇒ 2.25 b=4.5⇒ b=2,w=2.5\begin{aligned} &1.25\,b+b=4.5 \quad\text{(from (1),(2))}\\ &\Rightarrow\ 2.25\,b=4.5\\ &\Rightarrow\ b=2,\quad w=2.5 \end{aligned}

So there are 250250 white and 200200 black balls.

4

Count the non-metallic balls. Non-metallic white =60%=60\% of 250250; non-metallic black =50%=50\% of 200200.

non-metallic white=0.60×250=150non-metallic black=0.50×200=100⇒ total=150+100=250\begin{aligned} &\text{non-metallic white}=0.60\times 250=150\\ &\text{non-metallic black}=0.50\times 200=100\\ &\Rightarrow\ \text{total}=150+100=250 \end{aligned}
Number of non-metallic balls=250\text{Number of non-metallic balls}=250
CAT 2021 Slot 2 QA Q1: A box has 450 balls, each either white or black, there being as many metallic white balls as metallic black ba — Solution | TheCATExam