Try this beautiful problem from the Pre-RMO, 2017 based on ratio.

## Problem on Ratio – PRMO 2017

In a sports team, the total number of boys and girls are in the ratio 4:3. On one day it was found that 8 boys and 14 girls were absent from the team and that the number of boys was the square of the number of girls, find the total number of members in the sports team.

- is 107
- is 42
- is 840
- cannot be determined from the given information

**Key Concepts**

Ratio

Integers

Algebra

## Check the Answer

But try the problem first…

Answer: is 42.

PRMO, 2017, Question 12

Elementary Algebra by Hall and Knight

## Try with Hints

First hint

here ratio = 4:3

\(\Rightarrow boys =4x, girls=3x\)

given \((4x-8)=(3x-14)^{2}\)

Second Hint

\(\Rightarrow 4x-8=9x^{2}-84x+196\)

\(\Rightarrow 9x^{2}+196-84x=4x-8\)

\(\Rightarrow 9x^{2}-88x+204=0\)

Final Step

\(x=\frac{88\pm\sqrt{88^{2}-4(9)(204)}}{18}\)

\(\Rightarrow x=6 or \frac{34}{9}\)

\(\Rightarrow 7x=42\) or non-integer

\(\Rightarrow 42 members.\)

## Other useful links

- https://www.cheenta.com/rational-number-and-integer-prmo-2019-question-9/
- https://www.youtube.com/watch?v=lBPFR9xequA

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