Try this beautiful problem from the Pre-RMO, 2017 based on ratio.
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.
Ratio
Integers
Algebra
Answer: is 42.
PRMO, 2017, Question 12
Elementary Algebra by Hall and Knight
here ratio = 4:3
\(\Rightarrow boys =4x, girls=3x\)
given \((4x-8)=(3x-14)^{2}\)
\(\Rightarrow 4x-8=9x^{2}-84x+196\)
\(\Rightarrow 9x^{2}+196-84x=4x-8\)
\(\Rightarrow 9x^{2}-88x+204=0\)
\(x=\frac{88\pm\sqrt{88^{2}-4(9)(204)}}{18}\)
\(\Rightarrow x=6 or \frac{34}{9}\)
\(\Rightarrow 7x=42\) or non-integer
\(\Rightarrow 42 members.\)
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