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Algebra Arithmetic Math Olympiad PRMO

Problem on Ratio | PRMO 2017 | Question 12

Try this beautiful problem from the Pre-RMO, 2017 based on ratio and proportion. You may use sequential hints to solve the problem.

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.

Source
Suggested Reading

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

\(\Rightarrow 42 members.\)

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