Try this beautiful problem from the PRMO, 2019 based on Rational Number and Integer.

Rational Number and Integer – PRMO 2019


let the rational number \(\frac{p}{q}\) be closest to but not equal to \(\frac{22}{7}\) among all rational numbers with denominator < 100, find p-3q.

  • is 107
  • is 14
  • is 840
  • cannot be determined from the given information

Key Concepts


Rational number

Algebra

Integer

Check the Answer


But try the problem first…

Answer: is 14.

Source
Suggested Reading

PRMO, 2019, Question 9

Higher Algebra by Hall and Knight

Try with Hints


First hint

|\(\frac{22}{7}-\frac{p}{q}\)|=|\(\frac{22q-7p}{7q}\)| then |22q-7p|=1 for smallest value

Second Hint

and q=99 then p=311

Final Step

p-3q=311-(3)(99)=311-297=14.

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