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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

But try the problem first…

Source

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.