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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

## 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.

## Other useful links

- https://www.cheenta.com/smallest-perimeter-of-triangle-aime-2015-question-11/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s

Google