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Olympiad workbook
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IOQM
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Basics of Mathematics

Olympiad workbook · IOQM · Basics of Mathematics

Let the rational number p / q be closest to but not equal to 22 / 7...

Let the rational number p/qp / q be closest to but not equal to 22/722 / 7 among all rational numbers with denominator <100< 100. What is the value of p−3qp-3 q ?

Correct answer

14

Step-by-step solution

We have to have ∣227−pq∣\left|\frac{22}{7}-\frac{\mathrm{p}}{\mathrm{q}}\right| to be as small as possible for q<100\mathrm{q} < 100. \left|\frac{22}{7}-\frac{p}{q}\right|=\left|\frac{22 q-7 p}{7 q}\right|shouldbesmallest.should be smallest.\Rightarrow|22 q-7 p|=1andandq=99 ⇒p=311p−3q=311−3×99=311−297=14\begin{aligned} & \Rightarrow p=311 \\ & p-3 q=311-3 \times 99=311-297=14\end{aligned}
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