2019/016) Let the rational number $\frac{p}{q}$ be closest to but not equal to$\frac{22}{7}$ among all rational numbers with denominator less than 100. What is the value of $p-3q$

Solution

We have $\lvert \frac{p}{q}-\frac{22}{7}\rvert$ as close to zero
Or $\lvert \frac{7 p- 22q}{7q}\rvert$
Above value is lowest when 7q is close to the highest and 7p-22q is close to the lowest( as both may not be at the same point)
q highest is when q = 99  this gives p = 311 and 7p-22q = 1 so this is lowest when q is highest
so p = 311, q = 99 and p - 3q = 311 - 3 * 99 = 14