<p>Let <img alt="$S$" height="14" src="https://latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png" width="14" /> be the set of all positive rational numbers <img alt="$r$" height="8" src="https://latex.artofproblemsolving.com/b/5/5/b55ca7a0aa88ab7d58f4fc035317fdac39b17861.png" width="8" /> such that when the two numbers <img alt="$r$" height="8" src="https://latex.artofproblemsolving.com/b/5/5/b55ca7a0aa88ab7d58f4fc035317fdac39b17861.png" width="8" /> and <img alt="$55r$" height="12" src="https://latex.artofproblemsolving.com/b/5/c/b5c299c8b92f7565809ac418662490059475f653.png" width="26" /> are written as fractions in lowest terms, the sum of the numerator and denominator of one fraction is the same as the sum of the numerator and denominator of the other fraction. The sum of all the elements of <img alt="$S$" height="14" src="https://latex.artofproblemsolving.com/a/d/2/ad28c83c99a8fd0dd2e2e594c9d02ee532765a0a.png" width="14" /> can be expressed in the form <img alt="$\frac{p}{q},$" height="36" src="https://latex.artofproblemsolving.com/8/2/3/8238e23ea62031f4c122beff87871a238dc39d8a.png" width="17" /> where <img alt="$p$" height="11" src="https://latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png" width="10" /> and <img alt="$q$" height="11" src="https://latex.artofproblemsolving.com/0/6/1/0615acc3725de21025457e7d6f7694dab8e2f758.png" width="8" /> are relatively prime positive integers. Find <img alt="$p+q.$" height="14" src="https://latex.artofproblemsolving.com/0/4/3/043f651be7ed631a58104b324d1aa6c6c843beba.png" width="44" /></p>

Question No: 395

Let $S$ be the set of all positive rational numbers $r$ such that when the two numbers $r$ and $55r$ are written as fractions in lowest terms, the sum of the numerator and denominator of one fraction is the same as the sum of the numerator and denominator of the other fraction. The sum of all the elements of $S$ can be expressed in the form $\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$

Editorial

Tags: Algebra

Question Details