2022 MMATHS Individual Round Problems/Problem 3

Problem

Luke and Carissa are finding the sum of the first $20$ positive integers by adding them one at a time. Luke forgets to add one number and gets an answer of $207$. Carissa adds a number twice by mistake and gets an answer of $225$. What is the sum of the number that Luke forgot and the number that Carissa added twice?

Solution 1

The sum of the first $20$ positive integers is $\frac {20 \cdot 21}{2} = 210$. This means that Luke forgot to add $3$ and Carissa added the number $15$ twice. Therefore, the sum of the number that Luke forgot and the number Carissa added twice is $\boxed {18}$.

~Arcticturn