Track / Stage 5 / 145 of 400 #745 of 1964

Problem 745

AIME late

Algebra Difficulty 5.2 Find the answer

(5 points) On Valentine's Day, every student in the school gave each female student a valentine. It turned out that the number of valentines was 30 more than the total number of students. How many valentines were given?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

Answer: 64

Solution: Let $x, y$ be the number of boys and girls in the school, respectively. According to the problem, $x y=x+y+30$ . Then $(x-1)(y-1)=31$ . Therefore, the numbers $x-1$ and $y-1$ are 1 and 31. The number of valentines is $2 \cdot 32=64$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.