Puzzle about number of students who solved the task and number of girls in a class

Question

During an exam teacher gave students only one very hard task do solve.

Statistics built up after the exam told that: number of boys who solved the task was larger by 1 than number of girls who did not solved the task.

Which group is bigger

• number of all students who solved the task
• number of girls in a class

My opinion:

The number of all students who solved the task. If none of girls would solve the task then only one boy would solve it and the rest of girls, no matter how many girls it would be still students who solved > no girls. If all girls solved the task then number of boys would be 1 so : 1 + no girls > no girls. I do not know what happens in the middle.

Answer 1

Well, let $B$, $G$, and $S$ represent the set of boys, girls, and students who solved the task, respectively. Then we have $$ |B \cap S| = |G \cap S'| + 1.$$ So we can add $|G \cap S|$ to both sides to get $$|S| = |B \cap S| + |G \cap S| = |G \cap S'| + |G \cap S| + 1 = |G| + 1 .$$ That is, the number of students who solved the task is one bigger than the number of girls.

Answer 2

Denote by $G_s,G_i,B_s,B_i$ the girls that solved it, girls that didn't solve, boys that solved and boys that didn't solve respectively.

We know $B_s=G_i+1$ therefore $B_s+G_s=G_i+G_s+1$. But $B_s+G_s$ is the number of people who solved it and $G_s+G_i$ is the number of girls. Therefore the number of people who solved it is larger than th enumber of girls, by 1.