How to find the no of Questions?

Question

Liz and Mary compete in solving problems. Each of them is given the same list of 100 problems. For any problem, the first of them to solve it gets 4 points, while the second to solve it gets 1 point. Liz solved 60 problems, and Mary also solved 60 problems. Together, they got 312 points. How many problems were solved by both of them?

Answer 1

We can use "algebra." Let $x$ be the number of problems solved by both. Each such problem contributes $4+1$ points to the combined score, for a total of $5x$.

In addition, $(60-x)+(60-x)$ problems were solved by one person alone. These contributed $(4)(2)(60-x)$ to the combined score. We are told that the combined score was $312$. It follows that $$5x+8(60-x)=312.$$ Solve this linear equation for $x$.

Once upon a time, before the advent of "algebra," the problem might have been solved as follows.

Let us guess that $60$ of the problems were solved by both of them. Then the total score would be $300$, which is $12$ short of $312$. Let us guess now that $59$ of the problems were solved by both of them. That would add $4+4$ points to the total score, and subtract $5$, for a net gain of $3$. Similarly, each problem taken out of the pool of problems solved by both adds $3$ to the total score. We need to add $12$ points, so we need to take $12/3=4$ problems out of the pool of problems solved by both. So $56$ of the problems were solved by both.