Question

Question #3686f

Answer 1

`5` multiple choice

Explanation:

The best thing to do when dealing with hard and annoying math problems is take away the fluff and identify all the important info:

• 60 points total
• 15 questions total
• Multiple choice questions are two points each
• Open-ended are five points each

So, what don't we know? We don't know how many open-ended and multiple choice questions there are, and that's what we're trying to find. And what do we do when we don't know something? Assign it a variable! We'll have the number of open-ended questions be `O` and the number of multiple-choice questions be `M`.

Because there are only two types of questions and 15 questions total, we know that the number of open-ended questions plus the number of multiple choice questions is 15:
`O+M=15`

And we also know that there are 60 points total. If multiple-choice questions are two points each, that means the total number of points for answering multiple choice questions correctly is `2M`. For example, if you get 10 multiple-choice questions right, the number of points you get is `2*10=20` points. Likewise, the number of points for open-ended questions is `5O`. The key is that there are 60 points in all, so the amount of points you get from multiple choice plus the amount of points you get from open-ended must be 60:
`2M+5O=60`

Let's see. It appears we have the system:
`O+M=15`
`5O+2M=60`

We are being asked for the number of multiple-choice questions, so we have to solve this system for `M`. To do this, first solve for `O` in terms of `M`:
`O+M=15->O=15-M`

Now substitute this for `O` in `5O+2M=60` and solve:
`5O+2M=60`
`5(15-M)+2M=60`
`75-5M+2M=60`
`-3M=-15`
`M=5`

That means there are 5 multiple-choice questions on the test, and `15-5=10` open-ended. This makes sense because if each open-ended question is 5 points, then you can get 50 points max (`10*5`), and if there are 5 multiple-choice questions, then you can get 10 points max (`5*2`). The total number of points you can get is therefore 60, which is what we were told at the beginning.