# Question #3686f

Apr 19, 2016

$5$ 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 $2 M$. For example, if you get 10 multiple-choice questions right, the number of points you get is $2 \cdot 10 = 20$ points. Likewise, the number of points for open-ended questions is $5 O$. 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:
$2 M + 5 O = 60$

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

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 $M$:
$O + M = 15 \to O = 15 - M$

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

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 ($10 \cdot 5$), and if there are 5 multiple-choice questions, then you can get 10 points max ($5 \cdot 2$). The total number of points you can get is therefore 60, which is what we were told at the beginning.