Question

A 40-question test has 108 possible points. There are m 4-point questions and n 2-point questions. How many of each type of questions are on the test

Answer 1

We need to solve the following system of equations based on the information given:

1. The total number of questions is 40:
\[
m + n = 40
\]

2. The total points from the questions amount to 108:
\[
4m + 2n = 108
\]

Now, we can solve this system of equations step by step.

First, we can express \( n \) in terms of \( m \) using the first equation:
\[
n = 40 - m
\]

Now we can substitute this expression for \( n \) into the second equation:
\[
4m + 2(40 - m) = 108
\]

Distributing \( 2 \) in the equation gives:
\[
4m + 80 - 2m = 108
\]

Now, combine like terms:
\[
2m + 80 = 108
\]

Next, subtract 80 from both sides:
\[
2m = 28
\]

Now divide by 2:
\[
m = 14
\]

Now that we have \( m \), we can substitute back to find \( n \):
\[
n = 40 - m = 40 - 14 = 26
\]

Thus, the number of each type of question on the test is:
- \( m = 14 \) (4-point questions)
- \( n = 26 \) (2-point questions)

We can verify the solution:
- Total number of questions: \( 14 + 26 = 40 \) (Correct)
- Total points: \( 4(14) + 2(26) = 56 + 52 = 108 \) (Correct)

So, the solution is:
- There are 14 questions worth 4 points each and 26 questions worth 2 points each.