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.