To solve this problem, we'll set it up as a system of linear equations where:
- Let [tex]\( mc \)[/tex] be the number of multiple-choice questions.
- Let [tex]\( sa \)[/tex] be the number of short-answer questions.
Based on the information provided, we can create the following equations:
1. We know that the total number of questions is 20. This means:
[tex]\[ mc + sa = 20 \][/tex]
That equation matches sentence C: "The number of multiple-choice questions plus the number of short-answer questions is 20."
2. We know that multiple-choice questions are worth 1 point each and short-answer questions are worth 3 points each. For a test worth a total of 50 points, we can write the equation:
[tex]\[ mc \times 1 + sa \times 3 = 50 \][/tex]
Now, let's solve this system of equations:
From equation 1:
[tex]\[ mc + sa = 20 \][/tex]
Let's solve for one variable in terms of the other. For this instance, we can solve for [tex]\( sa \)[/tex]:
[tex]\[ sa = 20 - mc \][/tex]
Now, let's substitute [tex]\( sa \)[/tex] in equation 2 with [tex]\( 20 - mc \)[/tex]:
[tex]\[ mc + 3 \times (20 - mc) = 50 \][/tex]
[tex]\[ mc + 60 - 3mc = 50 \][/tex]
[tex]\[ 60 - 2mc = 50 \][/tex]
[tex]\[ 2mc = 60 - 50 \][/tex]
[tex]\[ 2mc = 10 \][/tex]
[tex]\[ mc = \frac{10}{2} \][/tex]
[tex]\[ mc = 5 \][/tex]
So, there are 5 multiple-choice questions.
We solved for the number of multiple-choice questions, but the original question asks which sentence represents the number of test questions. Based on our equations and the solution, the correct sentence is:
C. The number of multiple-choice questions plus the number of short-answer questions is 20.
