Students are given 3 minutes for each multiple-choice question and 5 minutes for each free-response question on a test. There are 15 questions on the test, and students are given 51 minutes to take it.
The system of equations shown can be used to find the number of multiple-choice questions, [tex]$m$[/tex], and the number of free-response questions, [tex]$f$[/tex], on the test.
[tex]\begin{array}{l}
m+f=15 \\
3 m+5 f=51
\end{array}[/tex]
How many multiple-choice questions are on the test?
A. 3
B. 5
C. 12
D. 14
Answer (2)
Express f in terms of m using the first equation: f = 15 − m .
Substitute this expression into the second equation: 3 m + 5 ( 15 − m ) = 51 .
Simplify and solve for m : 3 m + 75 − 5 m = 51 A rr − 2 m = − 24 A rr m = 12 .
The number of multiple-choice questions is 12 .
Explanation
Analyze the problem We are given a system of two equations with two variables, m and f , representing the number of multiple-choice and free-response questions, respectively. Our goal is to find the value of m . The equations are:
m + f = 15
3 m + 5 f = 51
Express f in terms of m We can solve this system of equations using substitution or elimination. Let's use substitution. From equation (1), we can express f in terms of m :
f = 15 − m
Substitute into the second equation Now, substitute this expression for f into equation (2):
3 m + 5 ( 15 − m ) = 51
Solve for m Simplify and solve for m :
3 m + 75 − 5 m = 51
− 2 m = 51 − 75
− 2 m = − 24
m = − 2 − 24
m = 12
Final Answer Therefore, there are 12 multiple-choice questions on the test.
Examples
Understanding systems of equations is crucial in many real-world scenarios. For instance, imagine you're planning a balanced diet. You need a certain number of calories and a specific amount of protein each day. Different foods have varying amounts of calories and protein. By setting up a system of equations, you can determine the exact quantities of each food to meet your nutritional goals. This same principle applies to budgeting, mixing chemicals, or even optimizing production processes.
There are 12 multiple-choice questions on the test, as determined by solving the system of equations. The equations represent the total number of questions and the time allotted for each type. Thus, the correct answer is option C, 12.
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