Question

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, m, and the number of free-response questions, f, on the test.

m + f = 15

3m + 5f = 51

How many multiple-choice questions are on the test?
3
5
12
14

Answer 1

option C is correct.

12 multiple-choice questions are on the test

Explanation:

Given: The system of equation :

m + f = 15 ......[1]

3m + 5f =51 .......[2]

where m is the number of multiple choice questions and f is the number of free response questions.

Now, multiply 3 to both sides of an equation in [1];

`3 \cdot (m+f) = 3\cdot 15`

Simplify:

3m + 3f = 45

Now, subtract 3f to both sides of an equation:

3m + 3f -3f = 45 -3f

Simplify:

3m = 45 -3f ......[3]

Substitute the equation [3] in [2];

45 -3f +5f = 51

Combine like terms;

45 + 2f = 51

Subtract 45 to both sides of an equation we get;

45+ 2f -45 =51-45

Simplify:

2f = 6

Divide by 2 to both sides of an equation:

`(2f)/(2) =(6)/(2)`

Simplify:

f = 3

Substitute the value of f = 3 in equation [1] solve for m;

m + 3 = 15

Subtracting 3 from both sides we get

m + 3 -3 = 15 - 3

Simplify:

m = 12

Therefore, the number of multiple-choice questions are on the test is; 12

Answer 2

m + f = 15 . . . (1)
3m + 5f = 51 . . . (2)

(1) * (5) => 5m + 5f = 75 . . . (3)
(3) - (2) => 2m = 24
m = 24/2 = 12

Therefore, there are 12 multiple choice question on the test.