Answer
(a) The system that models the situation is:
$m+r=20
\\2m+6r=60$
(b) There are $15$ multiple choice questions and $5$ extended response questions.
Work Step by Step
(a)
Let
$m =$ number of multiple choice questions
$r =$ number of extended response questions.
Knowing a total of $20$ questions:
$m + r = 20$ (Equation 1)
Knowing 60 minutes taken, with each multiple choice taking 2 minutes, and extended response taking 6 minutes.
$2m + 6r = 60$ (Equation 2)
Thus, the system of equations that model the relationship between the number of multiple choice questions and the number of extended-response questions is:
$m+r=20
\\2m+6r=60$
Solve for $m$ in Equation 1 by subtracting $r$ to both sides:
$m = 20 - r$
Substitute $20 - r$ to $m$ in Equation 2:
$2m + 6r=60
\\2(20-r) + 6r = 60$
Distribute 2:
$40 - 2r + 6r = 60$
Collect like terms:
$40 + 4r = 60$
Subtract $40$ to both sides:
$4r = 20$
Divide both sides by $4$ :
$r = 5$
Substitute $r = 5$ into Equation 1:
$m+r=20
\\m +5 = 20
\\m=20-5
\\m = 15$
Thus, $m = 15$ , $r = 5$
(b)
Therefore, there are:
$15$ multiple choice questions
$5$ extended response questions.
