Question

on a test consisting of only true/false questions and multiple-choice questions, true/false questions are worth 2 points each and multiple-choice questions are worth 6 points each. if the test contains 20 questions and is worth a total of 100 points, which of these is a correct statement?
a. the test contains 10 multiple-choice questions.
b. the test contains 15 multiple-choice questions.
c. the test contains 10 true/false questions.
d. the test contains 15 true/false questions.

Explanation:

Step1: Define variables

Let $x$ = number of true/false questions, $y$ = number of multiple-choice questions.

Step2: Set up equations

Total questions: $x + y = 20$
Total points: $2x + 6y = 100$

Step3: Simplify point equation

Divide $2x + 6y = 100$ by 2:
$x + 3y = 50$

Step4: Solve for $y$

Subtract $x + y = 20$ from $x + 3y = 50$:
$(x + 3y) - (x + y) = 50 - 20$
$2y = 30$
$y = 15$

Step5: Solve for $x$

Substitute $y=15$ into $x + y = 20$:
$x = 20 - 15 = 5$

Step6: Verify options

Check which option matches $y=15$ (multiple-choice) and $x=5$ (true/false).

Answer:

B. The test contains 15 multiple-choice questions.