A test has twenty questions worth 100 marks in total. There are two types of questions. Multiple choice questions are worth 3 marks each and essay questions are worth 11 marks each. How many multiple choice questions does the exam have?
15
This problem requires us to determine the number of multiple choice questions (MCQs) in a test, given the total number of questions, total marks, and the marks awarded for each type of question.
Setting Up the Equations for the Test
Let's define the variables:
- Let $m$ represent the number of multiple choice questions.
- Let $e$ represent the number of essay questions.
We are given the following information:
- The total number of questions is 20.
- Each MCQ is worth 3 marks.
- Each essay question is worth 11 marks.
- The total marks for the test is 100.
From this information, we can set up two equations:
- Equation for the total number of questions:
$$m + e = 20$$
- Equation for the total marks:
$$3m + 11e = 100$$
Solving for the Number of Multiple Choice Questions
We can use the first equation to express $e$ in terms of $m$:
$$e = 20 - m$$
Now, substitute this expression for $e$ into the second equation:
$$3m + 11(20 - m) = 100$$
Distribute the 11:
$$3m + 220 - 11m = 100$$
Combine the terms involving $m$:
$$(3 - 11)m + 220 = 100$$
$$-8m + 220 = 100$$
Subtract 220 from both sides of the equation:
$$-8m = 100 - 220$$
$$-8m = -120$$
Divide both sides by -8 to solve for $m$:
$$m = \frac{-120}{-8}$$
$$m = 15$$
Verification
To ensure our answer is correct, let's find the number of essay questions and check if the total marks match.
Number of essay questions, $e = 20 - m = 20 - 15 = 5$.
Total marks = (Number of MCQs × Marks per MCQ) + (Number of Essay Questions × Marks per Essay Question)
Total marks = ($$15 \times 3$$) + ($$5 \times 11$$)
Total marks = $$45 + 55$$
Total marks = $$100$$
The calculated total marks match the given total marks, confirming our result.
Conclusion
Therefore, the exam has 15 multiple choice questions.
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