Answer:
(a) 7 essays and 29 multiple questions
(b) Your friend is incorrect
Step-by-step explanation:
Represent multiple choice with M and essay with E.
So:
[tex]M + E= 36[/tex] --- Number of questions
[tex]2M + 6E = 100[/tex] --- Points
Solving (a): Number of question of each type.
Make E the subject of formula in [tex]M + E= 36[/tex]
[tex]E = 36 - M[/tex]
Substitute 36 - M for E in [tex]2M + 6E = 100[/tex]
[tex]2M + 6(36 - M) = 100[/tex]
[tex]2M + 216 - 6M = 100[/tex]
Collect Like Terms
[tex]2M - 6M = 100 - 216[/tex]
[tex]-4M = - 116[/tex]
Divide both sides by -4
[tex]M = \frac{-116}{-4}[/tex]
[tex]M = 29[/tex]
Substitute 29 for M in [tex]E = 36 - M[/tex]
[tex]E = 36 - 29[/tex]
[tex]E = 7[/tex]
Solving (b): Can the multiple questions worth 4 points each?
It is not possible.
See explanation.
If multiple question worth 4 points each, then
[tex]2M + 6E = 100[/tex] would be:
[tex]4M + xE = 100[/tex]
Where x represents the number of points for essay questions.
Substitute 7 for E and 29 for M.
[tex]4 * 29 + x * 7 = 100[/tex]
[tex]116 + 7x = 100[/tex]
Subtract 116 from both sides
[tex]116-116 + 7x = 100 -116[/tex]
[tex]7x = 100-116[/tex]
[tex]7x = -16[/tex]
Make x the subject
[tex]x = -\frac{16}{7}[/tex]
Since the essay question can not have worth negative points.
Then, it is impossible to have the multiple questions worth 4 points
Your friend is incorrect.
The test contained 29 multiple choice questions and 7 essay questions.
Let x represent the number of multiple questions and y represent the number of essay questions.
Given that there are a total of 36 questions, hence:
x + y = 36 (1)
Also, the test is worth a total of 100 points, hence:
2x + 6y = 100 (2)
Solving equation 1 and 2 simultaneously gives:
x = 29, y = 7
Therefore the test contained 29 multiple choice questions and 7 essay questions.
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