Math problem.?

Question

Someone asked me this question, and I didn't get it. I really want to know. Please help.

You are taking a test in which matching questions are worth 10 points and fill in questions are worth 15 points. It takes 3 minutes for each matching question and 5 minutes for each fill in question. The total time allowed is 60 minutes and you do exactly 18 questions. your score is 195 points by using the entire 60 minutes. Find how many questions of each type you answered correctly.

How many matching questions did you answer correctly?_____________
How many fill in questions did you answer correctly?______________

Answer 1

The previous answer has the correct solution, however the method used is flawed. (The variables in the third equation used above cannot be assumed to be the same as the variables in the first two equations. It just so happened to turn out that way, as you got all the questions you answered correct as shown below. To see that this is true, try the problem again with everything the same except you score 180.)

Let M be the number of matching questions answered.
and F be the number of fill in questions answered.

M+F = 18
3M + 5F = 60

multiply the first equation by 3:

3M + 3F = 54
3M + 5F = 60

subtract the first equation from the second:

2F = 6
F = 3, and since M+F=18,
M = 15

Let m be the number of matching questions that are correct.
Let f be the number of fill in questions that are correct.

15f + 10m = 195

f could only be 0,1,2 or 3

f can't be 0, since then 10m=195 or m = 19.5 and m must be an integer
f can't be 1, since then 10m = 180, or m = 18. But m must be ≤ 15.
f can't be 2, since then 10m = 165 or m = 16.5 and m must be an integer.

So f = 3 and 10m = 150 or m = 15.
You got all the questions you answered correct.

Answer 2

Let x be the number of matching questions
and y be the number of fill in questions.

Although there are only two variables, I could setup three equations from the questions.

x + y = 18 (total number of questions)
3x + 5y = 60 (total time)
10x + 15y = 195 (total points)

use any two equations, solve for x and y

10x + 15y = 195
9x + 15y = 180
x = 15
y = 3

Answer 3

Ans. Matching questions=15
Fill in questions= 3