Your teacher is giving you a test worth 100 points containing 40 questions. There are 2-point and 4-point questions on the test. How many of each type of question are on the test?

Two things are important to realize in this problem:

• There are 40 questions on the test, each worth two or four points.
• The test is worth 100 points.

The first thing we must do to solve the problem is give a variable to our unknowns. We do not know how many questions are on the test - specifically, how many two and four point questions. Let's call the number of two point questions #t# and the number of four point questions #f#. We know that the total number of questions is 40, so:
#t+f=40#
That is, the number of two point questions plus the number of four point questions gives us the total number of questions, which is 40.

We also know that the test is worth 100 points, so:
#2t+4f=100#
This is to say that the number of 2 point questions you get right times 2, plus the number of 4 point questions you get right times 4, is the total number of points - and the maximum you can get is 100.

We now have a system of equations:
#t+f=40#
#2t+4f=100#

I've decided to solve this system through substitution, but you could solve it by graphing and should get the same result. Begin by solving for either variable in the first equation (I solved for #t#):
#t=40-f#

Now plug this in for #t# in the second equation:
#2t+4f=100#
#2(40-f)+4f=100#

And solve for #f#:
#80-2f+4f=100#
#2f=20#
#f=10#

The number of four point questions is #10#. The number of two point questions can be determined from #t=40-f#:
#t=40-f#
#t=40-10=30#

So there are 10 four point questions and 30 two point questions.