What would the equations be in this?

Question

There are 24 questions on a test. each question is worth either 4 points or 5 points. The total number of points is 100. How many type of questions are on the test?

Answer 1

Let:

   x = number of 4-point questions

   y = number of 5-point questions

 

Translate:

"There are 24 questions"     means      x + y = 24     [eq1]

 

"The total number of points is 100"    means

   (points for 4-pt questions) + (points for 5-point questions) = 100

                4x                       +           5y                            = 100    [eq2]

 

You may solve by substitution or elimination.  Let's use substitution.

Solve eq1 for one of the variables, say x:

      x = 24 - y

Put that in place of x  in eq2:

     4(24 - y) + 5y = 100

     96 - 4y + 5y = 100                    [distribute]

     96 + y = 100                   [collect terms]

      y = 4                        [subtract 96 from both sides]

 

Use that value in either equation to solve for y (eq1 is easier):

     x + y = 24     [eq1]

     x + 4 = 24

    x = 20                [subtract 4 from both sides]

 

Checking (very important):

  Does x + y = 24   ?

        20 + 4 = 24   ? 

          24 = 24   ?yes

  Does 4x + 5y = 100   ?

        4*20 + 5*4 = 100   ?

            80 + 20 = 100   ?

              100 = 100   ?yes