Question

An instructor wants to write a test with 25 questions where each question is worth 3, 4, or 5 points based on difficulty. He wants the number of 3-point questions to be 4 less than the number of 4-point questions, and he wants the quiz to be worth a total of 100 points. How many 3, 4, and 5 point questions are there?

Answer 1

3-point questions = 7

4-point questions = 11

5-point questions = 7

Explanation:

Let x, y and z be number of 3-point, 4-point and 5-point questions respectively.

We are told that number of total questions in the quiz is 25.

`x+y+z=25...(1)`

We are also told that number of 3-point questions is 4 less than the number of 4-point questions.

`y=x+4...(2)`

The quiz is worth 100 points.

`3x+4y+5z=100...(3)`

Substituting equation 2 in equation 1 we will get,

`x+x+4+z=25`

`2x+z=25-4`

`z=21-2x`

Upon substituting value of y and z in equation 3 we will get,

`3x+4(x+4)+5(21-2x)=100`

`3x+4x+16+105-10x=100`

`7x+121-10x=100`

`7x-10x=100-121`

`-3x=-21`

`x=(21)/(3) =7`

Therefore, number of 3-point questions is 7.

Now we will find y by substituting x=7 in equation 2.

`y=7+4=11`

Therefore, number of 4-point questions will be 11.

Now we will substitute value of x and y in equation 1 to find our z.

`7+11+z=25`

`18+z=25`

`z=25-18=7`

Therefore, number of 5-point questions is 7.