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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?

Respuesta :

Answer:

3-point questions = 7

4-point questions = 11

5-point questions = 7

Step-by-step 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.

[tex]x+y+z=25...(1)[/tex]

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

[tex]y=x+4...(2)[/tex]

The quiz is worth 100 points.

[tex]3x+4y+5z=100...(3)[/tex]

Substituting equation 2 in equation 1 we will get,

[tex]x+x+4+z=25[/tex]

[tex]2x+z=25-4[/tex]

[tex]z=21-2x[/tex]

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

[tex]3x+4(x+4)+5(21-2x)=100[/tex]

[tex]3x+4x+16+105-10x=100[/tex]

[tex]7x+121-10x=100[/tex]

[tex]7x-10x=100-121[/tex]

[tex]-3x=-21[/tex]

[tex]x=\frac{21}{3} =7[/tex]

Therefore, number of 3-point questions is 7.

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

[tex]y=7+4=11[/tex]

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.

[tex]7+11+z=25[/tex]

[tex]18+z=25[/tex]

[tex]z=25-18=7[/tex]

Therefore, number of 5-point questions is 7.  

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