A 24 question mathematics exam is worth 100 points. There are fill-in-the-blank questions worth 2 points each, multiple choice questions worth 6 points each, and long answer questions worth 5 points each. The number of fill-in-the-blank questions plus the number of multiple choice questions is equal to the number of long answer questions. How many of each question is on the exam?

If there are

x fill-in
y multiple-choice
z long-answer

they have given us the following facts:

x+y+z = 24
2x+6y+5z = 100
x+y = z

Now you can solve for x,y,z

Let's break down the problem into variables and equations.

Let's assume the number of fill-in-the-blank questions is "x."
Let's assume the number of multiple-choice questions is "y."
Let's assume the number of long-answer questions is "z."

Given that x + y = z (the number of fill-in-the-blank questions plus the number of multiple-choice questions is equal to the number of long-answer questions).

We also know that:
The score for fill-in-the-blank questions is 2 points each.
The score for multiple-choice questions is 6 points each.
The score for long-answer questions is 5 points each.

Since the exam is worth 100 points, we can set up the following equation:
2x + 6y + 5z = 100.

Now we have two equations:
x + y = z,
2x + 6y + 5z = 100.

To solve this system of equations, we can either use substitution or elimination.

Let's use substitution:
Rearrange the first equation to solve for z in terms of x and y:
z = x + y.

Substitute this value of z into the second equation:
2x + 6y + 5(x + y) = 100.

Simplify and solve for x and y:
2x + 6y + 5x + 5y = 100,
7x + 11y = 100.

Since there are no fractional solutions, start by trying different values for y and solve for x.

Let's try y = 2:
7x + 11(2) = 100,
7x + 22 = 100,
7x = 78,
x = 11.14.

Since x is not a whole number, let's try another value for y.

Let's try y = 3:
7x + 11(3) = 100,
7x + 33 = 100,
7x = 67,
x ≈ 9.57.

Again, x is not a whole number. Let's try y = 4.

7x + 11(4) = 100,
7x + 44 = 100,
7x = 56,
x = 8.

Now we have a whole number solution for x.

Substitute this value of x back into the equation z = x + y:
z = 8 + y, or z = y + 8.

To check if these values satisfy the given equations, substitute them into the second equation:
2x + 6y + 5z = 100,
2(8) + 6y + 5(y + 8) = 100,
16 + 6y + 5y + 40 = 100,
11y = 44,
y = 4.

Now that we have the values of x and y, substitute them back into the first equation to find z:
x + y = z,
8 + 4 = z,
z = 12.

Therefore, the exam has 8 fill-in-the-blank questions, 4 multiple-choice questions, and 12 long-answer questions.