Question

Mr. Martin’s math test, which is worth 100 points, has 29 problems. Each problem is worth either 5 points or 2 points.

Let x be the number of questions worth 5 points and let y be the number of questions worth 2 points.

x + y = 29, 5x + 2y = 100

How many problems of each point value are on the test?

10 problems worth 5 points and 25 problems worth 2 points
14 problems worth 5 points and 15 problems worth 2 points
16 problems worth 5 points and 13 problems worth 2 points
15 problems worth 5 points and 13 problems worth 2 points

Answer 1

We are given the following equations based on the problem statement:

1. \( x + y = 29 \) (the total number of problems)
2. \( 5x + 2y = 100 \) (the total points)

We can solve this system of equations step by step.

First, from the first equation, we can express \( y \) in terms of \( x \):
\[
y = 29 - x
\]

Now, we substitute \( y \) into the second equation:
\[
5x + 2(29 - x) = 100
\]
Simplifying the equation:
\[
5x + 58 - 2x = 100
\]
\[
3x + 58 = 100
\]
Subtracting \( 58 \) from both sides:
\[
3x = 42
\]
Dividing by \( 3 \):
\[
x = 14
\]

Now, we can find \( y \):
\[
y = 29 - x = 29 - 14 = 15
\]

Therefore, there are \( 14 \) problems worth \( 5 \) points and \( 15 \) problems worth \( 2 \) points.

So the answer is:
**14 problems worth 5 points and 15 problems worth 2 points.**