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

Question

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.