Gretchen and Ezia received equal scores on a test made up of multiple choice questions and an essay. Gretchen got 18 multiple choice questions correct and received 19 points for her essay. Ezia got 15 multiple choice questions correct and received 31 points for her essay.

Question

How many points was each multiple choice question worth?
Enter your answer in the box.

Answer 1

Let the number of points for each multiple choice question be \( x \).

Gretchen's total score can be calculated as follows: \[ \text{Gretchen's Score} = 18x + 19 \]

Ezia's total score can be calculated as: \[ \text{Ezia's Score} = 15x + 31 \]

Since both received equal scores, we can set the two expressions equal to each other: \[ 18x + 19 = 15x + 31 \]

To solve for \( x \), first, we will get all terms involving \( x \) on one side and constant terms on the other side: \[ 18x - 15x = 31 - 19 \] \[ 3x = 12 \]

Now, divide by 3: \[ x = 4 \]

Thus, the points for each multiple choice question is \( \boxed{4} \).

To verify, we can calculate both Gretchen's and Ezia's scores using \( x = 4 \):

Gretchen's score: \[ 18 \times 4 + 19 = 72 + 19 = 91 \]

Ezia's score: \[ 15 \times 4 + 31 = 60 + 31 = 91 \]

Both scores are equal, confirming that our solution is correct. Each multiple choice question is worth \( \boxed{4} \) points.