Question

Ms. Sanders wrote a test. Part A had true/false questions, each worth 6 points. Part B had multiple choice questions, each worth 4 points. She made the number of points for Part A equal the number of points for Part B. It was the least number of points for which this was possible.
Answer the following questions.

Answer 1

To find the least number of points for which the total points for Part A (true/false questions) is equal to the total points for Part B (multiple choice questions), we start by defining the variables:

Let:
- \( a \) = the number of true/false questions
- \( b \) = the number of multiple choice questions

The total points for Part A is given by:
\[
\text{Points for Part A} = 6a
\]
The total points for Part B is given by:
\[
\text{Points for Part B} = 4b
\]

Setting the total points equal for both parts gives us the equation:
\[
6a = 4b
\]

To simplify, we can divide both sides of the equation by 2:
\[
3a = 2b
\]

We can rearrange this to express \( b \) in terms of \( a \):
\[
b = \frac{3}{2}a
\]

This shows that \( b \) must be a multiple of \( 3 \) to ensure \( a \) is an integer. Thus, we can set \( a = 2k \) for some integer \( k \), which gives us:
\[
b = \frac{3}{2}(2k) = 3k
\]

Now, we can express the total points in terms of \( k \):
\[
\text{Total points} = 6a = 6(2k) = 12k
\]
\[
\text{Total points} = 4b = 4(3k) = 12k
\]

Thus, the total points is \( 12k \). Since we are looking for the least number of points, we set \( k = 1 \):
\[
\text{Total points} = 12(1) = 12
\]

Now, substituting \( k = 1 \) back to find \( a \) and \( b \):
\[
a = 2k = 2(1) = 2
\]
\[
b = 3k = 3(1) = 3
\]

Thus, the least number of points for which the total points for Part A is equal to the total points for Part B is:
\[
\text{Total points} = 12
\]

So, the answers to your questions are:
- The total points for Part A = 12 points.
- The total points for Part B = 12 points.
- The least number of questions that Ms. Sanders made in Part A = 2 (true/false questions).
- The least number of questions that Ms. Sanders made in Part B = 3 (multiple choice questions).