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. How many points was each part worth?

Answer 1

Let \( a \) represent the number of true/false questions in Part A, and let \( b \) represent the number of multiple choice questions in Part B. The total points for Part A can be represented as:

\[
\text{Points for Part A} = 6a
\]

And for Part B, the total points can be expressed as:

\[
\text{Points for Part B} = 4b
\]

We know that the points for Part A and Part B are equal:

\[
6a = 4b
\]

To simplify this equation, we can divide both sides by 2:

\[
3a = 2b
\]

This can be rearranged to express \( b \) in terms of \( a \):

\[
b = \frac{3}{2}a
\]

Since \( b \) must be a whole number (as it represents the number of questions), \( a \) must be even. Let \( a = 2k \) for some integer \( k \). Then we substitute into the equation for \( b \):

\[
b = \frac{3}{2}(2k) = 3k
\]

Now substituting back to find the points, we calculate the total points for Part A and Part B. The points from Part A are:

\[
6a = 6(2k) = 12k
\]

The points from Part B are:

\[
4b = 4(3k) = 12k
\]

Both parts yield \( 12k \) points. To find the least number of points for which this is possible, we choose \( k = 1 \):

\[
\text{Points for each part} = 12(1) = 12
\]

Thus, both Part A and Part B are worth:

\[
\boxed{12} \text{ points}
\]

To summarize, Part A has \( a = 2 \) true/false questions (worth \( 12 \) points total), and Part B has \( b = 3 \) multiple choice questions (also worth \( 12 \) points total).