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).
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