The problem you shared involves two points on a coordinate grid: Carol's location at $(7, 13)$ and Marita's location at $(3, 5)$. Let's solve each part of the question:

Part (a): Finding the Midpoint

The midpoint between two points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated using the midpoint formula: $\text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Substituting the given coordinates: $\text{Midpoint} = \left(\frac{7 + 3}{2}, \frac{13 + 5}{2}\right)$ $\text{Midpoint} = \left(\frac{10}{2}, \frac{18}{2}\right)$ $\text{Midpoint} = (5, 9)$

So, the coordinates of the midpoint between Carol's and Marita's locations are $(5, 9)$.

Part (b): Finding the Point 1/3 the Distance from Carol to Marita

To find the point that is $\frac{1}{3}$ the distance from Carol to Marita, we use the section formula. If a point divides the line segment joining $(x_1, y_1)$ and $(x_2, y_2)$ in the ratio $m:n$, the coordinates are given by: $\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)$

For $\frac{1}{3}$ the distance from Carol to Marita, the ratio $m:n$ is $1:2$ (since 1 part to Carol and 2 parts to Marita).

Substituting the values: $\text{Point} = \left(\frac{1 \times 3 + 2 \times 7}{1+2}, \frac{1 \times 5 + 2 \times 13}{1+2}\right)$ $\text{Point} = \left(\frac{3 + 14}{3}, \frac{5 + 26}{3}\right)$ $\text{Point} = \left(\frac{17}{3}, \frac{31}{3}\right)$ $\text{Point} = \left(5.67, 10.33\right)$

So, the coordinates of the point that is $\frac{1}{3}$ the distance from Carol to Marita are approximately $(5.67, 10.33)$.

Would you like more details or have any questions about these calculations?

Here are 5 related questions you might find interesting:

1. How do you derive the formula for the midpoint of two points on a coordinate plane?
2. What is the geometric interpretation of the section formula?
3. How can the concept of dividing a line segment be applied in real-world scenarios?
4. How would the midpoint formula change if the coordinate plane were three-dimensional?
5. How does the distance formula relate to finding points along a line segment?

Tip: When dealing with ratios in geometry, always ensure you correctly understand which segment or part you are measuring from. This helps avoid mistakes, especially when dividing segments in non-trivial ratios.