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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 \((5.67, 10.33)\).

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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.

Carol and Marita are hiking in a state park and decide to take separate trails. The map of the park is set up on a coordinate grid. Carol's location is at the point (7, 13) and Marita is at (3, 5). a. Find the coordinates of the point midway between their locations. b. Find the coordinates of the point 1/3 the distance from Carol to Marita.

Solve problems involving finding the midpoint and a point one-third the distance between two locations on a coordinate grid. This problem covers key concepts in coordinate geometry, including the midpoint formula and section formula. Suitable for students in Grades 8-10.

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