To determine the location on the ribbon where Genevieve should make her cut, we'll follow a structured approach based on the information and steps provided in the problem statement.
1. Understand the Ribbon Length:
- The total length of the ribbon is 60 inches.
- However, 2 inches of the ribbon are frayed at one end, so we effectively have \( 60 - 2 = 58 \) inches of usable ribbon.
2. Identify the Ratio:
- The desired cutting ratio is 2:3.
3. Set Up the Formula:
- The formula to find the cutting point along the ribbon is:
[tex]\[
x = \left( \frac{m}{m+n} \right) (x_2 - x_1) + x_1
\][/tex]
- Here, \( m = 2 \) (the first part of the 2:3 ratio) and \( n = 3 \) (the second part of the 2:3 ratio).
- \( x_1 = 2 \) (the starting point marked from the frayed end).
- \( x_2 = 60 \) (the total length of the ribbon).
4. Calculate the Cutting Point:
- Plug the values into the formula:
[tex]\[
x = \left( \frac{2}{2+3} \right) (60 - 2) + 2
\][/tex]
- Simplify the fraction \( \left( \frac{2}{5} \right) \):
[tex]\[
x = \left( \frac{2}{5} \right) \times 58 + 2
\][/tex]
- Perform the multiplication:
[tex]\[
\left( \frac{2}{5} \right) \times 58 = 23.2
\][/tex]
- Add 2 to the result:
[tex]\[
x = 23.2 + 2 = 25.2
\][/tex]
Therefore, the cut should be made 25.2 inches from the starting point marked 2 inches from the frayed end. Rounding to the nearest tenth, we still get 25.2 inches as the location for the cut. Hence, the answer is 25.2 inches.
