Answer :
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.
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.
