Let's walk through the problem step-by-step to determine where Genevieve should make her cut on the 60-inch ribbon, given the ratio of [tex]\(2:3\)[/tex] and the 2 inches of frayed ribbon at one end.
Firstly, we need to account for the frayed 2 inches at the start:
1. The original length of the ribbon is 60 inches.
2. The 2 frayed inches mean we effectively start at the 2-inch mark.
3. Therefore, the usable part of the ribbon is from 2 inches to 60 inches.
4. To find the length of the usable part:
[tex]\[ 60 \, \text{inches} - 2 \, \text{inches} = 58 \, \text{inches} \][/tex]
Now, we need to divide this 58-inch length into a ratio of [tex]\(2:3\)[/tex]. We are given that the cut should divide the total usable length according to the ratio of [tex]\(2:3\)[/tex].
The formula to find where to cut, given a ratio and the start point, is:
[tex]\[ x = \left( \frac{m}{m+n} \right) \left( x_2 - x_1 \right) + x_1 \][/tex]
Where:
- [tex]\( m = 2 \)[/tex]
- [tex]\( n = 3 \)[/tex]
- [tex]\( x_1 = 2 \, \text{inches} \)[/tex] (start point)
- [tex]\( x_2 = 60 \, \text{inches} \)[/tex] (end point)
First, subtract the start point from the end point to get the usable length:
[tex]\[ 58 \, \text{inches} = 60 \, \text{inches} - 2 \, \text{inches} \][/tex]
Next, plug in the given values into the formula:
[tex]\[ x = \left( \frac{2}{2+3} \right) \times 58 + 2 \][/tex]
[tex]\[ x = \left( \frac{2}{5} \right) \times 58 + 2 \][/tex]
[tex]\[ x = 0.4 \times 58 + 2 \][/tex]
[tex]\[ x = 23.2 + 2 \][/tex]
[tex]\[ x = 25.2 \][/tex]
So, the cut should be made at the [tex]\(25.2 \, \text{inches}\)[/tex] mark on the ribbon.
Therefore, Genevieve should make her cut at:
[tex]\[ \boxed{25.2 \, \text{inches}} \][/tex]
The correct answer is:
[tex]\[ 25.2 \, \text{inches} \][/tex]
