To determine the point where Genevieve should cut the ribbon given the frayed part and the desired ratio, we need to follow a series of steps to implement the ratio mathematically. Let's walk through them:
1. Understand the Ratio and Lengths:
- Total ribbon length is 60 inches.
- The ratio given is [tex]\(2:3\)[/tex].
- The frayed part is 2 inches from one end.
2. Calculate the Total Parts of the Ratio:
- Combine the parts of the ratio: [tex]\(2 + 3 = 5\)[/tex].
3. Calculate the Effective Length of the Ribbon:
- Since the first 2 inches are frayed, the effective length of the ribbon to be cut is:
[tex]\( \text{Effective length} = 60 - 2 = 58 \)[/tex] inches.
4. Determine the Cut Point:
- To locate the cut point [tex]\(x\)[/tex], use the formula for dividing the ribbon in the given ratio, modified to account for the starting point instead of starting from zero:
[tex]\[
x = \left(\frac{2}{2+3}\right) \times (58) + 2
\][/tex]
5. Compute the Cut Point Numerically:
- Start with:
[tex]\[
x = \left(\frac{2}{5}\right) \times 58 + 2
\][/tex]
- Calculate the fraction:
[tex]\[
\frac{2}{5} = 0.4
\][/tex]
- Multiply with the effective length:
[tex]\[
0.4 \times 58 = 23.2
\][/tex]
- Add back the frayed part:
[tex]\[
x = 23.2 + 2 = 25.2
\][/tex]
6. Round to the Nearest Tenth:
- The calculated cut point is 25.2, which is already rounded to the nearest tenth.
Thus, Genevieve should make her cut at 25.2 inches on the ribbon. The correct answer is:
25.2 in.
