To determine where Genevieve should make her cut on the ribbon, we need to use the given length of the ribbon, the ratio in which she wants to cut it, and account for the frayed inches. Let's break down the problem step-by-step.
1. Identify the given values:
- Total length of the ribbon: [tex]\( 60 \)[/tex] inches.
- The ratio [tex]\( m:n \)[/tex] in which the ribbon needs to be cut: [tex]\( 2:3 \)[/tex].
- Frayed portion of the ribbon: [tex]\( 2 \)[/tex] inches.
2. Calculate the effective length of the ribbon:
- Since [tex]\( 2 \)[/tex] inches are frayed, the effective length of the ribbon that Genevieve can use is [tex]\( 60 - 2 = 58 \)[/tex] inches.
3. Determine the total ratio:
- The total ratio is given by [tex]\( m + n = 2 + 3 = 5 \)[/tex].
4. Use the formula to find the exact location for the cut:
- We will utilize the formula:
[tex]\[
x = \left(\frac{m}{m+n}\right) (x_2 - x_1) + x_1
\][/tex]
where:
- [tex]\( m \)[/tex]: part of the ratio that corresponds to the first piece ([tex]\( 2 \)[/tex]).
- [tex]\( n \)[/tex]: part of the ratio that corresponds to the second piece ([tex]\( 3 \)[/tex]).
- [tex]\( x_1 \)[/tex]: starting point after the frayed part ([tex]\( 2 \)[/tex] inches).
- [tex]\( x_2 \)[/tex]: end point of the ribbon ([tex]\( 60 \)[/tex] inches).
5. Plug the values into the formula:
- First, calculate the fraction:
[tex]\[
\frac{m}{m+n} = \frac{2}{5}
\][/tex]
- Then determine the effective portion we will consider:
[tex]\[
(x_2 - x_1) = (60 - 2) = 58 \text{ inches}
\][/tex]
- Substitute these into the formula:
[tex]\[
x = \left( \frac{2}{5} \right) \times 58 + 2
\][/tex]
6. Perform the calculations:
[tex]\[
x = \left( \frac{2}{5} \right) \times 58 + 2
\][/tex]
[tex]\[
x = 23.2 + 2
\][/tex]
[tex]\[
x = 25.2
\][/tex]
Therefore, after rounding to the nearest tenth, the exact location where Genevieve should make her cut is at [tex]\( \boxed{25.2} \)[/tex] inches.
