Standing wave question

Hi everyone,
I just have a question about the correct procedure to follow to complete a question.

The question is: A source of frequency of 60 Hz is used to make waves in a rope 3.0m long. It takes 0.10 s for the waves to travel from one fixed end of the rope to the other. How many loops are in the standing wave in the rope?

I got the same answer as the textbook, however, the procedure to get my answer was different.

The textbook did: v=fλ=(60Hz)x(3.0m)=180m/s
v=Δd/Δt or Δd=v(Δt)=180m/s(0.10s)=18m
Since each wavelength is 3.0m long, the number of wavelengths is 18m/3m=6
Because there are two loops for every wavelength, the are 12 loops.

The way I did it: v=Δd/Δt=(3m)/(0.10s)=30m/s
λ=v/f=(30m/s)/(60Hz)=0.5m
Because the string is 3m long, 0.5mx __ =3.0m
3.0m/0.5m=6 Thus, there are 6 wavelengths in the 3.0m of string but because 1 wavelength = 2 loops, there are 12 loops.

So, I have two questions

1) Is the way I solved this question accurate and reliable? Or did I just get lucky that this method happened to work with these numbers.

2) The question states that the string is 3.0m long. Then the textbook stated it as the wavelength (v=fλ=(60Hz)x(3.0m)=180m/s). Can someone please help me understand why this is okay? I thought it was the length of the string or distance from the start to the end.

Thank you!
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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