Question

The harmonic mean of two numbers, a and b, equals 2/1/a+1/b. As you vary the length of a violin or guitar string, its pitch changes. If a full length string is 1 unit long, then many lengths that are simple fractions produce pitches that harmonize, or sound pleasing together. The harmonic mean relates two lengths that produce harminous sounds. Find the harmonic mean for the string lengths of 1 and 1/2.
A.3/2
B.2/3
C.2/5
D.1

Answer 1

The harmonic mean of \(\rm a\) and \(\rm b\) is \(\rm \dfrac{2}{\frac1a+\frac1b}\)

So then the harmonic mean of \(\rm \color{orangered}{1}\) and \(\rm \color{royalblue}{\frac12}\) is \(\large\rm \dfrac{2}{\frac{1}{\color{orangered}{1}}+\frac{1}{\color{royalblue}{1/2}}}\)

We have some scary fraction business going on here, ya? :d

Answer 2

When you divide by a fraction,
you can rewrite it as multiplication by the reciprocal of the second fraction,
1/(1/2) = 1*(2/1) = 2

\[\large\rm \dfrac{2}{\frac{1}{\color{orangered}{1}}+\frac{1}{\color{royalblue}{1/2}}}\qquad=\qquad \frac{2}{\frac11+2}\]Simplify a little further to get your answer.