Fourth Proportional
In the proportion \[ \frac{a}{b} = \frac{c}{x} \] the number \(x \) is called the fourth proportional to the numbers \(a, \;b, \;c \).
Question 1
Let \( x \) be the fourth proportional to the numbers \( 3, 4, 5 \), and let \( y \) be the geometric mean of \(x\) and 15. Based on this, what is the harmonic mean of \( y \) and 15?
\[
\text{A)} 11 \quad
\text{B) } 12 \quad
\text{C) } 13 \quad
\text{D) } 14 \quad
\text{E) } 15
\]
Solution:
\[ \frac{3}{4} = \frac{5}{x} \Rightarrow x = \frac{20}{3} \]
\[ y = \sqrt{ 15x} = \sqrt{ 15 \cdot \frac{20}{3} } = 10 \]
Accordingly, the harmonic mean of y and 15 is,
\[ \frac{ 2 \cdot 15 \cdot y }{ 15 + y } = \frac{2 \cdot 15 \cdot 10 }{15 +10 } = 12 \]
\(\textbf{Answer: B} \)
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