To solve the problem of finding the ratio \(a:b\) given that the geometric mean (G) and harmonic mean (H) of two numbers are \(16\) and \(\frac{64}{5}\) respectively, we need to follow these steps:
- Recall the formula for the geometric mean of two numbers \(a\) and \(b\): \(G = \sqrt{ab}\).
- Using the given value of the geometric mean: \(\sqrt{ab} = 16\), square both sides to find \(ab\): \(ab = 16^2 = 256\).
- Recall the formula for the harmonic mean of two numbers \(a\) and \(b\): \(H = \frac{2ab}{a+b}\).
- Using the given value of the harmonic mean: \(\frac{64}{5} = \frac{2ab}{a+b}\). Substituting \(ab = 256\) into this equation, we get: \(\frac{64}{5} = \frac{2 \times 256}{a+b}\).
- Simplify the equation: \(\frac{64}{5} = \frac{512}{a+b}\).
- Cross-multiply to solve for \(a+b\): \(64(a+b) = 512 \times 5\).
- Calculate: \(64(a+b) = 2560\).
- Divide both sides by 64 to find \(a+b\): \(a+b = \frac{2560}{64} = 40\).
- We now have two equations:
- Let \(a = 40 - b\). Substitute \(a = 40 - b\) into the equation \(ab = 256\): \((40 - b)b = 256\).
- Expand the equation: \(40b - b^2 = 256\).
- Rearrange into standard quadratic form: \(b^2 - 40b + 256 = 0\).
- Use the quadratic formula: \(b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}\), where \(A = 1\), \(B = -40\), \(C = 256\).
- Calculate: \(b = \frac{40 \pm \sqrt{1600 - 1024}}{2} = \frac{40 \pm \sqrt{576}}{2} = \frac{40 \pm 24}{2}\).
- We get two values for \(b\):
- \(b = \frac{64}{2} = 32\)
- \(b = \frac{16}{2} = 8\)
- If \(b = 32\), then \(a = 40 - 32 = 8\). Thus, \(a:b = 8:32 = 1:4\).
- Alternatively, If \(b = 8\), then \(a = 40 - 8 = 32\). Thus, \(a:b = 32:8 = 4:1\).
Therefore, the ratio \(a:b\) is \(4:1\), which is the correct answer.