If A, g and H are respectively arithmetic ,
geometric and harmomic means between a and b
both being unequal and positive, then
`A = (a + b)/(2) rArr a + b = 2A`
`G = sqrt(ab) rArr G^(2) = ab `
`H = (2ab)/(a+ b ) rArr G^(2) = AH`
On the basis of above information answer the following questions .
If the geometric mean and harmonic means of two number are `16` and ` 12(4)/(5)` , then the ratio of one number to the other is

Options

A. `1: 4`
B. `2:4`
C. `1:3`
D. `2:3`

Answer

To find the ratio of one number to the other given the geometric mean (G) and harmonic mean (H), we can follow these steps:

### Step 1: Use the Geometric Mean
Given that the geometric mean \( G = 16 \), we can express this as:
\[ G = \sqrt{ab} \implies \sqrt{ab} = 16 \]
Squaring both sides gives:
\[ ab = 16^2 = 256 \]

### Step 2: Use the Harmonic Mean
The harmonic mean \( H = 12 \frac{4}{5} \) can be converted to an improper fraction:
\[ H = 12 \frac{4}{5} = \frac{60 + 4}{5} = \frac{64}{5} \]
The formula for the harmonic mean is:
\[ H = \frac{2ab}{a + b} \]
Substituting \( H \) and \( ab \) into the equation gives:
\[ \frac{64}{5} = \frac{2 \cdot 256}{a + b} \]

### Step 3: Solve for \( a + b \)
Cross-multiplying to solve for \( a + b \):
\[ 64(a + b) = 5 \cdot 512 \]
Calculating \( 5 \cdot 512 \):
\[ 5 \cdot 512 = 2560 \]
Thus:
\[ 64(a + b) = 2560 \]
Dividing both sides by 64:
\[ a + b = \frac{2560}{64} = 40 \]

### Step 4: Forming the Equations
Now we have two equations:
1. \( ab = 256 \)
2. \( a + b = 40 \)

Let \( a \) and \( b \) be the roots of the quadratic equation \( x^2 - (a + b)x + ab = 0 \):
\[ x^2 - 40x + 256 = 0 \]

### Step 5: Solving the Quadratic Equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = -40, c = 256 \):
\[ x = \frac{40 \pm \sqrt{(-40)^2 - 4 \cdot 1 \cdot 256}}{2 \cdot 1} \]
Calculating the discriminant:
\[ (-40)^2 - 4 \cdot 1 \cdot 256 = 1600 - 1024 = 576 \]
Thus:
\[ x = \frac{40 \pm \sqrt{576}}{2} = \frac{40 \pm 24}{2} \]
Calculating the two possible values:
1. \( x = \frac{64}{2} = 32 \)
2. \( x = \frac{16}{2} = 8 \)

### Step 6: Finding the Ratio
Now we have \( a = 32 \) and \( b = 8 \). The ratio \( \frac{a}{b} \) is:
\[ \frac{a}{b} = \frac{32}{8} = 4 \]
Thus, the ratio of one number to the other is:
\[ \text{Ratio } a : b = 32 : 8 = 4 : 1 \]

### Final Answer
The ratio of one number to the other is \( 1 : 4 \).

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