Understanding the AM-GM Ratio for Positive Numbers

Let the two positive numbers be \(a\) and \(b\). We are given information about the ratio of their Arithmetic Mean (AM) to their Geometric Mean (GM).

The Arithmetic Mean (AM) of \(a\) and \(b\) is defined as:

\( \text{AM} = \frac{a+b}{2} \)

The Geometric Mean (GM) of \(a\) and \(b\) is defined as:

\( \text{GM} = \sqrt{ab} \)

Setting up the Equation from the Given AM-GM Ratio

The question states that the ratio of the AM to the GM of these two positive numbers is 5 : 3. We can write this as an equation:

\( \frac{\text{AM}}{\text{GM}} = \frac{5}{3} \)

Substitute the formulas for AM and GM into this ratio:

\( \frac{\frac{a+b}{2}}{\sqrt{ab}} = \frac{5}{3} \)

Simplify the left side:

\( \frac{a+b}{2\sqrt{ab}} = \frac{5}{3} \)

Solving for the Ratio a : b

Now, we need to solve this equation to find the ratio \(a:b\). Let's cross-multiply:

\( 3(a+b) = 10\sqrt{ab} \)

To eliminate the square root, we square both sides of the equation:

\( (3(a+b))^2 = (10\sqrt{ab})^2 \)

\( 9(a+b)^2 = 100(ab) \)

Expand the term \((a+b)^2\) on the left side:

\( 9(a^2 + 2ab + b^2) = 100ab \)

Distribute the 9:

\( 9a^2 + 18ab + 9b^2 = 100ab \)

Move all terms to one side to form a quadratic equation in terms of \(a\) and \(b\):

\( 9a^2 + 18ab + 9b^2 - 100ab = 0 \)

\( 9a^2 - 82ab + 9b^2 = 0 \)

To find the ratio \(a:b\), we can divide the entire equation by \(b^2\) (since \(b\) is a positive number, \(b \ne 0\)):

\( \frac{9a^2}{b^2} - \frac{82ab}{b^2} + \frac{9b^2}{b^2} = 0 \)

\( 9\left(\frac{a}{b}\right)^2 - 82\left(\frac{a}{b}\right) + 9 = 0 \)

Let \(x = \frac{a}{b}\). The equation becomes a standard quadratic equation in terms of \(x\):

\( 9x^2 - 82x + 9 = 0 \)

We can solve this quadratic equation by factoring. We look for two numbers that multiply to \(9 \times 9 = 81\) and add up to \(-82\). These numbers are \(-81\) and \(-1\).

Rewrite the middle term \(-82x\) as \(-81x - x\):

\( 9x^2 - 81x - x + 9 = 0 \)

Factor by grouping:

\( 9x(x - 9) - 1(x - 9) = 0 \)

\( (9x - 1)(x - 9) = 0 \)

Set each factor equal to zero to find the possible values for \(x\):

• \( 9x - 1 = 0 \implies 9x = 1 \implies x = \frac{1}{9} \)
• \( x - 9 = 0 \implies x = 9 \)

Since \(x = \frac{a}{b}\), the possible ratios for \(a:b\) are:

• \( \frac{a}{b} = \frac{1}{9} \implies a:b = 1:9 \)
• \( \frac{a}{b} = 9 \implies \frac{a}{b} = \frac{9}{1} \implies a:b = 9:1 \)

Both of these ratios result in the same AM to GM ratio of 5:3. Looking at the given options, 9:1 is one of the choices.

Thus, the ratio \(a:b\) is equal to 9 : 1.

Revision Table: Key Concepts for AM and GM

Additional Information: Implications of AM > GM

The fact that the ratio of AM to GM is 5:3, which is greater than 1 (\(5/3 \approx 1.67\)), is consistent with the AM-GM inequality for positive numbers. The AM-GM inequality states that for non-negative numbers, the AM is always greater than or equal to the GM (\( \text{AM} \ge \text{GM} \)). Equality holds only when the two numbers are equal (\(a=b\)). Since the ratio is strictly greater than 1, we know that \(a\) and \(b\) must be different numbers.

Our solution yielded two reciprocal ratios, 9:1 and 1:9. This duality arises because the equation \( \frac{a+b}{2\sqrt{ab}} = \frac{5}{3} \) is symmetric with respect to \(a\) and \(b\). If you swap \(a\) and \(b\), the AM and GM remain the same, and thus their ratio also remains the same. If \(a:b = 9:1\), then \(b:a = 1:9\). The problem asks for \(a:b\), and 9:1 is provided as an option, corresponding to one of these possibilities.

The method involved transforming the ratio equation into a quadratic equation in terms of \(a/b\), which is a common technique when dealing with ratios involving sums and products of variables.