Animated Solution for Mathematics - Sequence and Series: Let the harmonic mean and geometric mean of two positive numbers be the ratio $4:5$. Then the two number are in the ratio \dots.

When we discuss the Harmonic Mean ($H.M.$) and the Geometric Mean ($G.M.$), we are exploring fundamental pillars of mathematical analysis. For two positive numbers $a$ and $b$, these means are defined as:

These values are not merely abstract formulas; they represent deep geometric relationships. They are two sides of the same coin, linked by the properties of segments on a line.

Our problem provides the ratio of the Harmonic Mean to the Geometric Mean as $4:5$. Mathematically, we express this as:

Substituting the definitions into this ratio, we obtain:

By simplifying the expression through the cancellation of the common factor $\sqrt{ab}$, we arrive at a more manageable form:

To solve for the ratio $a:b$, we first invert the equation to isolate the structure of a perfect square:

We now apply the rule of Componendo and Dividendo, which states that if $\frac{x}{y}=\frac{p}{q}$, then $\frac{x+y}{x-y}=\frac{p+q}{p-q}$. Applying this to our equation:

This simplifies elegantly to:

Taking the square root of both sides of the equation, we obtain:

Cross-multiplying to solve for the relationship between $\sqrt{a}$ and $\sqrt{b}$ yields:

Rearranging the terms, we find:

Squaring both sides, we reach the final result:

The ratio of the two numbers is $4:1$.