Understanding Mean and Third Proportion

This problem involves finding two unknown numbers based on their mean proportion and third proportion. We are given the values for both and need to find the numbers first, then calculate their average.

Defining Proportion Concepts

Let the two numbers be $a$ and $b$.

• Mean Proportion: The mean proportion of two numbers $a$ and $b$ is a number $m$ such that $a:m :: m:b$. This means $\frac{a}{m} = \frac{m}{b}$, leading to $m^2 = ab$, or $m = \sqrt{ab}$.
• Third Proportion: The third proportion of two numbers $a$ and $b$ is a number $x$ such that $a:b :: b:x$. This means $\frac{a}{b} = \frac{b}{x}$, leading to $ax = b^2$, or $x = \frac{b^2}{a}$. (Note: If the question implies the third proportion of b and a, it would be $a:b::b:x$, so $x = b^2/a$. If it were third proportion to a and b, it would be $a:b::b:x$, giving $x=b^2/a$. Some definitions use $a:b::x:b$, giving $xb=ab$, so $x=a$. However, the standard definition is sequential proportion $a:b::b:x$). We will use the standard definition $x = \frac{b^2}{a}$.

Setting up Equations

Given information:

• Mean proportion of the two numbers is 9. So, $\sqrt{ab} = 9$.
• The third proportion is 243. So, $\frac{b^2}{a} = 243$.

From the mean proportion, squaring both sides gives:

\( ab = 9^2 \)

\( ab = 81 \quad (Equation\ 1) \)

From the third proportion:

\( \frac{b^2}{a} = 243 \quad (Equation\ 2) \)

Solving for the Numbers

We have a system of two equations with two variables, $a$ and $b$. We can solve this system.

From Equation 1, we can express $a$ in terms of $b$:

\( a = \frac{81}{b} \)

Substitute this expression for $a$ into Equation 2:

\( \frac{b^2}{\frac{81}{b}} = 243 \)

\( \frac{b^2 \cdot b}{81} = 243 \)

\( \frac{b^3}{81} = 243 \)

Multiply both sides by 81:

\( b^3 = 243 \times 81 \)

We can express 243 and 81 as powers of 3:

\( 243 = 3 \times 81 = 3 \times 9 \times 9 = 3 \times 3^2 \times 3^2 = 3^{1+2+2} = 3^5 \)

\( 81 = 9 \times 9 = 3^2 \times 3^2 = 3^4 \)

So,

\( b^3 = 3^5 \times 3^4 \)

\( b^3 = 3^{5+4} \)

\( b^3 = 3^9 \)

Take the cube root of both sides:

\( b = (3^9)^{1/3} \)

\( b = 3^{9/3} \)

\( b = 3^3 \)

\( b = 27 \)

Now substitute the value of $b$ back into the expression for $a$:

\( a = \frac{81}{b} = \frac{81}{27} \)

\( a = 3 \)

The two numbers are 3 and 27.

Calculating the Average

The average of the two numbers $a$ and $b$ is given by \(\frac{a+b}{2}\).

Average \( = \frac{3 + 27}{2} \)

Average \( = \frac{30}{2} \)

Average \( = 15 \)

The average of the two numbers is 15.

Revision Table: Proportion and Average Calculation

Additional Information: Types of Proportion

Understanding different types of proportion is key to solving such problems.

• Direct Proportion: Two quantities $x$ and $y$ are directly proportional if their ratio is constant, i.e., \(\frac{x}{y} = k\). As one quantity increases, the other increases proportionally.
• Inverse Proportion: Two quantities $x$ and $y$ are inversely proportional if their product is constant, i.e., \(xy = k\). As one quantity increases, the other decreases proportionally.
• Continued Proportion: Numbers $a, b, c$ are in continued proportion if $a:b :: b:c$, which means $\frac{a}{b} = \frac{b}{c}$. Here, $b$ is the mean proportion between $a$ and $c$, and $c$ is the third proportion to $a$ and $b$.
• Fourth Proportion: For three numbers $a, b, c$, their fourth proportion $d$ is such that $a:b :: c:d$. This means $\frac{a}{b} = \frac{c}{d}$, so $d = \frac{bc}{a}$.

The problem uses the concepts of mean proportion (related to continued proportion) and third proportion directly.