Finding the Mean Proportion of Numbers

The question asks us to find the mean proportion of two numbers, 64 and 144. Understanding the concept of mean proportion is key to solving this problem.

What is Mean Proportion?

The mean proportion of two positive numbers, say 'a' and 'b', is a number 'x' such that the ratio of 'a' to 'x' is equal to the ratio of 'x' to 'b'. In symbolic form, this is written as:

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

This relationship implies that the square of the mean proportion ('x') is equal to the product of the two numbers ('a' and 'b'). Mathematically:

\( x^2 = ab \)

To find the mean proportion 'x', we take the square root of the product of the two numbers:

\( x = \sqrt{ab} \)

Calculating the Mean Proportion of 64 and 144

We are given the two numbers: a = 64 and b = 144.

Using the formula \( x = \sqrt{ab} \), we substitute the given values:

\( x = \sqrt{64 \times 144} \)

First, we calculate the product of 64 and 144:

\( 64 \times 144 = 9216 \)

Now, we find the square root of the product:

\( x = \sqrt{9216} \)

To find the square root of 9216, we can either use a calculator or prime factorization. Alternatively, we can find the square roots of the factors separately because \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \) (for non-negative a and b):

\( x = \sqrt{64} \times \sqrt{144} \)

We know that \( \sqrt{64} = 8 \) and \( \sqrt{144} = 12 \).

So, substitute these values back into the equation:

\( x = 8 \times 12 \)

\( x = 96 \)

Thus, the mean proportion of 64 and 144 is 96.

Let's check if this fits the proportion definition:

\( \frac{64}{96} = \frac{2 \times 32}{3 \times 32} = \frac{2}{3} \)

\( \frac{96}{144} = \frac{2 \times 48}{3 \times 48} = \frac{2}{3} \)

Since \( \frac{64}{96} = \frac{96}{144} \), our calculation is correct.

Summary of Mean Proportion Calculation

• Identify the two numbers (a and b).
• Use the formula \( x = \sqrt{ab} \).
• Calculate the product \( ab \).
• Find the square root of the product.

For 64 and 144:

• a = 64, b = 144
• \( x = \sqrt{64 \times 144} \)
• \( 64 \times 144 = 9216 \)
• \( x = \sqrt{9216} = 96 \)

The mean proportion is 96.

Revision Table: Mean Proportion

Additional Information: Means and Proportions

Besides the mean proportion (also known as geometric mean), there are other types of means and concepts related to proportion:

• Arithmetic Mean: For two numbers 'a' and 'b', the arithmetic mean is \( \frac{a+b}{2} \).
• Harmonic Mean: For two numbers 'a' and 'b', the harmonic mean is \( \frac{2}{\frac{1}{a} + \frac{1}{b}} = \frac{2ab}{a+b} \).
• Direct Proportion: Two quantities are in direct proportion if an increase in one quantity leads to a proportional increase in the other, and vice versa. Their ratio is constant.
• Inverse Proportion: Two quantities are in inverse proportion if an increase in one quantity leads to a proportional decrease in the other, and vice versa. Their product is constant.

The mean proportion specifically relates to geometric progression where 'a', 'x', and 'b' form a geometric sequence (a, ax/a, ax²/a = a, x, b where the common ratio is r = x/a and b = x*r = x*(x/a) = x²/a, leading to x²=ab).