Finding the Mean Proportional Between Two Numbers

The mean proportional between two positive numbers, say \(a\) and \(b\), is defined as the square root of their product. It is often represented by \(x\) such that \(a : x :: x : b\), which means \(\frac{a}{x} = \frac{x}{b}\). Cross-multiplying gives \(x^2 = a \times b\). Therefore, \(x = \sqrt{a \times b}\).

In this problem, we need to find the mean proportional between 0.08 and 0.32.

Let \(a = 0.08\) and \(b = 0.32\).

Step-by-Step Calculation of Mean Proportional

To find the mean proportional, we use the formula:

Mean Proportional \(x = \sqrt{a \times b}\)

Substitute the given values:

\(x = \sqrt{0.08 \times 0.32}\)

First, calculate the product of 0.08 and 0.32:

\(0.08 \times 0.32 = 0.0256\)

Now, find the square root of 0.0256:

\(x = \sqrt{0.0256}\)

To find the square root of 0.0256, we can think of it as \(\sqrt{\frac{256}{10000}}\). The square root of 256 is 16, and the square root of 10000 is 100.

So, \(\sqrt{0.0256} = \frac{16}{100} = 0.16\).

Thus, the mean proportional between 0.08 and 0.32 is 0.16.

Understanding the Mean Proportional Result

The mean proportional 0.16 lies between 0.08 and 0.32. It satisfies the proportion: \(0.08 : 0.16 :: 0.16 : 0.32\). Let's check this:

• \(\frac{0.08}{0.16} = \frac{8}{16} = \frac{1}{2}\)
• \(\frac{0.16}{0.32} = \frac{16}{32} = \frac{1}{2}\)

Since both ratios are equal to \(\frac{1}{2}\), the proportion holds true, confirming that 0.16 is indeed the mean proportional between 0.08 and 0.32.

Revision Table: Key Concepts for Mean Proportional

Additional Information: Ratio and Proportion Fundamentals

Ratio and proportion are fundamental concepts in mathematics used to compare quantities and establish relationships between them.

• Ratio: A ratio expresses how much of one quantity there is compared to another quantity. It can be written as a:b or a/b, where b is not zero. Ratios can be simplified like fractions. For example, the ratio 10:5 is the same as 2:1.
• Proportion: A proportion states that two ratios are equal. If a/b = c/d, then a, b, c, and d are in proportion. Here, a and d are called the 'extremes', and b and c are called the 'means'. In a proportion, the product of the extremes is equal to the product of the means (ad = bc). This property is very useful in solving problems involving proportions.
• Continuous Proportion: Three quantities a, b, and c are said to be in continuous proportion if a:b :: b:c, which means a/b = b/c. In this case, b is called the mean proportional between a and c. The relationship is \(b^2 = ac\), so \(b = \sqrt{ac}\). This is the formula used to find the mean proportional.

Understanding these concepts helps in solving various problems related to ratios, proportions, scaling, and similarity.