Finding the Mean Proportional: Step-by-Step Guide

The question asks us to find the mean proportional between the numbers 2 and 98. Understanding what the mean proportional is key to solving this problem.

What is Mean Proportional?

The mean proportional (also known as the geometric mean) between two numbers, say \(a\) and \(c\), is a number \(b\) such that the ratio of \(a\) to \(b\) is equal to the ratio of \(b\) to \(c\). This can be written as:

\[ \frac{a}{b} = \frac{b}{c} \]

By cross-multiplying, we get:

\[ b \times b = a \times c \]

\[ b^2 = ac \]

To find \(b\), we take the square root of both sides:

\[ b = \sqrt{ac} \]

So, the mean proportional between two numbers is the square root of their product.

Calculating the Mean Proportional Between 2 and 98

Using the formula \(b = \sqrt{ac}\), where \(a = 2\) and \(c = 98\), we can calculate the mean proportional.

Step 1: Identify the two numbers

The given numbers are 2 and 98.

• Number 1 (\(a\)) = 2
• Number 2 (\(c\)) = 98

Step 2: Calculate the product of the two numbers

Multiply the two numbers together:

\[ ac = 2 \times 98 \]

\[ ac = 196 \]

Step 3: Find the square root of the product

Take the square root of the product (196) to find the mean proportional (\(b\)):

\[ b = \sqrt{196} \]

\[ b = 14 \]

Therefore, the mean proportional between 2 and 98 is 14.

Revision Table: Key Concepts

Additional Information on Proportion

A proportion is a statement that two ratios are equal. For example, \(a:b = c:d\).

• In the proportion \(a:b = b:c\), \(b\) is the mean proportional between \(a\) and \(c\). This is also called a continuous proportion.
• In the proportion \(a:b = c:d\), \(a\) and \(d\) are called extremes, and \(b\) and \(c\) are called means. The property of proportion states that the product of the extremes is equal to the product of the means (\(ad = bc\)).
• The concept of mean proportional is closely related to the geometric mean, especially for two numbers. For more than two numbers, the geometric mean is the \(n\)-th root of their product.

Understanding the relationship between ratio, proportion, and different types of proportionals like mean, third, and fourth proportional is essential for solving problems in this area of mathematics.