Understanding Proportionality Concepts

This problem requires us to calculate two specific types of proportionals: the mean proportional and the third proportional, and then find the difference between them. Let's break down each part.

Calculating the Mean Proportional

The mean proportional between two numbers, say \(a\) and \(b\), is a number \(x\) such that \(a:x :: x:b\). This can be written as a fraction:

\[ \frac{a}{x} = \frac{x}{b} \]Combining this, we get \(x^2 = ab\), which means the mean proportional \(x\) is the square root of the product of the two numbers: \(x = \sqrt{ab}\).

In this question, we need the mean proportional between 1.8 and 3.2.

Let the mean proportional be \(M\).

\[ M = \sqrt{1.8 \times 3.2} \]First, calculate the product:

\[ 1.8 \times 3.2 = 5.76 \]Now, find the square root of the product:

\[ M = \sqrt{5.76} \]To find the square root of 5.76, we can think of the square root of 576, which is 24. Since there are two decimal places in 5.76, there will be one decimal place in its square root.

\[ M = 2.4 \]

So, the mean proportional between 1.8 and 3.2 is 2.4.

Calculating the Third Proportional

The third proportional to two numbers, say \(a\) and \(b\), is a number \(c\) such that \(a:b :: b:c\). This means the ratio of the first to the second is equal to the ratio of the second to the third:

\[ \frac{a}{b} = \frac{b}{c} \]To find \(c\), we can rearrange the equation:

\[ ac = b^2 \]\[ c = \frac{b^2}{a} \]

In this question, we need the third proportional to 5 and 3. Here, \(a=5\) and \(b=3\).

Let the third proportional be \(T\).

\[ T = \frac{3^2}{5} \]Calculate the square of the second number:

\[ 3^2 = 9 \]Now, divide by the first number:

\[ T = \frac{9}{5} \]To express this as a decimal:

\[ T = 1.8 \]

So, the third proportional to 5 and 3 is 1.8.

Finding the Difference

The question asks for the difference between the mean proportional and the third proportional.

Difference = Mean Proportional - Third Proportional

Difference = \(2.4 - 1.8\)

Difference = \(0.6\)

The difference is 0.6.

Conclusion

The mean proportional between 1.8 and 3.2 is 2.4.

The third proportional to 5 and 3 is 1.8.

The difference between the mean proportional and the third proportional is \(2.4 - 1.8 = 0.6\).

Revision Table: Proportionality Basics

Additional Information on Proportionality

Proportionality is a fundamental concept in mathematics, especially in ratios and proportions. It describes the relationship between quantities where their ratios are constant. Understanding mean, third, and fourth proportionals is crucial for solving various problems related to similar figures, scaling, and other areas.

• A proportion is an equality of two ratios. For example, \(a:b = c:d\) is a proportion, also written as \(\frac{a}{b} = \frac{c}{d}\).
• In a proportion \(a:b = c:d\), \(a\) and \(d\) are called the extremes, and \(b\) and \(c\) are called the means. The product of the means equals the product of the extremes (\(bc = ad\)).
• The mean proportional is also known as the geometric mean of two numbers. It lies between the two numbers in a continuous proportion (\(a:x :: x:b\)).
• The third proportional extends the concept of continuous proportion to three numbers. If \(a, b, c\) are in continuous proportion, then \(a:b :: b:c\), and \(c\) is the third proportional to \(a\) and \(b\).
• The fourth proportional involves four numbers in a proportion (\(a:b :: c:d\)). Here, \(d\) is the fourth proportional to \(a, b\), and \(c\).

These concepts are frequently tested in competitive exams and form the basis for more advanced topics in algebra and geometry.