Find the third proportional of two numbers 6 and 24.

Question

Prepp Team · Accepted Answer

The question asks us to find the third proportional of two numbers, 6 and 24. Understanding the concept of the third proportional is key to solving this problem.

Understanding the Third Proportional

When we talk about the third proportional of two numbers, say 'a' and 'b', it means we are looking for a number 'x' such that the ratio of the first number to the second number is equal to the ratio of the second number to the third proportional. In other words, a, b, and x are in continuous proportion. This relationship is written as:

\(a : b :: b : x\)

This can also be expressed as a fraction:

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

Here, 'a' is the first term, 'b' is the second term (also the mean proportional), and 'x' is the third proportional.

Finding the Third Proportional of 6 and 24

In this problem, the two numbers are 6 and 24. So, we have:

First number (a) = 6
Second number (b) = 24
Third proportional (x) = ?

Using the relationship \(a : b :: b : x\), we can substitute the given values:

\(6 : 24 :: 24 : x\)

Writing this as a fraction gives us:

\(\frac{6}{24} = \frac{24}{x}\)

Solving for the Third Proportional

To find the value of x, we can cross-multiply the terms in the equation:

\(6 \times x = 24 \times 24\)

\(6x = 576\)

Now, to isolate x, divide both sides of the equation by 6:

\(x = \frac{576}{6}\)

Performing the division:

\(x = 96\)

So, the third proportional of 6 and 24 is 96.

Verification

We can check if the proportion holds true with our calculated third proportional, 96:

\(6 : 24 :: 24 : 96\)

As fractions:

\(\frac{6}{24} = \frac{1}{4}\)

\(\frac{24}{96}\)

To simplify \(\frac{24}{96}\), we can divide both numerator and denominator by their greatest common divisor. 24 divides 96 (24 * 4 = 96). So,

\(\frac{24}{96} = \frac{24 \div 24}{96 \div 24} = \frac{1}{4}\)

Since \(\frac{6}{24} = \frac{1}{4}\) and \(\frac{24}{96} = \frac{1}{4}\), the proportion is true:

\(\frac{6}{24} = \frac{24}{96}\)

This confirms that 96 is indeed the third proportional of 6 and 24.

Step	Description	Calculation
1	Identify the given numbers (a, b) and the unknown third proportional (x).	a = 6, b = 24, x = ?
2	Set up the continuous proportion relationship.	\(a : b :: b : x\)
3	Write the proportion as an equation using fractions.	\(\frac{a}{b} = \frac{b}{x}\)
4	Substitute the values of a and b.	\(\frac{6}{24} = \frac{24}{x}\)
5	Cross-multiply to solve for x.	\(6 \times x = 24 \times 24\)
6	Simplify the equation.	\(6x = 576\)
7	Divide to find x.	\(x = \frac{576}{6}\)
8	Calculate the result.	\(x = 96\)

Revision Table: Third Proportional Calculation

Concept	Formula	Example (a=6, b=24)
Third Proportional (x) of a and b	If \(a : b :: b : x\), then \(x = \frac{b^2}{a}\)	\(x = \frac{24^2}{6} = \frac{576}{6} = 96\)

Additional Information: Ratio and Proportion Concepts

Understanding ratios and proportions is fundamental in mathematics. Here are some related concepts:

Ratio: A ratio compares two quantities. It is expressed as \(a:b\) or \(\frac{a}{b}\).
Proportion: A proportion is an equality between two ratios. If \(a:b = c:d\), it's a proportion. This means \(\frac{a}{b} = \frac{c}{d}\). Here, a and d are called extremes, and b and c are called means. The product of extremes equals the product of means (\(ad = bc\)).
Mean Proportional: For two numbers 'a' and 'c', the mean proportional 'b' is the number such that \(a : b :: b : c\). This means \(\frac{a}{b} = \frac{b}{c}\), leading to \(b^2 = ac\), so \(b = \sqrt{ac}\). In our third proportional case (\(a:b::b:x\)), 'b' is the mean proportional between 'a' and 'x'.
Fourth Proportional: For three numbers a, b, and c, the fourth proportional 'd' is the number such that \(a : b :: c : d\). This means \(\frac{a}{b} = \frac{c}{d}\), leading to \(ad = bc\), so \(d = \frac{bc}{a}\).

The question specifically asks for the third proportional, which involves a continuous proportion where the middle term is repeated, as shown in the solution above.