What is the third proportional to 16 and 24 ?

Question

Prepp Team · Accepted Answer

Understanding the Third Proportional Concept

The question asks for the third proportional to two numbers, 16 and 24. When three numbers are in continued proportion, the ratio of the first to the second number is equal to the ratio of the second number to the third number. If we have three numbers, say \(a\), \(b\), and \(c\), they are in continued proportion if \(a : b = b : c\). In this case, \(c\) is called the third proportional to \(a\) and \(b\). The second number, \(b\), is called the mean proportional between \(a\) and \(c\).

For this specific question, the two given numbers are 16 and 24. Let the third proportional be \(x\). According to the definition of continued proportion, we can set up the relationship:

\(16 : 24 = 24 : x\)

Setting up the Equation to Find the Third Proportional

We can write this proportion as a fraction equation:

\(\frac{16}{24} = \frac{24}{x}\)

Solving for the Third Proportional

To find the value of \(x\), we can use cross-multiplication. Multiply the numerator of the first fraction by the denominator of the second fraction, and set it equal to the product of the denominator of the first fraction and the numerator of the second fraction.

\(16 \times x = 24 \times 24\)

\(16x = 576\)

Now, to isolate \(x\), divide both sides of the equation by 16:

\(x = \frac{576}{16}\)

Performing the division:

\(x = 36\)

So, the third proportional to 16 and 24 is 36.

Let's check the proportion with the calculated value:

\(16 : 24 = 24 : 36\)

The ratio \(16 : 24\) simplifies to \(2 : 3\) (dividing both by 8).

The ratio \(24 : 36\) simplifies to \(2 : 3\) (dividing both by 12).

Since \(2 : 3 = 2 : 3\), the value \(x=36\) is correct.

Summary of the Calculation

Step	Description	Calculation
1	Set up the proportion	\(16 : 24 = 24 : x\)
2	Write as equation	\(\frac{16}{24} = \frac{24}{x}\)
3	Cross-multiply	\(16x = 24 \times 24\)
4	Calculate RHS	\(16x = 576\)
5	Solve for \(x\)	\(x = \frac{576}{16}\)
6	Final Answer	\(x = 36\)

The third proportional to 16 and 24 is 36.

Revision Table: Types of Proportion

Type	Definition	Example (a, b, c)	Relationship
Fourth Proportional	If \(a : b = c : d\), then \(d\) is the fourth proportional to \(a, b, c\).	3, 5, 6, 10	\(3 : 5 = 6 : 10\) (\(\frac{3}{5} = \frac{6}{10}\))
Third Proportional	If \(a : b = b : c\), then \(c\) is the third proportional to \(a\) and \(b\). (Continued proportion)	4, 6, 9	\(4 : 6 = 6 : 9\) (\(\frac{4}{6} = \frac{6}{9}\))
Mean Proportional	If \(a : b = b : c\), then \(b\) is the mean proportional between \(a\) and \(c\).	4, 6, 9	\(6 = \sqrt{4 \times 9}\)

Additional Information on Proportions and Ratios

Understanding ratios and proportions is fundamental in mathematics. A ratio is a comparison of two quantities. A proportion is an equality of two ratios.

Ratio: A ratio like \(a : b\) can be written as a fraction \(\frac{a}{b}\). The order matters.
Proportion: A proportion like \(a : b = c : d\) means that the ratio \(a : b\) is equivalent to the ratio \(c : d\). This can be written as \(\frac{a}{b} = \frac{c}{d}\).
Product of Extremes and Means: 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 extremes is always equal to the product of the means, i.e., \(ad = bc\). This is the basis of cross-multiplication used in solving proportions.
Continued Proportion: When a proportion involves only three terms, where the middle term is repeated, it is called a continued proportion, like \(a : b = b : c\). Here, the product of extremes equals the square of the mean term: \(ac = b^2\). This is why the mean proportional \(b\) is the square root of the product of \(a\) and \(c\), i.e., \(b = \sqrt{ac}\). For the third proportional \(c\), the formula derived is \(c = \frac{b^2}{a}\), which matches our calculation \(\frac{24^2}{16}\).