Understanding Proportions: Third and Fourth Proportion

This question asks us to find the ratio between two different types of proportions: the third proportion and the fourth proportion. Let's first understand what these terms mean.

What are Proportions?

A proportion is a statement that two ratios are equal. For example, if \(a, b, c, d\) are four quantities, they are in proportion if the ratio of \(a\) to \(b\) is equal to the ratio of \(c\) to \(d\). This is written as \(a:b :: c:d\) or \(\frac{a}{b} = \frac{c}{d}\). Here, \(a\) and \(d\) are called the extreme terms, and \(b\) and \(c\) are called the mean terms.

Calculating the Third Proportion

The third proportion applies to three quantities where the ratio of the first to the second is equal to the ratio of the second to the third. If we have two numbers, say \(a\) and \(b\), and \(x\) is their third proportion, then \(a, b, x\) are in continued proportion. This means:

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

To find \(x\), we rearrange the formula:

\(x = \frac{b^2}{a}\)

In this question, we need to find the third proportion of 5 and 12. Here, \(a=5\) and \(b=12\). Let the third proportion be \(x\).

Using the formula:

\(x = \frac{12^2}{5}\)

\(x = \frac{144}{5}\)

So, the third proportion of 5 and 12 is \(\frac{144}{5}\).

Calculating the Fourth Proportion

The fourth proportion applies to four quantities where the ratio of the first to the second is equal to the ratio of the third to the fourth. If we have three numbers, say \(a, b, c\), and \(y\) is their fourth proportion, then \(a, b, c, y\) are in proportion. This means:

\(\frac{a}{b} = \frac{c}{y}\)

To find \(y\), we rearrange the formula:

\(y = \frac{b \times c}{a}\)

In this question, we need to find the fourth proportion of 5, 8, and 9. Here, \(a=5\), \(b=8\), and \(c=9\). Let the fourth proportion be \(y\).

Using the formula:

\(y = \frac{8 \times 9}{5}\)

\(y = \frac{72}{5}\)

So, the fourth proportion of 5, 8, and 9 is \(\frac{72}{5}\).

Finding the Ratio of the Proportions

The question asks for the ratio of the third proportion of 5 and 12 with the fourth proportion of 5, 8 and 9.

Ratio = (Third Proportion) : (Fourth Proportion)

Ratio = \(\frac{144}{5} : \frac{72}{5}\)

To simplify the ratio, we can multiply both terms by 5:

Ratio = \(144 : 72\)

Now, we can simplify the ratio by dividing both terms by their greatest common divisor, which is 72.

\(\frac{144}{72} = 2\)

\(\frac{72}{72} = 1\)

So, the ratio is \(2 : 1\).

Ratio = (Third Proportion) : (Fourth Proportion) = \(\frac{144}{5} : \frac{72}{5} = 144 : 72 = 2 : 1\).

Revision Table: Proportions Key Concepts

Additional Information: Types of Proportions

Besides the third and fourth proportion, the concept of mean proportion is also important when studying proportions.

• Mean Proportion: If three numbers \(a, b, c\) are in continued proportion, then the middle number \(b\) is called the mean proportion between \(a\) and \(c\). The relationship is \(\frac{a}{b} = \frac{b}{c}\), which gives \(b^2 = ac\), so \(b = \sqrt{ac}\). For example, the mean proportion between 4 and 9 is \(\sqrt{4 \times 9} = \sqrt{36} = 6\), because \(4:6 = 6:9\) (\(\frac{4}{6} = \frac{2}{3}\) and \(\frac{6}{9} = \frac{2}{3}\)).
• Continued Proportion: When three or more numbers are in proportion such that the ratio of the first to the second is equal to the ratio of the second to the third, and so on, they are said to be in continued proportion. For example, \(a, b, c, d\) are in continued proportion if \(\frac{a}{b} = \frac{b}{c} = \frac{c}{d}\).

Understanding these definitions and formulas is crucial for solving problems involving proportions.