Understanding Proportions and Ratios Concepts

This problem requires us to find the ratio between two values derived from proportions: the fourth proportional of three given numbers and the third proportional of two given numbers. Let's break down how to find each of these values.

Calculating the Fourth Proportional

When four numbers, say \(a, b, c,\) and \(d\), are in proportion, they satisfy the relationship \(a:b :: c:d\). This can be written as the equality of two ratios: \(\frac{a}{b} = \frac{c}{d}\). Here, \(d\) is called the fourth proportional to \(a, b,\) and \(c\).

We are given the numbers 12, 16, and 6, and we need to find the fourth proportional. Let the fourth proportional be \(x\). So, we have the proportion:

\(12 : 16 :: 6 : x\)

Writing this as fractions, we get:

\(\frac{12}{16} = \frac{6}{x}\)

To solve for \(x\), we can cross-multiply:

\(12 \times x = 16 \times 6\)

\(12x = 96\)

Now, divide both sides by 12:

\(x = \frac{96}{12}\)

\(x = 8\)

So, the fourth proportional of 12, 16, and 6 is 8.

Calculating the Third Proportional

When three numbers, say \(a, b,\) and \(c\), are in continued proportion, they satisfy the relationship \(a:b :: b:c\). This means the ratio of the first two terms is equal to the ratio of the second and third terms. This can be written as the equality of two ratios: \(\frac{a}{b} = \frac{b}{c}\). Here, \(c\) is called the third proportional to \(a\) and \(b\), and \(b\) is called the mean proportional between \(a\) and \(c\).

We are given the numbers 4 and 6, and we need to find the third proportional. Let the third proportional be \(y\). The continued proportion is:

\(4 : 6 :: 6 : y\)

Writing this as fractions, we get:

\(\frac{4}{6} = \frac{6}{y}\)

To solve for \(y\), we can cross-multiply:

\(4 \times y = 6 \times 6\)

\(4y = 36\)

Now, divide both sides by 4:

\(y = \frac{36}{4}\)

\(y = 9\)

So, the third proportional of 4 and 6 is 9.

Determining the Final Ratio

The question asks for the ratio between the fourth proportional of 12, 16, 6 and the third proportional of 4, 6.

• Fourth proportional = 8
• Third proportional = 9

The ratio is simply the first value compared to the second value:

Ratio = (Fourth proportional) : (Third proportional)

Ratio = 8 : 9

This ratio cannot be simplified further, as 8 and 9 have no common factors other than 1.

Summary of Proportion Calculations

Here is a summary of the calculated values:

The ratio of these results is 8 : 9.

Revision Table: Key Proportion Definitions

Understanding different types of proportionals is crucial:

Additional Information on Ratios and Proportions

Ratios and proportions are fundamental concepts in arithmetic and algebra. They are used to compare quantities and understand relationships between them.

• Ratio: Expresses how many times one number contains another. It can be written as \(a:b\) or as a fraction \(\frac{a}{b}\). Ratios can be simplified by dividing both parts by their greatest common divisor.
• Proportion: A statement that two ratios are equal. For example, \(a:b = c:d\). This indicates that \(a\) is to \(b\) as \(c\) is to \(d\). Proportions are useful for solving problems involving scaling, recipes, mixtures, and more.
• Finding proportionals (third, fourth, mean) involves setting up the correct proportional relationship based on the given numbers and solving for the unknown term. Always ensure the terms are placed in the correct order according to the definition of the specific proportional you are looking for.