Understanding the Third Proportional

The question asks for the third proportional to 7 and 63. Let's first understand what a third proportional is. If three quantities, say \(a\), \(b\), and \(c\), are in continued proportion, it means that the ratio of the first to the second is equal to the ratio of the second to the third. This is written as \(a : b :: b : c\).

In this continued proportion, \(c\) is called the third proportional to \(a\) and \(b\). The relationship can be expressed as a fraction:

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

Here, \(a\) and \(b\) are the two given numbers, and we need to find \(c\).

Calculating the Third Proportional to 7 and 63

Given the two numbers are 7 and 63, we can set up the proportion using the definition of the third proportional. Let \(a = 7\) and \(b = 63\). Let the third proportional be \(c\). The proportion is:

\( \frac{7}{63} = \frac{63}{c} \)

To find the value of \(c\), we can cross-multiply:

\( 7 \times c = 63 \times 63 \)

Now, we need to isolate \(c\) by dividing both sides of the equation by 7:

\( c = \frac{63 \times 63}{7} \)

We can simplify this calculation by dividing 63 by 7 first:

\( \frac{63}{7} = 9 \)

So, the equation becomes:

\( c = 9 \times 63 \)

Now, we perform the multiplication:

\( 9 \times 63 = 567 \)

Therefore, the third proportional to 7 and 63 is 567.

Step-by-Step Calculation

Here are the steps to find the third proportional:

1. Identify the given numbers: \(a = 7\), \(b = 63\).
2. Set up the proportion for the third proportional: \( \frac{a}{b} = \frac{b}{c} \).
3. Substitute the values: \( \frac{7}{63} = \frac{63}{c} \).
4. Cross-multiply: \( 7c = 63 \times 63 \).
5. Solve for \(c\): \( c = \frac{63 \times 63}{7} \).
6. Simplify: \( c = 9 \times 63 \).
7. Calculate the final value: \( c = 567 \).

The third proportional is 567.

Revision Table: Key Concepts in Proportion

Additional Information on Proportions

A proportion is a statement that two ratios are equal. For example, \(a:b = c:d\) or \( \frac{a}{b} = \frac{c}{d} \). In this proportion, \(a\) and \(d\) are called the extremes, and \(b\) and \(c\) are called the means. A fundamental property of proportion is that the product of the extremes is equal to the product of the means (cross-multiplication property): \(ad = bc\).

When a proportion is in the form \(a:b :: b:c\), it is called a continued proportion. In a continued proportion, the middle term (b) is the mean proportional between \(a\) and \(c\), and \(c\) is the third proportional to \(a\) and \(b\).

Understanding different types of proportionals (mean, third, fourth) is crucial for solving various problems in ratios and proportions. The method involves setting up the correct proportional relationship based on the definition and then using algebraic techniques to solve for the unknown term.