The third proportional to 4 and 6 is:
9
Understanding the Third Proportional Concept
The question asks us to find the third proportional to the numbers 4 and 6. Let's first understand what a third proportional means in the context of ratios and proportions.
When three numbers, say \(a\), \(b\), and \(c\), are in continued proportion, it means that the ratio of the first number to the second is equal to the ratio of the second number to the third. This can be written as:
\( a : b :: b : c \)
Mathematically, this relationship is expressed as:
\( \frac{a}{b} = \frac{b}{c} \)
In this relationship, \(c\) is called the third proportional to \(a\) and \(b\).
Calculating the Third Proportional
In our question, the two given numbers are 4 and 6. We need to find the third proportional to these numbers. Let the third proportional be \(x\). According to the definition of continued proportion, the numbers 4, 6, and \(x\) are in continued proportion.
So, we can set up the proportion as:
\( 4 : 6 :: 6 : x \)
Writing this in fractional form, we get:
\( \frac{4}{6} = \frac{6}{x} \)
To solve for \(x\), we can cross-multiply:
\( 4 \times x = 6 \times 6 \)
\( 4x = 36 \)
Now, divide both sides by 4 to find the value of \(x\):
\( x = \frac{36}{4} \)
\( x = 9 \)
Therefore, the third proportional to 4 and 6 is 9.
Verifying the Third Proportional
Let's check if 4, 6, and 9 are in continued proportion:
Ratio of the first to the second: \( \frac{4}{6} = \frac{2}{3} \)
Ratio of the second to the third: \( \frac{6}{9} = \frac{2}{3} \)
Since \( \frac{4}{6} = \frac{6}{9} \), the numbers 4, 6, and 9 are indeed in continued proportion, and 9 is the third proportional to 4 and 6.
| Concept | Definition/Formula | Example (using 4 and 6) |
|---|---|---|
| Ratio | Comparison of two quantities by division (\(a:b\) or \(a/b\)) | Ratio of 4 to 6 is \(4:6\) or \(4/6 = 2/3\) |
| Proportion | Equality of two ratios (\(a:b :: c:d\) or \(a/b = c/d\)) | \(4:6 :: 2:3\) is a proportion because \(4/6 = 2/3\) |
| Continued Proportion | When three numbers \(a, b, c\) satisfy \(a:b :: b:c\) | 4, 6, 9 are in continued proportion because \(4:6 :: 6:9\) |
| Third Proportional | In \(a:b :: b:c\), \(c\) is the third proportional to \(a\) and \(b\). Formula: \(c = b^2/a\) | Third proportional to 4 and 6 is \(6^2 / 4 = 36 / 4 = 9\) |
Revision Table: Key Proportion Terms
| Term | Description |
|---|---|
| Ratio | Compares two quantities. |
| Proportion | States that two ratios are equal. |
| Mean Proportional | If \(a:b :: b:c\), \(b\) is the mean proportional between \(a\) and \(c\). (\(b = \sqrt{ac}\)) |
| Third Proportional | If \(a:b :: b:c\), \(c\) is the third proportional to \(a\) and \(b\). (\(c = b^2/a\)) |
| Fourth Proportional | If \(a:b :: c:d\), \(d\) is the fourth proportional to \(a, b, c\). (\(d = (b \times c) / a\)) |
Additional Information on Proportions
Proportions are a fundamental concept in mathematics with many real-world applications, from scaling recipes to designing buildings and understanding relationships between different quantities.
- Understanding the relationship between quantities is key to solving proportion problems.
- Always set up the proportion correctly based on whether you are looking for a mean, third, or fourth proportional.
- Cross-multiplication is a common method to solve equations involving proportions.
- Ensure units are consistent when dealing with real-world problems involving proportions.
Important Questions from Third Proportional
Find the third proportional to 6 and 12.
If p is the third proportional to 3, 9, then what is the fourth proportional to 6, p, 4?
What is the third proportional to 10 and 25?
What is the third proportional to 10 and 20?
The third proportional to (x2 - y2) and (x - y) is:
