Find the 3 rd proportional to 9 and 21.
49
Finding the Third Proportional to 9 and 21
The question asks us to find the third proportional to the numbers 9 and 21. Let's understand what a third proportional means in the context of ratios and proportions.
What is a Third Proportional?
When three numbers are in continued proportion, the third number is called the third proportional to the first two. If we have three numbers, say $a$, $b$, and $c$, they are in continued proportion if the ratio of the first to the second is equal to the ratio of the second to the third. This can be written as:
$\frac{a}{b} = \frac{b}{c}$
In this proportion, $c$ is the third proportional to $a$ and $b$. The number $b$ is called the mean proportional between $a$ and $c$.
In this problem, the given numbers are 9 and 21. We are looking for a third number, let's call it $x$, such that 9, 21, and $x$ are in continued proportion. Following the definition, this means:
$\frac{9}{21} = \frac{21}{x}$
Calculating the Third Proportional
To find the value of $x$, we can solve the proportion $\frac{9}{21} = \frac{21}{x}$. We can do this by cross-multiplication.
Multiplying the numerator of the first fraction by the denominator of the second fraction, and the denominator of the first fraction by the numerator of the second fraction, we get:
$9 \times x = 21 \times 21$
$9x = 441$
Now, to find $x$, we need to divide both sides of the equation by 9:
$x = \frac{441}{9}$
Performing the division:
$441 \div 9 = 49$
So, the value of the third proportional $x$ is 49.
Thus, the numbers 9, 21, and 49 are in continued proportion because $\frac{9}{21} = \frac{3}{7}$ and $\frac{21}{49} = \frac{3 \times 7}{7 \times 7} = \frac{3}{7}$. The ratios are equal.
| Concept | Explanation |
|---|---|
| Third Proportional | For numbers $a$ and $b$, the third proportional $x$ satisfies $a:b = b:x$. |
| Proportion for 9 and 21 | $9:21 = 21:x$ or $\frac{9}{21} = \frac{21}{x}$ |
| Equation | $9x = 21 \times 21$ |
| Solving for x | $x = \frac{441}{9} = 49$ |
The third proportional to 9 and 21 is 49.
Revision Table: Key Concepts of Proportion
| Term | Definition | Example (using $a, b, c, d$) |
|---|---|---|
| Ratio | Comparison of two quantities by division ($a:b$ or $\frac{a}{b}$) | $3:4$ |
| Proportion | Equality of two ratios ($a:b = c:d$) | $2:3 = 4:6$ |
| Extremes | The first and fourth terms in a proportion ($a$ and $d$ in $a:b = c:d$) | 2 and 6 in $2:3 = 4:6$ |
| Means | The second and third terms in a proportion ($b$ and $c$ in $a:b = c:d$) | 3 and 4 in $2:3 = 4:6$ |
| Product of Extremes = Product of Means | Property of proportion: $a \times d = b \times c$ | $2 \times 6 = 3 \times 4 \implies 12 = 12$ |
| Continued Proportion | When $a, b, c$ are in proportion $a:b = b:c$ | $4:6 = 6:9$ |
| Third Proportional | The third term $c$ in a continued proportion $a:b = b:c$ | 9 is the third proportional to 4 and 6 |
| Mean Proportional | The middle term $b$ in a continued proportion $a:b = b:c$ ($b = \sqrt{ac}$) | 6 is the mean proportional between 4 and 9 ($6 = \sqrt{4 \times 9} = \sqrt{36}$) |
Additional Information: Applications of Proportion
Proportions are widely used in various fields. Understanding proportions helps in solving problems related to:
- Scaling recipes in cooking.
- Calculating distances on maps using scale.
- Mixing ingredients for construction or chemistry.
- Solving problems in geometry involving similar shapes.
- Determining quantities in finance and business ratios.
The concept of the third proportional is a specific case of proportion that is useful in understanding relationships between three numbers in a sequence where the ratio between the first two is the same as the ratio between the second and third.
Important Questions from Third Proportional
Find the third proportion to 16 and 24.
Find the third proportional to 16 and 24.
What is the third proportional to the numbers 9 and 45?
Find the third proportional to 6 and 12.
The third proportional to 4 and 6 is:
