Find the third proportional to 8 and 20.
50
Understanding Third Proportional Concepts
The question asks us to find the third proportional to two numbers, 8 and 20. To solve this, we first need to understand what a third proportional is.
When three numbers, let's say $a$, $b$, and $c$, are in a continued proportion, it means that the ratio of the first number to the second number is equal to the ratio of the second number to the third number. In mathematical terms, this is expressed as:
\[ \frac{a}{b} = \frac{b}{c} \]
In this relationship, $c$ is called the third proportional to $a$ and $b$. The number $b$ is called the mean proportional between $a$ and $c$.
Finding the Third Proportional to 8 and 20
We are given the first two numbers, $a=8$ and $b=20$. We need to find the third proportional, which we will call $x$. Using the definition of continued proportion, we can set up the following relationship:
\[ \frac{8}{20} = \frac{20}{x} \]
Now, we can solve this equation for $x$ using cross-multiplication. Cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction and setting them equal.
\[ 8 \times x = 20 \times 20 \]
Simplify the right side of the equation:
\[ 8x = 400 \]
Now, to isolate $x$, divide both sides of the equation by 8:
\[ x = \frac{400}{8} \]
Performing the division:
\[ x = 50 \]
So, the third proportional to 8 and 20 is 50.
Step-by-Step Calculation Summary
| Step 1 | Identify the given numbers (a and b) and the unknown third proportional (x). | $a=8$, $b=20$, $c=x$ |
| Step 2 | Set up the proportion for continued proportion. | $\frac{a}{b} = \frac{b}{c} \implies \frac{8}{20} = \frac{20}{x}$ |
| Step 3 | Perform cross-multiplication. | $8 \times x = 20 \times 20$ |
| Step 4 | Simplify the equation. | $8x = 400$ |
| Step 5 | Solve for x by dividing both sides by 8. | $x = \frac{400}{8}$ |
| Step 6 | Calculate the final value of x. | $x = 50$ |
Therefore, the third proportional to 8 and 20 is 50.
Revision Table: Proportion Concepts
| Concept | Definition | Example |
|---|---|---|
| Ratio | A comparison of two quantities by division (e.g., $a:b$ or $\frac{a}{b}$). | Ratio of 8 to 20 is $8:20$ or $\frac{8}{20}$. |
| Proportion | An equality of two ratios (e.g., $a:b :: c:d$ or $\frac{a}{b} = \frac{c}{d}$). | $2:4 :: 3:6$ because $\frac{2}{4} = \frac{1}{2}$ and $\frac{3}{6} = \frac{1}{2}$. |
| Continued Proportion | When three numbers $a, b, c$ are such that the ratio of the first to the second is equal to the ratio of the second to the third ($a:b :: b:c$). | 8, 20, 50 are in continued proportion because $\frac{8}{20} = \frac{2}{5}$ and $\frac{20}{50} = \frac{2}{5}$. |
| Third Proportional | In a continued proportion $a:b :: b:c$, $c$ is the third proportional to $a$ and $b$. | 50 is the third proportional to 8 and 20. |
| Mean Proportional | In a continued proportion $a:b :: b:c$, $b$ is the mean proportional between $a$ and $c$. Calculated as $b = \sqrt{ac}$. | 20 is the mean proportional between 8 and 50, because $20 = \sqrt{8 \times 50} = \sqrt{400} = 20$. |
Additional Information on Proportionality
Proportionality is a fundamental concept in mathematics used to describe relationships between quantities. It appears in various areas like geometry, physics, and everyday calculations.
- Direct Proportion: Two quantities are in direct proportion if an increase in one quantity results in a proportional increase in the other, and vice versa. Their ratio is constant. For example, the cost of apples is directly proportional to the number of apples purchased.
- Inverse Proportion: Two quantities are in inverse proportion if an increase in one quantity results in a proportional decrease in the other, and vice versa. Their product is constant. For example, the time taken to travel a distance is inversely proportional to the speed of travel.
- Applications: Understanding proportionality helps in solving problems related to scaling, ratios, rates, and percentages. Finding the third proportional is one specific application within the concept of continued proportion, often encountered in competitive exams.
Being comfortable with setting up and solving proportions is a key skill for tackling many mathematical problems efficiently.
Important Questions from Third Proportional
What is the third proportional to the numbers 9 and 45?
Find the 3 rd proportional to 9 and 21.
The third proportional to 0.9 and 0.3 is:
Third proportional of 9 and 24 is:
Find the third proportional of two numbers 6 and 24.
