The fourth proportional to the numbers 5, 6 and 8 is:
9.6
Understanding the Fourth Proportional
The question asks us to find the fourth proportional to the numbers 5, 6, and 8. When four numbers, say $a$, $b$, $c$, and $d$, are in proportion, it means that the ratio of the first two numbers is equal to the ratio of the last two numbers. This relationship is written as $a:b :: c:d$. Mathematically, this means $\frac{a}{b} = \frac{c}{d}$.
In this problem, the given numbers are 5, 6, and 8. We need to find the fourth number, let's call it $x$, such that 5, 6, 8, and $x$ are in proportion. So, we can write the proportion as:
\(5 : 6 :: 8 : x\)
Using the definition of proportion, this can be written as a fraction equation:
\(\frac{5}{6} = \frac{8}{x}\)
Calculating the Fourth Proportional
To find the value of $x$, we can use cross-multiplication. We multiply the numerator of the first fraction by the denominator of the second fraction and set it equal to the denominator of the first fraction multiplied by the numerator of the second fraction.
\(5 \times x = 6 \times 8\)
This simplifies to:
\(5x = 48\)
Now, to isolate $x$, we divide both sides of the equation by 5:
\(x = \frac{48}{5}\)
Performing the division:
\(x = 9.6\)
So, the fourth proportional to the numbers 5, 6, and 8 is 9.6.
Comparing with Options
Let's compare our calculated value with the given options:
- Option 1: 9.8
- Option 2: 9.6
- Option 3: 9
- Option 4: 9.5
Our calculated value, 9.6, matches Option 2.
| Numbers in Proportion | Equation | Result |
|---|---|---|
| 5, 6, 8, x | \(\frac{5}{6} = \frac{8}{x}\) | \(x = 9.6\) |
Revision Table: Key Concepts in Proportion
| Concept | Description | Example |
|---|---|---|
| Ratio | Comparison of two quantities of the same kind, expressed as \(a:b\) or \(\frac{a}{b}\). | Ratio of 10 apples to 5 oranges is not a simple ratio unless specified type. Ratio of 10 apples to 5 apples is \(10:5 = 2:1\). |
| Proportion | An equality of two ratios (\(a:b = c:d\) or \(\frac{a}{b} = \frac{c}{d}\)). \(a, b, c, d\) are called terms. \(a\) and \(d\) are extreme terms, \(b\) and \(c\) are mean terms. | \(2:3 = 4:6\) is a proportion because \(\frac{2}{3} = \frac{4}{6}\). |
| Fourth Proportional | In \(a:b :: c:d\), \(d\) is the fourth proportional to \(a, b, c\). Calculated as \(d = \frac{b \times c}{a}\). | Fourth proportional to 2, 3, 4 is \(\frac{3 \times 4}{2} = 6\). |
| Third Proportional | If \(a, b, c\) are in proportion, where \(a:b :: b:c\), then \(c\) is the third proportional to \(a\) and \(b\). Calculated as \(c = \frac{b^2}{a}\). This applies when there are only three distinct numbers forming a continuous proportion. | Third proportional to 3 and 6 is \(\frac{6^2}{3} = \frac{36}{3} = 12\). (\(3:6 :: 6:12\)) |
| Mean Proportional | If \(a, b, c\) are in proportion, where \(a:b :: b:c\), then \(b\) is the mean proportional between \(a\) and \(c\). Calculated as \(b = \sqrt{a \times c}\). | Mean proportional between 4 and 9 is \(\sqrt{4 \times 9} = \sqrt{36} = 6\). (\(4:6 :: 6:9\)) |
Additional Information on Proportion and Ratio
Ratios and proportions are fundamental concepts in mathematics with many real-world applications, such as scaling maps, mixing ingredients, and calculating speeds. Understanding how to find missing terms in a proportion is a key skill.
In a proportion \(a:b = c:d\), the product of the means equals the product of the extremes. This means \(b \times c = a \times d\). This property is very useful for solving problems involving proportions, as we used it to find the fourth proportional in this question ($6 \times 8 = 5 \times x$).
Proportion problems can involve different scenarios:
- Direct Proportion: As one quantity increases, the other increases at the same rate. Example: If you buy more apples, the total cost increases proportionally.
- Inverse Proportion: As one quantity increases, the other decreases proportionally. Example: If more workers are assigned to a task, the time taken to complete it decreases.
The problem of finding the fourth proportional is a classic example of direct proportion involving four terms.
Important Questions from Fourth Proportional
What is the fourth proportional to the numbers 50, 35 and 20?
If 162 ∶ x ∶∶ x ∶ 338, find the positive value of x.
The fourth proportional of 10, 15 and 30 is:
The fourth proportional to 3, 12, 14 is:
What is the fourth proportional of 21, 56 and 27?
