The sum of the fourth proportional of 4, 5, 16 and the mean proportional of 4, 16 is
30
Calculate the Sum of Fourth and Mean Proportionals
This problem asks us to find the sum of two values: the fourth proportional of three given numbers and the mean proportional of two other given numbers. Let's break down each part.
Finding the Fourth Proportional
The fourth proportional of three numbers, say $a$, $b$, and $c$, is a number, let's call it $x$, such that the ratio of the first two numbers is equal to the ratio of the third and the fourth numbers. This can be written as a proportion:
\(\frac{a}{b} = \frac{c}{x}\)
In this question, the numbers are 4, 5, and 16. So, $a=4$, $b=5$, and $c=16$. We need to find the fourth proportional, $x$.
Setting up the proportion:
\(\frac{4}{5} = \frac{16}{x}\)
To solve for $x$, we can cross-multiply:
\(4 \times x = 5 \times 16\)
\(4x = 80\)
Now, divide both sides by 4:
\(x = \frac{80}{4}\)
\(x = 20\)
So, the fourth proportional of 4, 5, and 16 is 20.
Finding the Mean Proportional
The mean proportional of two numbers, say $a$ and $b$, is a number, let's call it $y$, such that the ratio of the first number to $y$ is equal to the ratio of $y$ to the second number. This can be written as a proportion:
\(\frac{a}{y} = \frac{y}{b}\)
In this question, the numbers are 4 and 16. So, $a=4$ and $b=16$. We need to find the mean proportional, $y$.
Setting up the proportion:
\(\frac{4}{y} = \frac{y}{16}\)
To solve for $y$, we cross-multiply:
\(y \times y = 4 \times 16\)
\(y^2 = 64\)
Now, take the square root of both sides to find $y$. Since mean proportional is usually positive, we take the positive root:
\(y = \sqrt{64}\)
\(y = 8\)
So, the mean proportional of 4 and 16 is 8.
Calculating the Sum
The question asks for the sum of the fourth proportional and the mean proportional. We found the fourth proportional to be 20 and the mean proportional to be 8.
Sum = Fourth Proportional + Mean Proportional
Sum = \(20 + 8\)
Sum = \(28\)
The sum of the fourth proportional of 4, 5, 16 and the mean proportional of 4, 16 is 28.
| Concept | Numbers Used | Calculation | Result |
|---|---|---|---|
| Fourth Proportional | 4, 5, 16 | \(\frac{4}{5} = \frac{16}{x} \implies x = \frac{5 \times 16}{4} = 20\) | 20 |
| Mean Proportional | 4, 16 | \(\frac{4}{y} = \frac{y}{16} \implies y^2 = 4 \times 16 = 64 \implies y = 8\) | 8 |
| Sum | 20, 8 | \(20 + 8 = 28\) | 28 |
Revision Table: Proportion Concepts
| Term | Definition | Formula (for numbers a, b, c) | Example |
|---|---|---|---|
| Ratio | Comparison of two quantities by division | a : b or \(\frac{a}{b}\) | Ratio of 4 to 5 is \(\frac{4}{5}\) |
| Proportion | An equality between two ratios | \(\frac{a}{b} = \frac{c}{d}\) | \(\frac{4}{5} = \frac{8}{10}\) |
| Fourth Proportional | The fourth term (x) in a proportion where a:b = c:x | \(\frac{a}{b} = \frac{c}{x} \implies x = \frac{bc}{a}\) | Fourth proportional of 4, 5, 16 is 20 (4/5 = 16/20) |
| Third Proportional | The third term (x) in a continued proportion where a:b = b:x | \(\frac{a}{b} = \frac{b}{x} \implies x = \frac{b^2}{a}\) | Third proportional of 4, 8 is 16 (4/8 = 8/16) |
| Mean Proportional | The middle term (y) in a continued proportion where a:y = y:b | \(\frac{a}{y} = \frac{y}{b} \implies y^2 = ab \implies y = \sqrt{ab}\) | Mean proportional of 4, 16 is 8 (\(\sqrt{4 \times 16} = \sqrt{64} = 8\)) |
Additional Information: Ratio and Proportion Basics
Ratio and proportion are fundamental concepts in mathematics used to compare quantities and express relationships between them.
- A ratio compares two numbers, often by division. For example, the ratio of 4 to 5 is written as 4:5 or \(\frac{4}{5}\).
- A proportion is a statement that two ratios are equal. For instance, 4:5 = 16:20 is a proportion because \(\frac{4}{5}\) is equal to \(\frac{16}{20}\).
- In a proportion \(\frac{a}{b} = \frac{c}{d}\), the terms $a$ and $d$ are called the 'extremes', and the terms $b$ and $c$ are called the 'means'. The property of cross-multiplication states that the product of the extremes equals the product of the means ($ad = bc$).
- Continued Proportion: Three numbers $a, b, c$ are in continued proportion if $a:b = b:c$. Here, $b$ is the mean proportional between $a$ and $c$, and $c$ is the third proportional to $a$ and $b$.
Understanding these definitions is key to solving problems involving proportionals like the fourth proportional or mean proportional.
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