In mathematics, a proportion is an equation stating that two ratios are strictly equivalent. When numbers or quantities share a constant ratio, they are said to be proportional.

Depending on how many distinct terms are involved, proportions are categorised into two fundamental structures:

Four-Term Proportion: An equality between two separate ratios:

\dfrac{a}{b} = \dfrac{c}{d} \quad \text{or} \quad a : b = c : d

In this expression, a and d are called the extremes, while b and c are called the means. The term d is the fourth proportional to a, b, and c.

Continued Proportion (Three Terms): An ongoing chain of ratios where the second term repeats as the denominator of the first ratio and the numerator of the second:

\dfrac{a}{b} = \dfrac{b}{c} \quad \text{or} \quad a : b = b : c

In this relationship:

b is known as the mean proportional (or geometric mean) between a and c.
c is known as the third proportional to a and b.

Why Proportions Matter

Real-World Applications: In cartography and architecture, proportions maintain scale factors across technical blueprints and maps. In photographic and screen aspect ratios (such as 16:9), proportions ensure images resize without distortion. In financial analysis, the mean proportional represents the exact compound growth rate across successive multi-year periods.
Academic Relevance: The mean and third proportionals connect basic arithmetic ratios directly to similar triangles, Euclid's geometric theorems, and the common ratio (r) of geometric sequences.

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Core Theory

1. The Cross-Multiplication Property (Product Rule)

For any four quantities a, b, c, d in proportion:

\dfrac{a}{b} = \dfrac{c}{d} \iff ad = bc

The Fundamental Rule of Proportion:

The product of the extremes always equals the product of the means:

\text{Extreme}_1 \times \text{Extreme}_2 = \text{Mean}_1 \times \text{Mean}_2

From this single identity, we derive the formulas for each proportional term.

2. The Fourth Proportional

Given three known quantities a, b, and c, the fourth proportional d completes the equality:

\dfrac{a}{b} = \dfrac{c}{d} \implies ad = bc \implies d = \dfrac{bc}{a}

In simple words: multiply the second and third values together, then divide by the first value.

3. The Mean Proportional (Geometric Mean)

When three positive quantities a, b, and c are in continued proportion:

\dfrac{a}{b} = \dfrac{b}{c}

Applying cross-multiplication:

b \times b = a \times c \implies b^2 = ac

Taking the square root of both sides gives the formula for the mean proportional b:

b = \sqrt{ac}

The mean proportional between two numbers is the positive square root of their product.
In statistics and sequences, this value is identically called the geometric mean.
If the numbers can be signed, $b = \pm\sqrt{ac}$ , though geometric lengths are strictly positive (b > 0).

4. The Third Proportional

In a continued proportion of three terms, the third proportional c is the final term that extends the ratio established by the first two terms a and b:

\dfrac{a}{b} = \dfrac{b}{c}

Cross-multiplying gives:

ac = b^2 \implies c = \dfrac{b^2}{a}

In simple words: take the second number, square it, and divide the result by the first number.

Worked Examples

Worked Example 1: Finding the Fourth Proportional

Find the fourth proportional to 4, 9, and 16.

Step 1: Set up the proportion equation

Let the fourth proportional be d:

\dfrac{4}{9} = \dfrac{16}{d}

Step 2: Apply the cross-multiplication rule

4 \times d = 9 \times 16

4d = 144

Step 3: Solve for d

d = \dfrac{144}{4} = 36

The fourth proportional is 36.

Worked Example 2: Finding the Third Proportional

Find the third proportional to 8 and 12.

Step 1: Set up the continued proportion equation

Let the third proportional be c:

\dfrac{8}{12} = \dfrac{12}{c}

Step 2: Cross-multiply

8 \times c = 12 \times 12

8c = 144

Step 3: Solve for c

c = \dfrac{144}{8} = 18

The third proportional is 18.

Worked Example 3: Finding the Mean Proportional

Determine the mean proportional between 4 and 25.

Step 1: Set up the continued proportion equation

Let the mean proportional be b:

\dfrac{4}{b} = \dfrac{b}{25}

Step 2: Cross-multiply

b^2 = 4 \times 25

b^2 = 100

Step 3: Take the positive square root

b = \sqrt{100} = 10

The mean proportional is 10.

Practice Questions & Solutions

Score:

Find the fourth proportional to $\text{[math]}$ , $\text{[math]}$ , and $\text{[math]}$ .

Solution

Set up the proportion:

$\text{[math]}$

Cross-multiply:

$\text{[math]}$

Divide by $\text{[math]}$ :

$\text{[math]}$

Find the third proportional to $\text{[math]}$ and $\text{[math]}$ .

Solution

Set up the continued proportion:

$\text{[math]}$

Cross-multiply:

$\text{[math]}$

Divide by $\text{[math]}$ :

$\text{[math]}$

Find the mean proportional between $\text{[math]}$ and $\text{[math]}$ .

Solution

Let the mean proportional be b:

$\text{[math]}$

Compute the product:

$\text{[math]}$

Evaluate the square root:

$\text{[math]}$

Calculate the fourth proportional to $\text{[math]}$ , $\text{[math]}$ , and $\text{[math]}$ .

Solution

Set up the proportion:

$\text{[math]}$

Cross-multiply:

$\text{[math]}$

Divide by $\text{[math]}$ :

$\text{[math]}$

Determine the third proportional to $\text{[math]}$ and $\text{[math]}$ .

Solution

Set up the third proportional formula $\text{[math]}$ :

$\text{[math]}$

Evaluate the numerator:

$\text{[math]}$

Divide by $\text{[math]}$ :

$\text{[math]}$

Calculate the mean proportional between $\text{[math]}$ and $\text{[math]}$ .

Solution

Apply the mean proportional formula $\text{[math]}$ :

$\text{[math]}$

Separate the square roots:

$\text{[math]}$

Since $\text{[math]}$ and $\text{[math]}$ :

$\text{[math]}$

The numbers $\text{[math]}$ , $\text{[math]}$ , and $\text{[math]}$ are in continued proportion. Find the positive value of $\text{[math]}$ .

Solution

Because $\text{[math]}$ , $\text{[math]}$ , and $\text{[math]}$ are in continued proportion:

$\text{[math]}$

Cross-multiply:

latex^2 = 4 \times 36[/latex]

$\text{[math]}$

Divide by $\text{[math]}$ :

$\text{[math]}$

Taking the positive square root gives:

$\text{[math]}$

Find the fourth proportional to the algebraic expressions $\text{[math]}$ , $\text{[math]}$ , and $\text{[math]}$ .

Solution

Let the fourth proportional be d:

$\text{[math]}$

Cross-multiply:

$\text{[math]}$

Divide both sides by $\text{[math]}$ (for $\text{[math]}$ ):

$\text{[math]}$

In a right-angled triangle, the perpendicular altitude drawn to the hypotenuse divides it into two segments of length $\text{[math]}$ and $\text{[math]}$ . Calculate the length of the altitude.

Solution

By the Altitude Theorem, the altitude h is the mean proportional between the two segments:

$\text{[math]}$

Find the third proportional to $\text{[math]}$ and $\text{[math]}$ .

Solution

Let the third proportional be c:

$\text{[math]}$

Factorise $\text{[math]}$ as $\text{[math]}$ :

$\text{[math]}$

Cancel the factor $\text{[math]}$ :

$\text{[math]}$

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Gianpiero Placidi

UK-based Chemistry graduate with a passion for education, providing clear explanations and thoughtful guidance to inspire student success.

Mean Proportion Formula: Steps and worked examples

Why Proportions Matter

Core Theory

1. The Cross-Multiplication Property (Product Rule)

2. The Fourth Proportional

3. The Mean Proportional (Geometric Mean)

4. The Third Proportional

Worked Examples

Worked Example 1: Finding the Fourth Proportional

Worked Example 2: Finding the Third Proportional

Worked Example 3: Finding the Mean Proportional

Practice Questions & Solutions

Ratio

Simple Interest

Percentage

Inverse Proportions

Direct Proportions

Proportions

Third Mean Proportional Formula

Ratio Formulas

Simple Interest Word Problems

Percentage Word Problems

Ratio Word Problems

Cancel reply

Mean Proportion Formula: Steps and worked examples

Why Proportions Matter

Core Theory

1. The Cross-Multiplication Property (Product Rule)

2. The Fourth Proportional

3. The Mean Proportional (Geometric Mean)

4. The Third Proportional

Worked Examples

Worked Example 1: Finding the Fourth Proportional

Worked Example 2: Finding the Third Proportional

Worked Example 3: Finding the Mean Proportional

Practice Questions & Solutions

Theory

Ratio

Simple Interest

Percentage

Inverse Proportions

Direct Proportions

Proportions

Third Mean Proportional Formula

Formulas

Ratio Formulas

Exercises

Simple Interest Word Problems

Percentage Word Problems

Ratio Word Problems

Cancel reply