Proportions

What is a proportion?

In a proportion, we’re comparing two fractions:

$$ \frac{a}{b} = \frac{c}{d} $$

The core rule for proportions is the means-extremes property: the product of the means equals the product of the extremes.

This relationship can be written as:

$$ a \cdot d = b \cdot c $$

Why is this useful? This property lets us find an unknown value when the other three terms are known.

Proportions aren’t just abstract math concepts; they have real-world applications across various fields. For example, they’re used on maps to show the scale between map distance and actual distance. You’ll also see proportions in unit conversions.

A practical example

Let’s say we want to solve a proportion where one of the terms, $ x $, is unknown.

$$ \frac{3}{4} = \frac{x}{8} $$

Here, 3 and 8 are the extremes, and 4 and x are the means. Applying the means-extremes property:

$$ 3 \cdot 8 = 4 \cdot x $$

This simplifies to:

$$ 24 = 4x $$

Now, dividing both sides by 4, we get the value of x:

$$ x = \frac{24}{4} = 6 $$

So, x equals 6. This straightforward calculation shows how useful proportions are for finding unknown values.

Example 2

The means-extremes property is also handy for checking if two ratios form a proportion.

Suppose we want to know if the ratios \(\frac{2}{5}\) and \(\frac{6}{15}\) are proportional.

To do this, you could calculate each ratio’s value to see if they’re the same, but that involves two divisions, which might not be convenient.

$$ \frac{2}{5} = \frac{6}{15} $$

Instead, it’s simpler to apply the means-extremes property.

Multiply the extremes and compare the result to the product of the means.

$$ 2 \cdot 15 = 5 \cdot 6 $$

Since the products are equal, the two ratios do indeed form a proportion.

$$ 30 = 30 $$

This approach lets you verify the proportion without any division.

Continuous Proportions

A proportion is referred to as continuous when the two middle terms are the same.

In general, a continuous proportion is written in the form \(a : b = b : c\), where \(b\) connects the two ratios and is known as the mean proportional. 

$$ a:b = b:c $$

Here’s an example of a continuous proportion:

$$ 3 : 9 = 9 : 27 $$

In this case, the number \(9\) serves as the mean proportional between \(3\) and \(27\).

Properties of Proportions

Proportions have several key properties that make them particularly useful:

• Fundamental Property: The product of the means equals the product of the extremes: $$ a \cdot d = b \cdot c $$ This is called the "fundamental" property because it serves as the basis for all other properties of proportions.

  For example, consider the proportion $$ 5 : 10 = 3 : 6 $$. Multiplying the extremes \( 5 \cdot 6 \) and the means \( 10 \cdot 3 \) gives the same result: $$ 5 \cdot 6 = 10 \cdot 3 $$ $$ 30 = 30 $$

• Inversion of Ratios: You can reverse the terms in each ratio without changing the proportion: $$ a : b = c : d \implies b : a = d : c $$ This is also known as the inversion property.

  For instance, in the proportion $$ 5 : 10 = 3 : 6 $$, flipping the numerator and denominator of each ratio yields another valid proportion: $$ 10 : 5 = 6 : 3 $$ $$ 2 = 2 $$

• Swapping Means or Extremes: You can interchange the means or the extremes while maintaining the validity of the proportion: $$ a : b = c : d \implies a : c = b : d $$ This is also called the permutation property.

  Consider the proportion $$ 5 : 10 = 2 : 4 $$. Interchanging the means results in another valid proportion: $$ 5 : 2 = 10 : 4 $$ $$ \frac{5}{2} = \frac{10}{4} $$ $$ \frac{5}{2} = \frac{5}{2} $$ The same holds true if the extremes are swapped: $$ 4 : 10 = 2 : 5 $$ $$ \frac{4}{10} = \frac{2}{5} $$ $$ \frac{2}{5} = \frac{2}{5} $$

• Term Addition: Adding the terms of each ratio does not affect the proportion: $$ a : b = c : d \implies (a + b) : a = (c + d) : c $$ $$ a : b = c : d \implies (a + b) : b = (c + d) : d $$ This is also referred to as the composition property.

  For example, take the proportion $$ 5 : 10 = 2 : 4 $$. Applying this property gives another valid proportion: $$ (5+10) : 5 = (2+4) : 2 $$ $$ 15 : 5 = 6 : 2 $$ $$ 3 = 3 $$ Similarly: $$ (5+10) : 10 = (2+4) : 4 $$ $$ 15 : 10 = 6 : 4 $$ $$ \frac{3}{2} = \frac{3}{2} $$

• Term Subtraction: Subtracting the terms of each ratio also preserves the proportion: $$ a : b = c : d \implies (a - b) : a = (c - d) : c $$ $$ a : b = c : d \implies (a - b) : b = (c - d) : d $$ This is also known as the decomposition property.

  For example, in the proportion $$ 10 : 5 = 4 : 2 $$, applying this property gives another valid proportion: $$ (10-5) : 10 = (4-2) : 4 $$ $$ 5 : 10 = 2 : 4 $$ $$ 2 = 2 $$ Similarly: $$ (10-5) : 5 = (4-2) : 2 $$ $$ 5 : 5 = 2 : 2 $$ $$ 1 = 1 $$

Mastering these properties will allow you to approach many mathematical problems with greater confidence and ease.