Properties of Ratios and Proportions

In mathematics, the properties of proportions are closely tied to the properties of fractions.

Fundamental Property of Proportions

In a proportion a:b = c:d, the product of the means equals the product of the extremes. $$ a:b = c:d \Longleftrightarrow a \cdot d = b \cdot c $$

This result follows directly from cross multiplication of equivalent fractions.

A proportion is an equality between equivalent fractions

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

Two fractions are equivalent if and only if the cross product of the numerator of the first and the denominator of the second equals the product of the numerator of the second and the denominator of the first.

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

This property is called "fundamental" because all other properties of proportions can be derived from it.

Example. Consider the proportion $$ 2:8 = 3:12 $$ Rewrite it as an equality between equivalent fractions $$ \frac{2}{8} = \frac{3}{12} $$ Since the fractions are equivalent, their cross products are equal. $$ 2 \cdot 12 = 3 \cdot 8 $$ $$ 24 = 24 $$

Addition Property

In a proportion a:b = c:d


• the sum of the antecedent and consequent (a+b) is to the antecedent (a) as the sum of the corresponding terms (c+d) is to the antecedent (c) $$ (a+b):a = (c+d):c $$
• the sum of the antecedent and consequent (a+b) is to the consequent (b) as the sum of the corresponding terms (c+d) is to the consequent (d) $$ (a+b):b = (c+d):d $$

Proof

Start from a proportion

$$ a:b = c:d $$

Rewrite it as an equality between two equivalent fractions

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

By the invariant property of equations, add 1 to both sides

$$ \frac{a}{b} + 1 = \frac{c}{d} + 1 $$

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

Rewrite the equality as a proportion to obtain one componendo formula

$$ (a+b):b = (c+d):d $$

To derive the second formula, apply invertendo and repeat the same steps

$$ a:b = c:d $$

Applying invertendo gives

$$ b:a = d:c $$

Rewrite as equivalent fractions

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

Add 1 to both sides

$$ \frac{b}{a} + 1 = \frac{d}{c} + 1 $$

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

Rewrite again as a proportion to obtain the second formula

$$ (a+b):a = (c+d):c $$

Example. Consider the proportion $$ 2:6 = 3:9 $$ Apply componendo.
A] From componendo, (a+b):a = (c+d):c is also a proportion $$ (2+6):2 = (3+9):3 $$ $$ 8:2 = 12:3 $$ Check using the cross product $$ 8 \cdot 3 = 12 \cdot 2 $$ $$ 24 = 24 $$ The identity holds, so (2+6):2 = (3+9):3 is a proportion.
B] From componendo, (a+b):b = (c+d):d is also a proportion $$ (2+6):6 = (3+9):9 $$ $$ 8:6 = 12:9 $$ Check using the cross product $$ 8 \cdot 9 = 6 \cdot 12 $$ $$ 72 = 72 $$ The identity holds, so (2+6):6 = (3+9):9 is a proportion.

Subtraction Property

In a proportion a:b = c:d


• the difference of the antecedent and consequent (a-b) is to the antecedent (a) as the difference of the corresponding terms (c-d) is to the antecedent (c) $$ (a-b):a = (c-d):c $$
• the difference of the antecedent and consequent (a-b) is to the consequent (b) as the difference of the corresponding terms (c-d) is to the consequent (d) $$ (a-b):b = (c-d):d $$

Proof

Start from a proportion

$$ a:b = c:d $$

Rewrite it as equivalent fractions

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

By the invariant property of equations, subtract 1 from both sides

$$ \frac{a}{b} - 1 = \frac{c}{d} - 1 $$

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

Rewrite as a proportion to obtain one dividendo formula

$$ (a-b):b = (c-d):d $$

To derive the second formula

$$ a:b = c:d $$

Apply invertendo

$$ b:a = d:c $$

Rewrite as equivalent fractions

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

Subtract 1 from both sides

$$ \frac{b}{a} - 1 = \frac{d}{c} - 1 $$

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

Rewrite as a proportion to obtain the second formula

$$ (a-b):a = (c-d):c $$

Example. Consider the proportion $$ 4:2 = 10:5 $$ Apply dividendo.
A] From dividendo, (a-b):a = (c-d):c is also a proportion $$ (4-2):4 = (10-5):10 $$ $$ 2:4 = 5:10 $$ Check using the cross product $$ 2 \cdot 10 = 4 \cdot 5 $$ $$ 20 = 20 $$ The identity holds, so (4-2):4 = (10-5):10 is a proportion.
B] From dividendo, (a-b):b = (c-d):d is also a proportion $$ (4-2):2 = (10-5):5 $$ $$ 2:2 = 5:5 $$ Check using the cross product $$ 2 \cdot 5 = 5 \cdot 2 $$ $$ 10 = 10 $$ The identity holds, so (4-2):2 = (10-5):5 is a proportion.

Permutation Property

In a proportion a:b = c:d, interchanging the means

or interchanging the extremes

produces another valid proportion

Proof

Start with a proportion

$$ a:b = c:d $$

A proportion is an equality between two equivalent fractions

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

By the invariant property of equations, multiply both sides by b/c

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

$$ \frac{ab}{bc} = \frac{bc}{cd} $$

Simplify

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

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

Rewriting this equality as a proportion gives the permutando form with the means interchanged

$$ a:c = b:d $$

To derive the second form, proceed in the same way

Consider again the original equality

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

Multiply both sides by d/a

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

$$ \frac{ad}{ab} = \frac{cd}{ad} $$

Simplify

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

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

Rewriting as a proportion gives the permutando form with the extremes interchanged

$$ d:b = c:a $$

Example. Consider the proportion $$ 3:9 = 4:12 $$ Apply permutando.
A] By permutando, a:c = b:d is also a proportion $$ 3:4 = 9:12 $$ Verify using the cross product $$ 3 \cdot 12 = 4 \cdot 9 $$ $$ 36 = 36 $$ The identity holds, so 3:4 = 9:12 is a proportion.
B] By permutando, d:b = c:a is also a proportion $$ 12:9 = 4:3 $$ Verify using the cross product $$ 12 \cdot 3 = 4 \cdot 9 $$ $$ 36 = 36 $$ The identity holds, so 12:9 = 4:3 is a proportion.

Inversion Property

In a proportion a:b = c:d, interchanging each antecedent with its corresponding consequent

produces another valid proportion

Proof

Start with a proportion

$$ a:b = c:d $$

A proportion is an equality between two equivalent fractions

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

Multiply both sides by d and simplify

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

$$ \frac{ad}{b} = c $$

Multiply both sides by b and simplify

$$ \frac{ad}{b} \cdot b = c \cdot b $$

$$ ad = bc $$

Multiply both sides by 1/a and simplify

$$ ad \cdot \frac{1}{a} = bc \cdot \frac{1}{a} $$

$$ d = \frac{bc}{a} $$

Multiply both sides by 1/c and simplify

$$ d \cdot \frac{1}{c} = \frac{bc}{a} \cdot \frac{1}{c} $$

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

Rewriting this equality as a proportion gives the invertendo form

$$ d:c = b:a $$

Example. Consider the proportion $$ 2:6 = 4:12 $$ Apply invertendo to interchange antecedents and consequents $$ 6:2 = 12:4 $$ Verify using the cross product $$ 6 \cdot 4 = 12 \cdot 2 $$ $$ 24 = 24 $$ The identity holds, so 6:2 = 12:4 is a proportion.

Sum of Antecedents and Consequents

In a proportion a:b = c:d, the sum of the antecedents (a+c) is to the sum of the consequents (b+d) as the first term (a) is to the second term (b) $$ (a+c):(b+d) = a:b $$

Example

Consider a proportion

$$ 4:2 = 10:5 $$

Apply the property

$$ (4+10):(2+5) = 4:2 $$

$$ 14:7 = 4:2 $$

Difference of Antecedents and Consequents

In a proportion a:b = c:d, the difference of the antecedents (a-c) is to the difference of the consequents (b-d) as the first term (a) is to the second term (b) $$ (a-c):(b-d) = a:b $$

Example

Consider a proportion

$$ 10:5 = 4:2 $$

Apply the property

$$ (10-4):(5-2) = 4:2 $$

$$ 6:3 = 4:2 $$

And so on.