What is Proportion? (With Meaning & Examples)

This post is also available in: हिन्दी (Hindi)

Proportion in math is a comparison of two numbers that represent things or people, and it is often written as a fraction or with a colon. The concept of proportion is related to ratio and fractions, which represents a relation between two values. 

We use proportions in solving many daily life problems such as in business while dealing with transactions or while cooking, etc. It establishes a relation between two or more quantities and thus helps in their comparison.

Let’s understand what is proportion and how it is used.

What is Proportion?

Proportion is an equation that defines that the two given ratios are equivalent to each other. In other words, the proportion states the equality of the two fractions or ratios.

For example, a car moving at a certain speed and covering a distance of $60 km$ in $1hr$ is the same as a car covering a distance of $300 km$ in $5 hr$. So, we can say that the numbers $60$, $1$, $300$, and $5$ are in proportion or $300 : 5 = 60 : 1$ or $\frac{300}{5} = \frac{60}{1}$.

A proportion involves two ratios. The meaning of proportion is that the two ratios are equal to each other. As in the above example, the two ratios $300: 5$ and $60: 1$ are equal. 

In general four numbers $a$, $b$, $c$, and $d$ are said to be in proportion, when $a:b = c:d$, or $\frac{a}{b} = \frac{c}{d}$, and the ratios $a:b$ and $c:d$ are called equivalent ratios.

Proportions are denoted using the symbol  “::” or “=”.

When the values $a$, $b$, $c$ and $d$ are in proportion, i.e., $a:b :: c:d$ or $\frac{a}{b} = \frac{c}{d}$, the numbers $a$ and $d$ are called extremes and the numbers $b$ and $d$ are called means.

Difference Between Ratio and Proportion

The following differences help in distinguishing between ratio and proportion.

Checking for Values in Proportion

Four numbers $a$, $b$, $c$ and $d$ are said to be in proportion, when $a:b = c:d$. To check whether the numbers $a$, $b$, $c$, and $d$ are in proportion, you can use any of the following methods.

• Fraction Method
• Multiplication Method

Fraction Method

The numbers $a$, $b$, $c$ and $d$ will be in proportion, if $\frac{a}{b}$ and \frac{c}{d}$ are equivalent fractions.

Let’s consider the following examples to understand the method.

The numbers $12$, $20$, $21$, $35$ are in proportion, since, $\frac{12}{20} = \frac{21}{35}$.

$\frac{12}{20} = $\frac{3 \times 4}{5 \times 4} = $\frac{3}{5}$ and $\frac{21}{35} = \frac{3 \times 7}{5 \times 7} \frac{3}{5}$.

The fractions $\frac{12}{20}$ and $\frac{21}{35}$ are called equivalent fractions.

The numbers $6$, $9$, $4$, $10$ are not in proportion, since, $\frac{6}{9} \ne \frac{4}{10}$.

$\frac{6}{9} = \frac{2 \times 3}{3 \times 3} = \frac{2}{3}$ and $\frac{4}{10} = \frac{2 \times 2}{5 \times 2} = \frac{2}{5}$.

Multiplication Method

The numbers $a$, $b$, $c$ and $d$ will be in proportion if the products of means is equal to the product of extremes.

To understand this method, let’s again consider the above examples.

The numbers $12$, $20$, $21$, $35$ are in proportion, since, $12 \times 35 = 420$ and $20 \times 21 = 420$.

And the numbers $6$, $9$, $4$, $10$ are not in proportion, since, $6 \times 10 = 60$ and $9 \times 4 = 36$.

Properties of Proportion

Proportion establishes an equivalent relation between two ratios. Following are the properties of proportion.

Addendo

The property states that if $a$, $b$, $c$, and $d$ are in proportion, i.e., $a : b = c : d$, then value of each ratio is $a + c : b + d$.

In other words, we can say that if $a$, $b$, $c$, and $d$ are in proportion, then $a : b = c : d = a + c : b + d$.

Let’s consider a proportion $2$, $3$, $4$, $6$ to understand the Addendo property.

$2 + 4 = 6$ and $3 + 6 = 9$

And, $\frac{6}{9} = \frac{2}{3} = \frac{4}{6}$.

Subtrahendo

The property states that if $a$, $b$, $c$, and $d$ are in proportion, i.e., $a : b = c : d$, then value of each ratio is $a – c : b – d$.

In other words, we can say that if $a$, $b$, $c$, and $d$ are in proportion, then $a : b = c : d = a – c : b – d$.

Let’s consider a proportion $5$, $10$, $1$, $2$ to understand the Subtrahendo property.

$5 – 1 = 4$ and $10 – 2 = 8$

And, $\frac{4}{8} = \frac{5}{10} = \frac{1}{2}$.

Dividendo

The property states that if $a$, $b$, $c$, and $d$ are in proportion, i.e., $a : b = c : d$, then $a – b : b = c – d : d$.

Let’s consider a proportion $3$, $2$, $6$, $4$ to understand the Dividendo property.

$\left(3 – 2 \right) : 2 = 1 : 2$ and $\left(6 – 4 \right) : 4 = 2:4 = 1:2$.

Componendo

The property states that if $a$, $b$, $c$, and $d$ are in proportion, i.e., $a : b = c : d$, then $a + b : b = c + d : d$.

Let’s consider a proportion $2$, $3$, $4$, $6$ to understand the Componendo property.

$\left(2 + 3 \right) : 3 = 5 : 3$ and $\left(4 + 6 \right) : 6 = 10:6 = 5:3$.

Alternendo

The property states that if $a$, $b$, $c$, and $d$ are in proportion, i.e., $a : b = c : d$, then $a : c = b: d$.

Let’s consider a proportion $2$, $3$, $4$, $6$ to understand the Alternendo property.

$2 : 4 = 1 : 2$ and $3 : 6 = 1:2$.

Invertendo

The property states that if $a$, $b$, $c$, and $d$ are in proportion, i.e., $a : b = c : d$, then $b : a = d : c$.

Let’s consider a proportion $2$, $3$, $4$, $6$ to understand the Invertendo property.

$3 : 2$ and $6 : 4 = 3:2$.

Componendo and Dividendo

The property states that if $a$, $b$, $c$, and $d$ are in proportion, i.e., $a : b = c : d$, then $\left(a + b \right) : \left(a – b \right) = \left(c + d \right) : \left(c – d \right)$.

Let’s consider a proportion $3$, $2$, $6$, $4$ to understand the Componendo and Dividendo property.

$\left(3 + 2 \right) : \left(3 – 2 \right) = 5 : 1$ and $\left(6 + 4 \right) : \left(6 – 4 \right) = 10:2 = 5:1$.

Examples

Ex 1: Check whether the numbers $27$, $33$, $54$, $66$ are in proportion or not.

Let $a = 27$, $b = 33$, $c = 54$, $d = 66$

$\frac{a}{b} = \frac{27}{33} = \frac{9 \times 3}{11 \times 3} = \frac{9}{11}$

And, $\frac{c}{d} = \frac{54}{66} = \frac{9 \times 6}{11 \times 6} = \frac{9}{11}$

Since, $\frac{27}{33} = \frac{54}{66} = \frac{9}{11}$, therefore, the numbers $27$, $33$, $54$, $66$ are in proportion.

Alternatively, it can be solved as

$a \times d = 27 \times 66 = 1782$ and $b \times d = 33 \times 54 = 1782$.

Since,  $27 \times 66 = 33 \times 54$, therefore, the numbers $27$, $33$, $54$, $66$ are in proportion.

Ex 2: Find the value of $x$, such that $2$, $x$, $3$, and $15$ are in proportion.

For $2$, $x$, $3$, and $15$ to be in proportion, $2 \times 15 = x \times 3 => 3x = 30 => x = \frac{30}{3} = 10$.

Conclusion

Proportion is an equation that defines that the two given ratios are equivalent to each other. In other words, the proportion states the equality of the two fractions or ratios. If any four numbers $a$, $b$, $c$ and $d$ are in proportion, then $a:b = c:d$.

Practice Problems

1. Check whether the following numbers are in proportion or not.
   • $1$, $2$, $3$, $4$
   • $2$, $3$, $4$, $6$
   • $2$, $3$, $4$, $8$
   • $1$, $5$, $10$, $15$
2. For what value of $x$, the following numbers will be in proportion?
   • $2$, $8$, $x$, $20$
   • $x$, $75$, $1$, $5$
   • $7$, $x$, $5$, $15$
   • $9$, $36$, $11$, $x$

Recommended Reading

• What is Ratio(Meaning, Simplification & Examples)
• Factors and Multiples (With Methods & Examples)
• Fractions On Number Line – Representation & Examples
• Reducing Fractions – Lowest Form of A Fraction
• Comparing Fractions (With Methods & Examples)
• Like and Unlike Fractions
• Improper Fractions(Definition, Conversions & Examples)
• How To Find Equivalent Fractions? (With Examples)
• 6 Types of Fractions (With Definition, Examples & Uses)
• What is Fraction?  – Definition, Examples & Types
• Mixed Fractions – Definition & Operations (With Examples)
• Multiplication and Division of Fractions
• Addition and Subtraction of Fractions (With Pictures)

FAQs