In this post we will discuss the** concept of proportions**, its definition, properties and some examples.

To understand proportion, you should have basic understanding of the concept of ratios and fractions.

Some experience of solving algebraic and rational expression will also be beneficial.

**What is Proportion?**

When the given **two or more ratios are equal to each other**, then the ratios are said to be in proportion.

**For example;**

Consider the ratio 1 : 3 and 2 : 6.

We know that ratios can be written in the form of fractions.

1 : 3 ⟹ 1 / 3

2 : 6 ⟹ 2 / 6

The fraction 2/6 can be further simplified.

Dividing numerator and denominator by 2, we get;

\mathtt{\frac{2\ \div \ 2}{6\ \div \ 2} \ \Longrightarrow \frac{1}{3}}

So the value of both the fraction is 1/3.

So we can express ratio as :

1 : 3 = 2 : 6

Hence, both the ratios are proportional to each other.

**Definition of Proportion**

Proportion shows the **comparison of two or more entities**.**If the entities are proportional** to each other, it means that the **increase/ decrease in value are always in same ratio**.

**Proportion Example**

In building construction, the quantity of cement and sand are proportional. For a wall, the cement and sand used is 40 kg and 80 kg respectively. Find the amount of sand used if cement used is 120 kg.

**Solution**

Ratio of cement and sand used = 40 kg : 80 kg

Cement : Sand = 40 : 80

Convert the ratio in form of fraction.

40 : 80 ⟹ 40 / 80

Simplifying the ratio.

Divide numerator and denominator by 40.

\mathtt{\frac{40\ \div \ 40}{80\ \div \ 40} \ \Longrightarrow \frac{1}{2} \ }

Hence, we have reduced the ratio to 1 : 2.

Its given that quantity of cement and sand are proportional.

It means that cement and sand will always maintain the ratio 1 : 2.

\mathtt{\frac{Cement}{Sand} \ \Longrightarrow \frac{1}{2} \ }

Now the cement used is 120 kg.

We have to find amount of sand used.

We know that cement & sand are proportional. It, thus, always maintain ratio of 1 : 2.

Putting the values in below equation;

\mathtt{\frac{120}{Sand} \ =\ \frac{1}{2} \ }\\\ \\ \mathtt{Sand\ =\ 120\ \times \ 2}\\\ \\ \mathtt{Sand\ =\ 240\ kg}

Hence when 120 kg of cement is used, the quantity of sand will be 240 kg.

**Conclusion**

If any given quantity are in proportion, it means that they are always in same ratio.

**How to represent a Proportion?**

There are **two symbols** through which proportion can be represented:

(a) Equal to sign ” = “

(b) Double colon ” : : “

**Example**

Here the ratios a : b and c : d are in proportion.

The ratios can be expressed as:

a : b = c : d or;

a : b : : c : d

Both the expression mean the same thing.

**Explain the difference between Ratios and Proportion**

Given below are some point which will state the difference between ratios and proportion.

The points will also develop clarity for the above two concepts.

(a) Ratio is comparison of two or more numbers.

Proportion tells the relation between two ratios.

(b) Ratio is just an expression which can be also written in fraction.

Proportion is an equation of ratio which can be solved using algebraic equation.

(c) Ratios are expressed with symbols ” : “

Proportions are expressed as ” = ” and ” : : “

**Examples of Proportion**

In order to solve the above questions, you should have good understanding of algebraic and rational expression. Brush up the concepts before moving on to understand below problems.

**Example 01**The ratios 2 : 7 and z : 14 are in proportion.

Find value of z

**Solution**

Proportional ratios are equal.

2 : 7 = z : 14

Converting ratios in form of fractions.

2 / 7 = z / 14

Solving the equation using cross multiplication.

In cross multiplication, we multiply denominator with opposite numerator.

\mathtt{2\ \times \ 14\ =\ 7\ z}\\\ \\ \mathtt{28\ =\ 7z}

Dividing numerator & denominator by 7

\mathtt{\frac{28}{7} \ =\ \frac{7\ z}{7}}\\\ \\ \mathtt{z\ =\ 4}

**Hence, the value of z is 7**

**Example 02**

The ratios 7 : 13 and 8 : y are in proportion.

Find the value of y.

**Solution**

Proportional ratios are equal to each other.

7 : 13 = 8 : y

Writing the ratios in form of fraction.

\mathtt{\frac{7}{13} \ =\ \frac{8}{y}}

Solving the equation using cross multiplication.

In cross multiplication, we multiply the denominator with the opposite numerator.

Solving the equation further..

\mathtt{7\ y\ =\ 13\ \times \ 8}\\\ \\ \mathtt{7\ y\ =\ 104}\\\ \\ \mathtt{y\ =\ \frac{104}{7}}

Hence,** the value of y is 104 / 7**

**Example 03**The ratios 5 : y and 28 : 35 are in proportion.

Find the value of y.

**Solution**

Proportional ratios are equal to each other.

5 : y = 28 : 35

Converting ratios in form of fractions

\mathtt{\frac{5}{y} \ =\ \frac{28}{35} \ }

Now we have to solve the algebraic equation.

Doing cross multiplication . . .

\mathtt{5\ \times \ 35\ =\ 28\ y}\\\ \\ \mathtt{175\ =\ 28\ y}\\\ \\ \mathtt{y\ =\ \frac{175}{28}}

Divide numerator and denominator by 7

\mathtt{y\ =\ \frac{175\div 7}{28\div 7}}\\\ \\ \mathtt{y\ =\ \frac{25}{4}}

Hence, **value of y is 25/4**.

**Proportion Terminologies**

Given below are some of the terminologies used in proportion.

Let the ratios a : b and c : d are proportional to each other.

Representing ratio in the form of fraction.

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

Here the elements “b” and “c” is called **means of the proportion**.

And elements ” a ” and ” d” is called **extremes of proportion**.

**Proportion Property**

The property of proportion says that **the product of mean** of proportion is** equal to product of extremes**.

This property is useful for finding if the given ratio is proportional or not.

**Example 01**

Check if the below ratio is proportional or not.

4 : 9 and 16 : 36

Solution

If the ratios are parallel then the products of ” means ” and ” extremes” are equal.

Product of Mean

⟹ 9 x 16

⟹ 144

Product of extremes

⟹ 4 x 36

⟹ 144

Product of Mean = Product of extremes

Hence, the ratios are proportional.

**Example 02**

Check if the below ratios are proportional or not.

7 : 11 and 14 : 10

Solution

Product of Mean

⟹ 11 x 14

⟹ 154

Product of Extremes

⟹ 7 x 10

⟹ 70

Products of means and extremes are not equal.

Hence, the ratios are not proportional.