Ratio and Proportion

Ration and proportion have an exciting relationship.

Ratio

A ratio is a comparison of 2 quantities. The numbers are mostly separated by a colon ‘:’, which is the sign of ratio in mathematics.

Formula

If two quantities x and y are in a ratio, they can be written in any of the 3 ways:  

• x:y
• ${\dfrac{x}{y}}$, or
• x to y

Ratios are always reduced to their reduced forms. For example, 12 chairs to 6 tables are expressed as:

12:8 = ${\dfrac{12}{8}=\dfrac{3}{2}}$

Thus, while writing a ratio, the two quantities should be the same type, and the units should be the same. Two or more ratios can be compared if they are reduced to their simplest forms.

Types

A ratio can be a part-to-part or part-to-whole ratio. While preparing tea:

• The ratio of tea leaves to the amount of sugar is a part-to-part ratio. It compares the amounts of two ingredients.

• The ratio of tea leaves to tea or sugar to tea is the part-to-whole ratio. It compares the amount of one ingredient to the sum of all ingredients. 

Equivalent ratios are ratios having the same value. Given a ratio, we can generate equivalent ratios by multiplying both the antecedent and consequent of the ratio by the same value.

Some other types of ratios are:

• The duplicate ratio of x:y is x²:y²
• The sub-duplicate ratio of x:y is ${x^{\dfrac{1}{2}}:y^{\dfrac{1}{2}}}$
• The triplicate ratio of x:y is x³:y³
• The sub-triplicate ratio of x:y is ${x^{\dfrac{1}{3}}:y^{\dfrac{1}{3}}}$

Comparison of Ratios

If (x:y) > (z:w), then ${\dfrac{x}{y} >\dfrac{z}{w}}$

If (x:y) < (z:w), then ${\dfrac{x}{y} <\dfrac{z}{w}}$

Solved Examples

Proportion

A proportion is an equality between two ratios or fractions.

Formula

If x, y, z, and w are in proportion, then x:y::z:w or ${\dfrac{x}{y}::\dfrac{z}{w}}$, here ${\dfrac{x}{y}, \dfrac{z}{w}}$ are equivalent ratios, and ‘::’ is the symbol of proportion. Thus, two equivalent ratios form a proportion.

The terms y and z are called mean terms, and x and w are extremes. Thus, x and y in the first ratio and w in the second ratio separately should be of the same kind.

Identifying Proportion by Formula

In proportion, the product of the means = the product of the extremes and thus is written as

y × z = x × w

For example, to find whether ratios 2:8 and 8:32 are in proportion, we will use the above concept

Using the proportional formula, we get

8 × 8 = 2 × 32

64 = 64

Thus, ratios 2:8 and 8:32 are in proportion.

Identifying Proportion by HCF Method

We can also identify proportion by the HCF method.

We need to express both in their lowest terms by finding their HCF and dividing both antecedent and consequent by their corresponding HCF.

The HCF of 2 and 8 is 2

Thus,

${\dfrac{2\div 2}{8\div 2}=\dfrac{1}{4}}$

The HCF of 8 and 32 is 8

${\dfrac{8\div 8}{32\div 8}=\dfrac{1}{4}}$

Since both the reduced forms of the fractions are equivalent, ratios 2:8 and 8:32 are in proportion.

The above concepts will help to find an unknown term if the two ratios that are in proportion are given.

Types

There are 3 types of proportions.

1. Direct Proportion

It establishes a direct relationship between two quantities. Thus, if one quantity increases, the other will also increase and vice-versa. When two variables, x and y, are directly proportional, it is written as x ∝ y.

For example, the amount of distance covered with time is directly proportional.

2. Inverse Proportion

It establishes an inverse relationship between two quantities. Thus, if one quantity increases, the other will decrease, and vice-versa. When two variables x and y are in direct proportion, then it is written as ${x\propto \dfrac{1}{y}}$.

For example, if the speed of walking has increased, the distance will get covered in a shorter time.

3. Continued Proportion

If x:y and z:w are in ratios, we convert the mean to a single term or digit to find the continued proportion for the two ratios. That is the LCM of the means.

The LCM of y:z is yz

Multiplying the first ratio by z and the second ratio by y, we have 

First ratio – xz:yz

Second ratio – yz:yw

Thus, the continued proportion can be written in the form of xz:yz:yw

There are other kinds of proportion as well used in mathematics. They are:
Third Proportion

If x:y = z:w, then z is the third proportion to x and y

Fourth Proportion

If x:y = z:w, then w is the fourth proportion to x, y, z

Mean Proportion

The mean proportion of x and y is the ${\sqrt{xy}}$

Solved Examples

Ratio and Proportion Rules and Tricks

• In a ratio, if both the antecedent (numerator) and the consequent (denominator) are multiplied or divided by the same digit number, except 0, then the ratio remains the same
• Componendo Rule: If ${\dfrac{x}{y}=\dfrac{z}{w}}$ then, ${\dfrac{x+y}{y}=\dfrac{z+w}{w}}$
• Dividendo Rule: If ${\dfrac{x}{y}=\dfrac{z}{w}}$ then, ${\dfrac{x-y}{y}=\dfrac{z-w}{w}}$
• Componendo and Dividendo Rule: If ${\dfrac{x}{y}=\dfrac{z}{w}}$ then, ${\dfrac{x+y}{x-y}=\dfrac{z+w}{z-w}}$
• Invertendo Rule: If ${\dfrac{x}{y}=\dfrac{z}{w}}$ then, ${\dfrac{y}{x}=\dfrac{2}{w}}$
• Alternendo Rule: If ${\dfrac{x}{y}=\dfrac{z}{w}}$ then, ${\dfrac{x}{z}=\dfrac{y}{w}}$
• If ${\dfrac{x}{y+z}=\dfrac{y}{2+x}=\dfrac{2}{x+y}}$and a+b+c ≠0, then a =b = c

Ratio vs Proportion

The key differences between them are:

Ratio	Proportion
Compares two quantities having the same unit	Expresses the relation between two ratios
Expressed as ‘:’ or ‘/’	Expressed as ‘::’ or ‘=’
It is an expression	It is an equation
Example: x:y = ${\dfrac{x}{y}}$	Example: x:y::z:w => ${\dfrac{x}{y}=\dfrac{z}{w}}$