Table of Contents

Last modified on August 17th, 2023

Ration and proportion have an exciting relationship.

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

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.

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 x^{2}:y^{2} - The
**sub-duplicate ratio**of x:y is ${x^{\dfrac{1}{2}}:y^{\dfrac{1}{2}}}$ - The
**triplicate ratio**of x:y is x^{3}:y^{3} - The
**sub-triplicate**ratio of x:y is ${x^{\dfrac{1}{3}}:y^{\dfrac{1}{3}}}$

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

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

**Write the given statements in the ratio form. Give the answers in simplified form****a. 4 days out of 5 days****b. 8 students out of 17 students****c. 16 bottles to 18 bottles**

Solution:

a. ${\dfrac{4}{5}}$

b. ${\dfrac{8}{17}}$

c. ${\dfrac{16}{18}=\dfrac{8}{9}}$

**Assume a:b = 2:3 and b:c = 5:7. Find the ratio of a:b:c**

Solution:

Given, a:b = 2:3 and b:c = 5:7

To make the term of ‘b’ common, we will multiply b:c by ${\dfrac{3}{5}}$

Thus, b:c = ${5\times \dfrac{3}{5}:7\times \dfrac{3}{5}}$

b:c = ${3:\dfrac{21}{5}}$

a:b:c = ${2:3:\dfrac{21}{5}}$ = 10:15:21

A proportion is an equality between two ratios or fractions.

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.

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.

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.

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 ${\sqrt{xy}}$

**Find if the ratios 2:5 and 4:10 are said to be in proportion.**

Solution:

Since,

2:5 = ${\dfrac{2}{5}}$ = 0.4

4:10 = ${\dfrac{4}{5}}$= 0.4

Since, 2:5 = 4:10 = 0.4, the ratios 2:5 and 4:10 are in proportion

**If Greg travels 15 km in 3 hours. What distance will he travel in 7 hours?**

Solution:

Let Greg travel x km in 7 hours

So, 3:7 = 15:x

x = ${\dfrac{15\times 7}{3}}$

= 35 km

Thus Greg will travel a distance of 35 km in 7 hours.

**A bag has \$1, \$2, and \$5 bills in the ratio of 5:9:4. Find the bag’s worth if the total number of bills is 72.**

Solution:

Number of \$1 bill = ${\dfrac{5}{18}\times 72=20}$

Number of \$2 bill = ${\dfrac{9}{18}\times 72=36}$

Number of \$5 bill = ${\dfrac{4}{18}\times 72=16}$

Thus, the total worth of the bag = (20 × 1) + (36 × 2) + (16 × 5) = $172

**Write the means and extremes of the proportion 1: 3:: 5: 15.**

Solution:

In the given proportion 1: 3:: 5: 15

The means – 3:5

The extremes – 1:15

**For dividing $1365, the fourth times the first share, thrice the second share, and twice the third share amounts the same. Find the value of the second share.**

Solution:

Total amount = \$1000

Let the share of A, B, and C be

4A:3B:2C

Thus,

A:B:C = ${\dfrac{1}{4}:\dfrac{1}{3}:\dfrac{1}{2}}$ = 3:4:6

Thus, the value of the 2^{nd} share is = ${\dfrac{4}{13}\times 1,365}$ = $420

- 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

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 => ${\dfrac{x}{y}=\dfrac{z}{w}}$ |

Last modified on August 17th, 2023