A ratio gives the relationship between two or more values (or) shows the relative size of two or more numbers.

Comparing two quantities of the same units is called a ratio.
A ratio of two quantities can be expressed as a fraction also. For example, the ratio 3:4 can be expressed as $\frac{3}{4}$ and vice versa.

If two given ratios are the same, they are said to be in proportion.

## What is a Ratio?

The magnitude of quantities relative to each other is called the ratio of two quantities. It is usually expressed as “p to q” or “p : q”.

### Examples on Ratios

Given below are some examples on ratios.

Example 1:

If a class of 40 students has 25 boys, then calculate the ratio of boys to girls?

Solution:

Total number of students in the class = 40
The number of boys in the class = 25
The number of girls in the class = 40 – 25 = 15
So, the ratio of ratio of boys to girls = 25 : 15

= 5 : 3

Example 2:

If Kelly runs a mile in 6 minutes 50 seconds and jimmy takes 7 minutes 50 seconds, then calculate the ratio of kelly's time to Jimmy's time?

Solution:

6 minutes 50 seconds = (6 x 60) + 50

= 360 + 50
= 410 seconds.

7 minutes 50 seconds = (7 x 60) + 50
= 420 + 50
=470 seconds.

Ratio of Kelly's time to Jimmy's time = 410 : 470
= 41 : 47

## Properties of Ratio

Ratios are used to compare two quantities with the same units. For example, consider the ratio of the weight of a man whose weight is 60kg and that of a dog whose weight is 30 kg. The ratio is 60:30 = 2:1

Given below are some properties of ratios:

• Ratios can also be represented as a fraction.
• Equivalent ratios are similar to equivalent fractions.
• Different ratios can be simplified and compared in order to find whether they are equivalent or not.

## Simplifying Ratios

Ratios can be simplified by following the procedure given below:

• Write the ratio in terms of a fraction.
• Find the greatest common divisor of the numerator and the denominator.
• Reduce both the numerator and the denominator by dividing them by the greatest common divisor.
• Write the ratio in its simplest form

### Examples on Simplifying Ratios:

Given below are some examples that explain how to simplify ratios.

Example 1:

Write the ratio 12:86 in the simplest form

Solution:

Following the steps;

12:86 = $\frac{12}{86}$

The greatest common divisor of 12 and 86 is 2. So, we divide the numerator and denominator by 2

Therefore, 12:86 = $\frac{12}{86}$ = $\frac{6}{43}$

Thus the reduced form of 12:86 is 6:43

Example 2:

Determine the simplest form of the ratio 28:96

Solution:

28:96 = $\frac{28}{96}$

= $\frac{7}{24}$ , by removing the greatest common factor, 4, from both the numerator and the denominator.

Hence, the reduced form(simplest form) of 28:96 is 7:24.

## Ratio Rates

A rate is a ratio that would compare two quantities with different units. Rates are used to provide information in a variety of situations. The rate at which people drive on an interstate is a ratio of the number of miles driven to the time that has passed. This refers to the speed that we are travelling at.

The vehicle gas mileage is taken as the ratio of the number of miles driven to the volume of gasoline consumed.

 Rate Fraction Driving speed Miles driven / elapsed time Gas mileage Miles driven / gas consumed Density Mass / volume

## How to do Ratios?

Comparing two quantities in terms of the number of times is called a ratio. As an example, consider the weight of a cow and a man.

Let the weight of the cow = 600 kg

and the weight of the man = 60 kg

We can say that the weight of the cow is 10 times that of the man. This is known as the ‘ratio of the weight of the cow and the weight of the man'. This is represented as 10: 1

Ratios can be expressed as fractions. Let us see how.

Ratio of the weight of the cow to that of the man = $\frac{600}{60}$ = $\frac{10}{1}$ , which is equal to 10:1

We can also see that the weight of the man is $\frac{1}{10}$ to that of the cow. Hence, the ratio of the weight of the man to that of the cow = 1:10

Quantities with the same units only can be compared.

For example, the weight of a man is 60kg and that of a grasshopper is 10g.

The ratio of the weight of the man to that of the grass hopper = $\frac{(60\times1000)}{10}$

$\frac{6000}{1}$ = 6000 : 1

We have converted the weight of the man into grams, so that both the quantities are in the same units.

### Equivalent Ratios

Equivalent ratios are same as equivalent fractions.

We know that $\frac{2}{3}$ = $\frac{4}{6}$

Hence, 2:3 is equivalent to 4:6. They are said to be equivalent ratios.

### Ratio Examples:

Given below are some examples on ratios

Example 1:

Find the ratio of 40cm and 8m.

Solution:

The two quantities should be of the same units for comparison. So, let us convert 8m to cm.

8m = 8 x 100cm = 800 cm

Hence, the ratio is $\frac{40}{800}$ = $\frac{1}{20}$

Therefore, the ratio is 1:20

Example 2:

Find an equivalent ratio of 3:5

Solution:

Ratio 3:5 = $\frac{3}{5}$

= $\frac{(3\times2)}{(5\times2)}$ = $\frac{6}{10}$

6:10 is an equivalent ratio of 3:5.

## Ratio Word Problems

We come across problems involving ratios in real life. We have to understand the problem thoroughly and formulate it in terms of ratios to successfully solve it.

A word problem consists of a series of expressions which have to be combined and converted to equations. Then, we have to solve these equations through appropriate methods.

### Steps in solving ratio word problems

• We have to translate the given sentence into equations
• When we deal with a word problem on ratios, we have to convert all quantities into the same unit.
• Then, solve the equation.

### Examples of Ratio Word Problems:

Given below are some examples on ratio word problems.

Example 1:

The perimeter of a rectangle is 18cm, and the ratio of its length to width is 1:2. Find the dimensions of the rectangle.

Solution:

We know that the perimeter of a rectangle = 2( l + w)

Given that the ratio of length and width is 1:2

Let l be the length of the rectangle, then, width = 2l

Thus, we have 2(l+2l) = 18

2(3l) = 18

6l = 18

l = 3

Hence, w = 6

Thus, the length = 3 cm and the width = 6 cm

Example 2:

In a class, the ratio of the number of boys to the total number of students is 2:3. Find the ratio of the number of girls to boys.

Solution:

The ratio of the number of boys to the total number of students is given as 2:3.

Hence, if we take 2x as the number of boys, there will be 3x students in the class. Therefore, the number of girls = 3x - 2x = x

Hence, the ratio of number of girls to the number of boys = 1x : 2x

= 1:2

## Ratios to Percentage

There are a few steps to be followed to convert a ratio into percentage:

• Convert the ratio into a fraction.
• Multiply the fraction by 100.
• The result we get is the required percentage.

### Examples to Convert Ratios to Percentage

Given below are some examples that explain how to convert ratios to percentage.

Example 1:

Convert 3:4 into percentage.

Solution:

3:4 = $\frac{3}{4}$

= $\frac{3}{4}$ x 100

= 3 x 25

= 75 %

3:4 = 75 %

## How to Scale a Drawing

Scale drawings are drawings or illustrations that are proportional in scale to the real structures they represent. Since drawings cannot be as large as the structure, the person doing a drawing would use scale drawings to depict the structure. The miniature versions of the actual structure shows the sizes, shapes, and arrangements of rooms and items like doors, windows etc and other important details of the structure. The scale drawings of these structures must be in exact proportion to the actual structure. Various scales are used for the purpose.

For example, we may take 1/6th of an inch to represent one yard and thus a six yard actual building structure would be depicted in 6 $\times$ $\frac{1}{6}$ = 1 inch long on paper. The measurements could also be in metric scales depending upon the requirement. Scale drawings are mainly used in engineering and architect.