What are ratios and proportions? How do you calculate them?

In mathematics, ratio and proportion are the terms that behave like a key foundation for the other concepts. They are widely used to compare distance, time, weights, and heights. These terms are also helpful in cooking by adding ingredients. 

In this lesson, we are going to explain the relationship, difference, and definitions of ratio and proportion. We will also learn how to use ratios and proportions in calculating mathematical problems.  

Ratio and proportion – introduction

In mathematics, the ratio is the comparison of 2 quantities by finding the quotient. It is denoted such as a : b or a / b. On the other hand, proportion is the equation of two ratios. It is denoted by double semi-colons “: :” and is written as a : b :: c : d. It can also be written as a/b : : c/d

1. Ratio

The ratio is the link or comparison of two objects or quantities which is found by taking the quotient of the first and the second quantity. The ratio can be written as if two quantities with the same kind and unit and the second quantity is not equal to zero then the division of u / v is known as the ratio of u & v.

Formula of ratio

The ratio formula is 

u : v ⇒ u / v

 where u is known as an antecedent (first quantity) and v is known as a consequent (second quantity).

 Proportion

The equality between two ratios is known as the proportion. If the quantities u & v are equivalent to the quantitates w & x then they are in proportion. It is denoted by a double semicolon sign (: :). 

This term is widely used to calculate the unknown quantity with the help of the other three values. If the two fractions or ratios are equal, then they are used to denote them in proportion. Let us take an example of the proportion in which two equivalent ratios are involved.

If the non-zero terms u : v :: w : x is 12 : 6 :: 24 : 12, then we have to calculate the ratios one by one to determine whether they are equivalent or not. The first ratio is 12 : 6  ⇒ 12/6 = 2 and the other ratio is 24 : 12  ⇒ 24/12 = 2.

Hence, both the ratios are equivalent so these ratios are proportional. There are two further types of the proportion such as 

1. Direct proportion 

2. Inverse proportion

1. Direct proportion

The proportion in which an increase in the first quantity causes an increase in the other quantity is known as a direct proportion or directly proportional. The decrease in the first quantity causes a decrease in the second quantity also referred to as the direct proportion. 

2. Inverse proportion

The proportion in which an increase in the first quantity causes a decrease in the other quantity is known as an inverse proportion or inversely proportional. The decrease in the first quantity causes an increase in the second quantity is also referred to as the inverse proportion. 

How to calculate ratios and proportions?

The calculation of ratios and proportions can be done easily by using their formulas. Let us take some examples to understand how to calculate the ratio and proportion manually. 

Example 1: For proportion

10 eggs are required to make 100 biscuits; how many eggs are required to make 1000 biscuits? 

Solution 

Step 1: First of all, take the given information about eggs & biscuits. 

Eggs required to make 100 biscuits = 10

Eggs required to make 1000 biscuits = y

Step 2: Now write the general expression of finding the proportion with two equivalent ratios. 

u : v :: w : x

Step 3: Now write the number of eggs and biscuits in the above expression of proportion to calculate the unknown value.

eggs : biscuits : : eggs : biscuits

10 : 100 : : y : 1000

It can also be written as 

10 : 100 = y : 1000

Step 4: Now calculate the unknown value y by making the ratios in the form of a fraction.  

10/100 = y/1000

Step 4: Now multiply by 1000 on both sides to calculate the unknown term.

1000 *10/100 = 1000 * y/1000

10 * 10 = 1 * y

100 = y

Alternately

Cross multiply the given expression.

10 * 1000 = y * 100

10000 = 100y

10000/100 = y

100 = y

Hence, 100 eggs are required to make 1000 biscuits. 

A proportions calculator is a helpful source for calculating the unknown value of any object. 

Example

In a water container, there are 24 fish. From these 24 fishes, 6 are red, 8 are yellow, and 10 are orange. Calculate the ratio of:

Solution 

Step 1: First of all, write the given number and colors of the fish. 

Total fishes = 24

Red fishes = 6

Yellow fishes = 8

Orange fishes = 10

Step 2: First of all, find the ratio of red fishes to yellow fishes.

Number of red fishes = 6

Number of yellow fishes = 8

The ratio will be written as:

Red fishes : yellow fishes

6 : 8

3 : 4

The above term can also be written as:

6/8

3/4

Step 3: Now determine the ratio of orange fishes to total fishes

Number of total fishes = 24

Number of orange fishes = 6

The term will be referring as a part to whole ratios

The ratio will be written as:

Orange fishes : Total fishes

10 : 24

5 : 12

The above term can also be written as:

10/24

5/12

Step 4: Now find the ratio of orange fishes to red fishes. 

Number of orange fishes = 10

Number of red fishes = 6

The ratio will be written as:

orange fishes : red fishes

10 : 6

5 : 3

The above term can also be written as:

5 / 3

Step 5: Now determine the ratio of red fishes to orange fishes

Number of red fishes = 6

Number of orange fishes = 10

The term will be referring as a part to whole ratios

The ratio will be written as:

red fishes : orange fishes

6 : 10

3 : 5

The above term can also be written as:

6/10

3/5

Wrap up

Now you can easily understand the concept of ratios and proportion with the help of the above discussion and examples. The ratio and proportion are discussed above with the help of examples to understand more accurately. 

The ratio is the comparison of the two-term and the proportion is nothing but the equation of two equivalent two fractions or ratios.

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