Ratio and Proportion: Defined and Explained with Calculations

Ratios and proportions are widely practiced concepts. We have to deal with them on a daily basis. This is why it is necessary to learn about them. 

In math, both of these concepts are greatly related to fractions. It is because the ratio is fundamentally a fraction. And proportion is the comparison of two fractions. 

It is no wonder if you are vaguely familiar with ratios. For example, you would have come across situations where the ratio of one group is higher than the other.

Similarly, You might have an idea about proportions already. For instance, you know that a motorcycle is faster than a bicycle in the same way a sports car is faster than a normal car.

In this blog post, we will discuss ratios and propositions with examples, so that by the end of this post you will have no confusion. 

What are Ratios?

Ratio can be defined as,

A ratio compares the quantity of one kind to the quantity of another kind. For ratios to exist, the quantities should have the same units. 

Ratios are represented as:

a:b or a/b

The symbols of both slash (/) and colon (:) are used to represent a ratio. The first value, here a, is known as antecedent. The second value, b in this case, is called the consequent.

Types of Ratios

There are two ways ratios exist.

• Part to part ratios

A ratio in which two different quantities are compared. For instance, in a cake mixture, the ratio of flour to oil or milk. At a party, the ratio of bottles of orange juice to the bottles of mango or strawberry juice.

• Part to whole ratios

A ratio in which a quantity is compared to the whole amount. For example, a ratio of males in a city to the total population of the city.  

What are Proportions?

A proportion can be defined as,

A proportion compares two ratios. Through a proportion, we get to know that in two ratios a:b and c:d, a is related to b in the same way as c is related to d. 

The proportion of two ratios is represented as:

a : b : : c : d   or   a : b = c : d

The double semicolon (: :) or equal to (=) tells that the two ratios mentioned are in equal proportion. The values a and d are known as extremes and the values b and c are known as means.

Parts of Proportion

In a proportion “a : b : : c : d”

• The mean between b and c is √ab.
• c is third proportional to a and b.
• d is fourth proportional to a, b and c.

What do Ratios and Proportions tell us?

In a ratio, one quantity is compared in terms of “times” with the other quantity. This means if A and B are in a ratio of 3:1, then A is 3 times the quantity of B.  

For example, In a cake recipe, flour is 2 times the quantity of oil. It can be represented in ratio as

2 : 1

A proportion tells a person that two ratios in proportion increase or decrease in a similar way. 

We can see this in an example. Previously we discussed that in a cake recipe, flour is two times the quantity of oil. This recipe is to make a one-pound cake. 

To make a 2-pound cake, increase the quantity of both in the same way, we will multiply two on both sides.

= 2 x 2 : 2 x 1

= 4 : 2

This ratio is in proportion to our previous ratio 2:1. It can be written as,

 2 : 1 :: 4 : 2

How to find Ratios and Proportions?

It will be easy to learn about ratios with the help of an example.

Example - Ratio

In a fruit basket, there are 15 fruits. Out of these 15 fruits, 6 are bananas and 3 are apples. Find the ratio of 

• Bananas to rest of the fruits
• Apples to total fruits
• Apples to bananas

Solution:

Step 1: For Bananas:

To find in which ratio bananas are present in the fruit basket, we will have to subtract bananas from the total number of fruits. It is a “part to part” ratio.

Total number of bananas = 6

Total fruits in the basket   = 15

Fruits - bananas  = 15 - 6 = 9

This means there are 9 more fruits in the basket. The ratio of bananas to the rest of the fruits is:

6 : 9

After dividing by 3: 

2 : 3

Step 2: For apples to the total fruits.

We will not need to subtract no. of apples from no. of fruits in this part. This is because we have to find the “part to the whole” ratio.

No. of apples = 3

No. of total fruits = 15       

The ratio is:

3 : 15            

After simplifying,

1 : 5

Step 3: Apples to bananas.

It is also a “part to part” ratio. 

No. of apples = 3

No. of bananas = 6

The ratio is,

3 : 6

Simplifying: 

1 : 2

A proportion can be formed by multiplying or dividing the ratio by the same number on both sides. After that writing the new ratio in proportion (with double semicolon i.e ::) the old ratio.

We usually determine whether a proportion exists or not. We can prove this in 3 ways. Let’s see how to do that with the help of an example.

Example - Proportion

Prove that the ratios 3:6 and 6:12 are in proportion.

Solution:

Step 1: By simplifying. 

3 : 6 : : 6 : 12

Dividing the first ratio by 3 and the second ratio by 6.

1 : 2 : : 1 : 2

Both ratios are equal, hence proportion exists.

Step 2: Cross multiplication.

Write the ratios in the fraction form and cross multiply the fractions.

3 x 12 = 6 x 6

36 = 36

Both sides are the same. Hence, the proportion is true.

Step 3: Decimals.

Solve the fractions for decimal numbers.

3/6 = 0.2

6/12 = 0.2

Both of the decimals are the same. Proportion exists.

Problems related to Ratios and Proportions

Under this heading, we will solve the problems related to ratios and proportions.

Problem No. 1

There are two black paint cans. In the first can we mix white paint into the black paint in a ratio of 6:9. In the second can we mix white in a ratio of 4:5. Which ratio is bigger?

Solution:

By bigger ratio we mean, In which fraction (can) antecedent (white paint) is in more quantity. It is very easy to find. Just solve the fractions and compare.

Step 1: Solve fractions.

6:9 = 6/9 = 0.6

4:5 = 4 /5 = 0.8 

Step 2: Compare.

 As 0.8 > 0.6, so the ratio 4:5 is greater.

You can also find this through the ratio calculator.

Problem No. 2

Find a in proportion, 4 : a : : 25 : 100.

Solution:

Step 1: Write the ratios in fractions and cross multiply.

 4 x 100 = a x 25

a = (4 x 100) / 25

a = 400 / 25

a = 16

 Try solving this problem using the proportional calculator.