Understanding Proportions with the help of Ratio

Ratios and proportions are widely known concepts. We have to deal with them on daily basis. Which makes is very essential to learn them properly. 

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. 

Definition – Ratios

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

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.

How many types of Ratios are there?

There are two types of ratio.

Compound Ratio

For two or more ratios, if we take antecedent as product of antecedents of the ratios and consequent as product of consequents of the ratios, then the ratio thus formed is called mixed or compound ratio. As, compound ratio of m : n and p :  q is mp : nq.

Duplicate Ratio

Duplicate ratio is the ratio of two equal ratios. For example, the duplicate ratio of the ratio x : y is the ratio x2 : y2.

Definition – Proportions

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  

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.

In a proportion “a : b :: c : d”

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

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.  

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 solve ratios?

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

Example:

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:

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

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

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:

Prove that the ratios 4:8 and 2:4 are in proportion.

Solution:

Write both ratios in proportion.

4 : 8 :: 2 : 4

Simplify both ratios by dividing with the common numbers. We get,

 2 : 4 :: 1 : 2

By simplifying more, we get,

1 : 2 :: 1 : 2

As we know, if both ratios are equal, proportion exists between both of them.

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

½ = ½ 

2 = 2 

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

Solve the fractions for decimal numbers.

4/8 = ½ = 0.5

2/4 = ½ = 0.5

Both of the decimal values are the identical which means that the proportion exists between these two ratios.

Under this heading, we will solve the problems related to ratios and proportions. Proportion calculator could be very handy in case you are looking to solve the proportions.

  1. 1.    There are two buckets of water. In the first bucket we mix oil into the water in a ratio of 5:8. In the second bucket, we mix another fluid in a ratio of 3:5. Which ratio is bigger?

Solution:

Step 1: Solve fractions and convert them into decimal values.

5:8 = 5/8 = 0.625

4:5 = 4/5 = 0.6

Step 2: Compare both values.

 As 0.625 > 0.6, so the ratio 5:8 is greater than 4:5.

  • 2.    Find x in proportion, 2 : x : : 20 : 50.

Solution:

Step 1: Convert the ratio in fractions and solve them to find the variable x.

 2 × 50 = 20x

100 = 20x

x = 100/20

x = 5