Ratio and Proportion

Ratio and proportion is one of the most important concepts in Mathematics. The concept of the ratio is used for the comparison of two quantities, whereas the concept of proportion is used to show equivalency between given ratios.

The ratio of two numbers a and b is given by a divided by b and represented by a/b or a:b

While the proportion is the relation between two ratios of a/b and c/d is represented by a:b::c:d or a/b=c/d.

Here in this article, we are going to discuss the concepts of ratio and proportion, then some tricks with examples.

Ratio and Proportion in Daily Life

Ratio and Proportion are the concepts used in Mathematics and in daily life. For instance, the concept of ratio is found in many daily life things like price (rupees/metres), speed (distance/time), etc. Similarly, you can also find concepts of proportion also in real life. 

Let's take an example; 

Weight and rate of potatoes: 1kg of potatoes costs 20 rupees, then 2kg of potatoes costs 40 rupees such that 1/20=2/40. There are many more examples which we will discuss later in this article.

Real-life Example of Ratio 

Whenever we compare two quantities, then we compare them using a ratio. For example, if there are two people who go to a nearby city on any day of the week. Let A travels at the speed of 50 Km/hr and B travels at the speed of 10 Km/hr. Then the ratio of both of the speeds will be 

Speed of A/ Speed of B = 50 / 10 = 5 /1  

So we can write it as 5:1. This ratio will be read as A is to B.

Ratio and Proportion

The ratio is used to compare any of the two quantities.  The ratio of two numbers a and b is denoted by the a/b or a:b. Also, When two or more such ratios become equal to each other, then they will be said to be in proportion. The concept of ratio and proportion is completely based on fractions. The ratio and proportion are some of the basic concepts of mathematics.

Ratio and proportion have a very important role in solving many real-life problems.  It is important when we compare heights, weights, distances, time, etc. Also, it is important while dealing with business transactions or while adding ingredients in cooking, and so on.

What are Ratio and Proportion?

Comparison of two quantities of the same kind by division is termed a ratio. When two ratios are equal they are said to be in proportion. However, the rate is also a type of ratio. Thus, a ratio can also be called a way to compare two or more quantities using the division method. If the price of a pen is 2 rupees per item then it compares money and quantity. 

The ratio of x to y can be  written can be represented in three forms :

• x to y

• x : y

• x/y

If a proportion is read as x is to y as z is to w. x/y = z/w where w & y are not equal to 0. And so, x : y: : z: w

A proportion is generally an equation according to which two ratios are equivalent.

Definition of Ratio

The ratio is the relationship between two quantities. It shows the amount of one quantity in the other one. It is used to compare two quantities of which magnitude is given to us. It follows the division method and hence we can write the ratio of a is to b as a divided by b. The ratio can also be defined as the comparison of two quantities of the same kind. This depicts how much a quantity is equal to another one. 

It is the fraction that is used to denote this comparison of two quantities. A ratio can be denoted by the symbol of ‘:’. 

Definition of Proportion

Proportion can be defined as the part of a share between two quantities compared to another one. Two ratios that are equivalent are called in proportion. It denotes that two ratios are equal to each other. It is a mathematical comparison of numbers.  Ratios are said to be directly proportional if increasing or decreasing one of them has the same effect on another one. These are denoted by the symbol of ‘::’. 

The proportion is of two types : 

• Direct Proportion

• Inverse Proportion

Direct Proportion

It is used to denote the direct relationship between two quantities. If one of these quantities increases, the other quantity also increases. Also if one of them decreases another one also decreases. 

Thus, a direct proportion is denoted as y ∝ x. For example, you must have noticed that if the speed of a motorcycle is increased, then it covers more distance in the same amount of time.

Inverse Proportion

Inverse proportion describes the indirect relationship between any two quantities.  

If one of the quantities from these increases, others start decreasing, and also if one decreases the other one starts increasing. So it is said to be inversely proportional.

Suppose there is a vehicle. If the speed of a vehicle is increased, then it will take less time to cover a distance.

Ratio and Proportion Formula

If two quantities let A and B are in ratio then the formula to find this ratio would be A/B. 

For example, the ratio of  8 and 16. So here the value of A is 8 and the value of B is 16.

According to which the ratio of these quantities would be\[ = \frac{8}{16} =  \frac{1}{2}\].

Similarly, if we are given that two ratios 2:3 and 5: x are in proportion then the value of x will be:

\[ \frac{2}{3} =  \frac{5}{x}\].

\[ \Rightarrow  x = \frac{15}{2} \]. 

NOTE:-  When the ratio is multiplied and divided by the same non-zero number, there will be no effect on the ratio.

Tricks for Ratio and Proportion

• If a/b = c/d, then ad = bc

• If a/b = c/d, then a/c = b/d

• If a/b = c/d, then b/a = d/c

• If a/b = c/d, then (a+b)/b = (c+d)/d

• If a/b = c/d, then (a-b)/b = (c-d)/d

• Componendo -Dividendo Rule = If a/b = c/d, then (a+b)/ (a-b) = (c+d)/(c-d)

• If a/b = b/c, then a/c = a2/b2

• If a/b = c/d, then a = c and b =d

• If a/(b+c) = b/(c+a) = c/(a+b) and a+b+ c ≠0, then a =b = c

Few Solved Examples

Question 1 

There are 30 girls and 35 boys in a class. Find the ratio of the no. of boys to the total no. of students.

Solution

The ratio of number of boys to total no. of students, 

35/75= 7/15 = 7:15

Question 2

Determine if the following are in proportion.

1. 15,45,40,120

2. 32,48,70,210

    Solutions

1. 15/45   = 1/3 

40/120 = 1/3 

Hence, 15/45 = 40/120 

Then these are in proportion.

2. 32/48   = 2/3  

70/210 = 1/3 

Hence 32:48 and 70:210 are not equal. These are not in proportion.

Question 3

If the cost of a clothing item is Rs 2170, find the cost of 5m cloth.

Solution

Cost of 7m cloth = 2170

Cost of 1m cloth = 2170/7 = 310

Cost of 5m cloth = 310*5 = 1550

Question 4

The weight of 72 books is 9 kg. What is the weight of 40 such books?

Solution

Weight of 72 books = 9kg

Weight of 1 book = 9/72kg = 1/8   

So, weight of 40 books = 1/8*40 = 5kg

Question 5

Determine the proportion of the given ratios. Also, write the means and extremes. 25cm:1m and rs40:rs160 

Solution

 25cm = 25/100 

= 0.25m

0.25/1 = 1/4

40/160 = 1/4

Hence, these terms are in proportion.

Means = 1m and rs 40

Extremes = 25cm and rs 160. 

Conclusion

The concept of ratio and proportion is easy to comprehend when you focus on their meanings and refer to the solved examples. Learn how to differentiate between a ratio and a proportion. This is how you can easily grab hold of the concept properly.