When two ratios are equal, we can say the ratios are in proportion. There are two types of proportion, direct proportion and inverse proportion. Two quantities are said to be in direct proportion if they increase or decrease together so that the ratio of their corresponding values remains constant.

When the value of one quantity increases concerning the decrease in other or vice-versa, we call it inversely proportional. Proportion determines application in solving many real-life problems such as cooking, business deals during money transactions, etc. Read on to learn more about the types of proportion and their application.

Ratio and Proportion

Ratio: The ratio is an arithmetic concept that is used to compare two or more quantities. It can be expressed as a fraction. It helps to determine how larger or smaller one quantity is to another when it is compared. It can be represented as \(a:b\) or in the fraction as \(\frac{a}{b}.\)

Example: If the cost of a pen is ₹\(28\) and the cost of a pencil is ₹\(7,\) find the ratio of their costs.
The ratio of the cost of the pen to the cost of the pencil is \(\frac{{28}}{7} = \frac{4}{1}\) which is represented by \(4:1.\)

Proportion: A proportion says that two ratios are equal. Proportion is a mathematical comparison of two quantities. Proportions are denoted by \(\left( {::} \right)\) symbol.
If \(a:b = c:d,\) then it is represented as
\(a:b::c:d\)

For example, the time taken by a car to cover \({\rm{60}}\,{\rm{kmph}}\) is equal to the time taken by it to cover the distance of \({\rm{150}}\,{\rm{km}}\) for \(5\) hours. Thus,
\(60\,{\rm{km}}/{\rm{hr}} = 150\,{\rm{km}}/5\,{\rm{hrs}}.\)

Continued Proportion

If the ratio between the first and the second term is equal to the ratio between the second and the third term of any three quantities then, they are said to be in continued proportion. If \(a,\,b,\,c\) are in continued proportion then,
\(a:b = b:c\) or, \(a:b::b:c\)

Thus,
\(\frac{a}{b} = \frac{b}{c} \Rightarrow {b^2} = ac \Rightarrow b = \sqrt {ac} \)
Here, \(c\) is called the third proportion, and \(b\) is called the mean proportion.

Similarly, when four quantities are in a continued proportion, the ratio between the first and second equals the ratio between the third and fourth.
If \(a,\,b,\,c,\,d\) are in continued proportion then,
\(a:b = c:d\) or, \(a:b::c:d\)
Here, \(d\) is called the fourth proportion.

Types of Proportion

Based on the different nature of the relationship between two or more quantities, the proportion can be classified into two different types.

1. Direct Proportion
2. Inverse Proportion

1. Direct Proportion

Direct proportion is the relation between two variables whose ratio is equal to a constant value. In other words, the direct proportion is a condition where an increase in one quantity causes a corresponding increase in the other quantity or a decrease in one quantity results in a reduction of the other quantity.

Examples of Direct Proportion

In our everyday life, we observe variations in the values of multiple quantities depending upon the variation in values of some other quantities.
For example, the amount of petrol you buy is directly proportional to the cost of the petrol that you have to pay. In this case, as mentioned above, two quantities are termed to exist in direct proportion.
Some more examples are:

1. Marks are directly proportional to performance.
2. The cost of the shirts is directly proportional to the number of shirts.
3. If a car is running at a constant speed, then the distance covered by the car and the time taken are directly proportional to each other.

Constant of Direct Proportionality

If two quantities \(a\) and \(b\) existing in direct proportion can be expressed as:
\(a \propto b\)
\(\frac{a}{b} = K\)
\(a = Kb\)
\(K\) is known as the non-zero constant of proportionality.

If \({a_1},\,{b_1}\) are the initial values of two quantities and \({a_2},\,{b_2}\) are the final values of these two quantities in direct proportion, then, they can be expressed as
\(\frac{{{a_1}}}{{{b_1}}} = \frac{{{a_2}}}{{{b_2}}} = K\)

Example: Given that \(y\) is directly proportional to \(x,\) and when \(x = 3,\) then \(y = 9.\) What is the constant of proportionality?
Solution: \(y\) is directly proportional to \(x\)
\( \Rightarrow y = Kx\)
Substituting \(x = 3\) and \(y = 9,\) we get
\(9 = K \times 3\)
\( \Rightarrow K = \frac{9}{3} = 3\)
Therefore, the constant of proportionality is \(3.\)

Direct Proportion Formula

The proportional symbol represents the direct proportion \(\left( \propto \right).\) For example, if two variables \(x\) and \(y\) are directly proportional to each other, this statement can be represented as \(x \propto y.\)

When we replace the proportionality sign \(\left( \propto \right)\) with an equal sign \(\left( = \right),\) the equation changes to \(x = K \times y\) or \(\frac{x}{y} = K,\) where \(K\) is called the non-zero constant of proportionality.

2. Inverse Proportion

When the value of one quantity increases concerning the decrease in other or vice-versa, we call it inversely proportional. If two quantities are in inverse variation, then we also say that they are inversely proportional to each other.

Examples of Inverse Proportion

In our everyday life, we observe inverse proportion relation between two quantities.
It means that the two quantities behave opposite. For example, to finish construction work, if we increase the number of labourers, the time taken to finish the job will reduce and vice versa.

So, the team strength is inversely proportional to the time taken to complete a job. In this case, two quantities are termed to exist in inverse proportion. Some more examples are:

1. Pressure is inversely proportional to the volume of a gas at a given temperature.
2. Speed is inversely proportional to the time taken to travel.

Constant of Inverse Proportionality

If two quantities \(a\) and \(b\) existing in inverse or indirect proportion can be expressed as:
\(a \propto \frac{1}{b}\)
\(ab = K\)
\(K\) is known as the non-zero constant of proportionality.

If \({a_1},\,{b_1}\) are the initial values of two quantities and \({a_2},\,{b_2}\) are the final values of these two quantities in indirect proportion, then, they can be expressed as
\({a_1}{b_1} = {a_2}{b_2} = K\)

For example, If \(y\) is inversely proportional to \(x,\) then it is the same thing as \(y\) is directly proportional to \(\frac{1}{x}.\)
\( \Rightarrow y \propto \frac{1}{x}\)
\( \Rightarrow y = \frac{k}{x}\)

Example: Given that \(y\) is inversely proportional to \(x,\) and when \(x = 3,\) then \(y = 9.\) What is the constant of proportionality?
Solution: \(y\) is inverse proportional to \(x\)
\( \Rightarrow y = \frac{k}{x}\)
Substituting \(x = 3\) and \(y = 9,\) we get
\( \Rightarrow 9 = \frac{k}{3} \Rightarrow k = 27\)
Therefore, the constant of proportionality is \(27.\)

Inverse Proportion Formula

We can represent inverse proportion by using \(\left( \propto \right)\) symbol. For example, if two variables \(x\) and \(y\) are inversely proportional to each other, this statement can be represented as \(x \propto \frac{1}{y}.\) When we replace the proportionality sign \(\left( \propto \right)\) with an equal sign \(\left( = \right),\) the equation changes to \(x = \frac{K}{y}\) or \(xy = K,\) where \(K\) is called the non-zero constant of proportionality.

Solved Examples – Proportion

Q.1. The fuel consumption of a car is \(10\) litres of diesel per \(150\,{\rm{km}}.\) What distance can the car cover with \(5\) litres of diesel?
Ans: The fuel consumed for every \({\rm{150}}\,{\rm{km}}\) covered \(10\) litres
Therefore, the car will cover \(\left( {150/10} \right)\,{\rm{km}}\) using \(1\) litre of fuel.
If \(1\) litre \( = \frac{{150}}{{10}}\,{\rm{km,}}\) then \(5\) litres \( = \left[ {\frac{{150}}{{10}} \times 5} \right]\,{\rm{km}}\)
\( = 45\,{\rm{km}}\)
So, the car can cover \(45\,{\rm{km}}\) using \(5\) litres of fuel.

Q.2. If \(40\) meters of cloth costs ₹\(1940,\) how many meters can be bought for ₹\(727.5\) ?
Ans: Let \(x\) meters of cloth be bought for ₹\(727.5.\) Then, we can write the given information as below.
\(40\) meters \( \to \) ₹\(1940\)
\(x\) meters \( \to \) ₹\(727.5\)
Less money will fetch fewer meters of cloth. So, it is a case of direct variation. Therefore, the ratio of the number of rupees \( = \) the ratio of the number of meters.
\( \Rightarrow \frac{{1940}}{{727.5}} = \frac{{40}}{x}\)
\( \Rightarrow x = \frac{{727.5 \times 40}}{{1940}}\)
\(x = 15\) meter
Hence, we can buy \(15\) meters of cloth for ₹\(727.5.\)

Q.3. Preethu types \(360\) words for half an hour. How many words would she type in \(6\) minutes?
Ans: Let Preethu types \(x\) words in \(6\) minutes. We can write the given information as
\(360\) words \( \to 30\) minutes
\(x\) words \( \to 6\) minutes
More words can be typed with more time. So, it is a case of direct proportion.
Therefore, the ratio of number of words \( = \) Ratio of number of minutes
\( \Rightarrow \frac{{360}}{x} = \frac{{30}}{6}\)
\( \Rightarrow x = \frac{{6 \times 360}}{{30}}\)
\( \Rightarrow x = 72\)
Therefore, Preethu types \(72\) words in \(6\) minutes.

Q.4. If \(a:b = 4:5\) and \(c:d = 3:7;\) find \(a:d.\)
Ans: Given, \(a:b = 4:5\) and \(c:d = 3:7.\)
Here, the means are \(5\) and \(3.\) To find the ratio \(a:d,\) we have to make the means the same.
For that, we need to find the LCM of the means.
So, LCM\( = 5 \times 3 = 15\)
Now, multiplying the first ratio by \(3\) and the second ratio by \(5,\) we have, \(12:15::15:35\)
Hence, \(a:d = 12:35.\)

Q.5. If \(2,\,x,\,8\) are in continued proportion, then find \(x.\)
Ans: We know, If \(a,\,b,\,c\) are in continued proportion then, \(a:b = b:c\) or, \(a:b::b:c.\) Thus, \(\frac{a}{b} = \frac{b}{c} \Rightarrow {b^2} = ac \Rightarrow b = \sqrt {ac} \)
Here, \(2,\,x,\,8\) are in continued proportion.
So, \(\frac{2}{x} = \frac{x}{8}\)
\( \Rightarrow {x^2} = 16\)
\( \Rightarrow x = \sqrt {16} = 4\)
Hence, \(x = 4\)

Summary

In this article, we have learned about the proportion that, when two ratios are equal is known as proportion. Proportion is a mathematical comparison of two quantities. We have discussed the continued proportion, the meaning of direct proportion, inverse proportion, their formula and solved some examples related to those.

Frequently Asked Questions (FAQs) – Proportion

Q.1. What is proportion?
Ans: A proportion says that two ratios are equal. Proportion is a mathematical comparison of two quantities. Proportions are denoted by the \(\left( {::} \right)\) symbol. If \(a:b = c:d.\) It is represented as \(a:b::c:d.\)

Q.2.What is the proportion by which the gratuity will be shared?
Ans: The percent of gratuity amount that an employee wants to distribute to his nominees in case of sudden death is called the proportion gratuity.
For example, if you have made your spouse and mother nominees, then you can allot \(50\% \) of the gratuity to your mother and \(50\% \) to your spouse in case of your sudden death.

Q.3. What is the proportion in statistics?
Ans: In statistics, the proportion means the population proportion. It is denoted by the Greek letter \(^{”}\mu {.^{”}}\) It is a parameter that expresses a percentage value related to a population.

Q.4. What does form a proportion mean?
Ans: To find a mean proportion, three quantities must be in continued proportion.
If \(a,\,b,\,c\) are in continued proportion then, \(a:b = b:c\) or, \(a:b::b:c.\)
Thus, \(\frac{a}{b} = \frac{b}{c}\)
\( \Rightarrow {b^2} = ac\)
\( \Rightarrow b = \sqrt {ac} \)
Here, \(b\) is called the mean proportion.

Q.5. How do you identify direct and inverse proportions?
Ans: We can identify the two quantities are in direct proportion when an increase in one quantity causes a corresponding increase in the other quantity or a decrease in one quantity results in a reduction of the other quantity.
When the value of one quantity increases concerning the decrease in other or vice-versa, we call it inversely proportional.

Q.6. How to calculate proportion?
Ans: If the ratio between the first and the second is equal to the ratio between the second and the third terms of any three quantities then, they are said to be in continued proportion.
If \(a,\,b,\,c\) are in continued proportion then, \(a:b = b:c\) or, \(a:b::b:c.\)
Thus, \(\frac{a}{b} = \frac{b}{c} \Rightarrow {b^2} = ac \Rightarrow b = \sqrt {ac} \)
Here, \(c\) is called the third proportion, and \(b\) is called the mean proportion.

Similarly, when four quantities are in a continued proportion, the ratio between the first and second equals the ratio between the third and fourth.
If \(a,\,b,\,c,\,d\) are in continued proportion then, \(a:b = c:d\) or, \(a:b::c:d.\)
Here, \(d\) is called the fourth proportion.

We hope this article on proportion has provided significant value to your knowledge. If you have any queries or suggestions, feel to write them down in the comment section below. We will love to hear from you. Embibe wishes you all the best of luck!