Direct Proportion: Definition, Symbol, Examples & More

A proportion tells that two ratios are equal. Four numbers are said to be in proportion if the ratio of the first two numbers is equal to the ratio of the last two numbers. There are two types of proportion, Direct Proportion and indirect/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. The symbol to represent the proportionality is \( \propto \).

In this article, we have provided complete details about direct proportion, symbols, examples, etc. Continue reading to know more.

Ratio and Proportion

Ratio:

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

Example: If the cost of television is \(₹ 35000\) and the cost of radio is \(₹ 14000\), find the ratio of their costs.

Solution: The ratio of the cost of television to the cost of radio is \(\frac{{₹35000}}{{₹14000}} = \frac{5}{2}\) which is represented by \(5:2\)

Proportion:

A proportion tells that two ratios are equal.

If two ratios \(p:q\) and \(r:s\) are equal, then \(p,q,r\) and \(s\) are said to be in proportion.

If \(a,{\rm{ }}b,{\rm{ }}c,\) and \(d\) are in proportion if \(a:b = c:d\).

It is represented as \(a:b::c:d\).

Definition of Direct Proportion

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

Sometimes, the word proportional is used without the word direct. They have the same meaning.

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, our earnings are directly proportional to how many hours we work.

Work more hours to get more pay, which means the increase in the value of one quantity also increases the value of another quantity. A decrease in the value of one quantity also decreases the value of the other quantity. In this case, as mentioned above, two quantities are termed to exist in direct proportion.

Some more examples are

1. The amount of heat transferred is directly proportional to the temperature change.
2. Marks are directly proportional to performance.
3. The cost of the groceries is directly proportional to its weight

Constant of 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 and \({a_2},{b_2}\) are the final values of quantities existing in direct proportion, then, they can be expressed as:

\(\frac{{{a_1}}}{{{b_1}}} = \frac{{{a_2}}}{{{b_2}}} = K\)

Example: \(y\) is directly proportional to \(x\), and when \(x = 4\) then \(y = 16.\) What is the constant of proportionality?

Solution: Given: \(y\) is directly proportional to \(x\)

\(\Rightarrow y = Kx\)

Substituting \(x = 4\) and \(y = 16\), we get

\(16 = K \times 4\)

\( \Rightarrow K = \frac{{16}}{4}\)

\( \Rightarrow K = 4\)

Therefore, the constant of proportionality is \(4\).

Direct Proportion Formula

The proportional symbol represents the direct proportion \(( \propto )\). 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 \(( \propto )\) with an equal sign \({\rm{( = )}}\), the equation changes to \(x = K \times y\) or \(\frac{x}{y} = K\), where \(K\) is called the non-zero constant of proportionality.

Inverse/Indirect Proportion

When the value of one quantity increases concerning the decrease in other or vice-versa, we call it inversely proportional. It means that the two quantities behave opposite. For example, speed and time are in inverse proportion with each other. As you increase the speed, the time is reduced to reach the destination. Inverse proportion is also called indirect proportion.

If two quantities are in inverse variation, then we also say that they are inversely proportional to each other.

Examples:

1. Pressure is inversely proportional to the volume of a gas at a given temperature.
2. The number of workers is inversely proportional to the time taken by them to complete the work.
3. Speed is inversely proportional to the time taken to travel.

If \(y\) is inversely proportional to \(x\), then it is the same thing as \(y\) is directly proportional to \(\frac{1}{x}\).

\( \Rightarrow y \propto \frac{1}{x}\)

\( \Rightarrow y = \frac{k}{x}\)

Solved Examples – Direct Proportion

Q.1. The fuel consumption of a car is \(15\) litres of diesel per \(100\;{\rm{km}}\). What distance can the car cover with \({\rm{5}}\) litres of diesel?
Ans: The fuel consumed for every \(100\;{\rm{km}}\) covered \(= 15\) litres.
Therefore, the car will cover \((100/15)\;{\rm{km}}\) using \(1\) litre of fuel.
If \(1\) litre \( = \frac{{100}}{{15}}\;{\rm{km}}\), then \(5\) litres \(= \left[ {\frac{{100}}{{15}} \times 5} \right]{\rm{km}}\)
\( = 33.3\;{\rm{km}}\)
So, the car can cover \({\rm{33}}.{\rm{3}}\,{\rm{km}}\) using \({\rm{5}}\) litres of fuel.

Q.2. If \({\rm{40}}\) 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. Seema types \(540\) words for half an hour. How many words would she type in \(6\) minutes?
Ans: Let Seema types \(x\) words in \(6\) minutes. We can write the given information as 
\(540\;{\rm{words}} \to 30\;{\rm{minutes}}\)
\(x\;{\rm{words}} \to 6\;{\rm{minutes}}\)
More words can be types with more time.  So, it is a case of direct proportion.
Therefore, the ratio of number of words = Ratio of number of minutes
\( \Rightarrow \frac{{540}}{x} = \frac{{30}}{6}\)
\( \Rightarrow x = \frac{{6 \times 540}}{{30}}\)
\( \Rightarrow x = 108\)
Therefore, Seema types \(108\) words in \(6\) minutes.

Q.4. An electric pole, \(8\) meters high, casts a shadow of \(6\) meters. Find the height of a tree that casts a shadow of \(18\) meters under similar conditions.
Ans: Let the height of the tree be \(a\) meters and shadow be \(b\) meters. We know that if the height of the pole increases, the length of shadow will also increase in the same proportion. Hence, we observe that the height of the tree and the length of its shadow exist in direct proportion. In other words height of the pole is directly proportional to the length of its shadow. Thus,
\(\frac{{{a_1}}}{{{b_1}}} = \frac{{{a_2}}}{{{b_2}}}\)
\( \Rightarrow \frac{8}{6} = \frac{x}{{18}}\)
\( \Rightarrow x = \frac{{8 \times 18}}{6}\)
\( \Rightarrow x = 24\) meters
So, the height of the tree is \(24\) meters

Q.5.The amount of extension in an elastic spring varies directly as the weight hanging on it. If the weight of \({\rm{150\;gm}}\) produces an extension of \({\rm{2}}{\rm{.9 \;cm}}\), then what weight would produce an extension of \({\rm{17}}{\rm{.4 \;cm}}\)?
Ans: Let the required weight be \(x{\rm{gm}}\). We can write the given information as 
\(150\;{\rm{gm}} \to 2.9\;{\rm{cm}}\)
\(x\;{\rm{gm}} \to 17.4\;{\rm{cm}}\)
The amount of extension in the spring varies directly as the weight hanging on it. So, it is in direct proportion.
Therefore, the ratio of weights = ratio of extensions
\( \Rightarrow 150:x = 2.9:17.4\)
\( \Rightarrow \frac{{150}}{x} = \frac{{2.9}}{{17.4}}\)
\( \Rightarrow \frac{{150}}{x} = \frac{1}{6}\)
\( \Rightarrow 150 \times 6 = 1 \times x\)
\( \Rightarrow x = 900\)
Therefore, a weight of \(900{\rm{gm}}\) would produce an extension of \({\rm{17}}{\rm{.4 \;cm}}\)

Summary

In this article, we have learned about ratios, the meaning of direct proportion, inverse proportion, the formula for direct proportion, the meaning of constant proportionality, and some examples of direct and inverse proportion. 

Frequently Asked Questions

We have provided some frequently asked questions about direct proportion here:

Q.1. What is the formula for inverse/indirect proportion?
Ans: If two quantities \(x\) and \(y\) are inversely proportional to each other, then, it will be denoted as \(x \propto \frac{1}{y}\) or \(x = \frac{k}{y}\).

Q.2. How do you find the direct proportion?
Ans: In mathematical statements, directly proportion can be found using the equation \(y = Kx\). This reads as \(“y\) varies directly as \(x”\) or \(“y\) is directly proportional to \(x”\).

Q.3. How do you identify direct and inverse proportions?
Ans: If two quantities \(a\) and \(b\) are directly proportional to each other, then it will be denoted as \(a \propto b\). If two quantities \(a\) and \(b\) are inversely proportional to each other, then, it will be denoted as \(a \propto \frac{1}{b}\).

Q.4. What does it mean to be in direct proportion?
Ans: If we want to say the two quantities are directly proportional, their ratio must be equal to a non-zero constant. Mathematically we can say that, if \(a\)  and \(b\) are directly proportional to each other, then, \(a \propto b \Rightarrow \frac{a}{b} = K\) or \(a = Kb\), where \(K\) is a non-zero constant.

Q.5. What are the types of proportion?
Ans: The two types of proportions are:
a. Direct Proportion
b. Inverse/Indirect Proportion

Now that you have a detailed article on direct proportion, we hope you study well. If you have any confusion regarding the same, do let us know about it in the comments section below and we will get back to you soon.