Proportion uses to solve various problems in everyday life, such as in business when dealing with transactions or in the kitchen, for example. It establishes a relationship between two or more quantities, making comparison easier. When the value of one item rises concerning a decrease in another or vice versa, two quantities are said to be inversely proportional. This means that in nature, these two quantities are opposed. For example, the time it takes to perform a task lowers as the number of workers completing it increases, and it increases as the number of workers drops. In this article, we will be covering Inverse Proportion in detail.

Proportionality

We study quantities that depend on one another in mathematics and physics, and these values are referred to as proportional to one another. In other words, two variables or quantities are proportional to one another, meaning that if one is changed, the other also changes by the same amount. Proportionality is the word for this feature of variables, and the sign for proportionality is proportionality is “\( \propto \)”.

There are two categories of proportionality of variables. They are:

1. Directly Proportional
2. Inversely Proportional

Directly Proportional

Variables or quantities that are directly proportional are those in which as one increases, the other increases as well. They are directly proportional when an increase in one quantity causes an increase in the other and vice versa.

Example: If the number of individuals visiting a resort increases, earning of the resort also increases and vice versa.

Direct Proportion Formula

According to the direction proportion formula, if amount \(y\) is directly proportional to quantity \(x\), we can say:

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

The formula for direct proportion is \(x=k y\), where \(k\) is proportionality constant.

Here, as \(x\) grows, \(y\) grows as well. And, as \(x\) drops, \(y\) decreases as well.

Inverse Proportion

Two quantities are inversely proportional when an increase in one causes a decrease in the other and vice versa. 

Consider the relationship between speed and time. Speed and travel time are inversely linked because the faster we travel, the less time we spend travelling, i.e., the faster we travel, the less time we spend travelling.

Inverse Proportion Examples

1. If we increase the bike’s speed, then the time taken to reach the destination decreases.
2. The brightness of the sunlight decreases as the distance from the sun increases.
3. More vehicles on the road mean there is less space on the road.
4. The lesser the number of workers, the more time it takes to finish the work and vice versa.
5. The number of people and the share each gets from a fixed quantity of something in inverse proportion.

Inverse Proportion Formula

The proportional relationship between two quantities is denoted by the symbol “\(\propto \)”. Let \(x\) and \(y\) be two different numbers. If \(y\) is inversely proportional to \(x\), it is the same as if \(y\) is directly proportional to \(\frac{1}{x}\). Then \(y\) is said to be inversely proportional to \(x\) and is expressed as \(y \propto \frac{1}{x}\) in mathematics.

Hence, we shall write, \(y=\frac{k}{x}\) where \(k\) is the proportionality constant, is the general equation for inverse proportion.

Two quantities \(x\) and \(y\) are said to vary inversely if there exists a relation of the type \(x y=k\) between them, where \(k\) is a positive constant number.

If \(x_{1}\) and \(x_{2}\) are two values of \(x\), and \(y_{1}\) and \(y_{2}\) are the corresponding values of \(y\). Then, \(x_{1} \times y_{1}=x_{2} \times y_{2}=\,\text {constant}\)

Therefore, \(\frac{x_{1}}{x_{2}}=\frac{y_{2}}{y_{1}}\)

An increase in one quantity leads to a decrease in the other quantity in the inverse proportion.

This result is important as it helps us to solve inverse variation problems.

Inverse Proportion Graph

When two quantities have an inverse relationship, when one rises, the other falls, we get a curved graph when we graph this relationship.

If \(y\) is inversely proportional to \(x\), it is the same as if \(y\) is directly proportional to \(\frac{1}{x}\). Then \(y\) is said to be inversely proportional to \(x\) and is expressed as \(y = k \times \frac{1}{x}.\)

Difference Between Direct and Inverse Proportional Graph

Applications of Inverse Proportion

Inversely proportionality is a term that is commonly utilized in everyday life. Inverse proportion helps to solve numerous problems in science, statistics, and other fields. In physics, the concept of inverse proportionality is used to create several formulas. Ohm’s law, the speed and time relationship, the wavelength and frequency of sound are only a few examples.

Solved Examples – Inverse Proportion

Q.1. A brick wall is being cemented by 12 men. The work is completed in 18 days. If 18 men do the same work, how long will it take to complete the work?
Ans: Let the number of days to complete the work \(18\) days be \(y\).
The more men, the less will be the time taken.
Therefore, the two quantities vary inversely.

Number of men \((x)\)	\(12\)	\(18\)
Number of days \((y)\)	\(18\)	\(y\)

So, \(\frac{x_{1}}{x_{2}}=\frac{y_{2}}{y_{1}}\)
\(\Longrightarrow \frac{12}{18}=\frac{y}{18}\)
\(\Rightarrow y=\frac{18 \times 12}{18}\)
\(\Rightarrow x=12\)
Therefore, the time taken is \(12\) days.

Q.2. If 40 workers can finish a job in 15 days, how many workers should be employed if the job is to be finished in 8 days.
Ans: Let the number of workers to be employed to finish the job in \(8\) days be \(x\). 
The more the number of workers, the less will be the time taken.
Therefore, the two quantities vary inversely.

Number of workers \((x)\)	\(40\)	\(x\)
Number of days \((y)\)	\(15\)	\(8\)

So, \(\frac{x_{1}}{x_{2}}=\frac{y_{2}}{y_{1}}\)
\( \Rightarrow \frac{x}{{40}} = \frac{{15}}{8}\)
\( \Rightarrow x = \frac{{40 \times 15}}{8}\)
\(\Rightarrow x=75\)
Therefore, \(75\) workers are required.

Q.3. 20 women can do a job in 5 days. In how many days can 25 women do the same job?
Ans: Let the number of days to complete the job is \(y\).
The more the number of workers, the less will be the time taken. 
Therefore, the two quantities vary inversely.

Number of women \((x)\)	\(20\)	\(25\)
Number of days \((y)\)	\(5\)	\(y\)

So, \(\frac{x_{1}}{x_{2}}=\frac{y_{2}}{y_{1}}\)
\(\Longrightarrow \frac{20}{25}=\frac{y}{5}\)
\(\Rightarrow y=\frac{20 \times 5}{25}\)
\(\Rightarrow x=4\)
Therefore, in \(4\) days, \(25\) women do the same job.

Q.4. 12 men can dig a pond in 8 days. How many men can dig it in 6 days?
Ans: Let the number of men dig a pond is \(x\).
The more the number of workers, the less will be the time taken.
Therefore, the two quantities vary inversely.

Number of men \((x)\)	\(12\)	\(x\)
Number of days \((y)\)	\(8\)	\(6\)

So, \(\frac{x_{1}}{x_{2}}=\frac{y_{2}}{y_{1}}\)
\( \Rightarrow \frac{{12}}{x} = \frac{6}{8}\)
\(\Rightarrow x=\frac{12 \times 8}{6}\)
\(\Rightarrow x=16\)
Therefore, in \(16\) men can dig the same job in \(6\) days.

Q.5. 20 kg rice lasts 30 days in a family of 8 people. If 2 guests stay with the family. How many days will 20 kg of rice last?
Ans: Let the time the rice lasts when the number of people is \(8+2=10\), be \(y\).

Number of people \((x)\)	\(8\)	\(10\)
Time the rice lasts \((y)\)	\(30\)	\(y\)

The more the people, the less the time the rice will last. Hence, the number of people and time the rice will last vary inversely.
We therefore have,
So, \(\frac{x_{1}}{x_{2}}=\frac{y_{2}}{y_{1}}\)
\(\Rightarrow \frac{8}{10}=\frac{y}{30}\)
\(\Rightarrow y=\frac{30 \times 8}{10}\)
\(\Longrightarrow y=24\)
Therefore, the rice will last in \(24\) days.

Summary

When two quantities are inversely proportional, that is, when an increase in one causes a decrease in the other and vice versa, they are called inversely proportional. When one variable decline in inverse proportion, the further increases in the same proportion. This article includes the definition of proportion, direct and indirect proportion, formulas, and graphs.

It helps for a better understanding of the inverse proportion. The outcome of this inverse proportion topic helps in solving different problems based on it.

Frequently Asked Questions (FAQs) – Inverse Proportion

Q.1. What is the formula of inverse proportion?
Ans: The proportional relationship between two quantities is denoted by the symbol “\(\propto\)”. Let \(x\) and \(y\) be two different numbers. If \(y\) is inversely proportional to \(x\), it is the same as if \(y\) is directly proportional to \(\frac{1}{x}\). Then \(y\) is said to be inversely proportional to \(x\) and is expressed as \(y \propto \frac{1}{x}\) in mathematics.
\(y=\frac{k}{x}, w\) here \(k\) is the proportionality constant, is the general equation for inverse variation.

Q.2. What is an inverse proportion?
Ans: When two quantities are inversely proportional, that is, when an increase in one causes a decrease in the other and vice versa, they are called inversely proportional. When one variable decline in inverse proportion, the further increases in the same proportion.

Q.3. What is the formula of direct and inverse proportion?
Ans: \(x \propto y\) is written when two quantities \(x\) and \(y\), are directly proportional (or vary directly). The symbol “\(\propto\)” means “is proportionate to”. \(y \propto \frac{1}{x}\) is written when two quantities \(x\) and \(y\) are in inverse proportion (or vary inversely).

Q.4. What is the difference between direct and inverse proportion?
Ans: If we split two matching amounts in a direct proportion, the ratio between them remains the same. On the other hand, in an inverse or indirect proportion, as one item increases, the other automatically declines.

Q.5. What is k in inverse proportion?
Ans: If \(y\) is inversely proportional to \(x\), it is the same as if \(y\) is directly proportional to \(\frac{1}{x}\). Then \(y\) is said to be inversely proportional to \(x\) and is expressed as \(y \propto \frac{1}{x}\) in mathematics.
Hence, \(y=\frac{k}{x}\), where \(k\) is the proportionality constant, is the general equation for inverse variation.

We hope this detailed article on the concept of inverse proportion helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!