Indirect Proportion - Definition, Formula, Examples

Indirect Proportion, also known as Inverse proportion describes a relationship between two quantities where one increases as the other decreases, and their product remains constant. It offers a simple way to understand how two changing values are linked when they move in opposite directions. 

This concept is often introduced alongside direct proportion, but the two behave very differently. While direct proportion involves quantities that rise or fall together, inverse proportion shows how one value must decrease to balance the increase of the other. In this blog post, we will understand the concept of inverse proportion in detail with examples. 

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What is Inverse Proportion?

Inverse proportion (or inverse variation) states a relationship between two quantities in which one increases exactly as the other decreases, so that their product always remains constant. In other words, if one value doubles, the other is halved.

Let’s say two variables, A and B, are inversely proportional. Then the relationship is written as:

A × B = Constant (k)

or

A ∝ 1/B

This means A is inversely proportional to B.

This implies:

• When A increases, B decreases

• When A decreases, B increases

The key point is that the product of the two variables always remains the same, regardless of their individual values.

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Key Characteristics of Inverse Proportion

Understanding the unique traits of inverse proportion helps differentiate it from other mathematical relationships and apply it correctly in problem-solving. Below are its defining features:

• Variables move in opposite directions: As one variable increases, the other decreases in the same ratio, and vice versa.

• Product remains constant: The product of the two variables is always equal to a fixed constant. Mathematically, x × y = k, where k is the constant of proportionality.

• Expressed using reciprocal relation: The inverse relationship is written as y ∝ 1/x or x ∝ 1/y, meaning one quantity is proportional to the reciprocal of the other.

• Graph is a curve (hyperbola): The graph of inverse proportion is a smooth curve called a hyperbola. It gets closer to both the x-axis and y-axis but never touches them. This shows that neither variable can ever be zero in an inverse proportion.

• Opposite of direct proportion: In direct proportion, both variables increase or decrease together. In inverse proportion, one increases while the other decreases.

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Inverse Proportion Formula

Inverse Proportion Formula

If two quantities A and B are in inverse proportion, then the formula of inverse proportion is:

A ∝ (1 / B)

Which gives, 

A = k / B

or

A × B = k

Here,
A and B are the two variables, and
k is the constant of proportionality.

For two cases of each variable, let's consider A₁ and A₂ are the values of A, and B₁ and B₂ are the corresponding values of B. Then,

A₁ × B₁ = A₂ × B₂ = k

         Or

A₁ / A₂ = B₂ / B₁

This formula is used to solve problems where one quantity increases and the other decreases in such a way that their product remains constant.

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Inverse Proportion Examples

Let’s explore some inverse proportion examples to understand the concept better.

Example 1: A parcel company assigns 36 workers to deliver all parcels in 12 days. How many workers are needed to finish the task in 1 day?

Solution

Let

A₁ = 36 (workers)

B₁ = 12 (days)

B₂ = 1 (day)

A₂ = ?

Using the product formula
A₁ × B₁ = A₂ × B₂

36 × 12 = A₂ × 1

A₂ = 432

The company will need 432 workers to finish the task in 1 day.

Example 2: A team of 5 painters can paint a house in 15 days. How many days will it take if 10 painters do the same job?

Let:

• A₁ = 5 (painters), A₂ = 10 (painters)

• B₁ = 15 (days), B₂ = ?
  
  

Using the ratio formula

A₁ / A₂ = B₂ / B₁

Substituting values

5 / 10 = B₂ / 15

B₂ = (5 × 15) / 10 = 7.5

Hence, 10 painters will complete the work in 7.5 days.

Example 3: A train takes 6 hours to reach a destination at a speed of 60 km/h. How long will it take if the speed is increased to 90 km/h?

Let:

A₁ = 6 (hours), A₂ = ? (hours)

B₁ = 60 (km/h), B₂ = 90 (km/h)

Using ratio formula
A₁ / A₂ = B₂ / B₁

Substituting values

6 / A₂ = 90 / 60

Cross-multiplying

6 × 60 = A₂ × 90

A₂ = 360 / 90 = 4

Answer: The train will take 4 hours at a speed of 90 km/h.

Graph of Inverse Proportion

The graph of an inverse proportion forms a curve known as a hyperbola. Unlike a direct proportion, which produces a straight line, an inverse proportion creates a smooth curve that bends toward both the x-axis and the y-axis but never touches them.

This curve represents the relationship between two quantities where the product of their values remains constant. 

Every point on the graph satisfies the equation A × B = k, where A and B are variables in inverse proportion and k is a constant.

For example, consider two values on the x-axis, A₁ and A₂, such that A₁ is smaller than A₂. 

Their corresponding values on the y-axis, B₁ and B₂, will follow the opposite pattern, where B₁ is greater than B₂. This shows that as one variable increases, the other decreases, yet their product stays the same.

The graph never touches either axis because neither A nor B can be zero. If one of the variables were zero, the product would become zero, which would contradict the condition that the product must be constant and non-zero.

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Difference Between Direct and Inverse Proportion

Direct and inverse proportion describe how two quantities are related, but they behave in completely opposite ways. The table below shows the main differences between them

 Real-Life Applications of  Inverse Proportion

Inverse proportion is not limited to mathematics. It has practical applications in daily life, industry, science, and engineering. Here are a few real-life examples of inverse proportion:

Speed and Time:  If the speed of a vehicle increases, the time to cover the same distance decreases.

Workforce and Project Time:  Increasing the number of workers decreases the time taken to complete a job.

Pressure and Volume in Physics:  According to Boyle’s Law, pressure and volume of a gas are inversely proportional at constant temperature.

Supply and Demand in Economics: As per the law of demand, price and demand are inversely related. This means that when the price of a product increases, the demand for it tends to decrease, and when the price decreases, the demand generally increases.

These examples highlight how understanding inverse proportion helps in interpreting real-world problems.

Also read: Dimensional Formula

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