Direct and Inverse Proportions Class 8 Notes CBSE Maths Chapter 11 (Free PDF Download)

Variations And Proportions:

When the values of two quantities depend on one another in a way, such that the change in one quantity causes change in the other, the two quantities are said to be in variation.

Direct Proportion:

• Two quantities $a$ and $b$ are said to be in direct proportion if they vary i.e., they increase or decrease together in such a manner that the ratio of their corresponding values remains constant. That is if $\dfrac{a}{b}=c$ [$c$ is a positive number, then $a$ and $b$ are said to change directly.

• To explain the condition, let ${{a}_{1}},{{a}_{2}}$ be the two values of $a$ and ${{b}_{1}},{{b}_{2}}$ be the values of $b$, then $\dfrac{{{a}_{1}}}{{{b}_{1}}}=\dfrac{{{a}_{2}}}{{{b}_{2}}}$.

• To represent two quantities related proportionately, we write $a\propto b$.

Examples Of Direct Proportion: 

• If the quantity of petrol in a car increases, the total distance covered also increases.

• If the radius of a circle increases, the area of that circle also increases.

Inverse Proportions: 

• The two quantities $a$ and $b$ are said to be in inverse proportion if an increment in one causes a proportional decrement in the other or if an decrement in one causes a proportional increment in the other that is, the product of their corresponding values remains constant. Which means, if $ab=c$ or $a=\dfrac{c}{b}$ [$c$ is a constant], then $a$ and $b$ are said to vary inversely. 

• To explain this case, let ${{a}_{1}},{{a}_{2}}$ be the two values of $a$ and ${{b}_{1}},{{b}_{2}}$ be the values of $b$, then ${{a}_{1}}{{b}_{1}}={{a}_{2}}{{b}_{2}}$

• To represent two quantities related inversely, we write $a\propto \dfrac{1}{b}$. 

For example:

As the speed of a vehicle(bike) increases, the time taken to cover a particular distance decreases.

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About Direct and Inverse Proportion

There are so many situations in our life where we see some direct or indirect relationship between two things, like

• If the number of things purchased is increasing then the bill amount to pay also increases. (direct).

• If the speed of the car will increase then the time to reach the destination will decrease. (indirect).

Any two physical quantities that vary is said to be proportional according to Mathematics if they are multiplicatively connected to each other by a constant term. For example, the more we eat, the more energy we gain, and the more we run, the more energy we lose.

There are two kinds of proportionalities in Mathematics.

When one quantity increases along with the other then it is called directly proportion whereas if one increases and the other decreases then it is inversely proportional.

• Direct proportion meaning: Two quantities are said to be directly proportional to each other if the ratio of their values is constant at any instant of time. The increase in one quantity results in the increase of the other. For example, the more we exercise, the more fit is our body.

• Inverse proportion: Two quantities are said to be inversely proportional to each other if the product of their values at any instant is a constant. The increase in one quantity results in a decrease in the other. Example: The more we eat junk, the less is our physical fitness.

The key difference between direct and inverse proportion are listed in the table below:

Difference Between Direct and Inverse Proportion

Symbol Of Proportion

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When two quantities say l and m are in proportion, then they are written as l ∝ m where ∝ represents “is proportion to”.

There are two methods to solve the problems related to direct proportion-

1. Tabular Method

2. Unitary Method

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Conclusion

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