Direct and Inverse Variation: Direct and inverse variation can be applied to our day-to-day lives. 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 and indirect/inverse. These are otherwise known as direct and inverse variations.

Two quantities are said to be in direct variations if they increase or decrease together, so 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 representing the proportionality is ∝.∝. Read through the article to learn more about inverse variation examples, direct and inverse variation formulas, and much more!

Direct and Inverse Variation: Ratio and Proportion

Ratio: An arithmetic concept used to compare two or more numbers is known as a ratio. It is a way of comparing quantities or numbers by division. A ratio can be expressed as a fraction. It assists us in identifying how larger or smaller one quantity is than another when it is compared. It can be represented as \(x:y\) or in fractions as \(\frac{x}{y}.\)

Example: If the cost of the refrigerator is \(₹32000\) and the cost of the radio is \(₹12000,\) find the ratio of their costs.

The ratio of the cost of the refrigerator to the cost of the radio is \(\frac{{{\rm{ ₹ }}32000}}{{ ₹ 12000}} = \frac{8}{3}\) which is represented by \(8:3\)

Proportion: A proportion tells that two ratios are equal. It is also known as variation. If two ratios \(a:b\) and \(c:d\) are equal, then \(a, b, c\) and \(d\) are in proportion.

If \(p,q,r,\) and \(s\) are in proportion if \(p:q=r:s.\)

It is represented as \(p:q∷r:s.\)

Direct and Inverse Variation: Direct Variation

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

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

Direct and Inverse Variation: Applications of Direct Variation

In our everyday life, we notice variations in the values of multiple quantities depending on the variation in values of some other quantities.

For example, our earnings are varied directly to how many hours we work.

Work more hours to urge more pay, which suggests the rise in the value of one quantity also increases the worth of another quantity. A decrease in the value of one quantity also decreases the worth of the opposite quantity. In this case, as mentioned above, two quantities are termed to exist in direct variation.

Some more examples are as follows:

1. The amount of heat transferred is directly proportional to the temperature change.
2. The marks of a student are directly proportional to his hard work.
3. The cost of the vegetables is directly proportional to their weight.
4. The number of family members is varied directly with the expenditures.  
5. The number of fruits is directly proportional to the price of fruits.
6. The number of selling products is directly proportional to profit.
7. The distance covered by the body is directly proportional to the speed of the body.
8. The acceleration produced in the car is directly proportional to the change in velocity.
9. The cost of the electricity bill is directly proportional to the number of fans running in the house.
10. The quantity of water filled in the tank is varied directly with the time for which the motor is switched on.
11. The bank balance is directly proportional to the amount of savings.

Direct and Inverse Variation: Inverse Variation 

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

If two quantities are in inverse variation, we can say that they are inversely proportional.

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

\( \Rightarrow a \propto \frac{1}{b}\)

\( \Rightarrow a = \frac{k}{b}\)

Direct and Inverse Variation: Applications of Inverse Variation

Let us look at the applications of Inverse Variation:

1. The bank balance is inversely proportional to spending.
2. The number of family members (who do not work) is inversely proportional to savings.
3. The working days required to complete the work are inversely proportional to the number of labourers.
4. The velocity of the body is inversely proportional to time.
5. The acceleration is inversely proportional to time.
6. The acceleration of the body is inversely proportional to the weight of the body.
7. The number of students who fail in the class is inversely proportional to several qualified teachers and hardworking.
8. The battery power is inversely proportional to the time for which it is used.
9. The ability to work hard is inversely proportional to the age of a man from youngness to the end of life.
10. The number of mistakes in work is inversely proportional to practice.

Direct and Inverse Variation: Solved Examples

Let us look at the solved examples of Direct and Inverse Variations:

Q.1: The fuel consumption of a bike is \(30\) litres of petrol per \({\rm{100}}\,{\rm{km}}{\rm{.}}\) What distance can the car cover with \(5\) litres of petrol?
Ans: The fuel consumed for every (\({\rm{100}}\,{\rm{km}}\)) covered\(= 30\) litres
Therefore, the bike will cover \({\rm{100/30}}\,{\rm{km}}\) using \(1\) litre of fuel.
If \(1\) litre\(=\frac{{{\rm{100}}}}{{30}}\,{\rm{km,}}\) then \(5\) litres\( = \left[ {\frac{{100}}{{30}} \times 5} \right]{\rm{km}}\)
\({\rm{ = 16}}{\rm{.66}}\,{\rm{km}}\)
So, the bike can cover \({\rm{16}}{\rm{.66}}\,{\rm{km}}\) using \(5\) litres of fuel.

Q.2: Hema types \(540\) words for half an hour. How many words would she type in \(6\) minutes?
Ans: Let Hema 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 variation.
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, Hema types \(108\) words in \(6\) minutes.

Q.3: If \(50\) meters of cloth costs \(₹1950,\) how many meters can be bought for \(₹728.5\)?
Ans: Let x meters of cloth be bought for \(₹728.5.\) Then, we can write the given information as below.
\(50\) meters \(→₹1950\)
\(x\) meters \(₹728.5\)
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.
\( \Rightarrow \frac{{1950}}{{728.5}} = \frac{{50}}{x}\)
\( \Rightarrow x = \frac{{728.5 \times 50}}{{1950}}\)
\(x = 18.68\) meter
Hence, we can buy \(18.68\) meters of cloth for \(₹728.5.\)

Q.4: A contractor took up the job of construction of a bridge in \(6\) months and employed \(120\) men for the same. Due to some emergency, he was asked to finish the work in \(4\) months. How many more men should he put on the job?
Ans: This is a case of inverse variation. When the time to complete the job decreases, the number of persons needed to complete the job increases.
To complete the job of construction in \(6\) months, \(120\) men are employed.
To complete the construction job in \(1\) month, \((120×6)\) men will be employed.
To complete the job of construction in \(4\) months, \(\left( {\frac{{120 \times 6}}{4}} \right)\) men will be employed.
Therefore, \(\left( {\frac{{120 \times 6}}{4}} \right)\) men, i.e., \(180\) men, have to be employed to complete the work.
He should employ \((180-20)=60\) more men to complete the job in \(4\) months.

Q.5: A hostel had provisions for \(60\) men for \(35\) days. After \(10\) days, \(15\) more men arrived in the hostel. How long will the remaining provisions last at the same rate?
Ans: After \(10\) days, the provisions would be sufficient for \(60\) men for \((35-10)=25\) days.
For \(1\) man, the provisions would be sufficient for \((25×60)\) days.
For \(75\) men, the provisions would be sufficient for \(\frac{{\left( {25 \times 60} \right)}}{{75}}\) days, i.e. \(20\) days
Hence, the provisions will last for \(20\) more days.

Direct and Inverse Variation: Summary

In this article, we have learnt about the meaning of direct variation, inverse variation, the formula for direct variation, the meaning of constant proportionality, and the application of direct and inverse variation in real life and solved some examples of direct and inverse variation.

FAQs About Direct and Inverse Variation

Here are the most frequently asked questions about Direct and Inverse Variations:

Q.1: How is an inverse variation used in real life?
Ans: There are many real-life examples of inverse variation. For example:
1. The acceleration of the body is inversely proportional to the weight of the body.
2. The battery power is inversely proportional to the time for which it is used.
3. The ability to work hard is inversely proportional to the age of a man from youngness to end of life.
4. The number of mistakes in work is inversely proportional to practice.

Q.2: What are some examples of direct variation in real life?
Ans: Below is some examples of direct variation in real life.
1. The number of selling products is directly proportional to profit.
2. The distance covered by the body is directly proportional to the speed of the body.
3. The acceleration produced in the car is directly proportional to the change in velocity.
4. The cost of the electricity bill is directly proportional to the number of fans running in the house.

Q.3: What is the direct variation?
Ans: Direct variation is the interrelation between two variables whose ratio is equal to a constant value. In other words, direct variation is a circumstance where an increase in one quantity causes a corresponding increase in the other quantity. A decrease in one quantity results in a decrease in the other quantity.

Q.4: What is the inverse variation?
Ans: When the value of one quantity increases regarding the decrease in other or vice-versa, we call it inverse variation. Inverse variation means that the two quantities behave opposite.

Q.5: What is the formula for direct variation?
Ans: If two variables \(a\) and \(b\) are directly proportional to each other, this statement can be represented as \(a \propto b.\) When we replace the proportionality sign \(( \propto )\) with an equal sign \((=),\) the equation changes to \(a=K×b\) or \(\frac{a}{b} = K,\) where \(K\) is called the non-zero constant of proportionality.

Now you are provided with all the necessary information on the applications of direct and inverse variation in real life and we hope this detailed article is helpful to you.

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