Two quantities are said to be in inverse proportion if an increase in the amount of the first quantity causes a proportionate decrease in the second quantity in such a manner that the product of the two quantities remains constant throughout the variation. Similarly in the case of inverse proportion, by decreasing the value of one quantity, the number of the second value increases. Let, the two inversely proportional quantities be denoted by the variables x and y. Then, the product of the two variables can be represented by a constant k. The concept of inverse proportion facilitates the user to easily estimate the amount or value of a missing entity, provided the basic data regarding the problem statement is already known. In simple words, if two entities vary inversely and are related to each other in such a way that a change in the value or amount of one entity corresponds to an inverse change in the value or amount of the other entity, then both the entities are said to be in inverse proportion. Mathematically, inverse proportionality is given as x ∝ 1/y; where x and y are two variables.

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## Examples of Inverse Proportion

Some of the real-life applications of inverse proportion are listed below:

### 1. Different Modes of Travelling and the Time

Suppose a working professional uses different modes of travelling every day to reach his office. Some of the modes of travelling that he uses include walking, running, cycling, and riding a bike. Suppose the office is 90 km away from his house. The time taken by him to reach his destination while walking at a speed of 3 km/hr is equal to 30 minutes. The time further reduces to 15 minutes if he chooses to run at a speed of 6 km/hr. Likewise, he would reach the office in 10 minutes, if he opts for cycling at a speed equal to 9 km/hr and in 2 minutes, if he rides a motorcycle at 45 km/hr. Here, the maximum time is consumed when he opts to walk to the office and the minimum time is consumed when he chooses to ride a bike. In this case, time tends to vary inversely according to the speed of travelling medium opted by him.

### 2. Number of People and the Time that is taken to complete a Particular Task

The number of people performing a particular task is inversely proportional to the time taken for completion. Let us say that 2 people take 6 days to paint the fence of a garden, then according to the inverse proportion, a team of 3 people would complete the same task in 4 days and a team of 4 people would need only 3 days for completion. Here, the product of the two variables, i.e., the number of people and number of days is equal to 12 and remains constant throughout the variation.

### 3. Speed of the Vehicle and Time Covered

Suppose you have to travel a 160 km distance to reach a particular destination. If you travel at a speed of 40 km/hr then you would reach your destination in 4 hours. Now, if you double the speed of the vehicle, then the time taken to reach the destination gets reduced to half. This means that travelling at a speed equal to 80 km/hr takes 2 hours to complete the journey. This is because speed and time are the two physical quantities that are inversely related to each other.

### 4. Number of Vehicles on the Road and Free Space on Road

The number of vehicles present on a road is typically inversely proportional to the empty space on a road. This is because more number of vehicles would cover more area of the road, thereby leaving less empty space and fewer number of vehicles would need comparatively less area of the road, thereby providing more empty or free space.

### 5. Distance and Brightness

The illumination produced by a light source varies indirectly with respect to the distance. Let us say that, a light source is turned on at the one end of a hall. If you observe the objects present within a 100-metre range of the light source, they tend to appear comparatively brighter than the objects present 500 metres away from the light source. This means that the brightness reduces as you move away from the light source and increases when you move towards the light source.

### 6. Number of Rows and Columns

Suppose you have 12 marbles that you wish to arrange in form of rows and columns. There are a number of ways to arrange them in this particular manner. An arrangement that consists of two columns would have six rows. Now, if you increase the number of columns to three, the number of rows reduces to four. Similarly, the same twelve marbles can be organised in a format that consists of four columns and three rows, and so on. In this case, you can easily observe that the product of the number of rows and columns is equal to 12 and remains constant. This means that the number of rows varies in inverse proportion to the number of columns.

### 7. Time and Freshness of a Food Item

When you pluck a fruit from a tree and store it in a basket, it begins to lose its freshness as time passes by. As time increases, the freshness of the fruit begins to decrease. This means that time and freshness of a flower are inversely proportional to each other. An increase in the value of one quantity tends to induce a proportionate decrease in the number of other entities.

### 8. Number of Pipes required and Time taken to Fill a Swimming Pool

Let us say that a person is able to completely fill a swimming pool with water in 4 hours by connecting two water pipes to it. Now, if the number of pipes connected to the swimming pool is increased to 4, then the time required to fill the pool gets reduced to 2 hours, provided the flow rate of the fluid through all pipes remain constant. This means that the two variables, i.e., the number of pipes and the time taken are inversely proportional to each other.

### 9. Number of Students and the amount of Food available in a Hostel Mess

The number of students residing in a particular hostel is inversely proportional to the time taken to consume a particular amount of food available in the hostel mess. For instance, 100 students consume 50 kg of flour in a week. Now, if the number of students increases to 200, then the same amount of floor gets consumed in 3.5 days. Here, one can easily observe that when you double the number of students, then the time taken to consume a particular amount of food reduces to half.

### 10. Surface Area of the Blade and the Pressure exerted by a Knife

The blade of a knife is tapered and is constructed in a wedge shape. Here, the surface area of the blade is inversely proportional to the sharpness of the knife or the pressure exerted by the knife. The more is the surface area of the edge of the knife blade, the less will be its sharpness. Similarly, the lesser is the surface area, the more will is the sharpness.

### 11. Cost and Number of Articles Purchased

Let us say, a child visits a stationary shop to purchase a few comic books. The price of one comic book is equal to INR 15 and the total money that the child has is equal to INR 100. This means that he would be able to purchase a total of 6 comic books. A few weeks later, he visits the same stationery shop and finds that the price of the comic book has now been increased to INR 25. He has INR 100 in hand and therefore, this time is able to purchase only four comic books. In this particular example, an inverse relationship between the cost and the number of articles purchased can be observed easily. As the price of an item increases, the number of items purchased decreases and vice versa.

### 12. Volume and Pressure

One of the best examples to demonstrate inverse proportionality is the relationship between volume and pressure. Let a container has multiple holes drilled along its length. When water or any other liquid is poured into the container, it begins to flow out through these holes. Water escaping through the hole that is located closest to the base experiences the maximum pressure, while water escaping through the hole present near the top or near the opening of the container would encounter minimum pressure. This means that pressure increases with a decrease in volume and pressure decrease with an increase in volume because the volume and pressure quantities are inversely related to each other.

### 13. Expenditure and Savings

The expenditure and savings are the two entities in finance that are inversely related to each other. This means that if you spend more amount of money, then the savings that you possess would be less.

### 14. Cost and Demand of an Item

The cost of an item is usually inversely proportional to its demand in the market. When the cost of an item reduces, more people opt to buy it, thereby increasing its demand in the market. Similarly, when the cost goes high, most people prefer not to buy the item; therefore, the demand drops low.

### 15. Difference in the Altitude of the Edges of a See-Saw

A see-saw is a long narrow iron or wooden beam fixed on a pivot in the middle. The seats present on the edges of the board display the inverse proportionality in real life in the easiest possible manner. When one end of the see-saw goes higher, then the other end drops down. The relationship between the altitude of both the edges is inversely proportional in nature. As the height of one edge of the board begins to increase, the altitude of the other end of the board tends to reduce proportionally and vice versa.

### 16. Squeezing a Toothpaste Tube and the Contents of the Tube

There exists an inverse relationship between the magnitude of force applied to squeeze a toothpaste tube and the amount of paste contained by it. If you squeeze a toothpaste tube with force, then the amount of paste left inside the tube begins to reduce. An increase in the magnitude of force applied to the tube causes a proportional decrease in the amount of the contents of the tube.

### 17. Charging of a Gadget and the Usage Time

The battery power of a gadget is inversely related to the time for which it is used. Suppose a gadget is charged to 98% before use. Let us say, after using it for one hour the battery drops down to 88%, after two hours the battery percentage is equal to 78%, the charging contained by the gadget after three hours is equal to 68%, and so on. In this case, with an increase in the value of the time for which the gadget is being used, a significant and proportional decrease in the battery percentage can be observed easily.

### 18. Acceleration of an Object with respect to Time

Let us take the example of a spinning top. When the pointed end of the spinning top is placed on the ground and the rope wrapped around the top is pulled quickly with force, the top begins to spin. It must be observed that the acceleration with which the to spins is maximum in the beginning and begins to drop gradually as time passes by. This means that with an increase in the value of time, the magnitude of acceleration decreases proportionally, thereby demonstrating an inverse relationship between the magnitude of the acceleration of an object and the time.

### 19. Length of a Pencil with respect to the Usage

Let us say a pencil is 15 cm long. After using it to write one page of an assignment, the height of the pencil gets reduced by 2 cm. This means that after writing 4 pages the pencil would be 7 cm in length. Likewise, after writing 7 pages, the pencil would be equal to 1 cm in length. The length of the pencil tends to reduce with an increase in usage. This means that there exists an inverse relationship between the length and the usage of the pencil.

“The time it takes to consume a meal is inversely proportional to the time it takes to prepare it.”

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