If two entities vary directly and are related to each other in such a way that a change in the value or amount of one entity corresponds to a proportionate change in the value or amount of the other entity, then both the entities are said to be in direct proportion. Let, the two directly proportional quantities be denoted by the variables x and y. Then, the ratio of the two variables can be represented by a constant k. The concept of direct 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, two quantities are said to be in direct proportion, if an increase or decrease in the amount of the first quantity causes a proportionate increase or decrease in the second quantity in such a manner that the ratio of the two quantities remains constant. Mathematically, direct proportionality is given as x ∝ y; where x and y are two variables.
Examples of Direct Proportion
Some of the real-life applications of direct proportion are listed below:
1. Food Preparation at Home
One of the best examples of direct proportion in real life is food preparation at home. Let us take an example of a family that consists of 4 members. The number of chapatis required for a one-time meal of the four family members is equal to 20. This roughly indicates that there are a total of five chapatis for each member. Suppose, one day two guests join the family for lunch. This means that the total number of people consuming the food increases from four to six, hence the number of chapatis required also increase from 20 to 30. Here, one can easily observe the application of direct proportion because the change in the number of people causes a proportionate change in the number of chapatis required.
2. Cost of an Object vs the Number of Objects Purchased
The relationship between the cost and quantity of an article is a prominent example of direct proportion. Let us assume that a child goes to a stationary shop and buys 3 pencils. The cost of one pencil is equal to 5 rupees; therefore, he pays 15 rupees for 3 pencils. One week later, the child goes to the same shop and buys 5 pencils and pays 75 rupees to the shopkeeper. If you observe the relationship between the money spent on the pencils and the number of pencils brought, you can easily observe that with an increase in the number of pencils the amount spent increases and similarly with a decrease in the number of pencils, the amount of money spent reduces proportionately.
3. Earning of a Worker per Day
Suppose, a worker is paid 500 rupees for one day work. This means that the wages earned in two days are equal to 1000 rupees. Similarly, the worker tends to earn 2000 rupees for four days of work and so on. One can easily observe the pattern and relationship between the number of days and the amount of money earned. With an increase in the number of days, the amount of money earned increases. This verifies the application of direct proportion in real life.
4. Food Requirement at a Hostel
Suppose that a college has two hostels. One of the hostels provides residence to 100 students, while the other hostel has a total strength of 250 students. The hostel with student strength equal to 100 tends to use 50 packets of milk every day to prepare tea and coffee. This means that the number of packets required by the second hostel daily for the preparation of tea and coffee should be equal to 125. This is because the requirement of the milk packets is directly proportional to the number of students residing in the hostel. It must be observed that the ratio of the number of milk packets with respect to the number of students, in this case, is equal to 1/2 and remains constant for both the hostels irrespective of any variation in the number of students.
5. Petrol Consumption and Distance Travelled
Let us say that a vehicle requires 2 litres of petrol to cover a distance equal to 30 km. Now, one can easily employ the unitary method to estimate the amount of petrol required for the vehicle to cover a distance of 60 km. Similarly, one can also calculate the distance that the vehicle can cover with 8 litres of fuel. If you analyse the relationship between the quantity of petrol and the distance travelled by the vehicle, you can easily observe that they are directly proportional to each other. Also, the ratio of both entities with respect to each other gives out a constant value.
6. Shadow and Height of Objects
At any particular time of the day, the height of an object is directly proportional to the length of the shadow cast by it on the ground. For instance, suppose that two poles stand across the opposite corners of a playground. One of the poles is 3m high, while the height of the second pole is unknown. The pole with a height equal to 3m casts a shadow that is 6.3m long. At the same time, the other pole casts a shadow that is 8.4m long. Now, with the help of the unitary method, the height of the second pole can be calculated easily. The height of the second pole comes out to be 4m. If you compare the heights of the poles and the lengths of the shadow cast by them, you can easily observe that they are directly proportional to each other. This means that with an increase in the height of the pole, the length of the shadow increases accordingly.
7. Age and Height of a Person
The age and height of a person tend to maintain a direct proportionality for the first few years of his/her life. With an increase in age, a significant and proportionate increase in the height of a person can be observed easily; however, the reverse is not possible as the age or height of a person cannot be reversed.
8. Time taken and Distance covered by a Vehicle
Travelling is yet another daily life activity that demonstrates and verifies the concept of direct proportion in real life. This is because while travelling in a vehicle, the time and distance entities tend to vary directly. For instance, a car takes 1 hour to cover a distance of 20 km at a particular speed. It can be estimated that after two hours the car would be able to cover a distance equal to 40 km, provided the vehicle maintains a constant speed. This means that with an increase in time, the value of distance covered increases proportionally. The ratio of distance and time is known as speed, which in this case remains constant throughout the journey.
9. Temperature and Flame
A gas stove generally has a knob to vary the flame of the burner. When the knob is rotated in a clockwise direction, the intensity of the flame increases. This causes a proportionate increase in heat and temperature. Similarly, on rotating the knob in a counter-clockwise direction, the intensity of the flame reduces, thereby reducing the temperature. This clearly demonstrates that the two entities are directly proportional to each other.
10. Farming and Land Available
Farming is yet another real-life activity that makes use of the concept of direct proportionality. Here, the area of the field and the production of crops vary directly. This means that if you increase the area of the field, the amount of crop harvested increases proportionally. Similarly, the lower area of the field corresponds to less production of the crop.
11. Number of Visitors and Earnings of a Restaurant
The total money earned by a particular restaurant tends to vary directly with respect to the number of customers or visitors. If the number of customers visiting a restaurant increases, the sales of the restaurant tend to shoot, thereby increasing the money earned by it. Similarly, when the customer count at the same restaurant reduces, the sales drop low and comparatively less money is earned. The ratio between the two entities remains constant as the change in values of the number of visitors and the money earned is proportional to each other.
12. Goods Manufactured at a Factory per Hour
Let us say that a goods manufacturing industry produces 25 articles in an hour, then the number of goods that can be manufactured in 2 hours is definitely equal to 50. Similarly, 100 articles are produced in 4 hours and so on. This example clearly establishes a directly proportional relationship between the number of articles manufactured and the time taken by the industry for production.
13. Scaling on Map and Distance between two Cities
Construction of the map of a city is a prominent application of direct proportionality in real life. The scaling on a map is done at a significantly high rate of precision. This means that the distance between the cities located on the map and the distance between the actual cities is finely defined and is proportional to each other. For instance, suppose the scale of the map is given as 1:20000000 and the distance measured between two cities located on the map with the help of a ruler is equal to 4 cm. Now, with the help of a directly proportional relationship of distance between the cities on the map and in real life, the actual distance between the cities can be calculated easily. By equating the scale of the map to the ratio of the distance between the cities on the map and in reality, the actual distance between the two cities comes out to be equal to 800km.
14. Number of Points Scored and Goals Made
The number of points scored and the number of goals made by a particular soccer team, both entities are directly proportional to each other. If a player scores two goals, his/her team earns two points. Similarly, if the players score four goals, the points earned by their team increases by two, and so on.
15. Work and Energy
Work and energy are two physical quantities that usually vary in direct proportion to each other. Energy is the capacity to do a particular work. This means that the more be energy, the more will be the amount of work done. Similarly lower be energy, the lesser will be the amount of work done.
16. Weight and Cost of Fruits
Fruits and vegetables are generally sold according to their weight. For instance, the cost of 1 kg of apples is equal to 80 rupees. Now, if you intend to buy 4 kg of apples, you are required to pay 320 rupees to the vendor. Similarly, the total cost to buy 2 kg of apples is equal to 160 rupees. This means that on increasing the weight of the fruits, the cost increases and on decreasing the weight of the fruits, the amount to be paid reduces proportionally. A constant ratio between the weight and price of the fruit is maintained throughout the process.
17. Temperature and the Melting Rate of an Ice cream
The rate at which ice cream melts is directly proportional to the temperature of the surrounding in which it is placed. If the temperature of the place is low, then the ice cream will melt at a lower rate. Similarly, if the temperature of the place is significantly high, then the ice cream tends to melt at a comparatively faster rate. This means that an increase or decrease in the value of one entity tends to cause a proportionate increase or decrease in the value of another entity.
18. Ingredients of a Recipe and the Number of Consumers
Suppose a recipe book is drafted in such a way that the dish cooked with its reference can be consumed by two people. The ingredients required to cook the same dish for more or fewer people can be transformed according to the concept of direct proportionality. For instance, one of the recipes to cook scrambled eggs published in the book requires 2 eggs, 4gm butter, and 40 ml of milk. Now, if you need to prepare scrambled eggs for four people, then you are required to calculate the value of the ingredients with the help of direct proportionality. Here, in this case, the value of ingredients gets doubled, when the number of consumers double. This means that the amount of ingredients used in a recipe is directly proportional to the number of people consuming it.
19. Key Powered Toys
There are a number of key powered toys available in the market. The basis of the working of such toys is the spring action. When you twist the key attached to one end of the spring present in the internal structure of the toy in a particular direction for some time, the spring gets tight. When the user releases the key, the spring tends to unwind and regain its original shape. At the same time, the mechanism attached to the other end of the spring tends to move and the toy begins to exhibit motion. Here, the number of turns provided to the spring is directly proportional to the time duration for which the toy operates. This means that the more be the number of turns, the longer will be the time of operation and vice versa.
20. Inflating a Balloon
The force required to inflate a balloon is directly proportional to the number of molecules of air pushed into the balloon as well as to the size of the balloon. This means that with an increase in the magnitude of applied force, the number of air molecules inside the balloon increase, which further causes the balloon to expand and change shape proportionally.
21. Fan Speed
The speed of most of the fans is adjustable and can be controlled with the help of a regulator. Generally, the ceiling fans installed in homes can operate on 4-5 speed modes. The speed changes gradually on rotating the knob of the fan regulator. Here, the rotation of the regulator knob is directly proportional to the variation in the speed of the motor or the rotation speed of the fan.
22. Cycling Speed and Force Applied
The relationship between the magnitude of mechanical force applied by a cyclist and the speed with which the cycle moves is a prominent example of direct proportionality. This is because on increasing the magnitude of applied force, the speed of the cycle increases proportionally. Similarly, when the rider slows down the pedalling speed, the overall speed of the cycle reduces proportionally.
23. Number of Students and Number of Benches in a Class
The number of benches installed in a classroom is always maintained proportional to the strength of the class. Let us say that a class consists of 40 students, then the number of two-seater benches required to accommodate all of the students of the class is equal to 20. If you increase the number of students, then the number of benches required should also increase accordingly.
24. Number of Visitors on a Site and Chances of Crash
The number of visitors on a particular site is directly proportional to the chances of a website crash. This is because as the number of people browsing a particular site increases, a proportional amount of load tends to build on the servers, thereby increasing the chances of a site crash. Similarly, less traffic on the site corresponds to less traffic and fewer chances of a server crash.