Direct and Inverse Proportions Definition Formula Properties and Solved Examples
Direct and Inverse Proportions introduces students to the possibility of relating two things or situations with one another. It helps to understand the impact of change in one item (x) on another (y) more efficiently.
Being one of the introductory chapters, students benefit from their understanding of concepts and theories and can apply the same in real life situations effectively. Several study material and chapter-based solutions are available these days for students to access and further improve their understanding of these vital concepts.
CBSE Class 8 chapter 13 Direct and Inverse Proportions – Variation
Suppose two objects, ‘x’ and ‘y’ depend on each other in a way such that an increase or decrease in the value of either of them affects the other. In such a case, the objects will be considered to be in variation.
CBSE Class 8 Chapter 13 Direct and Inverse Proportions – Direct variation
It can be best described as the situation, wherein –
An increase in ‘x’ leads to an increase in ‘y’.
A decrease in ‘x’ leads to a decrease in ‘y’.
Notably, the ratio of respective values has to be same.
To elaborate -
‘x’ and ‘y’ will be in direct proportion if only
x/y =k(constant),
or
x=ky.
Also, in such a condition, if y1, y2 represent the values of y corresponding to the values x1, x2 respectively then x1/y2 = x2/y1.
The direct proportion between two objects is represented by the sign –
∝
Notably, there are several methods which can be used to solve problems based on direct proportion.
CBSE Class 8 Chapter 13 Direct and Inverse Proportions – Methods of Direct Proportion
There are 2 distinct methods for solving problems based on direct proportion, namely –
Tabular method
In this method, the ratio is constant. It means if one ratio is mentioned, the value of the others can also be found.
x1/y1 = x2/y2 = x3/y3 = xn/yn
Example: 4-litre milk costs Rs.200. Tabulate the cost of milk of 2l, 3l, 5l and 8l.
Sol: Suppose, x litre of milk costs Rs.Y
It is a given that as the volume increases, the cost will increase too.
x1/y1 = 4/200
x1/y1 = x2/y2
4/200=2/y2
4y2 = 2x200
y2 = (2x200)/4
y2 = 100
Therefore, 2 litre of milk costs Rs.100.
x1/y1 = x3/y3
4/200=3/y3
4y3= 3x200
y3 = (3x200)/4
y3 = 150
Therefore, 3 litre of milk costs Rs.150.
x1/y1 = x4/y4
4/200=5/y4
4y4= 5x200
y4 = (5x200)/4
y4 = 250
Therefore, 4 litre of milk costs Rs.250.
x1/y1 = x5/y5
4/200= 8/y5
4y5= 8x200
y5 = (8x200)/4
y5= 400
Therefore, 5 litre of milk costs Rs.400.
Unitary Method
When two quantities ‘x’ and ‘y’ are said to be in direct proportion, the relation would be expressed as –
k= x/y or,
x = ky
Example: If Sam gets Rs.2000 for 4 hours of work, how many hours will he have to work to earn Rs.60,000.
Sol:
k= Number of hours/salary of a worker
= 4/2000
= 1/500
By using this relation, x = ky
x = 1/500 x 60000 = 12
Therefore, Sam has to work for 12 hours to earn Rs.60000.
CBSE Class 8 Chapter 13 – Inverse Proportion
Typically, two quantities, says, ‘x’ and ‘y’ are said to be in inverse proportion when –
An increase in ‘x’ leads to a decrease in ‘y’.
A decrease in ‘y’ leads to an increase in ‘x’.
It must be noted that the respective values of the ratio must be the same.
‘x’ and ‘y’ will be inversely proportional when k=xy
In such a condition, y1, y2 are values of y corresponding to values of x1, x2. Notably, when two quantities x and y are considered to be in inverse proportion, they are expressed as ∝ 1/y.
Example: If it takes 15 artists to make a statue in 48 hours, how many artists would be required to complete the same in 30 hours?
Sol: Let y be the number of required artists.
It is a given that as the number of artists will be increased, the time taken to complete the work will decrease. Resultantly, the number of hours and artists are in an inverse proportion.
48 x 15 = 30 x y (x1y1 = x2y2)
Therefore,
y = (48 x 15)/30 = 24
Hence, the statue will be completed in 30 hours by 24 artists.
You can learn about the concepts of direct and inverse proportions in detail from subject experts by joining our free live online classes. Our compact study materials, including, exercise-based solutions, will further help you to improve your understanding of the chapter and related concepts much better.
Download the Vedantu App now to improve your learning experience significantly!
FAQs on Direct and Inverse Proportions Complete Guide with Concepts and Applications
1. What is direct proportion in Maths?
Direct proportion is a relationship where two quantities increase or decrease together at the same rate. In direct proportion, if one variable doubles, the other also doubles.
- It is written as y ∝ x.
- The equation form is y = kx, where k is the constant of proportionality.
- Example: If 1 pen costs $2, then 5 pens cost $10.
2. What is inverse proportion?
Inverse proportion is a relationship where one quantity increases while the other decreases in such a way that their product remains constant. In inverse proportion, if one variable doubles, the other becomes half.
- It is written as y ∝ 1/x.
- The equation form is y = k/x.
- Example: If 4 workers finish a job in 6 days, 8 workers will finish it in 3 days.
3. What is the formula for direct proportion?
The formula for direct proportion is y = kx, where k is the constant of proportionality.
- To find k, use k = y/x.
- If x = 4 and y = 20, then k = 20/4 = 5.
- The equation becomes y = 5x.
4. What is the formula for inverse proportion?
The formula for inverse proportion is y = k/x, where k is a constant.
- To find k, multiply the two variables: k = xy.
- If x = 5 and y = 12, then k = 60.
- The equation becomes y = 60/x.
5. How do you know if a relationship is direct or inverse proportion?
A relationship is direct if the ratio is constant and inverse if the product is constant.
- Direct proportion: y/x remains constant.
- Inverse proportion: xy remains constant.
- If both variables increase together, it is usually direct; if one increases while the other decreases, it is usually inverse.
6. What is the difference between direct and inverse proportion?
The main difference is that in direct proportion variables change in the same direction, while in inverse proportion they change in opposite directions.
- Direct proportion: y = kx
- Inverse proportion: y = k/x
- Graph of direct proportion is a straight line through the origin; inverse proportion is a curved graph (hyperbola).
7. Can you give an example of direct proportion?
An example of direct proportion is the relationship between distance and time at constant speed.
- If speed = 60 km/h, then distance = 60 × time.
- In 2 hours: distance = 120 km.
- In 5 hours: distance = 300 km.
8. Can you give an example of inverse proportion?
An example of inverse proportion is the relationship between speed and time when distance is fixed.
- If distance = 120 km, then time = 120/speed.
- At 60 km/h, time = 2 hours.
- At 40 km/h, time = 3 hours.
9. How do you solve direct proportion problems step by step?
To solve direct proportion problems, first find the constant of proportionality and then use it to calculate the unknown value.
- Step 1: Use given values to find k = y/x.
- Step 2: Write the equation y = kx.
- Step 3: Substitute the new value of x to find y.
- Example: If 3 books cost $9, then k = 3, so 7 books cost $21.
10. Where is direct and inverse proportion used in real life?
Direct and inverse proportion are used in everyday calculations involving price, speed, work, and resources.
- Direct proportion: Cost and quantity, wages and hours worked.
- Inverse proportion: Speed and travel time, number of workers and time taken.
- These proportional relationships help in solving real-world maths problems efficiently.
