What Are Direct and Inverse Proportions? Definitions and 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.
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FAQs on Direct and Inverse Proportions Explained
1. What is meant by direct proportion in Maths? Please provide an example.
Direct proportion describes a relationship between two quantities where as one quantity increases, the other quantity also increases at a constant rate, and vice versa. The ratio between the two quantities always remains the same. This constant ratio is known as the constant of proportionality (k). For instance, if you buy more pens, the total cost increases. If 1 pen costs ₹10, 5 pens will cost ₹50. Here, the number of pens and the total cost are in direct proportion.
2. What is inverse proportion? Explain with a real-world example.
Inverse proportion describes a relationship where an increase in one quantity causes a proportional decrease in another quantity. In this case, the product of the two quantities remains constant. A classic example is the relationship between speed and travel time. If you increase your driving speed, the time taken to cover a fixed distance decreases. If you double your speed, you halve your travel time.
3. What is the fundamental difference between direct and inverse proportion?
The fundamental difference lies in how the two quantities relate to each other's changes:
- In a direct proportion, both quantities move in the same direction. If one increases, the other increases. If one decreases, the other also decreases. Their ratio is constant (x/y = k).
- In an inverse proportion, the quantities move in opposite directions. If one increases, the other decreases. Their product is constant (x × y = k).
4. What are the formulas used for solving problems on direct and inverse proportions?
The formulas are essential for setting up and solving problems:
- Direct Proportion: If (x₁, y₁) and (x₂, y₂) are two pairs of values in direct proportion, the formula is x₁/y₁ = x₂/y₂.
- Inverse Proportion: If (x₁, y₁) and (x₂, y₂) are two pairs of values in inverse proportion, the formula is x₁y₁ = x₂y₂.
Using the correct formula is the most critical step in solving these problems accurately.
5. How can I identify whether a word problem involves direct or inverse proportion?
To identify the type of proportion, ask yourself a simple logical question: "If I increase the first quantity, what will happen to the second quantity?"
- If the second quantity also increases, it is a case of direct proportion (e.g., more workers, more work done).
- If the second quantity decreases, it is a case of inverse proportion (e.g., more workers, less time to finish the same job).
This simple check helps in applying the correct logical framework and formula.
6. What is the role of the 'constant of proportionality' (k) in these topics?
The constant of proportionality, denoted by 'k', is the unchanging value that defines the relationship between the two quantities. It represents the core of the proportional relationship.
- In direct proportion (x/y = k), 'k' is the constant ratio. For example, in the relationship between distance and time at a constant speed, 'k' would be the speed itself (km/hr).
- In inverse proportion (x × y = k), 'k' is the constant product. For example, if a fixed amount of work needs to be done, 'k' could represent the total man-hours required for the job.
Understanding 'k' helps in grasping the underlying principle beyond just solving for an unknown variable.
7. In which real-life situations is the concept of direct and inverse proportion applied?
These concepts are used widely in everyday life:
- Direct Proportion Examples: Calculating the cost of groceries based on quantity, scaling a recipe up or down, estimating fuel needed for a trip, and converting currencies.
- Inverse Proportion Examples: Planning a project timeline based on the number of workers, determining the effect of speed on travel time, and understanding the relationship between the pressure and volume of a gas in physics.
8. What is a common mistake students make when solving inverse proportion problems?
The most common mistake is incorrectly setting up the equation for inverse proportion as if it were a direct proportion. Students often write x₁/y₁ = x₂/y₂ out of habit. However, for an inverse relationship, the product of the quantities is constant, not the ratio. The correct formula to use is x₁y₁ = x₂y₂. Always double-check if the quantities are moving in opposite directions before writing the equation.
