Direct Proportion in Mathematics
Last Updated : 11 Dec, 2024
Direct proportion or direct variation is a mathematical relationship between two variables where they change in such a way that an increase in one variable leads to a corresponding increase in the other and vice - versa.
Example, if you have a situation where the number of hours worked is directly proportional to the amount earned, and you earn Rs. 100 for every hour worked, the equation representing this direct proportion would be:
Earnings (y) = 100 × Hours Worked (x)
This means that for every additional hour worked (increase in "x"), you will earn an additional Rs. 100 (increase in "y").
Examples of Direct Proportions in Real-Life
There are many situations in our daily life where the variation in one quantity brings a variation in the other. Let's consider some example for better understanding:
- The more you use electricity, the cost will more.
- The more you deposit money, the interest provided is more.
- The growth of plants can be directly proportional to the amount of water they receive.
- The number of cookies you can make is directly proportional to the amount of ingredients you use.
- The bill value in a restaurant is directly proportional to the amount of food we consume.
Direct Proportion Formula
Suppose we have two quantities x and y, the direct Proportion between them can be shown like this:
x = k.y
OR
x/y = k
Where k is a constant value.
If x1 and y1 are the initial values of any two quantities that are directly proportional to each other and x2 and y2 are the final values of those quantities. Then according to the direct proportionality relationship,
x1/y1 = k and x2/y2 = k
Where,
- x1 and x2 are the values of variable x,
- y1 and y2 are the values of variable y, and
- k is the constant of Proportionality.
Direct Proportion Symbol
In mathematics direct Proportion is represented using the symbol "∝". Let's say two quantities X and Y are directly proportional to each other, then mathematical expression used to show this relation will be
X ∝ Y
Direct Proportion Equation
Thus, x and y are in direct proportion, if x/y = k, where k is a constant
x1/y1 = x2/y2 = x3/y3 = . . . = k
Where,
- x1, x2, . . . are the values of variable x,
- y1, y2, . . . are the values of variable y, and
- k is the constant of Proportionality.
Direct Proportion Graph
A direct proportion graph, also known as a direct variation graph or a linear proportion graph, represents a relationship between two variables that are directly proportional to each other. The graph to represent the direct proportions is always a straight line representing constant increase in both variable. 
Read more about Direct and Inverse Proportions.
Note: In direct proportion, as one variable increases, the other variable increases proportionally. In inverse proportion, as one variable increases, the other variable decreases proportionally.
Read More,
Examples of Direct Proportion with Solution
Question 1: If x and y are directly proportional, find the values of x1, x2 and y1 in the table given below:
Solution:
Since x and y are directly proportional, we have:
3/ 36 = x1 / 60 = x2/96 = 10/ y1
Now, 3 / 36 = x1/ 60
⇒ x1 = (1 / 12 ) × 60 = 5
3/ 36 = x2 / 96
⇒ 1/12 = x2/96
⇒ x2 = (1/12) × 96 =8.
3/ 36 ⇒ 10 / y1
⇒ 1/12 = 10/y1
⇒ y1 × 1 = (12 × 10) = 120.
Hence, x1= 5, x2 = 8, y1= 120. Ans.
Question 2: If the Weight of 9 sheets of thick paper is 30 grams. how many sheets of the same paper would weight 5/4 kilograms.
Solution:
Let the required number of sheets be x.
5/4 kg = 5/4 × 1000 gm = 1250 gms.
thus we have,
| Number of Sheets | 9 | x |
|---|
| weight of Sheets (in gms) | 30 | 1250 |
|---|
More is the weight, more is the number of sheets. So, it is a case of direct proportion.
Hence, 9/30 = x/1250
⇒ 3/10 = x/1250
⇒ x = (3/10) × 1250
⇒ x = 375.
Hence the required number of sheets is 375.
Question 3: A car covers 432 km in 36 litres of petrol. How much distance would it cover in 25 litres of petrol?
Solution:
Let the required distance be x km, Then we have:
| Quantity of petrol (in litres) | 36 | 25 |
|---|
| Distance (in km) | 432 | x |
|---|
Less is the quantity of petrol consumed, less is the distance covered. So, it is a case of direct proportion.
36 / 432 = 25/ x
⇒ 1/ 12 = 25/ x
⇒ x × 1 = 12 × 25 = 300.
Hence the required distance is 300 km.
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