Direct Proportion in Mathematics

Direct Proportion, also known as direct variation, describes the direct relationship between two variables. In this direct relationship, when one variable increases in value, the other variable also increases, and conversely, when one variable decreases, the other variable decreases as well. This means that as one variable goes up or down, the other follows suit, maintaining a consistent and proportional change. 

In simple words, Direct Proportion is the relation between two quantities where the ratio of the two is equal to a constant value. Direct Proportion is represented by a proportional Symbol i.e.,∝. In this article, we are going to learn direct proportion or direct variation in detail. It is an important concept of mathematics.

What is Direct Proportion?

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 a decrease in one variable leads to a corresponding decrease in the other. In other words, if two quantities are in direct proportion, when one quantity doubles, the other also doubles, and when one quantity is halved, the other is also halved.

Direct Proportion Meaning

In Direct Proportion between two variables, an increase in one variable results in a proportional increase in the other variable. For 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”).

Direct Proportion Definition

Two Quantities are said to be in direct proportion or direct variation if whenever the value of x increases (or decreases), then the value of y increases (or decreases) in such a way that the ratio x/y remains constant.

Direct Proportion Symbol

Direct Proportion Symbol is given as “x ∝ y”. Let’s say two quantities x and y are directly proportional to each other, then mathematical expression used to show this relation is given in the image below:

Example of Direct Proportion

There are many situations in our daily life where the variation in one quantity brings a variation in the other. Let’s consider and example for better understanding:

Suppose we are going to a shop to buy some pencils. And the cost of 2 pencils is Rs.8. then the cost of 4 pencils will also increase and will be Rs.16. So this is the example of direct proportion that means the cost will increase if the count of the pencils are increased.

Direct Proportions in Real-Life

Some other examples of direct proportion are:

• Speed is directly proportional to distance.
• Energy is directly proportional to work.
• Temperature is directly proportional to heat.
• 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 amount of food we consume is directly proportional to how hungry we are.

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 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.

Some examples that can be represented with the help of above mentioned equation are:

• More articles will cost more.
• More money deposited will get more interest.
• more distance covered, more petrol consumed.

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. In a direct proportion, as one variable increases, the other variable also increases at a constant rate.

Direct and Inverse Proportion

The proportion is said to be a direct proportion between two values when one is a multiple of the other while The value is said to be inversely proportional when one value increases, and the other decreases or vice versa.

Read more about Direct and Inverse Proportions.

Difference between Direct Proportion and Inverse Proportion

The key differences between Direct Proportion and Inverse Proportion are given in the following table:

Property	Direct Proportion	Inverse Proportion
Relationship	When two variables change in the same direction	When two variables change in opposite directions
Formula	y = kx (where k is a constant)	y = k/x (where k is a constant)
Graph	A straight line passes through the origin (0,0)	A hyperbola
Example	The more hours you work, the more money you earn	The more people sharing a pizza, the smaller the slice each person gets
Symbol	∝ (proportional to)  e.g. a ∝ b	∝ (inversely proportional to) e.g. a ∝ 1/b
Equation	y = kx	xy = k

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,

• Direct Variation Formula
• Ratio and Proportion

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:

x	3	x1	x2	10
y	36	60	96	y1

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.

Practice Problems on Direct Proportion

Problem 1: If it takes 4 hours for 2 workers to complete a task, how many hours will it take for 6 workers to complete the same task?

Problem 2: A car travels 180 miles in 3 hours. How many miles will it travel in 5 hours if it maintains the same speed?

Problem 3: If a bakery uses 5 cups of flour to make 20 cookies, how many cups of flour will they need to make 60 cookies?

Problem 4: The cost of 8 notebooks is $16. What is the cost of 12 notebooks if they are priced at the same rate?

Problem 5: A garden hose can fill a 200-gallon tank in 10 minutes. How long will it take to fill a 300-gallon tank using the same hose?

Problem 6: A printer can print 30 pages in 6 minutes. How many pages can it print in 12 minutes?

Direct Proportion: FAQs

1. What Is Direct Proportion?

Direct proportion is a mathematical connection where an increase in one variable results in a proportionate increase in another. It’s symbolized as y = kx, where “k” is the constant.

2. What is the symbol of Direct Proportion?

The symbol used to represent direct proportionality in mathematical notation is the proportional symbol “∝.”

3. What is a Direct Proportion Equation?

The general form of a direct proportion equation is y = kx.

4. How Is Direct Proportion Represented Mathematically?

Direct proportion is represented mathematically as y = kx, where “y” and “x” are the two related variables, and “k” is the constant of proportionality that relates them.

5. What Does the Constant of Proportionality (k) Represent?

The constant of proportionality (k) represents the factor by which one variable changes when the other changes by one unit. It quantifies the strength of the direct proportionality relationship.

6. Can Direct Proportion Have a Constant of Proportionality Equal to Zero?

Direct proportion cannot have a constant of proportionality equal to zero; it would indicate no proportional relationship.

7. Can the Constant of Proportionality (k) Be Negative?

The constant of proportionality (k) can be negative, indicating an inverse relationship.

8. What Are Some Real-Life Examples of Direct Proportion?

Real-life examples of direct proportion include speed and time, earnings and working hour, storage and quantity, etc.

9. What is the Difference Between Direct Proportion and Inverse Proportion?

In inverse proportion, one variable increases while the other decreases. Direct proportion sees both variables move in the same direction.

10. Can a Direct Proportion Relationship Change Over Time?

A direct proportion relationship remains constant as long as the factors affecting it remain the same.