Direct Proportion: Formula, Graph & Examples

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A Direct Proportion is used to identify the relationship of two directly proportional quantities and is denoted by "∝". By adding a proportionality constant, the proportionality symbol is removed.

Proportion is a term that specifies the relationship between two digits.
When two numbers are multiplicatively related by a constant, they are said to be proportional.
The product or ratio of any two quantities can alternatively be defined as the proportional relationship between them.
Direct proportion is also known as direct variation.
In these cases, if the value of one variable increases, then the value of the other variable decreases.
The preparation of food at home is a real-life example of direct proportion.
It can be explained as the number of chapatis required is directly proportional to the number of people.
The concept of proportionality can be explained mathematically as:

x ∝ y

Table of Content

Direct Proportion
Direct Proportion Symbol
Direct Proportion Formula
Direct Proportion Graph
Difference between Direct Proportion and Inverse Proportion
Things to Remember
Sample Questions

Key Terms: Direct Proportion, Variables, Proportion, Constant, Inverse Proportion, Curve, Graph, Straight Line, Constant of Proportionality

Direct Proportion

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Direct Proportion is a form of relationship of two quantities in which increase in one number will causes increase in other and decrease in one causes decrease in other.

Direct proportion is a form of comparison between two numbers.
The ratio of the two numbers is equivalent to constant values.
∝ symbol is used to represent proportion variables.
The same symbol is used to represent inversely proportion variables.
Ratio of numbers is known as coefficient of proportionality or proportionality constant.
The inverse of the ratio is known as constant of normalization or normalizing constant.

Real-Life Examples of Direct Proportion

Example: The following are some real-life examples of direct proportionality:

The overall amount of money spent is directly proportionate to the quantity of food items purchased.
The amount of labour done is related to the quantity of workers.
In relation to a fixed time, speed is proportional to distance.

Direct Proportion Symbol

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The symbol used for direct proportion is "∝". Consider the numbers x and y, where x represents the quantity of candies and y represents the total money that is spent. We will have to pay more money if we buy more candy; we will have to pay less money if we buy fewer candies.

Due to this, we can state that x and y are proportional in the same manner.
It is represented as x∝ y.

Read More:

Chapter Related Concepts
Relation Between HCF and LCM	Binary Addition	Operations on Rational Numbers
Addition And Subtraction Of Integers	Number Line	Integers As Exponents

Direct Proportion Formula

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The direct proportion formula can be explained as: If the work y is in direct proportion to the labourers x, we can state y =kx for a constant k, according to the direct proportion formula.

The simple version of the concept direct proportion equation is y=kx.

If Y ∝ x

y = kx, for a constant k

where, k refers to a constant of proportionality.
Increase in Y causes increase in x
Decrease in Y causes decrease in x

Example of Direct Proportion Formula

Ques. A 7-meter-high electric pole poses a 9-meter-long shadow. Calculate the height of a tree that produces a 18-meter shadow under similar conditions.

Ans. Let's count the tree's height as x metres. We know that if the height of the pole rises, the length of the shadow will also rise in the same proportion. As a result, we can see that the height of the tree and the length of its shadow are directly proportional. In other words, the length of a pole's shadow is directly proportional to its height. Thus,

X1/Y1 = X2/Y2

7/9 = X/18

X = 14 meters

Direct Proportion Graph

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The direct proportion graph is represented in by a straight line. The slope of the direct proportion graph is pointing upwards. Suppose there are two points each marked on x-axis and y-axis.

In this the value of x1 is less than x2 and similarly the value of y1 is less than y2.

Direct Proportion Graph

Difference between Direct Proportion and Inverse Proportion

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The difference between direct proportion and indirect proportion are as follows:

Direct Proportion	Indirect Proportion
When a rise or decrease in one quantity causes an increase or decrease in the other, the two quantities are directly proportional.	When one quantity increases, the other decreases, and vice versa, two quantities are said to be in inverse proportion.
The direct proportion graph is a straight line	The inverse proportion graph is a curve
It is represented by: y ∝ x.	It is represented by: y ∝ 1/x.

Direct Proportion and Inverse Proportion

Things to Remember

Direct proportion states the relationship between two digits, which signifies an increase in one causes an increase in the other.
Italian scientist Galileo Galilei discovered the concept of direct proportion.
Mathematically it is represented as: x∝ y.
It is used in many real-life applications, such as fuel prices and exchange rates.
Another example includes the amount of labour done is related to the quantity of workers.
The graph of two variables in direct proportion is represented by a straight line.

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Mean, Median and Mode	Types of Relation	Mode of Grouped Data
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Sample Questions

Ques. A 7-meter-high electric pole poses a 5-meter-long shadow. Calculate the height of a tree that produces a 10-meter shadow under similar conditions. (2 marks)

X1/Y1 = X2/Y2
7/5 = X/10
X = 14 meters

Ques. A map has a scale of 1:20000000 on it. On the map, the distance between two cities is 4 cm. Calculate the actual distance. (2 marks)

Ans. Map distance = 4 cm.

Let take the actual distance as x cm,
then 1:20000000 = 4:x.
1/20000000 = 4/x
⇒ x = 80000000 cm = 800 km

Ques. lf a car travels 150 km in 5 hours, then what is the time taken for the car to travel 700 km. (3 marks)

Ans. Distance covered and time estimated are directly proportional to one another.

Given,

The distance covered in step 1 is x1 = 150 km
The distance covered in step 2 is x2 = 700 km
The time taken in step 1 is y1 = 5 hours
Time taken in step 2 is y2 = ?
The proportionality relationship can be determined as:
x1/y1 = x2y2
150/5= 700/y2
y2 = 700/150 ? 5
y2=23.33

So, the time taken by the car to travel 700 km is 23.33 hrs

Ques. In a situation given, x and y are stated directly proportional to each other, Now keeping this in mind, complete the table given below. (5 marks)

X	4	5	12	6
Y	6

Ans. From the table x1 = 4, y1 = 6, x2 = 5, x3 = 12, x4 = 6

y2 = ? y3 = ? y4 = ?
Case 1: To find y2
x1/y1= x2y2
4/6= 5/y2
y2= 5/4? 6
y2=7.5
Case 2: To find y3
x1/y1= x3/y3
4/6= 12/y3
y3 = 12/4? 6
y3= 18
Case 3: To find y4
x1/y1= x4/y4
4/6= 6/y4
y4 = 6/4 ? 6
y4 =9

Hence, the completed table is here;

X	4	5	12	6
Y	6	7.5	18	9

Ques. Khushi has Rs. 400/- with her. If she can buy 5 kgs of ghee for Rs. 2180 then, how much ghee can she buy with the money she has. (4 marks)

Ans. Total money for 5 kg ghee is x1 = Rs. 2180/-

Ghee purchased with Rs. 2180/- is y1 = 5 kg

Money with Khushi is x2 = Rs. 400/-

Ghee purchased with Rs. 400/- is y2 = ?

The money and amount of ghee she bought stated directly proportional to one another.

The direct proportionality relationship can be determined as:

x1/y1= x2/y2

2180/5= 400/y2

y2= 400/2180 ? 5

y2=0.917

Khushi can buy 0.917 kgs of ghee with Rs. 400.

Ques. A car consumes 15 gallons of diesel for every 100 kilometres, then determined, with 5 litres of diesel, how much distance a car covers. (3 marks)

Ans. Fuel consumed for every 100 km covered is 15 liters

Therefore, the car will travelled (100/15) km using 1 liter of the fuel

If 1 liter => (100/15) km

Now What about 5 liters of diesel

= {(100/15) × 5} km

= 33.3

Therefore, the car can travel 33.3 km consuming 5 liters of the fuel.

Ques. y varies directly with the square of x. If y = 5 when x = 3 then determine the value of y when x is 9. (4 marks)

Ans. In this case, y varies with the square of x rather than directly with x. The problem-solving concept remains the same. As x² grows larger, so does y, which we may characterise with the equation y = kx2, where k is the proportionality constant. Now, calculate for the proportionality constant using the provided x and y values.

y = kx²
5 = k(3)²
5 = 9k
5/9 = k
Now that the constant of proportionality is explained, we can state the direct variation relationship of our variables as y = 5/9 x2
To calculate what y is when x = 9, substitute into the equation.
y = 5/9 (9)2
y = 5/9 (81)
y = 45

Ques. lf a car travels 180 km in 9 hours, then what is the time taken for the car to travel 360 km. (3 marks)

Ans. Distance covered and time estimated are directly proportional to one another.

Given,

The distance covered in step 1 is x1 = 180 km
The distance covered in step 2 is x2 = 360 km
The time taken in step 1 is y1 = 9 hours
Time taken in step 2 is y2 = ?
The proportionality relationship can be determined as:
x1/y1 = x2y2
180/9= 360/y2
y2 = 360 x 9 /180
y2=18

So, the time taken by the car to travel 360 km is 18hrs

Ques. Let us assume that y varies directly with x, and y = 64 when x = 8. Using the direct proportion formula, find the value of y when x = 80. (3 marks)

Ans. As you know the direct proportion formula which goes as y = kx

Substitute the given x and y values, and solve for k.
64 = k × 8
k = 64/8 = 8
The direct proportion equation is: y = 8x
Now, substitute x = 80 and find y.
y = 8 × 80 = 640

Ques. If the cost of 10 pounds of apples is $100, what will be the cost of 20 pounds of apples. (3 marks)

Ans. Since it is given in the question that,

Weight of apples = 10 lb
Cost of 10 lb apples = $100
Let us consider the weight by x parameter and cost by y parameter.
To find the cost of 20 lb apples, we will use the direct proportion formula.
y=kx
10 = k × 100 (on substituting the values)
k = 10
Now putting the value of k = 10 when x = 20 we have,
The cost of 20 lb apples = 10 × 20
y =200

Ques. A map has a scale of 1:40000000 on it. On the map, the distance between two cities is 4 cm. Calculate the actual distance. (2 marks)

Ans. Map distance = 4 cm.

Let take the actual distance as x cm,
then 1:40000000 = 4:x.
1/40000000 = 4/x
⇒ x = 160000000 cm = 1600 km

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Direct Proportion: Formula, Graph & Examples

Direct Proportion

Real-Life Examples of Direct Proportion

Direct Proportion Symbol

Direct Proportion Formula

Example of Direct Proportion Formula

Direct Proportion Graph

Difference between Direct Proportion and Inverse Proportion

Things to Remember

Sample Questions

CBSE X Related Questions

1.
In a right triangle ABC, right-angled at A, if $\sin B = \dfrac{1}{4}$, then the value of $\sec B$ is

2.
In the adjoining figure, $\triangle CAB$ is a right triangle, right angled at A and $AD \perp BC$. Prove that $\triangle ADB \sim \triangle CDA$. Further, if $BC = 10$ cm and $CD = 2$ cm, find the length of AD.

3.
\(\alpha, \beta\) are zeroes of the polynomial \(3x^2 - 8x + k\). Find the value of \(k\), if \(\alpha^2 + \beta^2 = \dfrac{40}{9}\)

4.
Given that $\sin \theta + \cos \theta = x$, prove that $\sin^4 \theta + \cos^4 \theta = \dfrac{2 - (x^2 - 1)^2}{2}$.

5.
Let $p$, $q$ and $r$ be three distinct prime numbers. Check whether $pqr + q$ is a composite number or not. Further, give an example for three distinct primes $p$, $q$, $r$ such that
(i) $pqr + 1$ is a composite number
(ii) $pqr + 1$ is a prime number

6.
If the zeroes of the polynomial $ax^2 + bx + \dfrac{2a}{b}$ are reciprocal of each other, then the value of $b$ is

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Direct Proportion: Formula, Graph & Examples

Direct Proportion

Real-Life Examples of Direct Proportion

Direct Proportion Symbol

Direct Proportion Formula

Example of Direct Proportion Formula

Direct Proportion Graph

Difference between Direct Proportion and Inverse Proportion

Things to Remember

Sample Questions

CBSE X Related Questions

1.In a right triangle ABC, right-angled at A, if $\sin B = \dfrac{1}{4}$, then the value of $\sec B$ is

2.In the adjoining figure, $\triangle CAB$ is a right triangle, right angled at A and $AD \perp BC$. Prove that $\triangle ADB \sim \triangle CDA$. Further, if $BC = 10$ cm and $CD = 2$ cm, find the length of AD.

3.\(\alpha, \beta\) are zeroes of the polynomial \(3x^2 - 8x + k\). Find the value of \(k\), if \(\alpha^2 + \beta^2 = \dfrac{40}{9}\)

4.Given that $\sin \theta + \cos \theta = x$, prove that $\sin^4 \theta + \cos^4 \theta = \dfrac{2 - (x^2 - 1)^2}{2}$.

5.Let $p$, $q$ and $r$ be three distinct prime numbers. Check whether $pqr + q$ is a composite number or not. Further, give an example for three distinct primes $p$, $q$, $r$ such that (i) $pqr + 1$ is a composite number (ii) $pqr + 1$ is a prime number

6.If the zeroes of the polynomial $ax^2 + bx + \dfrac{2a}{b}$ are reciprocal of each other, then the value of $b$ is

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
In a right triangle ABC, right-angled at A, if $\sin B = \dfrac{1}{4}$, then the value of $\sec B$ is

2.
In the adjoining figure, $\triangle CAB$ is a right triangle, right angled at A and $AD \perp BC$. Prove that $\triangle ADB \sim \triangle CDA$. Further, if $BC = 10$ cm and $CD = 2$ cm, find the length of AD.

3.
\(\alpha, \beta\) are zeroes of the polynomial \(3x^2 - 8x + k\). Find the value of \(k\), if \(\alpha^2 + \beta^2 = \dfrac{40}{9}\)

4.
Given that $\sin \theta + \cos \theta = x$, prove that $\sin^4 \theta + \cos^4 \theta = \dfrac{2 - (x^2 - 1)^2}{2}$.

5.
Let $p$, $q$ and $r$ be three distinct prime numbers. Check whether $pqr + q$ is a composite number or not. Further, give an example for three distinct primes $p$, $q$, $r$ such that
(i) $pqr + 1$ is a composite number
(ii) $pqr + 1$ is a prime number

6.
If the zeroes of the polynomial $ax^2 + bx + \dfrac{2a}{b}$ are reciprocal of each other, then the value of $b$ is