
Arpita Srivastava Content Writer
Content Writer
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
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 ProportionExample: The following are some real-life examples of direct proportionality:
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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.
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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 FormulaQues. 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 |
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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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Class 8 Maths Related Concepts | ||
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Types of Relation | ||
Binary Operations |
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)
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
- 7/5 = 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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