Proportionality is one of the important topics of Mathematics, and thus, a better understanding of this topic is required for the students, not only in order to get good marks but also for future mathematics endeavours. But there is one more thing that the students need to have in order to master the topic of Direct Proportion, and that one thing is an easy and understandable explanation of the same topic.
An Overview of the Direct Proportion
When the two numbers or the two quantities, say x and y are multiplicatively related by a fixed number, that is to say, a constant; then such relationship is regarded as the Proportionality relationship. It also means that if there is a proportional relationship between the two quantities, x and y then the ratio between the same is going to be constant.
For understanding it in an even simpler manner, let us assume you own a grocery shop. Now if the number of people who buy groceries from your store increases, then your profits are also going to increase, and if the numbers of buyers decrease, then your profits are going to decrease as well. From this, we can say that there is a direct relationship between both the quantities, that is to say, numbers of buyers and your profits are directly related.
Direct Proportion Meaning
Proportion is a concept of Mathematics that gives the relation between any two mathematical quantities. Two quantities are said to be proportional if they are multiplicatively connected by a constant. The proportional relationship between any two quantities can also be defined as the quantities whose product or ratio is constant. Two quantities are said to be directly proportional if their ratio is constant.
If the product of any two quantities is a constant, then those two quantities are said to be inversely proportional. The two quantities which are directly proportional are related by a Direct Proportion symbol ‘∝’. The symbol for proportionality is removed by adding a Direct Proportion constant.
Direct proportion definition
Direct Proportion meaning is explained as follows. Two measurable quantities are said to be directly proportional if the increase in one quantity results in the increase of the other quantity and vice versa. Indirect variation, the ratio of two measurable quantities is constant. For example, if x and y are the two measurable quantities that are directly proportional to each other, then the direct proportion definition is Mathematically written as x ∝ y. If the direct proportion symbol is to be removed, a proportionality constant is added and the direct proportion symbol is replaced by an equal sign.
x = k y
\[\frac{x}{y}\] = k
In the above equation, ‘k’ is a proportionality constant.
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,
\[\frac{x1}{y1}\] = K and \[\frac{x2}{y2}\] = k
So, we can infer that the ratio of initial values and the final values of any two quantities varying directly are equal and constant.
\[\frac{x1}{y1}\] = K and \[\frac{x2}{y2}\] = k
Direct Proportion Example
Marks scored is directly proportional to the performance in the test.
Temperature is directly proportional to heat.
Energy is directly proportional to work.
Speed is directly proportional to distance.
Earning is directly proportional to the amount of work done.
The amount of food we consume is directly proportional to how hungry we are.
These are just a few real-world Direct Proportion examples.
Direct Variation Example Problems
1. In one of the real situations of direct proportion examples, a bus travels 150 km in 5 hours. What is the time taken by the bus to travel 700 km?
Solution:
Distance travelled and time taken are directly proportional to each other.
In the given question, the distance travelled in case 1 is x1 = 150 km
The distance travelled in case 2 is x2 = 700 km
The time taken in case 1 is y1 = 5 hours
Time taken in case 2 is y2 =?
The proportionality relationship can be stated as:
\[\frac{x1}{y1}\] = \[\frac{x2}{y2}\]
\[\frac{150}{5}\] = \[\frac{700}{y2}\]
y2 = \[\frac{700}{150}\] × 5
y2 = 23.33
So, the time taken by the bus to travel 700 km is 23.33 hrs
2. Given that a and b are directly proportional to each other, complete the table given below.
x | 4 | 5 | 12 | 6 |
y | 6 |
Solution:
From the table x1 = 4, y1 = 6, x2 = 5, x3 = 12, x4 = 6
y2 = ? y3 = ? y4 = ?
Case 1: To find y2
\[\frac{x1}{y1}\] = \[\frac{x2}{y2}\]
\[\frac{4}{6}\] = \[\frac{5}{y2}\]
y2 = \[\frac{5}{4}\] × 6
y2 = 7.5
Case 3: To find y4
\[\frac{x1}{y1}\] = \[\frac{x4}{y4}\]
\[\frac{4}{6}\] = \[\frac{6}{y4}\]
y4 = \[\frac{6}{4}\] × 6
Y4 = 9
So, the completed table is as below:
x | 4 | 5 | 12 | 6 |
y | 6 | 7.5 | 18 | 9 |
3. Sumanth has Rs. 400/- with him. If he can purchase 5 kgs of ghee for 2180, how much ghee can he purchase with the amount he has?
Solution:
Total amount for 5 kg ghee is x1 = Rs. 2180/-
Ghee purchased with Rs. 2180/- is y1 = 5 kg
Amount with Sumanth is x2 = Rs. 400/-
Ghee purchased with Rs. 400/- is y2 = ?
The money and amount of ghee purchased are directly proportional to each other.
The direct proportionality relationship can be written as:
\[\frac{x1}{y1}\] = \[\frac{x2}{y2}\]
\[\frac{2180}{5}\] = \[\frac{400}{y2}\]
y2 = \[\frac{400}{2180}\] × 5
y2 = 0.917
Sumanth can purchase 0.917 kgs of ghee with Rs. 400.
Fun Quiz:
1.Time and work are directly proportional to each other. Is this statement true?
Yes,
No
2.Which of the following are directly proportional measurements?
Current flow and resistance
Volume and temperature
Mass and weight
3. From the given figure, identify the graphs that indicate direct proportion definition.
Conclusion
This is all about the explanation of the concept of direct proportion and how it is used to solve problems. Focus on how this concept has been used to derive the formula and used in the solved examples.
