What is the Symbol used for Direct and Inverse Proportion?
To show how quantities are related to each other, you use Direct or Inverse Proportion or a Proportional symbol. When two quantities X and Y increase together or decrease together, they are said to be Directly Proportional or they are in Direct Proportion with each other. It is also known as a Direct variation. The ratio of these values will remain constant. But when quantities X and Y are Inversely Proportional to each other or in the Inverse Proportion, one quantity decreases when the other quantity increases, or when one quantity increases the other quantity decreases. It is also known as Inverse variation. The ratio of these values varies Inversely.
Direct and Inverse Proportion Signs:
When two quantities X and Y are Directly Proportional to each other, we say “X is Directly Proportional to Y” or “Y is Directly Proportional to X”. When two quantities X and Y are Inversely Proportional to each other, we say that “X is Inversely Proportional to Y” or “Y is Inversely Proportional to X”.
Properties of Direct and Indirect Proportion
Direct Proportion:
When one quantity increases the other quantity increases too.
When one quality decreases the other quantity decreases too.
The corresponding ratios always remain constant.
It is also called a Direct variation.
Example:
Let’s say: X is Directly Proportional to Y here. Relate X and Y if the value of X = 8 and Y = 4.
Solution:
We know, X ∝ Y
Or we can also write it as X = kY, where k = is a constant Proportionality.
8 = k x 4
k = 2.
Hence the relating equation between the two variables would be X = 2Y.
Indirect Proportion:
When one quantity increases the other quantity decreases too.
When one quantity decreases the other quantity increases too.
The corresponding ratios always vary Inversely.
It is also called an Indirect variation.
Example:
Let’s say: X is Inversely Proportional to Y here. Relate X and Y if the value of X = 815 and Y = 3.
Solution:
Let’s consider X1X2 to be the components of X and Y1Y2 to be the components of y.
Then,
\[\frac {X_1} {X_2}\] = \[\frac {Y_1} {Y_2}\]
Or
\[\frac {X_1} {X_2}\] = \[\frac {Y_1} {Y_2}\]
The statement “X is Inversely Proportional to Y” can be written as X ∝ 1/Y.
Let’s say X = \[\frac {15}{Y}\]
Since we have the value of one variable, the other can be figured out easily.
Take Y = 3.
Therefore,
X = \[\frac {15}{3}\]
X = 5
Since we now know X’s value is 5, the value of Y can be found.
5 = \[\frac {15}{Y}\]
Y = 3
How to Write Direct and Indirect Proportion Equations?
Step 1: You will have to write down the Proportional symbol
Step 2: With the help of the constant of Proportionality, convert the symbol into an equation
Step 3: Next, you will have to figure out the constant of Proportionality with the information that is given to you
Step 4: Now substitute the constant value in an equation
Examples of Direct and Indirect Proportion
Example 1: 45 km/hr is the uniform speed of the train at which it is moving. Find:
(i) the distance covered by it in 10 minutes
(ii) the time required to cover 100 km
Solution:
Consider,
the distance covered in 10 minutes = a
The time taken to cover 100 km = b
(i) Considering,
\[\frac {45}{60}\] = \[\frac {a}{10}\]
a = \[\frac {(45 \times 10)}{60}\]
a = 7.5 km
Therefore the distance covered in 10 minutes - 7.5 kilometres
(ii) Considering,
\[\frac {45}{60}\] = \[\frac {100}{b}\]
a = \[\frac {(100 \times 60)}{40}\]
a = 150 minutes
Therefore the time is taken to cover 100 kilometres - 150 minutes.
Example 2:
Let’s say: X is Directly Proportional to Y here. Relate X and Y if the value of X = 100 and Y = 25.
Solution:
We know, X ∝ Y
Or we can also write it as X = kY, where k = is a constant Proportionality.
100 = k x 25
k = 4.
Example 2:
The value of X1 = 4, X2 = 10, Y1 = 8. Find the value of Y2 if the values X and Y are varying Directly.
Solution:
Since X are Y are varying Directly with each other:
\[\frac {X_1}{X_2}\] = \[\frac {Y_1}{Y_2}\]
\[\frac {4}{10}\] = \[\frac {8}{Y_2}\]
Y2 = \[\frac {(8 \times 10)}{4}\]
Y2 = 20
How to discern Whether it is a Direct or Indirect Proportion?
If it is the case of Direct Proportion, then equivalent fractions will be formed as the ratio between the matching quantities would stay the same if they were to be divided. However, if it is a case of Indirect or Inverse Proportion, then if one quantity increases, the other will decrease and vice versa.
Quiz Time!
Try and solve these questions:
X is Directly Proportional to Y here. Relate X and Y if the value of X = 50 and Y = 5.
X is Inversely Proportional to Y here. Relate X and Y if the value of X = 49 and Y = 7.
The cost of 17 books is Rs. 400. How much would be the cost of 5 books?
Application of Direct and Inverse Proportion in Daily Life
Direct and Inverse Proportion can be quite handy and useful to one even in their everyday life because there are innumerable quantities in our day-to-day life that tend to share a Direct and an Inverse relationship. Here are a few examples of the same:
Examples of Direct Proportion in everyday life:
If you go to the grocery store to buy vegetables, then the number of vegetables you’re planning on buying (in kg) would be Directly Proportional to its price.
Another example of Direct Proportion is that the amount of work done in a business firm is Directly Proportional to the number of workers present in the firm. The more people present to do the work, the more work is going to get completed.
Examples of Inverse Proportion in everyday life:
If you’re driving a car and you increase its speed, you will take less time to reach your destination and if you drive at a slower speed, you will take much more time to reach your destination.
The balance in your bank is also Inversely Proportional to your expenditure.
FAQs on Direct and Inverse Proportion MCQs
1. What is Proportionality?
To show how quantities are related to each other, you use direct or inverse proportion or a proportional symbol. When two quantities X and Y increase together or decrease together, they are said to be directly proportional or they are in direct proportion with each other. When quantities X and Y are inversely proportional to each other or in the inverse proportion, one quantity decreases when the other quantity increases or when one quantity increases the other quantity decreases. It is also known as inverse variation.
2. What is direct and Indirect Proportionality?
When two quantities X and Y increase together or decrease together, they are said to be directly proportional or they are in direct proportion with each other. It is also known as a direct variation. The ratio of these values will remain constant. But when quantities X and Y are inversely proportional to each other or in the inverse proportion, one quantity decreases when the other quantity increases or when one quantity increases the other quantity decreases. It is also known as inverse variation. The ratio of these values varies inversely.
3. What is a Direct Proportionality Symbol?
The direct proportionality symbol is donated by “∝” and the indirect proportionality symbol is denoted by “1/∝”.
4. What does Proportionality mean?
In Mathematics, the term Proportionality refers to the quality of any relationship that renders it to be in the same ratio always. In other words, as the name itself suggests, Proportionality is all about “maintaining a proper balance”. For example, the number of oranges present in a crop is Proportional to the number of trees in the orchard, the ratio of Proportionality being the average number of oranges per tree.
There are two main types of Proportional methods, namely Direct Proportion and Indirect or Inverse Proportion.
5. What are the characteristics of Direct and Indirect Proportion?
When it comes to Direct Proportion, as one particular quantity increases, the other one also ends up increasing. Similarly, if one quantity is decreasing, then the other will also decrease. And, the corresponding ratios are always constant. It is also known as a Direct variation.
However, in the case of Indirect Proportion, if one quantity increases, then the other will decrease, and if one quantity decreases, then the other will increase. And, the corresponding ratios continue to vary Inversely. It is also referred to as an Indirect variation.
6. How can you apply the concept of Proportion in daily life?
The concept of Proportion is about maintaining balance and keeping the ratio of things the same. The importance of this concept, thus allows us to utilise the idea in more ways than one. For instance, if you go grocery shopping, you could use the concept of Proportionality to compare the prices of the things you ought to buy. Another example could be that of Inverse Proportion - the more the road is occupied with cars and bikes and other such vehicles, the less amount of space is likely to be vacant on the road.
7. What are the rules of Proportion?
A ratio is more of an expression, whereas a Proportion refers to an equation. So, a relationship wherein the ratio of the things involved always remains the same or constant is considered to be Proportionate. One of the most important rules when it comes to Proportion is that the product of the means is equal to the product of the extremes. Another imperative thing to remember is to always write down the symbol of Proportionality while writing down either Direct or Indirect Proportion equations.
8. How does Vedantu go on about solving complex Maths problems?
If you go through the free study material that Vedantu provides access to, you will see that for every mathematical concept, there is a brief definition of it followed by an example that shows the detailed solution of the question, and there are also follow-up questions or tables as per the requirement. All these measures ensure that the student comprehends the concept thoroughly and practises it enough to get it right. These little exercises are extremely helpful for the students.
