An Overview of Proportion
An equation known as proportion shows that the two ratios given are equal to one another. In other words, the proportion declares that the two ratios or fractions are equal. In terms of proportion, two sets of given numbers are said to be directly proportional if they increase or decrease in the same ratio.
Symbol of Proportion: The proportion is denoted by the symbol ‘=’ or ‘::’.
Example: When two ratios are equal, they are said to be in proportion. For instance, the time it takes a train to travel 60 kilometres per hour is equal to the time it needs to travel 300 kilometres in 5 hours. For example, 300 km/5 hours at 50 km/h. There are two types of proportion:
Direct Proportion
Indirect Proportion
Basics of Proportion
Let us first consider a,b,c and d as four quantities of proportion. So, we may write the proportion as a:b::c:d or a:b=c:d. Let us start explaining the proportions from these representations. Here:
a and d are the first and fourth term, respectively, known as extremes terms.
b and c are the second and third term, respectively, known as the mean terms.
Product of extremes = Product of mean.
Proportion
Fourth Proportion
In the representation a:b=c:d, the letter d is known as the fourth proportion to a,b and c. For example, if 6,7 and 8,9 are in proportion, then 9 will be the fourth proportion to 6, 7 and 8.
Fourth Proportional Formula
If we consider a, b, c and d are in proportion, then a:b::c:d. To calculate the fourth term that is d,
\[\dfrac{a}{b} = \dfrac{c}{d}\]
Therefore,
\[d = \dfrac{{b \times c}}{a}\]
Mean Proportion
Here in the proportion a:b::c:d, b and c are known as the mean terms. Thus, the square root of the product of these two is the mean proportional between the two terms of a ratio in a proportional.
Third Proportion
The second term of the mean terms serves as the third proportional of a proportion. If a:b = c:d, for instance, the term "c" is the third proportional to "a" and "b."
How to Find Fourth Proportional
In the above example, we have seen the explanation and formula for the calculation of the fourth proportion. Here let us understand it with the help of examples:
1. Find the fourth Proportion to 10, 20, 30.
Explanation: Let the fourth proportion to 10, 20 and 30 be x.
Then, 10:20:: 30: x
\[ \Rightarrow 10 \times x = 20 \times 30\](Product of extreme= product of mean)
\[ \Rightarrow x = \dfrac{{20 \times 30}}{{10}}\]
\[\Rightarrow x = 60\]
Hence, the fourth proportion to 10, 20, 30 is 60.
Summary
The equality of two ratios is referred to as proportion. The ratio symbol (::) is used to denote proportions, which aid in solving problems involving ambiguous quantities. To put it another way, proportion is an expression or equation that shows that two ratios or fractions are equal.
Solved Questions
1.What will be the fourth term if the first three terms of the proportion are 6, 12, and 24.
Ans: Let the fourth term be x.
Then, 6:12 :: 24:x
\[ \Rightarrow x = \dfrac{{12 \times 24}}{6}\]
\[ \Rightarrow x = 48\]
Thus, the fourth term is 48.
2.Find out if the ratios 4:8 and 12:24 are in proportion.
Ans: For the given ratio :
\[ \Rightarrow \dfrac{4}{8} = \dfrac{1}{2} = 0.5\] & \[\dfrac{{12}}{{24}} = \dfrac{1}{2} = 0.5\]
Hence, both the ratios are in proportion.
3.Find the value of x so that the given four numbers are in proportion:
4, 10, 24, and x
Ans: So here, the product of mean = product of extreme
\[\Rightarrow x = \dfrac{{10 \times 24}}{4} = 60\]
Hence, the value of x is 60.
Learning by Doing
Find out if the ratios 9:36 and 4:24 are in proportion.
What is the proportion?
FAQs on What is Proportion?
1. What exactly is a proportion in Maths?
In Maths, a proportion is a statement that says two ratios are equal. Think of it as a balanced relationship. For example, if the ratio of boys to girls in one class is 1:2, and it's 2:4 in another, these two ratios are in proportion because they both represent the same value.
2. What is the basic rule or formula to check for proportion?
The fundamental rule for proportion is that the product of the means equals the product of the extremes. If you write a proportion as a : b :: c : d, the outer values (a and d) are the 'extremes' and the inner values (b and c) are the 'means'. The formula is: a × d = b × c. If this equation is true, the ratios are in proportion.
3. Can you give a simple real-life example of proportion?
Certainly. Imagine you are baking a cake. The recipe says you need 2 eggs for every 3 cups of flour. If you want to make a larger cake using 6 cups of flour, you would need 4 eggs. The ratios 2:3 and 4:6 are in proportion, ensuring the cake tastes the same.
4. What is the main difference between a ratio and a proportion?
The main difference is simple: a ratio is a way to compare two quantities (like 2 apples to 3 oranges, or 2:3). A proportion, on the other hand, is an equation that states two ratios are equal to each other (like 2:3 = 4:6). In short, a proportion is made up of two equal ratios.
5. What are the different types of proportion students learn about?
Students primarily learn about two main types of proportion:
- Direct Proportion: This is when two quantities increase or decrease together at the same rate. For example, the more kilometres you drive, the more fuel you use.
- Inverse Proportion: This is when one quantity increases while the other decreases. For example, the faster you drive, the less time it takes to reach your destination.
6. How can you quickly check if two ratios, like 5:10 and 15:30, are in proportion?
The quickest way is to use the cross-multiplication method. Write the ratios as fractions: 5/10 and 15/30. Now, multiply the numerator of the first fraction by the denominator of the second (5 × 30 = 150). Then, multiply the denominator of the first by the numerator of the second (10 × 15 = 150). Since both products are 150, the ratios are in proportion.
7. Why is the 'product of means equals product of extremes' rule so useful?
This rule is incredibly useful because it turns a proportion problem into a simple equation. It provides a straightforward method to find a missing number in a proportion. For instance, in 3:6 :: 5:x, you can easily find 'x' by setting up the equation 3 × x = 6 × 5. This avoids complex fraction work and gives a reliable way to solve for the unknown value.
8. What does 'fourth proportional' mean and how do you find it?
The 'fourth proportional' is the fourth term in a proportion when you already know the first three. If you have three numbers a, b, and c, the fourth proportional 'x' is the number that makes the statement a:b :: c:x true. To find it, you use the rule: a × x = b × c, and then solve for x.
