Ratio and Proportion are explained majorly based on fractions.
When a fraction is represented in the form of a:b, then it is a ratio whereas a
proportion states that two ratios are equal. Here, a and b are any two integers. The ratio andproportion are the two important concepts,
and it is the foundation to understand the various concepts in mathematics as well as in science.
In our daily life, we use the concept of ratio and proportion such as in business while dealing with money or while cooking any dish, etc.
Sometimes, students get confused with the concept of ratio and proportion. In this article, the students get a clear vision of these two concepts
with more solved examples and problems.
For example, ⅘ is a ratio and the proportion statement is 20/25 =
⅘. If we solve this proportional statement, we get:
20/25 = ⅘ 4:5
20 x 5 = 25 x 4
100 = 100
, the ratio defines the relationship between two quantities such as a:b, where b is
Therefore
not equal to 0. Example: The ratio of 2 to 4 is represented as 2:4 = 1:2. And the
statement is said to be in proportion here. The application of proportion can be seen
in direct proportion.
The definition of ratio and proportion is described here in this section. Both concepts are an important part of Mathematics. In real
life also, you may find a lot of examples such as the rate of speed (distance/time) or price (rupees/meter) of a material, etc, where
the concept of the ratio is highlighted.
Proportion is an equation that defines that the two given ratios are equivalent to each
other. For example, the time taken by train to cover 100km per hour is equal to the time
taken by it to cover the distance of 500km for 5 hours. Such as 100km/hr = 500km/5hrs.
The ratio should exist between the quantities of the same kind
While comparing two things, the units should be similar
There should be significant order of terms
The comparison of two ratios can be performed, if the ratios are equivalent
like the fractions
Proportion is an equation that defines that the two given ratios are equivalent to each other. In other words, the proportion states
the equality of the two fractions or the ratios. In proportion, if two sets of given numbers are increasing or decreasing in the same
ratio, then the ratios are said to be directly proportional to each other.
For example, the time taken by train to cover 100km per hour is equal to the time taken by it to cover the distance of 500km for 5
hours. Such as 100km/hr = 500km/5hrs.
Ratio and proportions are said to be faces of the same coin. When two ratios are equal in value, then they are said to be
in proportion. In simple words, it compares two ratios. Proportions are denoted by the symbol ‘::’ or ‘=’.
The proportion can be classified into the following categories, such as:
Direct Proportion
Inverse Proportion
Continued Proportion
Now, let us discuss all these methods in brief:
Direct Proportion
The direct proportion describes the relationship between two quantities, in which the increases in one quantity, there is an increase
in the other quantity also. Similarly, if one quantity decreases, the other quantity also decreases. Hence, if “a” and “b” are two
quantities, then the direction proportion is written as a∝b.
Inverse Proportion
The inverse proportion describes the relationship between two quantities in which an increase in one quantity leads to a decrease
in the other quantity. Similarly, if there is a decrease in one quantity, there is an increase in the other quantity. Therefore, the
inverse proportion of two quantities, say “a” and “b” is represented by a∝(1/b).
Continued Proportion
Consider two ratios to be a: b and c: d.
Then in order to find the continued proportion for the two given ratio terms, we convert the means to a single term/number. This
would, in general, be the LCM of means.
For the given ratio, the LCM of b & c will be bc.
Thus, multiplying the first ratio by c and the second ratio by b, we have
First ratio- ca:bc
Second ratio- bc: bd
Thus, the continued proportion can be written in the form of ca: bc: bd
Ratio and Proportion Formula
Now, let us learn the Maths ratio and proportion formulas here.
Ratio Formula
Assume that, we have two quantities (or two numbers or two entities) and we
have to find the ratio of these two, then the formula for ratio is defined as;
a: b ⇒ a/b
where a and b could be any two quantities.
Here, “a” is called the first term or antecedent, and “b” is called the second
term or consequent.
Example: In ratio 4:9, is represented by 4/9, where 4 is antecedent and 9 is
consequent.
If we multiply and divide each term of ratio by the same number (non-zero), it
doesn’t affect the ratio.
Example: 4:9 = 8:18 = 12:27
Proportion Formula
Now, let us assume that, in proportion, the two ratios are a:b & c:d. The two terms ‘b’ and ‘c’ are called ‘means or mean
term,’ whereas the terms ‘a’ and ‘d’ are known as ‘extremes or extreme terms.’
a/b = c/d or a : b :: c : d
Example: Let us consider one more example of a number of students in a classroom. Our first ratio of the number of girls to boys is
3:5 and that of the other is 4:8, then the proportion can be written as:
3 : 5 :: 4 : 8 or 3/5 = 4/8
Here, 3 & 8 are the extremes, while 5 & 4 are the means.
Note: The ratio value does not affect when the same non-zero number is multiplied or divided on each term.
Important Properties of Proportion
The following are the important properties of proportion:
Addendo – If a : b = c : d, then a + c : b + d
Subtrahendo – If a : b = c : d, then a – c : b – d
Dividendo – If a : b = c : d, then a – b : b = c – d : d
Componendo – If a : b = c : d, then a + b : b = c+d : d
Alternendo – If a : b = c : d, then a : c = b: d
Invertendo – If a : b = c : d, then b : a = d : c
Componendo and dividendo – If a : b = c : d, then a + b : a – b = c + d : c – d
Difference Between Ratio and Proportion
To understand the concept of ratio and proportion, go through the difference between ratio and proportion given here.
Example 1:
Are the ratios 4:5 and 8:10 said to be in Proportion?
Solution:
4:5 :: 8:10
Since both the ratios are equal, they are said to be in proportion.
Example 2:
Are the two ratios 8:10 and 7:10 in proportion?
Solution:
8:10= 8/10= 0.8 and 7:10= 7/10= 0.7
Since both the ratios are not equal, they are not in proportion.
Example 3:
Given ratio are-
a:b = 2:3
b:c = 5:2
Find a: b: c.
Solution:
Multiplying the first ratio by 5, second by 3 and third by 6, we have
a:b = 10: 15
b:c = 15 : 6
In the ratio’s above, all the mean terms are equal, thus
a:b:c= 10:15:6
Example 4:
Check whether the following statements are true or false.
a] 12 : 18 = 28 : 56
b] 25 people : 130 people = 15kg : 78kg
Solution:
a] 12 : 18 = 28 : 56
The given statement is false.
12 : 18 = 12 / 18 = 2 / 3 = 2 : 3
28 : 56 = 28 / 56 = 1 / 2 = 1 : 2
They are unequal.
b] 25 persons : 130 persons = 15kg : 78kg
The given statement is true.
25 people : 130 people = 5: 26
15kg : 78kg = 5: 26
They are equal.
Example 5:
The earnings of Rohan is 12000 rupees every month and Anish is 191520 per year. If the monthly expenses of every person are
around 9960 rupees. Find the ratio of the savings.
Solution:
Savings of Rohan per month = Rs (12000-9960) = Rs. 2040
Yearly income of Anish = Rs. 191520
Hence, the monthly income of Anish = Rs. 191520/12 = Rs. 15960.
So, the savings of Anish per month = Rs (15960 – 9960) = Rs. 6000
Thus, the ratio of savings of Rohan and Anish is Rs. 2040: Rs.6000 = 17: 50.
Example 6:
Twenty tons of iron is Rs. 600000 (six lakhs). What is the cost of 560 kilograms of iron?
Solution:
1 ton = 1000 kg
20 tons = 20000 kg
The cost of 20000 kg iron = Rs. 600000
The cost of 1 kg iron = Rs{600000}/ {20000}
= Rs. 30
The cost of 560 kg iron = Rs 30 × 560 = Rs 16800
Example 7:
The dimensions of the rectangular field are given. The length and breadth of the rectangular field are 50 meters and 15 meters.
What is the ratio of the length and breadth of the field?
Solution:
Length of the rectangular field = 50 m
Breadth of the rectangular field = 15 m
Hence, the ratio of length to breadth = 50: 15
⇒ 50: 15 = 10: 3.
Thus, the ratio of length and breadth of the rectangular field is 10:3.
Example 8:
Obtain a ratio of 90 centimetres to 1.5 meters.
Solution:
The given two quantities are not in the same units.
Convert them into the same units.
1.5 m = 1.5 × 100 = 150 cm
Hence, the required ratio is 90 cm: 150 cm
⇒ 90: 150 = 3: 5
Therefore, the ratio of 90 centimetres to 1.5 meters is 3: 5.
Example 9:
There exists 45 people in an office. Out of which female employees are 25 and the remaining are male employees. Find the ratio of
a] The count of females to males.
b] The count of males to females.
Solution:
Count of females = 25
Total count of employees = 45
Count of males = 45 – 25 = 20
The ratio of the count of females to the count of males
= 25 : 20
=5:4
The count of males to the count of females
= 20 : 25
=4:5
Example 10:
Write two equivalent ratios of 6: 4.
Answer:
Given Ratio : 6: 4, which is equal to 6/4.
Multiplying or dividing the same numbers on both numerator and denominator, we will get the equivalent ratio.
⇒(6×2)/(4×2) = 12/8 = 12: 8
⇒(6÷2)/(4÷2) = 3/2 = 3: 2
Therefore, the two equivalent ratios of 6: 4 are 3: 2 and 12: 8
Example 11:
Out of the total students in a class, if the number of boys is 5 and the number of girls is 3, then find the ratio between girls and
boys.
Solution:
The ratio between girls and boys can be written as 3:5 (Girls: Boys). The ratio can also be written in the form of factor like 3/5.
Example 12:
Two numbers are in the ratio 2 : 3. If the sum of numbers is 60, find the numbers.
Solution:
Given, 2/3 is the ratio of any two numbers.
Let the two numbers be 2x and 3x.
As per the given question, the sum of these two numbers = 60
So, 2x + 3x = 60
5x = 60
x = 12
Hence, the two numbers are;
2x = 2 x 12 = 24
and
3x = 3 x 12 = 36
24 and 36 are the required numbers.
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