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 and proportion are the two important concepts, and it is the foundation to understand the various concepts in mathematics as well as in science.

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.

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.

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.

Proportion refers to the equality of two ratios. Two equivalent ratios are always in proportion. Proportions are denoted by the symbol : : and they help us to solve for unknown quantities. In other words, proportion is an equation or statement that is used to depict that the two ratios or fractions are equivalent. Four non-zero quantities, a, b, c, d are said to be in proportion if a : b = c : d. Now, let us consider the two ratios - 3 : 5 and 15 : 25. Here, 3 : 5 can be expressed as 3:5 = 3/5 = 0.6 and 15:25 can be expressed as 15:25 = 15/25 = 3/5 = 0.6. Since both the ratios are equal, we can say that these two are proportional.

The proportion can be classified into the following categories, such as:

Direct Proportion

Inverse Proportion

Continued Proportion

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 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

The formula for ratio is expressed as a : b ⇒ a/b, where,

a = the first term or antecedent.

b = the second term or consequent.

For example, ratio 2 : 7 is also represented as 2/7, where 2 is the antecedent and 7 is the consequent.

Ratio

The ratio of two quantities a and b in the same units, is the fraction and we write it as a : b.
In the ratio a : b, we call a as the first term or antecedent and b, the second term or consequent.
Eg. The ratio 5 : 9 represents 5
with antecedent = 5, consequent = 9.
Rule: The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio.
Eg. 4 : 5 = 8 : 10 = 12 : 15. Also, 4 : 6 = 2 : 3.
Proportion
The equality of two ratios is called proportion. If a : b = c : d, we write a : b :: c : d and we say that a, b, c, d are in proportion.
Here a and d are called extremes, while b and c are called mean terms.
Product of means = Product of extremes.
Thus, a : b :: c : d (b x c) = (a x d).
Fourth Proportional: If a : b = c : d, then d is called the fourth proportional to a, b, c.
Third Proportional: a : b = c : d, then c is called the third proportion to a and b.
Mean Proportional: Mean proportional between a and b is ab.
Compounded Ratio:The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf).
Duplicate Ratios: Duplicate ratio of (a : b) is (a2 : b2).
Sub-duplicate ratio of (a : b) is (a : b).
Triplicate ratio of (a : b) is (a3 : b3).
Sub-triplicate ratio of (a : b) is (a1/3 : b1/3).

Variations
We say that x is directly proportional to y, if x = ky for some constant k and we write, x y. We say that x is inversely proportional to y, if xy = k for some constant k and

Example 1
Suppose person A and person B started a partnership business and decided to divide the profit between them in a ratio of 2:4. By the end of the financial year, the total profit would be rs. 10,000. What will be their part of profit?
Solution
Their profit is to be divided into a ratio of 2:4.
So we can find the profit of each one of them by:
A= 10,000 x (2/6) = 3333.33
B = 10,000 x (4/6) = 6666.67
Therefore, their respective profit will be 3333.33 and 6666.67.

Example 2
Find a:b:c if the given ratios are as follows:
a:b = 2:3
b:c = 5:2
c:d = 1:4
Solution
If we multiply the first ratio by 5, the second ratio by 3, and the third ratio by 6, we will have:
a:b = 10:15
b:c = 15:6
c:d = 6:24
The ratios above have equal mean scores. Therefore, a:b:c:d = 10:15:6:24.

Example 3
In a handwriting competition, there are 5 boys and 3 girls. What will be the ratio between girls and boys?
Solution
The ratio between girls and boys will be 3 is to 5. We can also write it as 3/5.

Example 4
If Sam in 2 hours covers a distance of 40 km. What distance will he cover in 8 hours?
Solution
Let the distance be x. With time, the distance also increases.
Therefore, 2:8 = 40:y
y = (40 x 8) / 2
= 160 km.
Sam can cover a distance of 160km in 8 hours.

Example 5
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.

Key Points to Remember:
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.

Definition of Proportion
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 ‘=’.

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: 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

Comparison of Ratios

If (a:b)>(c:d) = (a/b>c/d)
The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf).

Duplicate Ratios
If a:b is a ratio, then:
a2:b2 is a duplicate ratio
√a:√b is the sub-duplicate ratio
a3:b3 is a triplicate ratio
Ratio and Proportion Tricks
Let us learn here some rules and tricks to solve problems based on ratio and proportion topics.
If u/v = x/y, then uy = vx
If u/v = x/y, then u/x = v/y
If u/v = x/y, then v/u = y/x
If u/v = x/y, then (u+v)/v = (x+y)/y
If u/v = x/y, then (u-v)/v = (x-y)/y
If u/v = x/y, then (u+v)/ (u-v) = (x+y)/(x-y), which is known as componendo -Dividendo Rule
If a/(b+c) = b/(c+a) = c/(a+b) and a+b+ c ≠0, then a =b = c
Ratio and Proportion Summary
Ratio defines the relationship between the quantities of two or more objects. It is used to compare the quantities of the same kind.
If two or more ratios are equal, then it is said to be in proportion. The proportion can be represented in two different ways. Either it can be represented using an equal sign or by using a colon symbol.(i.e) a:b = c:d or a:b :: c:d
If we multiply or divide each term of the ratio by the same number, it does not affect the ratio.
For any three quantities, the quantities are said to be in continued proportion, if the ratio between the first and second quantity is equal to the ratio between the second and third quantity.
For any four quantities, they are said to be in continued proportion, if the ratio between the first and second quantities is equal to the ratio between the third and fourth quantities

Ratio And Proportion Examples

Example 1:
Are the ratios 4:5 and 8:10 said to be in Proportion?
Solution:
4:5= 4/5 = 0.8 and 8: 10= 8/10= 0.8
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 area:b = 2:3
b:c = 5:2
c:d = 1:4
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
c:d = 6 : 24
In the ratio’s above, all the mean terms are equal, thus
a:b:c:d = 10:15:6:24
Example 4:
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 5:
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 6:
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 7:
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 8:
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 9:
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.