Ratio and Proportion- Concept Explained by Shubham Sharma

Ratio and Proportion are important concepts in aptitude, commonly tested in competitive exams and various problem-solving situations. They are used to express the relationship between two or more quantities and solve problems involving comparisons, scaling, or distribution of quantities.
1. Ratio
A ratio is a way to compare two or more quantities by expressing them as fractions or in the form of "a : b." It indicates how much of one quantity there is compared to another.
Key Points:
Definition: A ratio is expressed as a : b, where "a" is the first quantity and "b" is the second quantity.
Read as: "a is to b" or "a to b."
Example: If a basket contains 3 apples and 2 oranges, the ratio of apples to oranges is 3:2.
Properties of Ratios:
Ratios can be simplified (just like fractions). If the ratio of apples to oranges is 6:4, it can be simplified to 3:2.
Ratios are dimensionless, meaning they compare quantities of the same kind (like apples to oranges) or different kinds, depending on the context.
Types of Ratios:
Direct Ratio: When two quantities are compared directly (e.g., the ratio of girls to boys in a class).
Inverse Ratio: When two quantities are inversely related (e.g., the ratio of time to speed in certain distance-related problems).
2. Proportion
Proportion refers to the equality of two ratios. It states that two ratios are equal and is represented as:
𝑎
𝑏
=
𝑐
𝑑
or
𝑎
:
𝑏
=
𝑐
:
𝑑
b
a
​
=
d
c
​
ora:b=c:d
This is called a proportion, and it means that the ratio of "a" to "b" is the same as the ratio of "c" to "d."
Key Points:
Definition: If a : b = c : d, then it is a proportion. This implies that the cross-multiplication holds true: a × d = b × c.
Example: If 2 apples cost $4, how much would 5 apples cost? We set up the proportion as:
2
4
=
5
𝑥
4
2
​
=
x
5
​
Cross-multiply to find x (the cost of 5 apples):
2
×
𝑥
=
4
×
5
⇒
𝑥
=
20
2
=
10
2×x=4×5⇒x=
2
20
​
=10
So, 5 apples would cost $10.
Types of Proportions:
Direct Proportion: When the two quantities increase or decrease together. If a is directly proportional to b, then a/b = constant (or a = k × b), where "k" is a constant.
Inverse Proportion: When one quantity increases as the other decreases. If a is inversely proportional to b, then a × b = constant (or a = k / b).
3. Solving Problems Using Ratio and Proportion
Problems involving ratio and proportion can be approached systematically. Here’s a basic outline of how to tackle these problems:
Steps for Solving Ratio Problems:
Understand the given quantities and the relationship between them.
Set up the ratio correctly by identifying what you are comparing.
Simplify the ratio if necessary (if the quantities are not in their simplest form).
Solve using basic operations (multiplication, division) to find unknown quantities.
Steps for Solving Proportion Problems:
Write the proportion by comparing the two ratios.
Use cross-multiplication to form an equation.
Solve for the unknown variable using simple algebraic steps.
4. Examples:
Example 1 - Ratio:
Problem: The ratio of boys to girls in a class is 3:4. If there are 24 girls, how many boys are there?
Solution: The ratio of boys to girls is 3:4. If there are 24 girls, we can set up the proportion:
3
4
=
𝑥
24
4
3
​
=
24
x
​
Cross-multiply:
3
×
24
=
4
×
𝑥
⇒
72
=
4
𝑥
⇒
𝑥
=
18
3×24=4×x⇒72=4x⇒x=18
So, there are 18 boys.
Example 2 - Proportion:
Problem: A car travels 120 miles in 3 hours. How far will it travel in 5 hours at the same speed?
Solution: Set up the proportion:
120
3
=
𝑥
5
3
120
​
=
5
x
​
Cross-multiply:
120
×
5
=
3
×
𝑥
⇒
600
=
3
𝑥
⇒
𝑥
=
200
120×5=3×x⇒600=3x⇒x=200
So, the car will travel 200 miles in 5 hours.
5. Applications of Ratio and Proportion:
Mixing Problems: When combining two or more ingredients in a fixed ratio (e.g., in cooking, medicine, or manufacturing).
Speed, Time, and Distance: To calculate travel times or distances based on given speeds.
Work and Wages: To determine the relationship between the amount of work done and the corresponding payment.
Scaling: Used in mapping, model-making, or resizing objects.
Conclusion:
In aptitude, ratio and proportion are essential tools for comparing quantities and solving practical problems involving relationships between them. Mastery of these concepts helps solve a wide range of problems, from everyday scenarios to complex competitive exam questions. By practicing and applying these concepts, you can improve your problem-solving speed and accuracy.