Ratio and Proportion – Complete Notes for Competitive Exams
Ratio and Proportion is an important topic in quantitative aptitude and mathematics for competitive examinations. Questions based on ratio, proportion, division of quantities, ages, partnership, mixture, work, income and expenditure often require a clear understanding of basic relationships rather than lengthy calculations.
These notes explain Ratio and Proportion from the fundamentals to exam-level applications. The formulas, examples and shortcuts below are written specifically for ExamMitra and are intended for original learning and revision.
Table of Contents
- 1. What is a Ratio?
- 2. Terms Used in Ratio
- 3. Simplifying a Ratio
- 4. Equivalent Ratios
- 5. Comparing Ratios
- 6. Ratio of Three or More Quantities
- 7. Combining Two or More Ratios
- 8. Division of a Quantity in a Given Ratio
- 9. Proportion
- 10. Fourth Proportional
- 11. Third Proportional
- 12. Mean Proportional
- 13. Direct Proportion
- 14. Inverse Proportion
- 15. Ratio and Percentage
- 16. Ratio and Fractions
- 17. Ratio and Ages
- 18. Ratio in Income and Expenditure
- 19. Ratio in Partnership
- 20. Ratio in Mixture Problems
- 21. Ratio in Work Problems
- 22. Important Shortcuts
- 23. Solved Examples
- 24. Common Mistakes
- 25. Quick Revision Sheet
- 26. Preparation Strategy
- 27. Frequently Asked Questions
1. What is a Ratio?
A ratio is a comparison between two quantities of the same kind using division.
For example, if a class has 20 boys and 30 girls, the ratio of boys to girls is:
The ratio tells us the relative size of one quantity compared with another. It does not directly tell us the actual quantities unless one of them is known.
2. Terms Used in Ratio
Consider the ratio:
- a is called the first term or antecedent.
- b is called the second term or consequent.
For the ratio 7 : 11, 7 is the antecedent and 11 is the consequent.
Important condition
The quantities being compared must have the same units.
For example:
= 200 cm : 50 cm
= 4 : 1
3. Simplifying a Ratio
To simplify a ratio, divide both terms by their greatest common factor.
HCF of 48 and 72 = 24.
48 : 72 = 2 : 3.
Decimal ratios
If a ratio contains decimals, multiply both terms by a suitable power of 10.
Multiply both terms by 10:
15 : 25 = 3 : 5
Fraction ratios
For ratios involving fractions, multiply both terms by the LCM of the denominators.
LCM of 2 and 4 = 4.
= 2 : 3
4. Equivalent Ratios
Two ratios are equivalent when they represent the same relationship.
where k is any non-zero number.
For example:
Equivalent ratios are particularly useful when comparing quantities or combining ratios.
5. Comparing Ratios
To compare two ratios, one convenient method is cross multiplication.
Suppose we want to compare:
Compare:
5 × 7 = 35
Since 36 is greater than 35:
6. Ratio of Three or More Quantities
Ratios can compare three or more quantities.
For example:
A : B : C = 20 : 30 : 40
= 2 : 3 : 4
The common factor must divide every term.
7. Combining Two or More Ratios
Suppose:
B : C = 4 : 5
The common term is B. Make the B terms equal.
LCM of 3 and 4 = 12.
B : C = 12 : 15
Therefore:
A : B : C = 8 : 12 : 15
8. Division of a Quantity in a Given Ratio
Suppose ₹700 is divided between A and B in the ratio 3 : 4.
Total parts:
Value of one part:
Therefore:
B = 4 × 100 = ₹400
General Formula
First share = Q × a/(a+b)
Second share = Q × b/(a+b)
9. Proportion
A proportion states that two ratios are equal.
or
a/b = c/d
This can also be written as:
Example
4 × 15 = 60
6 × 10 = 60
Therefore, the ratios are proportional.
10. Fourth Proportional
If:
then x is called the fourth proportional to a, b and c.
Using cross multiplication:
4 : 6 = 8 : x
4x = 48
x = 12
11. Third Proportional
If:
then x is called the third proportional to a and b.
3 : 6 = 6 : x
3x = 36
x = 12
12. Mean Proportional
The mean proportional between two positive numbers a and b is the number x such that:
Therefore:
x = √ab
x = √(4 × 9)
x = √36
x = 6
13. Direct Proportion
Two quantities are in direct proportion when an increase in one quantity produces a corresponding increase in the other, or a decrease produces a corresponding decrease.
or
x/y = constant
Example
If 5 notebooks cost ₹100, then 10 notebooks will cost ₹200, assuming the price per notebook remains unchanged.
10 notebooks → ₹200
Number of notebooks doubles, so cost also doubles.
Direct proportion method
14. Inverse Proportion
Two quantities are in inverse proportion when an increase in one quantity causes a decrease in the other in such a way that their product remains constant.
or
xy = constant
Example
If 6 workers complete a job in 10 days, increasing the number of workers can reduce the required number of days, assuming all workers work at the same rate.
If 12 workers are used:
12 × days = 60
days = 5
15. Ratio and Percentage
Ratios can be converted into percentages.
If A : B = a : b, then A as a percentage of B is:
A is what percent of B?
= 3/5 × 100
= 60%
Percentage increase using ratio
If the old value and new value are in the ratio:
then percentage increase is:
16. Ratio and Fractions
A ratio can be represented as a fraction:
If the ratio of boys to girls is 2 : 3, then:
If the total number of students is known, the ratio can be used to find each group.
Boys : Girls = 2 : 3
Total parts = 5
Boys = 50 × 2/5 = 20
Girls = 50 × 3/5 = 30
17. Ratio and Ages
Ratio questions involving ages are common in competitive examinations.
Basic method
If the present ages of A and B are in the ratio 3 : 5, represent their ages as:
B = 5x
If their total age is 40:
8x = 40
x = 5
A = 15 years
B = 25 years
Important point about ages
The difference between two people's ages remains constant, although their ratio changes over time.
18. Ratio in Income and Expenditure
Ratios are frequently used in questions involving income, expenditure and savings.
Let income = 5x and expenditure = 3x.
Saving = 5x - 3x = 2x.
Therefore:
Income : Expenditure : Saving = 5 : 3 : 2
Example
If income is ₹30,000 and expenditure is ₹24,000:
= ₹6,000
19. Ratio in Partnership
In partnership questions, profit is generally divided according to the product of:
Thus:
B invests ₹30,000 for 8 months.
A's share ratio = 20,000 × 12 = 240,000
B's share ratio = 30,000 × 8 = 240,000
Therefore:
Profit ratio = 1 : 1
20. Ratio in Mixture Problems
Ratio is useful when two or more substances are mixed in different quantities.
For example, if milk and water are mixed in the ratio 5 : 2, then:
Water = 2 parts
Total = 7 parts
If the mixture contains 35 litres:
Water = 35 × 2/7 = 10 litres
Changing a mixture ratio
When some quantity is removed and another substance is added, carefully track the amount of each component. Do not apply the final ratio directly to the original quantity.
21. Ratio in Work Problems
Work problems often use ratios to compare the efficiency of workers.
If A and B have efficiencies in the ratio 3 : 2, then for the same amount of time:
For the same amount of work, time is inversely proportional to efficiency.
Time ratio = 2 : 3
22. Important Shortcuts
Shortcut 1: Ratio from difference
If two quantities are in the ratio a : b and their difference is D:
Then multiply by a and b to get the quantities.
Difference in parts = 5 - 3 = 2
One part = 24/2 = 12
Numbers = 36 and 60
Shortcut 2: Ratio from sum
If two quantities are in the ratio a : b and their sum is S:
Second quantity = S × b/(a+b)
Shortcut 3: If one quantity changes
Suppose:
If A becomes twice its original value:
Shortcut 4: Same ratio multiplier
If:
and B = 35, then:
1 part = 5
A = 4 × 5 = 20
23. Solved Examples
Example 1: Simplification
Simplify 84 : 126.
HCF = 42.
84 : 126 = 2 : 3.
Example 2: Division of money
Divide ₹1,200 in the ratio 5 : 7.
Total parts = 12.
One part = 1200/12 = 100.
First share = 5 × 100 = ₹500.
Second share = 7 × 100 = ₹700.
Example 3: Finding a missing value
If 7 : 9 = x : 45, find x.
7/9 = x/45
9x = 315
x = 35.
Example 4: Ratio from total
The ratio of two numbers is 4 : 7 and their sum is 99. Find the numbers.
Total parts = 11.
One part = 99/11 = 9.
Numbers = 36 and 63.
Example 5: Ratio from difference
Two numbers are in the ratio 5 : 8 and their difference is 27. Find them.
Difference in parts = 3.
One part = 27/3 = 9.
Numbers = 45 and 72.
Example 6: Age ratio
The present ages of A and B are in the ratio 2 : 3. Their total age is 50 years. Find their ages.
2x + 3x = 50
5x = 50
x = 10.
A = 20 years.
B = 30 years.
Example 7: Direct proportion
If 8 kg of rice costs ₹480, what will 15 kg cost at the same rate?
Cost per kg = 480/8 = ₹60.
Cost of 15 kg = 15 × 60 = ₹900.
Example 8: Inverse proportion
10 workers complete a task in 18 days. How many days will 15 workers require if their efficiency is the same?
Workers × Days = constant.
10 × 18 = 15 × d
180 = 15d
d = 12.
Example 9: Income and expenditure
A person's income and expenditure are in the ratio 7 : 5. If the income is ₹35,000, find the expenditure and saving.
7 parts = ₹35,000
1 part = ₹5,000.
Expenditure = 5 × 5,000 = ₹25,000.
Saving = ₹35,000 - ₹25,000 = ₹10,000.
Example 10: Partnership
A invests ₹40,000 for 6 months and B invests ₹30,000 for 8 months. Find their profit ratio.
A = 40,000 × 6 = 240,000.
B = 30,000 × 8 = 240,000.
Profit ratio = 1 : 1.
Example 11: Mean proportional
Find the mean proportional between 9 and 16.
x = √(9 × 16)
= √144
= 12.
Example 12: Three quantities
If A : B = 2 : 3 and B : C = 6 : 7, find A : B : C.
A : B = 4 : 6.
B : C = 6 : 7.
Therefore:
A : B : C = 4 : 6 : 7.
24. Common Mistakes
- Do not compare quantities having different units without conversion.
- Do not confuse antecedent with consequent.
- Do not add ratio terms before checking what the question asks.
- In direct proportion, both quantities move in the same direction.
- In inverse proportion, one quantity increases while the other decreases.
- For partnership, use capital multiplied by time.
- For age problems, use a common multiplier such as x.
- When dividing a total in a ratio, always calculate the total number of parts.
- When combining ratios, make the common term equal first.
- Do not assume a ratio represents actual values unless sufficient information is provided.
25. Quick Revision Sheet
| Concept | Formula / Rule |
|---|---|
| Ratio | a : b = a/b |
| Proportion | a : b = c : d, therefore ad = bc |
| Fourth proportional | x = bc/a |
| Third proportional | x = b²/a |
| Mean proportional | √ab |
| Division in a:b | First share = Total × a/(a+b) |
| Ratio from sum | Use total parts |
| Ratio from difference | One part = Difference / difference of ratio terms |
| Direct proportion | x/y = constant |
| Inverse proportion | xy = constant |
| Partnership | Profit ratio = Capital × Time |
| Age problems | Represent ages as ratio terms × x |
| Mixture | Use component ratio and total parts |
26. Preparation Strategy
For competitive examinations, Ratio and Proportion becomes much easier when the basic relationships are understood instead of memorising many separate formulas.
Step 1: Master ratio basics
Practise simplifying ratios involving whole numbers, decimals, fractions and different units.
Step 2: Practise division problems
Learn to divide a total amount quickly using the sum of ratio parts.
Step 3: Learn proportion
Become comfortable with cross multiplication and finding missing terms.
Step 4: Understand direct and inverse proportion
Before calculating, decide whether the two quantities move in the same direction or opposite directions.
Step 5: Connect ratio with other topics
Ratio is used extensively in:
- Percentage
- Profit and Loss
- Average
- Mixture and Alligation
- Time and Work
- Partnership
- Age Problems
- Speed, Time and Distance
- Income and Expenditure
Step 6: Use a short calculation method
In an examination, avoid writing unnecessary steps. Identify the ratio structure first, then use the shortest reliable calculation.
27. Frequently Asked Questions
What is the basic formula of ratio?
The basic form is a : b, which represents a/b.
What is proportion?
Proportion means equality between two ratios, such as a : b = c : d.
What is the difference between ratio and proportion?
A ratio compares two quantities, whereas a proportion states that two ratios are equal.
How do I divide a number in a ratio?
Add the ratio terms to find the total number of parts, divide the total quantity by those parts, and multiply by the required ratio term.
What is direct proportion?
In direct proportion, both quantities increase or decrease together while their ratio remains constant.
What is inverse proportion?
In inverse proportion, an increase in one quantity corresponds to a decrease in the other while their product remains constant.
Why is Ratio and Proportion important for competitive exams?
It is a fundamental concept used directly in quantitative aptitude and indirectly in topics such as percentage, partnership, mixture, work, ages and speed-time problems.
Final Revision
Ratio and Proportion is not just a separate mathematics chapter. It is a foundation for many other quantitative aptitude topics. If you can simplify ratios, combine ratios, divide quantities, identify direct or inverse relationships and use cross multiplication confidently, many exam questions become much faster to solve.
For preparation, focus first on the basic concepts and then move toward mixed questions involving ages, partnership, mixtures, work and percentage. Regular practice will improve both calculation speed and accuracy.
Ratio compares quantities.
Proportion compares ratios.
Direct proportion means same-direction change.
Inverse proportion means opposite-direction change.
For division, use total parts.
For partnership, use capital × time.
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