Ratio and proportion are basic but very useful topics in mathematics. We use them whenever we compare two quantities, divide something in a fixed share, check whether two relations are equal, or solve questions based on speed, work, money, marks, mixture, map, recipe, population, and percentage. In competitive exams, ratio and proportion questions are common because they test both calculation and understanding.
The best way to learn this chapter is to understand the meaning first. A ratio compares two quantities. A proportion tells us that two ratios are equal. If this difference is clear, most questions become simple.
Ratio
The ratio of two quantities a and b in the same unit is the fraction a/b. It is written as a:b. In the ratio a:b, a is called the first term or antecedent, and b is called the second term or consequent.
For example, if there are 12 boys and 8 girls in a class, the ratio of boys to girls is 12:8. This can be simplified by dividing both terms by 4. So, 12:8 = 3:2. It means for every 3 boys, there are 2 girls.
A ratio has no unit when both quantities are in the same unit. If one quantity is in metres and another is in centimetres, first convert both into the same unit, then find the ratio.
Important rule of ratio
Multiplying or dividing both terms of a ratio by the same non-zero number does not change the ratio.
Example: 4:5 = 8:10 = 12:15. These ratios look different, but their value is the same because both terms are multiplied by the same number.
Example: 24:36 can be simplified by dividing both terms by 12. So, 24:36 = 2:3.
How to simplify a ratio
To simplify a ratio, find the highest common factor of the terms and divide each term by it.
Example: Simplify 45:60.
The highest common factor of 45 and 60 is 15.
45 ÷ 15 = 3 and 60 ÷ 15 = 4.
So, 45:60 = 3:4.
Ratio with different units
If two quantities are in different units, convert them into the same unit first.
Example: Find the ratio of 2 metres to 50 centimetres.
2 metres = 200 centimetres.
Ratio = 200:50 = 4:1.
So, the ratio is 4:1.
Ratio in three quantities
A ratio can compare more than two quantities. For example, if the marks of three students are 30, 45, and 60, their ratio is 30:45:60. Divide all terms by 15, so the simplified ratio is 2:3:4.
Example: Divide Rs. 900 among A, B, and C in the ratio 2:3:4.
Total parts = 2 + 3 + 4 = 9.
Value of one part = 900 ÷ 9 = 100.
A gets 2 × 100 = Rs. 200.
B gets 3 × 100 = Rs. 300.
C gets 4 × 100 = Rs. 400.
Proportion
When two ratios are equal, they are said to be in proportion. If a:b = c:d, we write it as a:b :: c:d. Here, a and d are called extremes, and b and c are called means.
Formula: Product of means = Product of extremes.
So, if a:b :: c:d, then b × c = a × d.
Example: Check whether 2:3 and 8:12 are in proportion.
Product of means = 3 × 8 = 24.
Product of extremes = 2 × 12 = 24.
Both products are equal, so 2:3 :: 8:12.
Finding missing term in proportion
When three terms are given and one term is missing, use the product of means and extremes.
Example: Find x if 5:8 :: 20:x.
Using b × c = a × d:
8 × 20 = 5 × x
160 = 5x
x = 32
So, the missing term is 32.
Third proportion
If a:b = b:c, then c is called the third proportion to a and b.
Example: 4:8 :: 8:16. Here, 16 is the third proportion to 4 and 8.
Example: Find the third proportion to 6 and 12.
Let the third proportion be x.
6:12 :: 12:x
12 × 12 = 6 × x
144 = 6x
x = 24
So, the third proportion is 24.
Fourth proportion
If a:b = c:d, then d is called the fourth proportion to a, b, and c.
Example: Find the fourth proportion to 3, 5, and 12.
Let the fourth proportion be x.
3:5 :: 12:x
5 × 12 = 3 × x
60 = 3x
x = 20
So, the fourth proportion is 20.
Mean proportion
The mean proportion between a and b is sqrt(ab).
If a:x :: x:b, then x is the mean proportion between a and b. In this case, x² = ab, so x = sqrt(ab).
Example: Find the mean proportion between 4 and 16.
Mean proportion = sqrt(4 × 16)
= sqrt(64)
= 8
So, 8 is the mean proportion.
Example: 4:8 :: 8:16 because 8² = 4 × 16. So, 8 is the mean proportion.
Continued proportion
When three numbers a, b, and c are in continued proportion, we write a:b :: b:c. Here, b is the mean proportion and c is the third proportion.
Example: 2, 4, and 8 are in continued proportion because 2:4 = 4:8.
Important ratio forms
Duplicate ratio of a:b is a²:b².
Sub-duplicate ratio of a:b is sqrt(a):sqrt(b).
Triplicate ratio of a:b is a³:b³.
Sub-triplicate ratio of a:b is a^(1/3):b^(1/3).
Example: Find the duplicate ratio of 3:4.
Duplicate ratio = 3²:4² = 9:16.
Example: Find the triplicate ratio of 2:3.
Triplicate ratio = 2³:3³ = 8:27.
Comparison of ratios
To compare two ratios, write them as fractions. a:b is greater than c:d if a/b > c/d.
Example: Compare 3:4 and 5:7.
3/4 = 0.75 and 5/7 is about 0.714.
So, 3:4 is greater than 5:7.
Another method is cross multiplication. For 3:4 and 5:7, compare 3 × 7 and 5 × 4. We get 21 and 20. Since 21 is greater than 20, 3:4 is greater.
Useful formula in proportion
If a/b = c/d, then (a + b)/(a – b) = (c + d)/(c – d), when the denominators are not zero.
This formula is useful in algebraic ratio questions. It comes from the property that equal ratios remain equal when the same type of operation is applied properly to numerator and denominator.
Direct proportion
x is directly proportional to y if x = ky for some constant k. We write x ∝ y. If y increases, x also increases in the same ratio. If y decreases, x also decreases in the same ratio.
Example: If 5 notebooks cost Rs. 100, find the cost of 8 notebooks.
More notebooks means more cost, so this is direct proportion.
Cost of 1 notebook = 100 ÷ 5 = Rs. 20.
Cost of 8 notebooks = 8 × 20 = Rs. 160.
So, 8 notebooks cost Rs. 160.
Inverse proportion
x is inversely proportional to y if xy = k for some constant k. We write x ∝ 1/y. If y increases, x decreases. If y decreases, x increases.
Example: 6 workers can complete a work in 10 days. How many days will 12 workers take?
More workers means fewer days, so this is inverse proportion.
Workers × Days = constant.
6 × 10 = 12 × x
60 = 12x
x = 5
So, 12 workers will complete the work in 5 days.
Ratio and fraction
A ratio can be written as a fraction, but both are used slightly differently. A fraction usually shows a part of a whole, while a ratio compares two quantities. For example, if there are 3 boys and 2 girls, the ratio of boys to girls is 3:2. The fraction of boys in the whole class is 3/(3 + 2) = 3/5.
This difference is important in exam questions. If the ratio of boys to girls is 3:2, then boys are not 3/2 of the class. Boys are 3/5 of the class because total parts are 5.
Ratio and percentage
Ratio and percentage are connected. A ratio can be converted into percentage by finding the total parts.
Example: The ratio of passed students to failed students is 7:3. Find the percentage of passed students.
Total parts = 7 + 3 = 10.
Passed part = 7.
Percentage of passed students = 7/10 × 100 = 70%.
So, 70% students passed.
Dividing money in a ratio
Money-sharing questions are common in school and government exams.
Example: Divide Rs. 2400 between A and B in the ratio 5:7.
Total parts = 5 + 7 = 12.
Value of one part = 2400 ÷ 12 = 200.
A gets 5 × 200 = Rs. 1000.
B gets 7 × 200 = Rs. 1400.
Ratio in age problems
Age questions often use ratios. The important point is that after some years, the same number is added to both ages. Before some years, the same number is subtracted from both ages.
Example: The present ratio of ages of A and B is 3:5. If their total age is 48 years, find their ages.
Total parts = 3 + 5 = 8.
One part = 48 ÷ 8 = 6.
A’s age = 3 × 6 = 18 years.
B’s age = 5 × 6 = 30 years.
Ratio in mixture problems
Mixture questions use ratio to compare the quantities of two or more items.
Example: A mixture contains milk and water in the ratio 4:1. If the mixture is 50 litres, find the quantity of milk and water.
Total parts = 4 + 1 = 5.
One part = 50 ÷ 5 = 10 litres.
Milk = 4 × 10 = 40 litres.
Water = 1 × 10 = 10 litres.
Ratio in speed and time
For the same distance, speed and time are inversely proportional. If speed increases, time decreases. If speed decreases, time increases.
Example: A car travels a fixed distance at 40 km/h in 6 hours. How much time will it take at 60 km/h?
Speed × Time = constant.
40 × 6 = 60 × x
240 = 60x
x = 4
So, the car will take 4 hours.
Ratio in work questions
In work questions, number of workers and number of days are usually inversely proportional, if the amount of work is fixed.
Example: 15 men can finish a work in 20 days. How many days will 10 men take?
Men × Days = constant.
15 × 20 = 10 × x
300 = 10x
x = 30
So, 10 men will take 30 days.
Government exam type examples
These examples are similar to questions asked in SSC, Railway, Banking, Police, State PSC, and other competitive exams.
Example 1
The ratio of two numbers is 5:8 and their sum is 78. Find the numbers.
Total parts = 5 + 8 = 13.
One part = 78 ÷ 13 = 6.
Numbers are 5 × 6 = 30 and 8 × 6 = 48.
Answer: 30 and 48.
Example 2
The ratio of income and expenditure of a person is 7:5. If his saving is Rs. 6000, find his income.
Income parts = 7, expenditure parts = 5.
Saving parts = 7 – 5 = 2.
2 parts = Rs. 6000.
1 part = Rs. 3000.
Income = 7 parts = 7 × 3000 = Rs. 21000.
Example 3
A bag contains coins in the ratio 1:2:3. If the total number of coins is 120, find each type of coin.
Total parts = 1 + 2 + 3 = 6.
One part = 120 ÷ 6 = 20.
The numbers are 20, 40, and 60.
Example 4
The ratio of marks of A and B is 4:5. If B gets 80 marks, find A’s marks.
5 parts = 80.
1 part = 16.
A’s marks = 4 × 16 = 64.
Example 5
If 18 men can complete a work in 12 days, how many men are needed to complete it in 8 days?
Men and days are inversely proportional.
18 × 12 = x × 8
216 = 8x
x = 27
Answer: 27 men.
Example 6
If the ratio of A:B is 2:3 and B:C is 4:5, find A:B:C.
A:B = 2:3 and B:C = 4:5.
Make B equal in both ratios. LCM of 3 and 4 is 12.
A:B = 8:12.
B:C = 12:15.
So, A:B:C = 8:12:15.
Example 7
If x:y = 3:4, find (x + y):(x – y).
Let x = 3k and y = 4k.
x + y = 7k.
x – y = -k.
So, (x + y):(x – y) = 7:-1.
If the question expects positive difference, use (y – x), then answer is 7:1.
Common mistakes
The first mistake is comparing quantities with different units. Always convert units before making a ratio.
The second mistake is using only one part instead of total parts. If boys:girls = 3:2, then total class parts are 5, not 3 or 2.
The third mistake is confusing direct and inverse proportion. Cost and quantity are usually direct. Workers and days are usually inverse for fixed work.
The fourth mistake is not simplifying the final ratio. In most exams, the answer should be in lowest terms.
Quick revision table
| Topic | Formula or idea |
|---|---|
| Ratio | a:b = a/b |
| Proportion | a:b :: c:d |
| Means and extremes | b × c = a × d |
| Third proportion | a:b = b:c |
| Mean proportion | sqrt(ab) |
| Direct proportion | x = ky, x ∝ y |
| Inverse proportion | xy = k, x ∝ 1/y |
Practice questions
- Simplify 36:48.
- Find the ratio of 3 metres to 75 centimetres.
- Divide Rs. 1500 in the ratio 2:3.
- Check whether 6:9 and 10:15 are in proportion.
- Find x if 7:9 :: 21:x.
- Find the third proportion to 5 and 10.
- Find the fourth proportion to 4, 7, and 20.
- Find the mean proportion between 9 and 25.
- If 8 workers finish a work in 15 days, how many days will 12 workers take?
- The ratio of two numbers is 3:7 and their difference is 48. Find the numbers.
Answers
- 3:4
- 4:1
- Rs. 600 and Rs. 900
- Yes, they are in proportion
- 27
- 20
- 35
- 15
- 10 days
- 36 and 84
More exam patterns from ratio and proportion
In many government job exams, ratio and proportion questions are not asked only as direct formulas. They are mixed with percentage, average, partnership, time and work, and simple comparison. So the student should not only memorize definitions. The better method is to identify what is being compared and what is constant.
Pattern 1: Ratio after increase or decrease
Example: The ratio of two numbers is 4:5. If 8 is added to both numbers, the ratio becomes 5:6. Find the numbers.
Let the numbers be 4x and 5x.
After adding 8, the numbers become 4x + 8 and 5x + 8.
According to the question, (4x + 8):(5x + 8) = 5:6.
So, (4x + 8)/(5x + 8) = 5/6.
6(4x + 8) = 5(5x + 8).
24x + 48 = 25x + 40.
x = 8.
The numbers are 4x = 32 and 5x = 40.
Pattern 2: Ratio and total difference
Example: The ratio of salaries of A and B is 9:7. If A earns Rs. 4000 more than B, find both salaries.
Difference in ratio parts = 9 – 7 = 2 parts.
2 parts = Rs. 4000.
1 part = Rs. 2000.
A’s salary = 9 × 2000 = Rs. 18000.
B’s salary = 7 × 2000 = Rs. 14000.
Pattern 3: Ratio and average
Example: The ratio of three numbers is 2:3:5 and their average is 40. Find the numbers.
If the average is 40, then the sum of the three numbers is 40 × 3 = 120.
Total ratio parts = 2 + 3 + 5 = 10.
One part = 120 ÷ 10 = 12.
The numbers are 24, 36, and 60.
Pattern 4: Ratio in partnership
In partnership questions, profit is usually divided in the ratio of investment multiplied by time. If time is the same, profit is divided in the ratio of investment only.
Example: A invests Rs. 5000 and B invests Rs. 8000 for the same time. If total profit is Rs. 2600, find B’s share.
Investment ratio = 5000:8000 = 5:8.
Total parts = 5 + 8 = 13.
One part = 2600 ÷ 13 = 200.
B’s share = 8 × 200 = Rs. 1600.
Pattern 5: Ratio with time
Example: A invests Rs. 6000 for 8 months and B invests Rs. 4000 for 12 months. Find the ratio of their profits.
A’s investment-time value = 6000 × 8 = 48000.
B’s investment-time value = 4000 × 12 = 48000.
Profit ratio = 48000:48000 = 1:1.
So, both get equal profit.
How to study this chapter
First learn how to simplify a ratio. Then learn how to divide a total amount in a given ratio. After that, practice proportion questions where one term is missing. Once these basics are clear, move to word problems based on age, money, work, speed, partnership, and mixture.
While solving any question, write the ratio parts clearly. If the sum is given, add the ratio parts. If the difference is given, subtract the ratio parts. If one person’s value is given, compare that value with that person’s ratio part. This small habit prevents most mistakes.
Frequently asked questions
What is ratio?
Ratio is a comparison of two quantities of the same kind. It is written as a:b or a/b.
What is proportion?
Proportion means equality of two ratios. If a:b = c:d, then a, b, c, and d are in proportion.
What are means and extremes?
In a:b :: c:d, a and d are extremes, while b and c are means.
What is the main formula of proportion?
The main formula is product of means = product of extremes. So, b × c = a × d.
What is direct proportion?
Direct proportion means two quantities increase or decrease together in the same ratio. It is written as x ∝ y.
What is inverse proportion?
Inverse proportion means one quantity increases when the other decreases. It is written as x ∝ 1/y.
How do we compare two ratios?
Convert both ratios into fractions or use cross multiplication. The greater fraction gives the greater ratio.
Why should units be the same in ratio?
A ratio compares quantities correctly only when both are in the same unit. For example, metres and centimetres should first be converted into one unit.
Conclusion
Ratio compares quantities, and proportion shows that two ratios are equal.
Practice small examples first; then exam questions on work, age, money, speed, and mixture become much easier.