Ratio and Proportion Notes for SSC Exams

Ratio and Proportion is a basic chapter that supports many other topics in SSC Quantitative Aptitude, such as partnership, mixture, ages, time and work and profit and loss. The chapter is simple, and the questions usually test whether you can set up the ratio correctly. These ratio and proportion notes for SSC explain all the rules with examples and practice questions.

What Is a Ratio?

A ratio compares two quantities of the same kind by division. The ratio of a to b is written as a : b or a/b, where a is the first term (antecedent) and b is the second term (consequent).

• The two quantities must be in the same unit before you write the ratio.
• A ratio has no unit.
• Multiplying or dividing both terms by the same number does not change the ratio (3 : 4 = 6 : 8).
• To compare ratios, convert them into fractions and compare.

Types of Ratio

Proportion and Its Rules

Four numbers a, b, c and d are in proportion when a : b = c : d. It is written a : b :: c : d.

• Product rule: a x d = b x c.
• Fourth proportional to a, b, c = bc / a.
• Third proportional to a and b = b squared / a.
• Mean proportional between a and b = square root of (ab).
• Componendo and dividendo: if a/b = c/d, then (a + b)/(a - b) = (c + d)/(c - d).

Dividing a Quantity in a Ratio and Combining Ratios

To divide a quantity in the ratio a : b, add the terms (a + b). The first share is a/(a + b) of the whole and the second share is b/(a + b). If a : b and b : c are given, make the common term equal. For A : B = 2 : 3 and B : C = 4 : 5, multiply the first by 4 and the second by 3 to get A : B : C = 8 : 12 : 15.

Direct and Inverse Proportion

• In direct proportion, when one quantity increases, the other also increases in the same ratio. Example: more articles cost more money.
• In inverse proportion, when one quantity increases, the other decreases. Example: more men finish the work in fewer days (men x days = constant).

Solved Examples

Example 1: Divide Rs 1,200 in the ratio 3 : 5.
Solution: Total parts = 8. First share = 1200 x 3 / 8 = Rs 450. Second share = 1200 x 5 / 8 = Rs 750.

Example 2: Two numbers are in the ratio 3 : 4 and their sum is 84. Find the numbers.
Solution: Sum of ratio terms = 7. Numbers = 84 x 3 / 7 and 84 x 4 / 7 = 36 and 48.

Example 3: Find the fourth proportional to 4, 6 and 10.
Solution: Fourth proportional = (6 x 10) / 4 = 15.

Example 4: Find the mean proportional between 4 and 36.
Solution: Mean proportional = square root of (4 x 36) = square root of 144 = 12.

Example 5: If A : B = 2 : 3 and B : C = 4 : 5, find A : B : C.
Solution: Make B equal: A : B = 8 : 12 and B : C = 12 : 15. So A : B : C = 8 : 12 : 15.

Example 6: Two numbers are in the ratio 5 : 7. If 6 is added to each, the ratio becomes 3 : 4. Find the numbers.
Solution: (5x + 6) / (7x + 6) = 3/4, so 20x + 24 = 21x + 18, giving x = 6. Numbers are 30 and 42.

Quick Tips to Solve Faster

• Always convert both quantities into the same unit before writing a ratio.
• In a ratio question, assume the numbers as ax and bx, and find x from the given data.
• To combine two ratios with a common term, multiply to make the common term equal.
• Use the product rule (ad = bc) to check any proportion quickly.
• In inverse proportion problems, the product (men x days or speed x time) remains constant.

Common Mistakes to Avoid

• Writing the ratio in the wrong order.
• Comparing quantities in different units, such as minutes and hours.
• Adding the same number to both terms and expecting the ratio to remain the same.
• Using direct proportion in an inverse proportion problem.
• Forgetting to take the square root while finding the mean proportional.

Quick Revision Points

• a : b :: c : d means ad = bc.
• Fourth proportional = bc / a, third proportional = b squared / a, mean proportional = square root of ab.
• Compounded ratio is the product of the ratios.
• Divide in the ratio a : b by using a/(a + b) and b/(a + b).
• Men x days = constant (inverse proportion).

Practice Questions With Answers

1. Divide Rs 900 in the ratio 2 : 3 : 4. Find the largest share.
   Answer: Rs 400 (900 x 4 / 9).
2. If a : b = 3 : 4 and b : c = 8 : 9, find a : c.
   Answer: 2 : 3 (a : b : c = 6 : 8 : 9).
3. Find the mean proportional between 9 and 25.
   Answer: 15 (square root of 225).
4. Two numbers are in the ratio 4 : 5 and their product is 180. Find the larger number.
   Answer: 15 (20x squared = 180, x = 3).
5. Find the fourth proportional to 3, 5 and 9.
   Answer: 15 (5 x 9 / 3).
6. A and B share Rs 64 in the ratio 5 : 3. How much does A get?
   Answer: Rs 40 (64 x 5 / 8).
7. Find the ratio of 25 minutes to 2 hours.
   Answer: 5 : 24 (25 : 120).
8. Find the compounded ratio of 2 : 3, 4 : 5 and 6 : 7.
   Answer: 16 : 35 (48 : 105).
9. 6 men can do a piece of work in 10 days. In how many days can 4 men do it?
   Answer: 15 days (6 x 10 / 4).
10. If x : y = 3 : 4, find (2x + y) : (x + 2y).
    Answer: 10 : 11 (Put x = 3, y = 4).

How to Practise This Chapter

1. Learn the formulas in the table until you can write them without looking.
2. Solve the solved examples again on your own, without reading the solution.
3. Attempt the practice questions above with a timer of 10 minutes.
4. Solve the matching SSC percentage practice paper to test yourself in exam style.
5. Check the previous year papers to see how SSC frames this topic.

Related Notes and Practice Papers

• Percentage Notes
• Profit and Loss Notes
• Average Notes
• Time and Work Notes
• SSC Profit and Loss Practice Paper

Explore More SSC Subjects

SSC exams test many subjects, so connect this topic with the related subjects below:

• Reasoning Notes
• Statistics Notes
• Indian Economy Notes
• English Notes
• General Awareness Notes
• All SSC subjects and notes

Frequently Asked Questions

What is a ratio in simple words?

A ratio is a way of comparing two quantities of the same kind by division. For example, the ratio of 6 to 9 is 2 : 3.

What is proportion?

Proportion is the equality of two ratios. If a : b = c : d, then a, b, c and d are in proportion.

How many questions come from ratio and proportion in SSC exams?

One or two questions come directly from this chapter, and it is also used in many other chapters.

What is the formula for the mean proportional?

The mean proportional between a and b is the square root of (a x b).

What is the third proportional?

The third proportional to a and b is b squared divided by a.

How do I divide an amount in a given ratio?

Add the ratio terms to get the total parts, divide the amount by the total parts and multiply by each term.

What is the difference between direct and inverse proportion?

In direct proportion both quantities increase or decrease together. In inverse proportion, one increases when the other decreases.

What is componendo and dividendo?

If a/b = c/d, then (a + b)/(a - b) = (c + d)/(c - d). It is used to simplify ratio equations quickly.

How can I practise ratio and proportion?

Learn the rules, solve the examples in these notes and attempt practice sets of partnership, ages and mixtures.