Ratio and proportion for AFCAT

A ratio compares two quantities of the same kind. On its own, ratio is a short topic in AFCAT, and its questions rarely take long. But the idea runs through much of the section: mixtures and alligation, problems on ages, partnership, time and work, and speed all use ratios. Getting fluent here pays off across the paper.

The core idea is the part method. A ratio of 3 : 5 means the quantities are 3 parts and 5 parts of some unknown size x. Nearly every ratio question comes down to finding the value of one part. This guide shows how, for each type of question, and explains why the methods work.

Basics

• Same units first. 50 paise : ₹2 is 50 : 200 = 1 : 4, not 50 : 2.
• Simplest form. Divide both terms by their HCF: 24 : 36 = 2 : 3.
• Multiplying or dividing both terms by the same number leaves the ratio unchanged. Adding or subtracting the same number does not; that is what "changing ratio" questions test.

Dividing an amount in a ratio

Add the parts, find one part, then multiply.

To divide ₹1,200 in the ratio 3 : 5: total parts = 8, one part = 1,200 ÷ 8 = 150, so the shares are ₹450 and ₹750.

The same works with three or more terms. ₹7,000 in the ratio 8 : 12 : 15 has 35 parts of ₹200 each.

Combining ratios

If A : B = 2 : 3 and B : C = 4 : 5, you cannot just write 2 : 3 : 5. The "3" and the "4" both describe B but on different scales.

Make the common term equal. B is 3 in the first ratio and 4 in the second; the LCM is 12. Multiply the first ratio by 4 and the second by 3:

• A : B = 8 : 12
• B : C = 12 : 15
• So A : B : C = 8 : 12 : 15

Why it works: multiplying both terms of a ratio by the same number does not change it, so you are only rewriting each ratio on a common scale for B.

Proportion and proportionals

Four numbers a, b, c, d are in proportion (a : b = c : d) when a × d = b × c. The product of the outer terms equals the product of the inner terms.

Duplicate and compound ratios

The duplicate ratio matters in geometry: if the sides of two similar figures are in the ratio a : b, their areas are in the duplicate ratio a² : b².

Worked examples

Example 1 (three-way division). A : B = 2 : 3 and B : C = 4 : 5. Divide ₹7,000 among A, B and C.

• Combined ratio: 8 : 12 : 15 (as above), a total of 35 parts.
• One part = 7,000 ÷ 35 = 200.
• Shares: A ₹1,600, B ₹2,400, C ₹3,000. Check: 1,600 + 2,400 + 3,000 = 7,000.

Example 2 (changing ratio). Two numbers are in the ratio 3 : 5. If 10 is added to each, the ratio becomes 5 : 7. Find the numbers.

• Let them be 3x and 5x. Then (3x + 10) ÷ (5x + 10) = 5/7.
• Cross-multiply: 21x + 70 = 25x + 50, so 4x = 20 and x = 5.
• The numbers are 15 and 25. Check: 25 : 35 = 5 : 7.

Example 3 (income and expenditure). The incomes of two people are in the ratio 5 : 3, and their expenditures in the ratio 9 : 5. Each saves ₹2,000 a month. Find their incomes.

• Incomes 5x and 3x; expenditures 9y and 5y.
• Savings: 5x − 9y = 2,000 and 3x − 5y = 2,000.
• The savings are equal, so 5x − 9y = 3x − 5y, which gives 2x = 4y, or x = 2y.
• Substitute: 3(2y) − 5y = 2,000, so y = 2,000 and x = 4,000.
• Incomes: ₹20,000 and ₹12,000. Check: expenditures are ₹18,000 and ₹10,000, and each saves ₹2,000.

Example 4 (coins). A bag has ₹1, 50-paise and 25-paise coins in the ratio 5 : 6 : 8 by number. The total value is ₹210. How many 50-paise coins are there?

• Convert the ratio by number into a ratio by value: 5 × ₹1 : 6 × ₹0.50 : 8 × ₹0.25 = 5 : 3 : 2.
• 10 parts = ₹210, so one part = ₹21. The 50-paise coins are worth 3 × 21 = ₹63.
• Number of 50-paise coins = 63 ÷ 0.5 = 126.
• Check: the counts are 105, 126 and 168, which is 21 × (5 : 6 : 8).

Example 5 (proportionals). Find the mean proportional of 0.08 and 0.18, and the third proportional to 9 and 12.

• Mean proportional = √(0.08 × 0.18) = √0.0144 = 0.12.
• Third proportional = 12² ÷ 9 = 144 ÷ 9 = 16.

Example 6 (partnership). A invests ₹40,000 for 12 months and B invests ₹60,000 for 6 months. The profit is ₹35,000. How is it shared?

• Profit is shared in the ratio of capital × time: 40,000 × 12 : 60,000 × 6 = 4,80,000 : 3,60,000 = 4 : 3.
• 7 parts = 35,000, so one part = 5,000.
• A ₹20,000, B ₹15,000.

Practice set

1. Divide ₹840 in the ratio 3 : 4.
2. A : B = 3 : 4 and B : C = 6 : 7. Find A : B : C.
3. Find the mean proportional of 4 and 25.
4. Two numbers in the ratio 2 : 3 have a difference of 15. Find them.
5. Find the fourth proportional to 3, 5 and 12.
6. What number must be subtracted from each of 15, 28, 20 and 38 so that the results are in proportion?
7. The salaries of A and B are in the ratio 2 : 3. If each gets a raise of ₹4,000, the ratio becomes 3 : 4. Find B's original salary.
8. Find the sub-duplicate ratio of 49 : 64.

Answers:

1. ₹360 and ₹480. 7 parts = 840, so one part = 120.
2. 9 : 12 : 14. Make B equal (LCM 12): 9 : 12 and 12 : 14.
3. 10. √(4 × 25) = √100.
4. 30 and 45. The difference is 1 part, so one part = 15.
5. 20. x = 5 × 12 ÷ 3.
6. 2. (15 − x)(38 − x) = (28 − x)(20 − x) gives 570 − 53x = 560 − 48x, so x = 2. Check: 13 : 26 = 18 : 36 = 1 : 2.
7. ₹12,000. (2x + 4,000) ÷ (3x + 4,000) = 3/4 gives 8x + 16,000 = 9x + 12,000, so x = 4,000 and B's salary = 3x.
8. 7 : 8. √49 : √64.

Common mistakes

• Combining ratios without matching the common term. Always make B equal first.
• Comparing quantities in different units. Convert paise to rupees, or minutes to hours, before writing the ratio.
• Treating a ratio by number as a ratio by value. In coin questions, multiply by each coin's value first.
• Adding the same number to both terms and expecting the ratio to hold. It changes; that is the whole point of those questions.

What to do next

• Solve ten "combine the ratios" questions until matching the common term is automatic.
• Practise five income-expenditure and five coin questions.
• Move on to averages and mixtures and alligation, which build directly on ratios.
• Revise percentage alongside: a ratio of 3 : 5 means the first is 37.5% of the total.