Ratio, Proportion and Partnership — Simple Methods with Examples

A ratio compares two quantities of the same kind. If a class has 20 boys and 30 girls, boys : girls = 20 : 30 = 2 : 3.

Think of a ratio as parts: for every 2 parts of boys there are 3 parts of girls.

1. The parts method

Example 1

Share ₹500 between A and B in the ratio 2 : 3.

Total parts = 5. One part = 500 ÷ 5 = 100. A = 2 × 100 = ₹200, B = ₹300.

This one idea solves most ratio questions: find the value of one part.

Example 2

The ratio of boys to girls is 4 : 5. There are 36 boys. How many girls?

4 parts = 36 → 1 part = 9 → girls = 5 × 9 = 45.

2. Combining two ratios

A : B = 2 : 3 and B : C = 4 : 5. Find A : B : C.

B is in both. Make B the same: multiply the first by 4 and the second by 3.

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

3. Proportion

Four numbers a, b, c, d are in proportion if a : b = c : d, i.e. a × d = b × c ("product of extremes = product of means").

• Mean proportional of a and b = √(ab). Of 4 and 16 → √64 = 8.
• Third proportional of a and b is c where a : b = b : c → c = b²/a. Of 4 and 6 → 36/4 = 9.

Example 3

Find x if 3 : 5 = x : 40.

3 × 40 = 5x → x = 24.

4. Adding to a ratio

Example 4

Two numbers are in the ratio 3 : 5. If 9 is added to each, the ratio becomes 3 : 4. Find the numbers.

Let them be 3k and 5k. (3k + 9)/(5k + 9) = 3/4 → 12k + 36 = 15k + 27 → k = 3. Numbers: 9 and 15.

5. Coins

Example 5

A bag has ₹1, 50 p and 25 p coins in the ratio 1 : 2 : 4. The total value is ₹90. How many 25 p coins are there?

Value per set of coins = 1 × ₹1 + 2 × ₹0.50 + 4 × ₹0.25 = 1 + 1 + 1 = ₹3. Number of sets = 90 ÷ 3 = 30. 25 p coins = 4 × 30 = 120.

6. Partnership

Business profit is shared in the ratio of money × time each partner invested.

Profit share ∝ Investment × Time

Example 6

A invests ₹40,000 for 12 months; B invests ₹60,000 for 6 months. The profit is ₹35,000. Find each share.

• A: 40,000 × 12 = 4,80,000
• B: 60,000 × 6 = 3,60,000
• Ratio = 480 : 360 = 4 : 3 → total 7 parts → 1 part = 5,000
• A = ₹20,000, B = ₹15,000

Example 7

A starts a business with ₹50,000. After 4 months B joins with ₹75,000. What is the profit ratio after 1 year?

A: 50,000 × 12 = 6,00,000. B: 75,000 × 8 = 6,00,000. Ratio = 1 : 1.

Common mistakes

• Forgetting to make the common term equal when combining ratios.
• In partnership, using only the investment and forgetting the time.
• Mixing units (₹ and paise, months and years). Convert first.

Quick recap

• Ratio = parts. Find one part, then everything else.
• To combine A : B and B : C, make B equal.
• a : b = c : d ⇔ ad = bc. Mean proportional = √(ab).
• Partnership: share ∝ money × time.