Lesson 1.3: Ratio, Proportion, Partnership

Introduction:
Ratio and proportion are frequently asked topics in SSC exams. Partnership problems often appear in arithmetic sections, especially in CHSL and MTS exams. This lesson teaches you shortcuts and formulas to solve them quickly.

1. Ratio

• Definition: Ratio compares two quantities.
  Example: If A:B = 3:5, A = 3 parts, B = 5 parts.

• Key Properties:

  • Ratio of two numbers = their simplified fraction form.

  • Multiplying/dividing both terms by the same number does not change the ratio.

• Formula:

Ratio=Quantity AQuantity B\text{Ratio} = \frac{\text{Quantity A}}{\text{Quantity B}}Ratio=Quantity BQuantity A​

• Example:
  A = 12, B = 20 → Ratio A:B = 12:20 = 3:5

2. Proportion

• Definition: Two ratios are said to be in proportion if they are equal.
  Example: 3:5 :: 6:10 (3/5 = 6/10)

• Key Concepts:

  • Direct Proportion: When one quantity increases, the other increases.

    x∝y  ⟹  x1y1=x2y2x \propto y \implies \frac{x_1}{y_1} = \frac{x_2}{y_2}x∝y⟹y1​x1​​=y2​x2​​

  • Inverse Proportion: When one quantity increases, the other decreases.

    x∝1y  ⟹  x1y1=x2y2x \propto \frac{1}{y} \implies x_1 y_1 = x_2 y_2x∝y1​⟹x1​y1​=x2​y2​

• Example:
  If 5 men can complete a work in 12 days, how many days will 10 men take?
  Solution (Inverse Proportion): 5 × 12 = 10 × x → x = 6 days

3. Partnership

• Definition: Partnership problems involve profit-sharing based on invested capital and time.

• Key Formula:

Share of Profit=Individual Capital × TimeTotal Capital × Time×Total Profit\text{Share of Profit} = \frac{\text{Individual Capital × Time}}{\text{Total Capital × Time}} \times \text{Total Profit}Share of Profit=Total Capital × TimeIndividual Capital × Time​×Total Profit

• Example 1:
  A invests ₹10,000 for 12 months, B invests ₹15,000 for 8 months. Total profit = ₹7,000. Find A’s share.

• A’s contribution = 10,000 × 12 = 1,20,000

• B’s contribution = 15,000 × 8 = 1,20,000

• Total = 2,40,000

• A’s share = (1,20,000 / 2,40,000) × 7,000 = ₹3,500

• Example 2 (Quick Trick): If capitals × time are equal, profit is divided equally.

4. SSC Exam Tips

• Always simplify ratios before solving problems.

• Use the unitary method to save time.

• For partnership problems, calculate Capital × Time for each partner directly.

• Practice word problems carefully; ratio/proportion appears in speed, work, age, and profit-loss problems.

5. Practice Questions

1. If A:B:C = 2:3:5 and total = 500, find each share.

2. 8 men can finish a work in 15 days. How many days will 12 men take?

3. A invests ₹12,000 for 10 months, B invests ₹15,000 for 8 months. Total profit = ₹8,000. Find B’s share.

4. Direct proportion: If 6 kg of sugar costs ₹180, what is the cost of 10 kg?

5. Inverse proportion: 5 machines can produce 300 items in 8 hours. How many items will 10 machines produce in 8 hours?