Ratio and Proportion

Overview

Ratio and Proportion forms a foundational topic in MP TET Mathematics, appearing consistently across Varg-1, Varg-2, and Varg-3 papers. This topic tests your ability to compare quantities, solve for unknowns in proportional relationships, and apply these concepts to real-life situations like partnership problems and variation.

Mastery here directly supports other arithmetic topics—percentages, time-work, time-distance, and mixtures all build on ratio-proportion logic. For MP TET, expect 2–4 questions combining direct application with word problems involving partnership distribution or direct/inverse variation scenarios.

The key to scoring well is understanding the underlying relationship between quantities rather than memorising formulas mechanically. Once you grasp that ratio compares "part to part" while proportion equates two ratios, most problems become straightforward.

Key Concepts

• **Ratio** expresses the comparative relation between two quantities of the same kind. Written as a : b or a/b, it has no unit and should always be expressed in lowest terms.

• **Proportion** states that two ratios are equal. If a : b = c : d, then a, b, c, d are in proportion. Here, a and d are called **extremes**, while b and c are called **means**.

• **Product of means = Product of extremes** is the fundamental property: if a : b :: c : d, then b × c = a × d.

• **Continued Proportion**: Three quantities a, b, c are in continued proportion if a : b = b : c. Here, b² = a × c, and b is the **mean proportional** between a and c.

• **Direct Variation**: When two quantities increase or decrease together in the same ratio (y ∝ x), they vary directly. Example: More hours worked → More wages earned.

• **Inverse Variation**: When one quantity increases as the other decreases such that their product remains constant (y ∝ 1/x), they vary inversely. Example: More workers → Fewer days to complete work.

• **Partnership**: When two or more people invest capital for business, profit is divided in the ratio of (Capital × Time) for each partner.

• **Compounded Ratio**: The compounded ratio of a : b and c : d is (a × c) : (b × d).

Formulas / Key Facts

| Concept | Formula / Rule | |---------|----------------| | Ratio of a to b | a : b = a/b (express in lowest terms by dividing by HCF) | | Proportion test | a : b :: c : d ⟹ a × d = b × c | | Mean proportional of a and c | √(a × c) | | Third proportional to a and b | b²/a | | Fourth proportional to a, b, c | (b × c)/a | | Direct variation | y = kx, where k is constant | | Inverse variation | xy = k, or y = k/x | | Partnership profit share | Profit of A : Profit of B = (Capital_A × Time_A) : (Capital_B × Time_B) | | Duplicate ratio of a : b | a² : b² | | Sub-duplicate ratio of a : b | √a : √b | | Triplicate ratio of a : b | a³ : b³ |

Worked Examples

**Example 1: Finding Fourth Proportional**

*Find the fourth proportional to 3, 5, and 12.*

Let the fourth proportional be x. 3 : 5 :: 12 : x Using product of means = product of extremes: 5 × 12 = 3 × x 60 = 3x x = 20

**Answer: 20**

---

**Example 2: Partnership Problem**

*A invests ₹6000 for 8 months and B invests ₹8000 for 6 months. If total profit is ₹2100, find each partner's share.*

Step 1: Calculate capital-time product A's investment = 6000 × 8 = 48000 B's investment = 8000 × 6 = 48000

Step 2: Ratio of profits A : B = 48000 : 48000 = 1 : 1

Step 3: Divide profit Total parts = 1 + 1 = 2 A's share = (1/2) × 2100 = ₹1050 B's share = (1/2) × 2100 = ₹1050

**Answer: A gets ₹1050, B gets ₹1050**

---

**Example 3: Inverse Variation**

*12 workers can complete a task in 15 days. How many days will 9 workers take to complete the same task?*

Since workers and days are inversely proportional: Workers₁ × Days₁ = Workers₂ × Days₂ 12 × 15 = 9 × Days₂ 180 = 9 × Days₂ Days₂ = 20

**Answer: 20 days**

---

**Example 4: Direct Variation**

*If 5 pens cost ₹35, what is the cost of 8 pens?*

Cost varies directly with number of pens. 5 pens → ₹35 1 pen → ₹7 8 pens → 8 × 7 = ₹56

**Answer: ₹56**

Common Mistakes

• **Confusing ratio order**: Writing a : b when the question asks for b : a. Always identify which quantity comes first as per the question.

• **Not simplifying ratios**: Leaving 12 : 18 instead of simplifying to 2 : 3. Always divide both terms by their HCF.

• **Mixing up direct and inverse variation**: Assuming more workers means more days (wrong). Check the real-world logic—more workers complete work faster, so it's inverse.

• **Ignoring time in partnership**: Calculating profit share using only capital amounts. Remember: share depends on Capital × Time, not just capital.

• **Adding instead of multiplying for compounded ratio**: For ratios 2 : 3 and 4 : 5, compounded ratio is (2 × 4) : (3 × 5) = 8 : 15, not 6 : 8.

• **Wrong cross-multiplication in proportion**: In a : b :: c : d, the correct relation is ad = bc, not ac = bd.

Quick Reference

• Ratio has no unit; always simplify to lowest terms.

• Proportion: Product of extremes = Product of means (ad = bc).

• Mean proportional of a and c = √(ac).

• Direct variation: both quantities move in same direction (y = kx).

• Inverse variation: product stays constant (xy = k).

• Partnership profit ratio = (Capital × Time) ratio.