⚖️ Ratio and Proportion

📘 Introduction

Ratio and proportion are foundational mathematical concepts used to express relationships between quantities. They are essential for solving real-world problems in finance, cooking, engineering, and competitive exams.

📐 Understanding Ratio

🔹 Definition

A ratio compares two or more quantities of the same kind by division. It shows how many times one value is contained in another.

- Notation: a : b or a/b

📝 Important Points

• Ratios compare quantities of the same unit.
• They can be simplified using the GCD.
• Ratios are dimensionless — units cancel out.

✅ Examples

• 12 boys and 8 girls → 12 : 8 = 3 : 2
• 300 km with 20 liters → 300 : 20 = 15 : 1

📊 Types of Ratios

• Part-to-Part: Compares two parts (e.g., red to blue balls).
• Part-to-Whole: Compares a part to the total (e.g., red balls to total balls).

🔄 Proportion: Equality of Ratios

Definition

Two ratios are in proportion when they are equal: a : b = c : d or a/b = c/d

📌 Key Properties

• Cross-Multiplication: a × d = b × c
• Means: b and c | Extremes: a and d
• Mean Proportional: If a : b = b : c → b² = a × c
• Third Proportional: a : b = b : c → c = b² / a
• Fourth Proportional: a : b = c : d → d = (b × c) / a

🧮 Solving Proportion Problems

• Identify Ratios – From the given data.
• Set Up the Proportion – With variables for unknowns.
• Use Cross Multiplication – a × d = b × c
• Check the Answer – Plug back and verify.

🔗 Advanced Concepts

🔹 Compound Ratios

If a : b and c : d, then compound ratio = (a × c) : (b × d)

Example: 3 : 4 and 5 : 2 → (3×5):(4×2) = 15 : 8

🔹 Continued Proportion

Three values a, b, c are in continued proportion if a/b = b/c

🔹 Direct and Inverse Proportion

• Direct: y ∝ x → y/x = constant
• Inverse: y ∝ 1/x → x × y = constant

Examples:
• Direct: Distance ∝ Time (if speed is fixed)
• Inverse: Speed ↑ → Time ↓ (for fixed distance)

🌍 Real-Life Applications

🍳 Cooking: Adjusting recipes
💰 Finance: Investment mixes, interest
🗺️ Maps: Scale conversions
🏗️ Construction: Cement-sand ratios
📊 Demographics: Population ratios
🚗 Travel: Speed-time-distance calculations

Example 1: Sharing Money

Three friends share $1200 in the ratio 2 : 3 : 5

• Total parts = 10
• 1 part = 1200 / 10 = 120
• Shares: 2×120 = 240, 3×120 = 360, 5×120 = 600

Example 2: Mixture Problem

Original ratio 3 : 2. After adding 20 L water → 3 : 5

• Let alcohol = 3x, water = 2x
• 3x / (2x + 20) = 3 / 5
• 5×3x = 3(2x + 20) → 15x = 6x + 60 → x = 20/3
• Total solution = 5x = 5×(20/3) = 100/3 ≈ 33.33 L

Example 3: Direct Proportion

• 5 workers = 12 days → 10 workers = ?

• 5×12 = 10×x → x = 6 days

Example 4: Inverse Proportion

• 8 machines = 15 hrs → 12 machines = ?

• 8×15 = 12×x → x = 10 hours

🧩 Practice Problems

1. Simplify 36 : 60 and find equivalent ratio with 9 as the first term.
2. If 4 pens cost $12, what is the cost of 7 pens?
3. Ratio of siblings’ ages = 5 : 7. Elder is 14. Find younger’s age.
4. Find x: 3 : 5 = x : 25
5. If 6 liters contains alcohol and water in 2 : 1, find alcohol content.

✅ Solutions

1. 36 : 60 = 3 : 5 → Multiply by 3 → 9 : 15
2. Cost per pen = $3 → 7×3 = $21
3. Let x = younger → x/14 = 5/7 → x = 10
4. 3/5 = x/25 → x = (3×25)/5 = 15
5. Total parts = 3 → Alcohol = (2/3)×6 = 4 liters

📌 Tips to Excel in Ratio and Proportion

• Always simplify ratios before solving.
• Use cross-multiplication to solve proportions quickly.
• Identify if the problem is direct or inverse proportion.
• Maintain consistent units throughout.
• Practice real-life word problems regularly.

🏁 Conclusion

Mastering ratio and proportion enhances your ability to solve a wide range of mathematical and practical problems. With consistent practice, logical approach, and conceptual clarity, you can tackle even complex ratio-based questions with confidence.