Ratio and Proportion

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Ratio and Proportion — Key Facts for NABE (Pakistan)

• Ratio: Comparison of two quantities; expressed as a:b or a/b
• Proportion: Equality of two ratios; a:b = c:d
• Direct Proportion: When one increases, the other also increases
• Inverse Proportion: When one increases, the other decreases
• Componendo-Dividendo: If a/b = c/d, then (a+b)/(a-b) = (c+d)/(c-d)
• ⚡ Exam tip: Ratio questions frequently appear in NABE with word problems

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Ratio and Proportion — NABE (Pakistan) Study Guide

What is a Ratio?

A ratio is a mathematical tool used to compare two quantities of the same kind. It tells us how many times one quantity is contained in another.

Formula: Ratio of a to b = a:b = a/b (where b ≠ 0)

Key Properties:

• Ratios should be expressed in simplest form (divide by GCD)
• Example: 15:25 simplifies to 3:5 (divide both by 5)
• Both quantities must have the same unit

Types of Ratios

1. Part-to-Part Ratio: Compares two different parts

   • Example: In a class, ratio of boys to girls = 3:2

2. Part-to-Whole Ratio: Compares one part to the whole

   • Example: Boys to total students = 3:5

3. Compound Ratio: Product of two or more simple ratios

   • Example: Compounding 2:3 and 4:5 gives (2×4):(3×5) = 8:15

What is Proportion?

A proportion states that two ratios are equal.

Formula: a:b = c:d → a/b = c/d → ad = bc

Key Terms:

• Extremes: ‘a’ and ‘d’ in a:b = c:d
• Means: ‘b’ and ‘c’ in a:b = c:d
• Product of Extremes = Product of Means

Direct Proportion

When two quantities are in direct proportion, they increase or decrease together at the same rate.

Formula: a ∝ b → a = kb (where k is constant) Example: If 5 books cost Rs. 250, what do 8 books cost?

• 5 books = Rs. 250, so 1 book = Rs. 50
• 8 books = 8 × 50 = Rs. 400

Inverse Proportion

When one quantity increases, the other decreases proportionally.

Formula: a ∝ 1/b → ab = k (constant) Example: If 12 workers complete a job in 20 days, how many days for 15 workers?

• 12 workers × 20 days = 240 worker-days
• 15 workers: 240/15 = 16 days

Important Theorems

Componendo and Dividendo: If a/b = c/d, then:

• (a+b)/b = (c+d)/d
• (a-b)/b = (c-d)/d
• (a+b)/(a-b) = (c+d)/(c-d)

Alternendo: If a:b = c:d, then a:c = b:d

NABE Exam Pattern

Questions typically involve:

• Simplifying ratios
• Finding missing values in proportions
• Word problems involving direct/inverse proportion
• Application of componendo-dividendo

Common Question Types:

1. “If a:b = 3:4 and b:c = 5:6, find a:b:c”
2. “A sum of money is divided in the ratio 2:3:5. If the largest share is Rs. 1500, find the total sum”
3. “If 8 men can build a wall in 15 days, how long will 12 men take?”

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Ratio and Proportion — Comprehensive NABE (Pakistan) Notes

Historical Context

The concept of ratio and proportion has been used since ancient civilizations. Euclid’s “Elements” (circa 300 BCE) contains formal treatments of ratios. The Babylonian and Indian mathematicians extensively used these concepts for astronomical calculations and everyday commerce.

Detailed Theory

1. Ratio Fundamentals

Definition: A ratio a:b represents the relationship between two quantities, indicating how many times the first quantity contains the second.

Conditions for Valid Ratios:

• Both quantities must be of the same kind (both lengths, both weights, etc.)
• The second quantity (denominator) must be non-zero
• Ratios are abstract numbers — they don’t carry units

Simplification Process:

Step 1: Express the ratio as a fraction
Step 2: Find the GCD of numerator and denominator
Step 3: Divide both by GCD
Step 4: Express in ratio form

Example: Simplify 72:108

• GCD of 72 and 108 = 36
• 72÷36 = 2, 108÷36 = 3
• Simplified ratio = 2:3

2. Proportion Fundamentals

Definition: Four quantities are in proportion when the ratio of first to second equals the ratio of third to fourth.

Continuous Proportion: Three quantities a, b, c are in continuous proportion when a:b = b:c

• Here, b is called the “mean proportional” between a and c
• b² = ac

Fourth Proportional: If a:b = c:x, then x is the fourth proportional to a, b, c

Third Proportional: If a:b = b:x, then x is the third proportional to a and b

3. Special Types of Ratios

Duplicate Ratio: Square of a ratio

• (a:b)² = a²:b²

Sub-duplicate Ratio: Square root of a ratio

• √(a:b) = √a:√b

Triplicate Ratio: Cube of a ratio

• (a:b)³ = a³:b³

Sub-triplicate Ratio: Cube root of a ratio

• ³√(a:b) = ³√a:³√b

Inverse Ratio: Reciprocal of a ratio

• Inverse of a:b is b:a

4. Word Problem Strategies

Dividing a Quantity in Given Ratio: If a sum S is divided in ratio a:b:c:

• First part = S × a/(a+b+c)
• Second part = S × b/(a+b+c)
• Third part = S × c/(a+b+c)

Example: Divide Rs. 8400 among A, B, C in ratio 3:4:5

• Total parts = 3+4+5 = 12
• A’s share = 8400 × 3/12 = Rs. 2100
• B’s share = 8400 × 4/12 = Rs. 2800
• C’s share = 8400 × 5/12 = Rs. 3500

Proportion in Mixtures: Alligation method for weighted averages:

• Cost of cheaper item: Rs. c per unit
• Cost of expensive item: Rs. d per unit
• Mean price: Rs. m per unit
• Ratio of quantities = (d-m):(m-c)

5. Partnership Problems

When two or more persons invest money in a business:

• Their profit/loss is divided in ratio of their investments × time
• Ratio of profits = (Investment₁ × Time₁):(Investment₂ × Time₂)

Example: A invests Rs. 5000 for 8 months, B invests Rs. 3000 for 12 months. If profit is Rs. 7000, how is it divided?

• A’s contribution = 5000 × 8 = 40,000
• B’s contribution = 3000 × 12 = 36,000
• Ratio = 40,000:36,000 = 10:9
• A’s profit = 7000 × 10/19 ≈ Rs. 3684
• B’s profit = 7000 × 9/19 ≈ Rs. 3316

6. Common Mistakes to Avoid

1. Unit Mismatch: Always convert to same units before comparing
2. Forgetting to Simplify: Always reduce ratios to lowest terms
3. Direct vs. Inverse Confusion: Identify relationship correctly
4. Calculation Errors in Cross-Multiplication: Double-check ad = bc

Practice Questions for NABE

1. If x:y = 3:4 and y:z = 5:6, find x:z
2. A mixture contains milk and water in ratio 5:3. If 8 liters of water is added, the ratio becomes 5:4. Find original quantity.
3. The incomes of two persons are in ratio 7:5 and their expenditures are in ratio 3:2. If each saves Rs. 1000, find their incomes.
4. 45 workers can complete a project in 48 days. After 20 days, 15 workers leave. How many additional days are needed?

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