Fractions and Rational Numbers

Overview

Fractions and rational numbers form the backbone of arithmetic at the upper-primary level and are tested extensively in MAHA TET Paper II. A strong grasp of this topic is essential not only for solving direct computation questions but also for understanding algebra, ratio-proportion, and data interpretation problems that build on these concepts.

For the TET examination, you must be comfortable performing all four operations (addition, subtraction, multiplication, division) on both fractions and rational numbers, converting between forms, and understanding how fractions relate to the broader number system. Questions typically test procedural fluency, conceptual understanding of negative rationals, and the ability to compare and order numbers on a number line.

Pedagogically, this topic illustrates how students transition from whole-number thinking to proportional reasoning—a shift that many learners find challenging. Understanding common misconceptions helps you both answer pedagogy-linked questions and teach effectively in the classroom.

Key Concepts

• **Fraction as part-whole and division**: A fraction a/b represents both "a parts out of b equal parts" and "a divided by b." Both interpretations matter for teaching.

• **Types of fractions**: Proper fractions (numerator < denominator), improper fractions (numerator ≥ denominator), and mixed numbers (whole + proper fraction) are interconvertible.

• **Equivalent fractions**: Multiplying or dividing both numerator and denominator by the same non-zero number yields an equivalent fraction. This principle underpins simplification and finding common denominators.

• **Rational numbers extend fractions**: A rational number is any number expressible as p/q where p and q are integers and q ≠ 0. This includes negative fractions and zero.

• **Additive inverse and multiplicative inverse**: For any rational number a/b, its additive inverse is −a/b (sum = 0) and its multiplicative inverse (reciprocal) is b/a (product = 1), provided a ≠ 0.

• **Density property**: Between any two rational numbers, infinitely many other rational numbers exist—a concept that distinguishes rationals from integers.

• **Representation on number line**: Rational numbers can be located on a number line by dividing unit segments into equal parts, reinforcing the ordering of positive and negative rationals.

Formulas / Key Facts

| Operation | Rule | |-----------|------| | Addition (like denominators) | a/c + b/c = (a + b)/c | | Addition (unlike denominators) | a/b + c/d = (ad + bc)/bd — then simplify | | Subtraction | a/b − c/d = (ad − bc)/bd | | Multiplication | a/b × c/d = ac/bd | | Division | a/b ÷ c/d = a/b × d/c (multiply by reciprocal) |

**Conversion facts**:

• Mixed to improper: 3 2/5 = (3 × 5 + 2)/5 = 17/5
• Improper to mixed: 17/5 = 3 remainder 2 → 3 2/5

**Properties of rational-number operations**:

• Addition and multiplication are commutative and associative.
• Multiplication distributes over addition: a(b + c) = ab + ac.
• Zero is the additive identity; one is the multiplicative identity.

**Standard form of a rational number**: p/q is in standard form when q > 0 and GCD(|p|, q) = 1.

Worked Examples

### Example 1: Adding fractions with unlike denominators

**Problem**: Simplify 3/4 + 5/6.

**Solution**: 1. Find LCM of denominators 4 and 6. LCM = 12. 2. Convert each fraction: 3/4 = 9/12; 5/6 = 10/12. 3. Add numerators: 9/12 + 10/12 = 19/12. 4. Express as mixed number if required: 19/12 = 1 7/12.

---

### Example 2: Dividing rational numbers

**Problem**: Evaluate (−7/9) ÷ (14/27).

**Solution**: 1. Rewrite as multiplication by reciprocal: (−7/9) × (27/14). 2. Multiply numerators and denominators: (−7 × 27)/(9 × 14) = −189/126. 3. Simplify by dividing numerator and denominator by GCD (63): −189 ÷ 63 = −3; 126 ÷ 63 = 2. 4. Answer: −3/2 or −1 1/2.

---

### Example 3: Comparing rational numbers

**Problem**: Which is greater, −5/8 or −3/4?

**Solution**: 1. Convert to like denominators. LCM of 8 and 4 is 8. 2. −5/8 stays −5/8; −3/4 = −6/8. 3. On the number line, −5/8 is to the right of −6/8 (closer to 0). 4. Therefore, −5/8 > −3/4.

Common Mistakes

| Wrong Thinking | Correct Fix | |----------------|-------------| | Adding numerators and denominators directly: 2/3 + 1/4 = 3/7. | Always find a common denominator first; correct answer is 11/12. | | Ignoring signs when multiplying negatives: (−2/5) × (−3/7) treated as negative. | Product of two negatives is positive: answer is +6/35. | | Forgetting to flip the second fraction when dividing: a/b ÷ c/d = a/b × c/d. | Division means multiply by the reciprocal: a/b × d/c. | | Assuming larger denominator means larger fraction: 3/8 > 3/5 because 8 > 5. | Larger denominator with same numerator means smaller value; 3/5 > 3/8. | | Leaving answers unsimplified or with negative denominator. | Always reduce to lowest terms and keep denominator positive (standard form). |

Quick Reference

• **LCM of denominators** is the key to adding or subtracting fractions.
• **Division = multiply by reciprocal**; never divide fractions directly.
• **Negative × Negative = Positive; Negative × Positive = Negative**.
• A rational number is in **standard form** when the denominator is positive and GCD of numerator and denominator is 1.
• Between any two rationals you can always find another rational (density).
• On a number line, **right means greater**—this holds for negative rationals too.