Fractions
Overview
Fractions form the backbone of numerical reasoning in primary mathematics and appear frequently in KAR TET Paper I. This topic tests your conceptual understanding of part-whole relationships, ability to perform operations with fractions, and skill in converting between different fraction forms. Mastery here directly supports later topics like ratio, proportion, percentage, and decimal arithmetic.
For the exam, expect questions on identifying fraction types, comparing and ordering fractions, performing basic operations (addition, subtraction, multiplication, division), and converting between mixed numbers and improper fractions. Word problems involving fractions in real-life contexts—sharing, measuring, cooking—are common. From a pedagogy standpoint, you must understand how children develop fraction sense and the typical misconceptions they carry.
Key Concepts
- Fraction as part-whole: A fraction a/b represents 'a' equal parts out of 'b' total equal parts. The denominator tells how many equal parts the whole is divided into; the numerator tells how many parts are taken.
- Proper fraction: Numerator is less than denominator (e.g., 3/7). Value is always less than 1.
- Improper fraction: Numerator is greater than or equal to denominator (e.g., 9/4). Value is 1 or greater.
- Mixed fraction (mixed number): A whole number combined with a proper fraction (e.g., 2¼). It represents a quantity greater than 1.
- Equivalent fractions: Different fractions representing the same value (e.g., 1/2 = 2/4 = 3/6). Multiply or divide both numerator and denominator by the same non-zero number.
- Decimal fractions: Fractions with denominators that are powers of 10 (10, 100, 1000…). These convert directly to decimal notation (e.g., 7/10 = 0.7, 25/100 = 0.25).
- Like and unlike fractions: Like fractions share the same denominator; unlike fractions have different denominators. Operations require converting unlike to like fractions first.
- Unit fraction: A fraction with numerator 1 (e.g., 1/5, 1/8). Fundamental building block for understanding all fractions.
Formulas / Key Facts
| Concept | Formula / Rule |
|---|---|
| Mixed to improper | a b/c = (a × c + b) / c |
| Improper to mixed | Divide numerator by denominator; quotient = whole part, remainder = new numerator |
| Equivalent fraction | a/b = (a × k) / (b × k) for any k ≠ 0 |
| Simplest form | Divide numerator and denominator by their HCF |
| Addition (like) | a/c + b/c = (a + b) / c |
| Addition (unlike) | Find LCM of denominators, convert, then add numerators |
| Subtraction | Same as addition but subtract numerators |
| Multiplication | a/b × c/d = (a × c) / (b × d) |
| Division | a/b ÷ c/d = a/b × d/c (multiply by reciprocal) |
| Fraction to decimal | Divide numerator by denominator |
| Decimal to fraction | Write decimal over appropriate power of 10, then simplify |
Key facts to remember:
- Multiplying a fraction by its reciprocal gives 1.
- Adding or subtracting fractions requires a common denominator; multiplication and division do not.
- 0.5 = 1/2, 0.25 = 1/4, 0.75 = 3/4, 0.125 = 1/8 — memorise these common conversions.
Worked Examples
Example 1: Convert 3 2/5 to an improper fraction
Step 1: Multiply whole number by denominator → 3 × 5 = 15 Step 2: Add numerator → 15 + 2 = 17 Step 3: Place over original denominator → 17/5
Answer: 17/5
Example 2: Add 2/3 + 5/6
Step 1: Find LCM of 3 and 6 → LCM = 6 Step 2: Convert 2/3 to equivalent fraction with denominator 6 → 2/3 = 4/6 Step 3: Add numerators → 4/6 + 5/6 = 9/6 Step 4: Simplify → 9/6 = 3/2 = 1 1/2
Answer: 1 1/2 (or 3/2)
Example 3: Divide 4/5 by 2/3
Step 1: Find reciprocal of divisor → reciprocal of 2/3 is 3/2 Step 2: Multiply → 4/5 × 3/2 = 12/10 Step 3: Simplify → 12/10 = 6/5 = 1 1/5
Answer: 6/5 or 1 1/5
Example 4: Convert 0.375 to a fraction in simplest form
Step 1: Write as fraction over power of 10 → 375/1000 Step 2: Find HCF of 375 and 1000 → HCF = 125 Step 3: Divide both by 125 → 375 ÷ 125 = 3, 1000 ÷ 125 = 8
Answer: 3/8
Common Mistakes
- Adding denominators when adding fractions Wrong: 1/4 + 1/3 = 2/7 Correct: Find common denominator first → 3/12 + 4/12 = 7/12
- Forgetting to simplify the final answer Wrong: Leaving 8/12 as the answer Correct: Simplify to 2/3 by dividing by HCF (4)
- Confusing "of" with addition in word problems Wrong: Treating "1/2 of 20" as 1/2 + 20 Correct: "Of" means multiply → 1/2 × 20 = 10
- Inverting the wrong fraction in division Wrong: a/b ÷ c/d = b/a × c/d Correct: Invert the divisor (second fraction) → a/b × d/c
- Treating decimal places incorrectly during conversion Wrong: 0.05 = 5/10 Correct: Two decimal places means denominator 100 → 0.05 = 5/100 = 1/20
- Comparing fractions without common denominators Wrong: Assuming 3/5 > 2/3 because 3 > 2 and 5 > 3 Correct: Convert to common denominator or cross-multiply → 3/5 = 9/15, 2/3 = 10/15, so 2/3 > 3/5
Quick Reference
- Proper: numerator < denominator; Improper: numerator ≥ denominator
- Mixed → Improper: (whole × denominator + numerator) / denominator
- Add/subtract fractions: make denominators same first
- Multiply fractions: straight across (num × num, den × den)
- Divide fractions: multiply by reciprocal of divisor
- Decimal to fraction: place over 10, 100, or 1000 based on decimal places; simplify