Fractions and Decimals
Overview
Fractions and decimals form the backbone of numerical computation at the primary level and appear extensively in MAHA TET Paper I Mathematics. This topic tests both content knowledge (can you solve problems involving fractions and decimals?) and pedagogical understanding (can you teach these concepts effectively to Classes I–V students?).
Mastery here is essential because fractions and decimals connect directly to ratio, proportion, percentages, and measurement—all high-weightage areas. Questions typically involve identifying fraction types, performing the four operations, converting between fractions and decimals, and comparing or ordering numbers.
Students must visualise fractions as parts of a whole, understand place value in decimals, and move fluently between representations. The ability to simplify, find equivalent fractions, and align decimal places during operations is non-negotiable.
Key Concepts
- Fraction as part-whole relationship: A fraction a/b represents 'a' equal parts out of 'b' total parts. The denominator tells how many equal parts the whole is divided into; the numerator tells how many parts are taken.
- Proper fraction: Numerator < Denominator (e.g., 3/5). Value is always less than 1.
- Improper fraction: Numerator ≥ Denominator (e.g., 7/4). Value is 1 or greater.
- Mixed fraction (mixed number): A whole number combined with a proper fraction (e.g., 1¾). Every improper fraction can be written as a mixed number and vice versa.
- Equivalent fractions: Different fractions representing the same value (e.g., 1/2 = 2/4 = 3/6). Multiply or divide numerator and denominator by the same non-zero number.
- Decimal as an extension of place value: Decimals use powers of 10. Tenths (0.1), hundredths (0.01), thousandths (0.001) sit to the right of the decimal point.
- Fraction-decimal relationship: Every fraction can be written as a decimal by dividing numerator by denominator. Terminating decimals have denominators whose only prime factors are 2 and 5.
- Like and unlike fractions: Like fractions share the same denominator; unlike fractions do not. Converting to like fractions (common denominator) is essential for addition and subtraction.
Formulas / Key Facts
| Concept | Formula / Procedure |
|---|---|
| Improper → Mixed | Divide numerator by denominator. Quotient = whole part, remainder = new numerator, denominator unchanged. Example: 11/4 = 2¾ |
| Mixed → Improper | (Whole × Denominator) + Numerator, over same denominator. Example: 3²/₅ = (3×5+2)/5 = 17/5 |
| Simplest form | Divide numerator and denominator by their HCF. Example: 12/18 → HCF = 6 → 2/3 |
| Addition (unlike) | Find LCM of denominators, convert to equivalent fractions, add numerators. |
| Subtraction (unlike) | Same as addition—LCM, convert, subtract numerators. |
| Multiplication | (a/b) × (c/d) = ac / bd. Simplify before or after. |
| Division | (a/b) ÷ (c/d) = (a/b) × (d/c). Multiply by reciprocal. |
| Decimal ↔ Fraction | 0.25 = 25/100 = 1/4. Count decimal places → denominator is 10, 100, 1000… |
| Decimal addition/subtraction | Align decimal points, then add/subtract column-wise. |
| Decimal multiplication | Multiply as whole numbers; count total decimal places in both factors and place decimal in product. |
| Decimal division | Move decimal in divisor to make it whole; shift decimal in dividend equally; then divide. |
Worked Examples
Example 1: Convert and Compare
Question: Arrange in ascending order: 5/6, 1.2, 7/8, 0.9
Solution:
Step 1 – Convert all to decimals for easy comparison.
- 5/6 = 5 ÷ 6 ≈ 0.833
- 7/8 = 7 ÷ 8 = 0.875
- 1.2 and 0.9 are already decimals.
Step 2 – Order: 0.833 < 0.875 < 0.9 < 1.2
Ascending order: 5/6, 7/8, 0.9, 1.2
Example 2: Fraction Operations
Question: Simplify 2³/₄ + 1²/₃ − 1/2
Solution:
Step 1 – Convert mixed numbers to improper fractions.
- 2³/₄ = 11/4
- 1²/₃ = 5/3
Step 2 – Find LCM of 4, 3, 2 → LCM = 12.
Step 3 – Convert each fraction:
- 11/4 = 33/12
- 5/3 = 20/12
- 1/2 = 6/12
Step 4 – Compute: 33/12 + 20/12 − 6/12 = 47/12
Step 5 – Convert to mixed number: 47 ÷ 12 = 3 remainder 11 → 3¹¹/₁₂
Example 3: Decimal Division
Question: Divide 4.56 by 0.8
Solution:
Step 1 – Make divisor a whole number: multiply both by 10.
- 4.56 × 10 = 45.6
- 0.8 × 10 = 8
Step 2 – Divide: 45.6 ÷ 8 = 5.7
Answer: 5.7
Common Mistakes
| Wrong Thinking | Correct Fix |
|---|---|
| Adding fractions by adding numerators and denominators separately (e.g., 1/2 + 1/3 = 2/5). | Find a common denominator first; 1/2 + 1/3 = 3/6 + 2/6 = 5/6. |
| Forgetting to convert mixed numbers before multiplying or dividing. | Always change mixed numbers to improper fractions, perform the operation, then convert back if needed. |
| Misaligning decimal points during addition/subtraction, leading to wrong place values. | Write numbers vertically with decimal points in a straight column; fill empty places with zeros. |
| Moving the decimal incorrectly in division—shifting only in divisor or dividend. | Whatever shift you apply to the divisor, apply the same shift to the dividend. |
| Believing larger denominator means larger fraction (e.g., thinking 1/8 > 1/4). | Larger denominator means smaller parts; compare by converting to like fractions or decimals. |
Quick Reference
- Proper < 1; Improper ≥ 1; Mixed = Whole + Proper.
- To add/subtract unlike fractions → LCM → equivalent fractions → operate on numerators.
- Multiply fractions straight across; divide by flipping the second fraction.
- Decimal places: tenths (1), hundredths (2), thousandths (3).
- Align decimals for +/−; count total places for ×; equalise divisor for ÷.
- Simplify fractions using HCF; check terminating decimals by prime factors 2 and 5 only.