Fractions and Decimals
Overview
Fractions and decimals form the backbone of numerical reasoning in primary and upper-primary mathematics. For UPTET, this topic appears consistently across both Paper I (Classes 1–5) and Paper II (Classes 6–8), testing your ability to perform operations, convert between forms, and solve word problems. Questions typically range from straightforward computation to application-based problems involving money, measurement, and comparison.
Mastery here is non-negotiable because fractions and decimals underpin later topics—ratio and proportion, percentage, profit-loss, and mensuration. A teacher must understand not just the procedures but also the conceptual models (part-whole, division interpretation) to explain these ideas effectively to children.
Key Concepts
- Fraction as part-whole: A fraction a/b represents 'a' equal parts out of 'b' total equal parts of a whole. The denominator tells how many equal parts; the numerator tells how many are taken.
- Proper fraction: Numerator < Denominator (e.g., 3/7). Value is always less than 1.
- Improper fraction: Numerator ≥ Denominator (e.g., 9/4). Value is 1 or greater.
- Mixed fraction: Combination of a whole number and a proper fraction (e.g., 2¼). Every improper fraction can be written as a mixed fraction and vice versa.
- Equivalent fractions: Fractions representing the same value (e.g., 1/2 = 2/4 = 3/6). Obtained by multiplying or dividing numerator and denominator by the same non-zero number.
- Decimal as a fraction with denominator 10, 100, 1000, etc.: 0.3 = 3/10; 0.47 = 47/100; 2.135 = 2135/1000.
- Place value in decimals: Tenths (1/10), hundredths (1/100), thousandths (1/1000) moving right from the decimal point.
- Like and unlike fractions: Like fractions have the same denominator; unlike fractions have different denominators. Converting to like fractions is essential before adding or subtracting.
Formulas / Key Facts
| Operation | Rule |
|---|---|
| Converting improper to mixed | Divide numerator by denominator → Quotient = whole part, Remainder = numerator of fractional part. E.g., 17/5 = 3 and 2/5. |
| Converting mixed to improper | (Whole × Denominator) + Numerator, keep same denominator. E.g., 4 and 3/7 = (4×7+3)/7 = 31/7. |
| Addition/Subtraction of fractions | Make denominators equal (LCM), then add/subtract numerators. |
| Multiplication of fractions | (a/b) × (c/d) = ac/bd. Simplify before or after multiplying. |
| Division of fractions | (a/b) ÷ (c/d) = (a/b) × (d/c). Multiply by the reciprocal. |
| Decimal to fraction | Write digits after point over 10, 100, etc., then simplify. 0.75 = 75/100 = 3/4. |
| Fraction to decimal | Divide numerator by denominator. 3/8 = 0.375. |
| Adding/Subtracting decimals | Align decimal points, then add/subtract column-wise. |
| Multiplying decimals | Multiply as whole numbers, count total decimal places in both factors, place decimal in product accordingly. |
| Dividing decimals | Shift decimal in divisor to make it whole; shift same places in dividend; then divide. |
Worked Examples
Example 1: Convert and Compare
Problem: Arrange in ascending order: 5/6, 7/9, 3/4.
Solution:
- Find LCM of denominators 6, 9, 4 → LCM = 36.
- Convert each: 5/6 = 30/36; 7/9 = 28/36; 3/4 = 27/36.
- Compare numerators: 27 < 28 < 30.
- Ascending order: 3/4 < 7/9 < 5/6.
Example 2: Mixed Fraction Operations
Problem: Simplify 3 and 2/5 + 2 and 3/4.
Solution:
- Convert to improper fractions: 3 and 2/5 = 17/5; 2 and 3/4 = 11/4.
- LCM of 5 and 4 = 20.
- 17/5 = 68/20; 11/4 = 55/20.
- Add: 68/20 + 55/20 = 123/20.
- Convert back: 123 ÷ 20 = 6 remainder 3 → 6 and 3/20.
Example 3: Decimal Division
Problem: Divide 4.56 by 0.8.
Solution:
- Make divisor a whole number: multiply both by 10 → 45.6 ÷ 8.
- Divide: 45.6 ÷ 8 = 5.7.
- Answer: 5.7.
Example 4: Word Problem
Problem: A rope 8.4 m long is cut into pieces of 0.7 m each. How many pieces?
Solution:
- Number of pieces = 8.4 ÷ 0.7.
- Shift decimals: 84 ÷ 7 = 12.
- Answer: 12 pieces.
Common Mistakes
| Wrong Thinking | Correct Fix |
|---|---|
| Adding fractions by adding numerators AND denominators separately (e.g., 1/2 + 1/3 = 2/5). | Find LCM of denominators first, convert to like fractions, then add only numerators. Correct: 1/2 + 1/3 = 3/6 + 2/6 = 5/6. |
| Forgetting to simplify final answers. | Always reduce fractions to lowest terms by dividing by HCF. |
| Placing decimal incorrectly in multiplication (counting places in only one factor). | Count decimal places in BOTH factors combined and place decimal that many places from the right in the product. |
| Treating 0.5 as smaller than 0.45 because "45 > 5". | Compare digit by digit from left OR convert to same number of decimal places: 0.50 vs 0.45 → 50 > 45, so 0.5 > 0.45. |
| In division, forgetting to shift decimal in dividend when shifting in divisor. | Always shift BOTH by the same number of places to keep the quotient unchanged. |
Quick Reference
- Proper: numerator < denominator; Improper: numerator ≥ denominator.
- Mixed to improper: (Whole × Denom) + Num over Denom.
- Fraction division = multiply by reciprocal.
- Decimal places rule: 0.3 × 0.02 = 0.006 (1 + 2 = 3 decimal places).
- To compare fractions, convert to like fractions or to decimals.
- 1/2 = 0.5; 1/4 = 0.25; 1/5 = 0.2; 3/4 = 0.75 — memorise these for speed.