Fractions and Decimals
Overview
Fractions and decimals form the backbone of numerical reasoning at the primary level and appear consistently in PSTET Paper I. This topic tests both your conceptual clarity and your ability to teach these ideas to young learners. Students in Classes III-V first encounter the idea that numbers can represent "parts of a whole," which is a significant cognitive leap from counting whole objects.
For PSTET, you must understand fractions (proper, improper, mixed), the concept of equivalent fractions, conversion between fractions and decimals, and basic operations. Equally important is recognising how children develop fractional thinking—through visual models, real-life contexts (cutting a roti, sharing sweets), and gradual abstraction. Expect 3-5 questions directly on this content, plus pedagogical questions on teaching strategies.
Mastery here also supports later topics like ratio, proportion, percentage, and measurement—all of which build on fractional understanding.
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Key Concepts
- **Fraction as part of a whole**: A fraction a/b means a equal parts out of b total equal parts. The denominator (b) tells how many equal parts; the numerator (a) tells how many parts are taken.
- **Types of fractions**: Proper fractions have numerator < denominator (3/4). Improper fractions have numerator ≥ denominator (7/4). Mixed numbers combine a whole and a fraction (1¾).
- **Equivalent fractions**: Different fractions that represent the same value. Multiply or divide both numerator and denominator by the same non-zero number: 1/2 = 2/4 = 3/6 = 50/100.
- **Simplest form (lowest terms)**: A fraction is in simplest form when HCF of numerator and denominator is 1. Example: 8/12 simplified = 2/3.
- **Comparing fractions**: With same denominator, compare numerators directly. With different denominators, convert to equivalent fractions with a common denominator (LCM), then compare.
- **Decimal as a fraction with denominator 10, 100, 1000...**: The decimal 0.25 means 25/100. Place value extends: tenths (1/10), hundredths (1/100), thousandths (1/1000).
- **Conversion between fractions and decimals**: Fraction to decimal—divide numerator by denominator. Decimal to fraction—write as fraction over appropriate power of 10, then simplify.
- **Visual and concrete models**: Area models (shaded parts of shapes), number lines, and fraction strips help children internalise fractional concepts before symbolic manipulation.
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Formulas / Key Facts
| Concept | Formula / Rule | |---------|----------------| | Equivalent fraction | a/b = (a × k)/(b × k) for any non-zero k | | Simplifying | a/b in lowest terms when HCF(a, b) = 1 | | Mixed to improper | Whole × Denominator + Numerator, over same denominator. Example: 2¾ = (2×4 + 3)/4 = 11/4 | | Improper to mixed | Divide numerator by denominator. Quotient = whole part, remainder = new numerator. Example: 11/4 = 2 remainder 3 = 2¾ | | Decimal place values | 0.abc = a/10 + b/100 + c/1000 | | Fraction to decimal | a/b = a ÷ b | | Decimal to fraction | 0.75 = 75/100 = 3/4 (simplify) | | Comparing fractions | Cross-multiply: a/b ? c/d → compare a×d with b×c |
**Must-remember facts:**
- 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75, 1/5 = 0.2, 1/8 = 0.125
- Like fractions have same denominator; unlike fractions have different denominators.
- Unit fraction: numerator is 1 (e.g., 1/3, 1/7).
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Worked Examples
### Example 1: Finding Equivalent Fractions **Problem:** Write three fractions equivalent to 2/5.
**Solution:** Multiply numerator and denominator by 2: (2×2)/(5×2) = 4/10 Multiply by 3: (2×3)/(5×3) = 6/15 Multiply by 4: (2×4)/(5×4) = 8/20
**Answer:** 4/10, 6/15, 8/20 (all equivalent to 2/5)
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### Example 2: Converting Mixed Number to Decimal **Problem:** Convert 3²/₅ to a decimal.
**Solution:** Step 1: Convert to improper fraction. 3²/₅ = (3 × 5 + 2)/5 = 17/5
Step 2: Divide numerator by denominator. 17 ÷ 5 = 3.4
**Answer:** 3.4
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### Example 3: Comparing Unlike Fractions **Problem:** Which is greater: 3/4 or 5/6?
**Solution:** Step 1: Find LCM of denominators 4 and 6. LCM = 12.
Step 2: Convert to equivalent fractions. 3/4 = (3×3)/(4×3) = 9/12 5/6 = (5×2)/(6×2) = 10/12
Step 3: Compare numerators. 10 > 9, so 10/12 > 9/12.
**Answer:** 5/6 > 3/4
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### Example 4: Decimal to Fraction **Problem:** Express 0.125 as a fraction in lowest terms.
**Solution:** Step 1: Write as fraction over power of 10. 0.125 = 125/1000
Step 2: Find HCF of 125 and 1000. Both divisible by 125. 125 ÷ 125 = 1 1000 ÷ 125 = 8
**Answer:** 1/8
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Common Mistakes
1. **Confusing numerator and denominator roles**
- Wrong thinking: "The bigger number on bottom means bigger fraction."
- Correct fix: Larger denominator means smaller parts. 1/8 < 1/4 because eighths are smaller pieces than fourths.
2. **Adding fractions by adding numerators AND denominators**
- Wrong: 1/2 + 1/3 = 2/5
- Correct: Find common denominator first. 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
3. **Forgetting to simplify equivalent fractions when comparing or converting**
- Wrong: Leaving 75/100 as final answer when simplest form is asked.
- Correct: Always check if HCF > 1 and reduce. 75/100 = 3/4.
4. **Misplacing decimal point during conversion**
- Wrong: 3/4 = 0.075 (dividing incorrectly or misplacing decimal)
- Correct: 3 ÷ 4 = 0.75. Count decimal places carefully; use long division.
5. **Treating decimals as separate whole numbers**
- Wrong thinking: 0.15 > 0.9 because "15 is greater than 9"
- Correct fix: Compare place by place. 0.15 = 15 hundredths, 0.9 = 90 hundredths. So 0.9 > 0.15.
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Quick Reference
- **Fraction = Numerator ÷ Denominator**; denominator can never be zero.
- **Equivalent fractions**: Multiply or divide top and bottom by the same number.
- **Mixed → Improper**: (Whole × Denominator) + Numerator, keep denominator.
- **Decimal place values**: Tenths (0.1), Hundredths (0.01), Thousandths (0.001).
- **Quick conversions**: 1/2 = 0.5 | 1/4 = 0.25 | 1/5 = 0.2 | 1/8 = 0.125
- **Comparing fractions**: Same denominator → compare numerators; different denominators → find LCM or cross-multiply.