Decimals and Fractions — Study Notes

Overview

Decimals and fractions are two different ways to represent parts of a whole, and mastery of both is essential for SSC GD Elementary Mathematics. This topic typically accounts for 3–5 direct questions in the exam, but the concepts appear indirectly in almost every other area—percentage, ratio, interest, and mensuration all require fluent conversion and calculation with decimals and fractions.

Students must be able to convert between decimal and fraction forms quickly, perform all four operations (addition, subtraction, multiplication, division) on both, and simplify answers to their lowest terms. Many errors occur when students mix operations or forget place values in decimals. The key is to build mechanical fluency: know the standard conversions (like 0.5 = 1/2, 0.25 = 1/4, 0.75 = 3/4) by heart and practice the algorithms until they become automatic.

In SSC GD, questions are straightforward but require accuracy under time pressure. You will see problems asking you to add mixed fractions, multiply decimals, or convert a recurring decimal to a fraction. A single calculation mistake costs marks, so systematic working and double-checking are non-negotiable.

Key Concepts

• **Fraction basics**: A fraction a/b has numerator a (top) and denominator b (bottom). Proper fractions have a < b; improper fractions have a ≥ b. A mixed number combines a whole number and a proper fraction (e.g. 2 1/3).

• **Decimal place value**: In 45.678, the 6 is in the tenths place (6/10), 7 in hundredths (7/100), and 8 in thousandths (8/1000). Understanding place value prevents errors when adding or comparing decimals.

• **Equivalent fractions**: Multiplying or dividing both numerator and denominator by the same nonzero number gives an equivalent fraction. Example: 2/3 = 4/6 = 6/9. Simplification means reducing to lowest terms by dividing by the greatest common divisor.

• **Conversion fraction to decimal**: Divide the numerator by the denominator. Example: 3/4 = 3 ÷ 4 = 0.75. Some fractions give terminating decimals (like 1/2 = 0.5), others give recurring decimals (like 1/3 = 0.333...).

• **Conversion decimal to fraction**: Write the decimal as a fraction over the appropriate power of 10, then simplify. Example: 0.36 = 36/100 = 9/25 after dividing numerator and denominator by 4.

• **Operations on fractions**: For addition/subtraction, find a common denominator, convert both fractions, then add/subtract numerators. For multiplication, multiply numerators and denominators directly. For division, multiply by the reciprocal of the divisor.

• **Operations on decimals**: Align decimal points for addition and subtraction. For multiplication, ignore decimals initially, multiply as whole numbers, then count total decimal places in factors and place the decimal in the product. For division, make the divisor a whole number by shifting decimals in both dividend and divisor.

• **Recurring decimals**: Represented by a bar over repeating digits. Example: 1/3 = 0.3̄ means 0.333... To convert a recurring decimal to a fraction, use algebraic techniques involving multiplication and subtraction.

Formulas / Key Facts

1. **Fraction to decimal**: Divide numerator by denominator → a/b = a ÷ b 2. **Decimal to fraction**: 0.d₁d₂...dₙ = (d₁d₂...dₙ)/10ⁿ, then simplify 3. **Add/subtract fractions**: a/b + c/d = (ad + bc)/bd; then simplify 4. **Multiply fractions**: a/b × c/d = ac/bd 5. **Divide fractions**: a/b ÷ c/d = a/b × d/c = ad/bc 6. **Mixed to improper**: N a/b = (N×b + a)/b 7. **Improper to mixed**: Divide numerator by denominator; quotient is whole part, remainder over denominator is fractional part 8. **Decimal multiplication**: Count total decimal places in factors; place decimal in product that many places from the right 9. **Recurring decimal to fraction (single digit)**: 0.d̄ = d/9; 0.4̄ = 4/9 10. **Recurring decimal (block)**: 0.d₁d₂̄ = (d₁d₂)/99; 0.27̄ = 27/99 = 3/11

Worked Examples

**Example 1: Add fractions with different denominators** Problem: 2/5 + 3/4 = ?

Step 1: Find LCM of denominators 5 and 4 → LCM = 20 Step 2: Convert to equivalent fractions → 2/5 = 8/20 and 3/4 = 15/20 Step 3: Add numerators → 8/20 + 15/20 = 23/20 Step 4: Convert to mixed number → 23/20 = 1 3/20 **Answer: 1 3/20**

**Example 2: Multiply decimal numbers** Problem: 2.5 × 3.2 = ?

Step 1: Ignore decimals temporarily → 25 × 32 Step 2: Multiply as whole numbers → 25 × 32 = 800 Step 3: Count decimal places in factors → 2.5 (1 place) + 3.2 (1 place) = 2 places total Step 4: Place decimal 2 places from right → 8.00 = 8.0 **Answer: 8.0 or 8**

**Example 3: Convert recurring decimal to fraction** Problem: Convert 0.5̄ (0.555...) to a fraction

Step 1: Let x = 0.555... Step 2: Multiply by 10 → 10x = 5.555... Step 3: Subtract original → 10x - x = 5.555... - 0.555... Step 4: Simplify → 9x = 5, so x = 5/9 **Answer: 5/9**

**Example 4: Divide fractions** Problem: 3/4 ÷ 2/5 = ?

Step 1: Change division to multiplication by reciprocal → 3/4 × 5/2 Step 2: Multiply numerators and denominators → (3 × 5)/(4 × 2) = 15/8 Step 3: Convert to mixed number → 15 ÷ 8 = 1 remainder 7, so 1 7/8 **Answer: 1 7/8 or 15/8**

Common Mistakes

**Mistake 1**: Adding denominators when adding fractions → 1/2 + 1/3 = 2/5 is **wrong**. **Fix**: You must find a common denominator first. 1/2 + 1/3 = 3/6 + 2/6 = 5/6.

**Mistake 2**: Misplacing the decimal point in multiplication → 2.5 × 4 = 100 instead of 10. **Fix**: Count decimal places carefully. 2.5 has 1 decimal place, 4 has 0, so the product 10.0 has 1 decimal place.

**Mistake 3**: Forgetting to simplify final fraction answers → leaving 6/8 instead of reducing to 3/4. **Fix**: Always divide numerator and denominator by their GCD. Here GCD(6,8) = 2, so 6/8 = 3/4.

**Mistake 4**: Converting decimal to fraction without simplifying → writing 0.5 as 5/10 and stopping there. **Fix**: After writing 5/10, reduce by dividing both by 5 to get 1/2. The exam expects simplified answers.

**Mistake 5**: Aligning decimals incorrectly when adding/subtracting → writing 3.5 + 12.45 as 3.5 + 12.45 without vertical alignment leads to wrong totals. **Fix**: Write one number above the other with decimal points in a vertical line: ``` 3.50 +12.45 ------ 15.95 ```

Quick Reference

• **Common conversions**: 1/2 = 0.5; 1/4 = 0.25; 3/4 = 0.75; 1/5 = 0.2; 1/8 = 0.125; 1/10 = 0.1
• **To add/subtract fractions**: Same denominator? Add/subtract numerators. Different? Find LCM first.
• **To multiply fractions**: Multiply straight across (numerators together, denominators together), then simplify.
• **To divide fractions**: Flip the second fraction (reciprocal) and multiply.
• **Decimal to fraction**: Write over power of 10 matching decimal places, then simplify → 0.75 = 75/100 = 3/4.
• **Recurring 0.d̄ = d/9**: One repeating digit divided by 9 → 0.7̄ = 7/9; two digits divided by 99 → 0.18̄ = 18/99 = 2/11.