Decimals and Fractions — Railway Group D Study Notes
Overview
Decimals and fractions form the backbone of numerical computation in Railway Group D exams. Almost every arithmetic topic — percentage, ratio, profit-loss, time-work — requires fluent conversion between these forms and accurate computation.
Mastery means speed and accuracy: you must convert 0.75 to 3/4 and vice versa in seconds, perform long division without hesitation, and handle recurring decimals confidently. Railway exams test this under time pressure with multi-step calculations, so mechanical proficiency is non-negotiable.
The good news: these are highly procedural. Practice 50–60 problems and you'll automate the steps, freeing mental bandwidth for reasoning in harder questions.
Key Concepts
- Fraction basics: A fraction a/b has numerator a and denominator b (b ≠ 0). Proper fraction: a < b. Improper fraction: a ≥ b. Mixed number: whole number plus proper fraction (e.g. 2 3/5).
- Decimal place value: In 12.345, 3 is tenths (1/10), 4 is hundredths (1/100), 5 is thousandths (1/1000). Every decimal is implicitly a fraction with denominator 10, 100, 1000, etc.
- Like and unlike fractions: Like fractions share the same denominator (1/7, 3/7). Unlike fractions have different denominators (1/3, 2/5) and require LCM to add or subtract.
- Equivalent fractions: Multiplying or dividing numerator and denominator by the same non-zero number yields an equivalent fraction. 2/3 = 4/6 = 6/9.
- Terminating vs recurring decimals: Fraction converts to terminating decimal if denominator (in lowest terms) has only factors 2 and/or 5. Otherwise recurring: 1/3 = 0.333..., written 0.3̅.
- Operations are reversible: Any decimal can be written as a fraction (0.6 = 6/10 = 3/5), any fraction can be divided to get decimal (3/8 = 0.375). Choose the form that simplifies computation.
- Simplification priority: Always reduce fractions to lowest terms before final answer. Use HCF of numerator and denominator.
- Alignment in decimal operations: For addition/subtraction, align decimal points vertically. For multiplication, ignore decimal points initially, then count total decimal places in factors and place in product.
Formulas / Key Facts
- Fraction to decimal: Divide numerator by denominator using long division.
- Decimal to fraction: Write decimal as fraction with denominator 10^n (n = number of decimal places), then simplify. Example: 0.125 = 125/1000 = 1/8.
- Mixed to improper: Whole × Denominator + Numerator over Denominator. Example: 2 3/5 = (2×5 + 3)/5 = 13/5.
- Improper to mixed: Divide numerator by denominator; quotient is whole part, remainder over denominator. Example: 17/5 = 3 2/5.
- Addition/subtraction of fractions: a/b ± c/d = (ad ± bc)/bd. Better: find LCM of denominators, convert, then add/subtract numerators.
- Multiplication of fractions: (a/b) × (c/d) = (a×c)/(b×d). Cancel common factors before multiplying to simplify.
- Division of fractions: (a/b) ÷ (c/d) = (a/b) × (d/c). Invert the divisor and multiply.
- Decimal addition/subtraction: Align decimal points, add/subtract column-wise, carry/borrow as needed.
- Decimal multiplication: Multiply as whole numbers, count total decimal places in both factors, place decimal in product from right.
- Decimal division: If divisor is decimal, multiply both dividend and divisor by 10^n to make divisor whole, then divide.
- Recurring decimal to fraction: For single-digit recurrence 0.a̅, fraction is a/9. For two-digit 0.ab̅, fraction is ab/99. For 0.a̅bc̅, use (abc - a)/(990).
Worked Examples
Example 1: Convert 2.36 to a fraction in lowest terms.
Step 1: Write as fraction over power of 10. 2.36 = 236/100
Step 2: Find HCF of 236 and 100. HCF = 4.
Step 3: Divide numerator and denominator by 4. 236 ÷ 4 = 59, 100 ÷ 4 = 25
Answer: 59/25 or 2 9/25
Example 2: Add 2/3 + 5/6
Step 1: LCM of 3 and 6 is 6.
Step 2: Convert to like fractions. 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
Example 3: Multiply 3.2 × 0.15
Step 1: Ignore decimals, multiply as whole numbers. 32 × 15 = 480
Step 2: Count decimal places: 3.2 has 1, 0.15 has 2. Total = 3.
Step 3: Place decimal 3 places from right in 480. 0.480 = 0.48
Answer: 0.48
Example 4: Divide 7/8 ÷ 3/4
Step 1: Invert the divisor. 3/4 becomes 4/3
Step 2: Multiply. (7/8) × (4/3) = (7×4)/(8×3) = 28/24
Step 3: Simplify. HCF of 28 and 24 is 4. 28 ÷ 4 = 7, 24 ÷ 4 = 6
Answer: 7/6 or 1 1/6
Example 5: Convert recurring decimal 0.7̅ to fraction.
For single-digit recurrence, fraction = digit/9. 0.7̅ = 7/9
Check by division: 7 ÷ 9 = 0.777...
Answer: 7/9
Common Mistakes
- Forgetting to align decimal points in addition/subtraction → Always write numbers one below the other with decimal points in a vertical line. Treat absent digits as zeros: 3.4 + 0.567 should be aligned as 3.400 + 0.567.
- Adding denominators when adding fractions → Never do (a/b) + (c/d) = (a+c)/(b+d). Correct method: find common denominator using LCM, convert, then add numerators only.
- Placing decimal incorrectly in multiplication → Count total decimal places in both factors combined, not in each separately. 1.2 × 0.3: total 2 places, so 36 becomes 0.36, not 0.036.
- Not simplifying final fraction answer → Examiners often mark unsimplified fractions wrong. Always find HCF and reduce. 15/20 must become 3/4.
- Division confusion: not inverting the second fraction → When dividing fractions, invert only the divisor (the second fraction) and multiply. Students often invert the first fraction or try to "divide across" numerators and denominators.
Quick Reference
- Decimal ↔ Fraction: 0.25 = 1/4, 0.5 = 1/2, 0.75 = 3/4, 0.2 = 1/5, 0.125 = 1/8, 0.375 = 3/8, 0.625 = 5/8.
- Recurring shortcut: 0.3̅ = 1/3, 0.6̅ = 2/3, 0.1̅ = 1/9, 0.16̅ = 1/6.
- Mixed ↔ Improper: 3 1/4 = 13/4; 22/7 = 3 1/7.
- Fraction division = multiply by reciprocal: a/b ÷ c/d = a/b × d/c.
- LCM method for addition: Convert unlike fractions to like fractions using LCM of denominators, then add/subtract.
- Simplify before multiplying fractions: Cancel common factors diagonally to avoid large numbers.
Word count: ~1190