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. Expect 3–5 direct questions on operations (addition, subtraction, multiplication, division) and at least 2–3 embedded in word problems.

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

1. **Fraction to decimal**: Divide numerator by denominator using long division.

2. **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.

3. **Mixed to improper**: Whole × Denominator + Numerator over Denominator. Example: 2 3/5 = (2×5 + 3)/5 = 13/5.

4. **Improper to mixed**: Divide numerator by denominator; quotient is whole part, remainder over denominator. Example: 17/5 = 3 2/5.

5. **Addition/subtraction of fractions**: a/b ± c/d = (ad ± bc)/bd. Better: find LCM of denominators, convert, then add/subtract numerators.

6. **Multiplication of fractions**: (a/b) × (c/d) = (a×c)/(b×d). Cancel common factors before multiplying to simplify.

7. **Division of fractions**: (a/b) ÷ (c/d) = (a/b) × (d/c). Invert the divisor and multiply.

8. **Decimal addition/subtraction**: Align decimal points, add/subtract column-wise, carry/borrow as needed.

9. **Decimal multiplication**: Multiply as whole numbers, count total decimal places in both factors, place decimal in product from right.

10. **Decimal division**: If divisor is decimal, multiply both dividend and divisor by 10^n to make divisor whole, then divide.

11. **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

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**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

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**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

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**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

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**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

1. **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.

2. **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.

3. **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.

4. **Not simplifying final fraction answer** → Examiners often mark unsimplified fractions wrong. Always find HCF and reduce. 15/20 must become 3/4.

5. **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.

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