Decimals and Fractions — RRB NTPC Study Notes

Overview

Decimals and fractions form the backbone of arithmetic computation in RRB NTPC exams. Expect 3–5 direct questions from this topic, plus its application in percentage, ratio, time-and-work, and data interpretation problems. Mastery here means speed and accuracy — a strong candidate converts between forms instinctively and performs operations without hesitation.

The syllabus demands fluency in interconversion (fraction ↔ decimal), all four arithmetic operations on both forms, simplification of complex expressions, and comparison of fractional/decimal quantities. Most errors stem from misplaced decimal points, incorrect denominator handling, and careless simplification. Treat this topic as foundational: once solid, it accelerates every other quantitative section.

Questions range from straightforward computation ("What is 0.125 + 3/8?") to multi-step word problems involving money, measurements, or mixture. The key is recognising when to keep numbers as fractions (for exact answers) versus decimals (for quick approximation).

Key Concepts

• **Fraction representation**: A fraction a/b represents division of a by b. The numerator (a) is the part; denominator (b) is the whole. Proper fraction: a < b. Improper fraction: a ≥ b. Mixed number: whole number + proper fraction (e.g., 2 3/4).

• **Decimal representation**: A decimal is another way to write fractions with denominators 10, 100, 1000, etc. The decimal point separates the whole part (left) from the fractional part (right). Each place to the right is one-tenth of the previous: tenths (0.1), hundredths (0.01), thousandths (0.001).

• **Conversion fraction → decimal**: Divide numerator by denominator. Example: 3/4 = 3 ÷ 4 = 0.75. Terminating decimals end (like 0.5, 0.125); non-terminating decimals repeat forever (like 1/3 = 0.333...).

• **Conversion decimal → fraction**: Write the decimal as a fraction with denominator 10, 100, 1000 (depending on decimal places), then simplify. Example: 0.6 = 6/10 = 3/5. For repeating decimals, use the algebraic method (less common in RRB but good to know).

• **Like and unlike fractions**: Fractions with the same denominator are like fractions (can add/subtract directly). Unlike fractions need a common denominator first (usually the LCM of denominators).

• **Simplification**: Always reduce fractions to lowest terms by dividing numerator and denominator by their HCF. For decimals, remove trailing zeros after the decimal point (e.g., 2.500 = 2.5).

• **Order of operations**: BODMAS applies to both fractions and decimals. Simplify brackets first, then division/multiplication (left to right), then addition/subtraction (left to right).

• **Comparing values**: To compare fractions, cross-multiply or convert to decimals. To compare decimals, align decimal points and compare digit-by-digit from left.

Formulas / Key Facts

• **Addition/subtraction of fractions**: Make denominators equal (LCM), then add/subtract numerators: a/b + c/d = (ad + bc)/bd. Always simplify the result.

• **Multiplication of fractions**: Multiply numerators together and denominators together: (a/b) × (c/d) = ac/bd. Cancel common factors before multiplying to simplify work.

• **Division of fractions**: Multiply by the reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c) = ad/bc.

• **Addition/subtraction of decimals**: Align decimal points vertically, then add/subtract as with whole numbers. Carry/borrow normally; decimal point in answer stays aligned.

• **Multiplication of decimals**: Multiply as whole numbers (ignore decimal points), then count total decimal places in both numbers and place the decimal in the product accordingly.

• **Division of decimals**: Convert divisor to a whole number by shifting decimal point right (shift dividend's decimal the same number of places), then divide normally.

• **Common fraction-decimal equivalents**: 1/2 = 0.5; 1/4 = 0.25; 3/4 = 0.75; 1/5 = 0.2; 1/8 = 0.125; 1/10 = 0.1; 1/3 = 0.333...; 2/3 = 0.666...; 1/6 = 0.1666...

• **Mixed number conversion**: Convert to improper fraction: a b/c = (ac + b)/c. Convert back: divide numerator by denominator; quotient is whole part, remainder over denominator is fraction part.

Worked Examples

**Example 1**: Simplify 0.75 + 5/8.

*Solution*: Convert to common form. 0.75 = 75/100 = 3/4. Now 3/4 + 5/8. LCM(4,8) = 8. Convert: 3/4 = 6/8. Add: 6/8 + 5/8 = 11/8 = 1 3/8 or 1.375.

**Example 2**: Compute 2.5 × 1.2.

*Solution*: Ignore decimals: 25 × 12 = 300. Count decimal places: 2.5 has 1, 1.2 has 1 → total 2 places. Result: 3.00 = 3.

**Example 3**: Divide 3/5 by 4/7.

*Solution*: Multiply by reciprocal: (3/5) × (7/4) = 21/20 = 1 1/20 or 1.05.

**Example 4**: Arrange in ascending order: 0.6, 5/8, 0.58.

*Solution*: Convert all to decimals: 0.6 = 0.600, 5/8 = 0.625, 0.58 = 0.580. Compare: 0.58 < 0.6 < 0.625. Order: 0.58, 0.6, 5/8.

**Example 5**: Simplify (2/3 + 1/6) ÷ 1/2.

*Solution*: First bracket: LCM(3,6) = 6. 2/3 = 4/6. So 4/6 + 1/6 = 5/6. Now divide: (5/6) ÷ (1/2) = (5/6) × 2 = 10/6 = 5/3 = 1 2/3.

Common Mistakes

• **Misplacing the decimal point in multiplication/division**: Students forget to count total decimal places or shift incorrectly. *Fix*: Always count decimal places before and after operation; double-check by estimation (e.g., 2.5 × 1.2 should be near 3, not 30).

• **Adding/subtracting fractions without common denominator**: Directly adding numerators and denominators (e.g., 1/2 + 1/3 ≠ 2/5). *Fix*: Always find LCM of denominators first, convert both fractions, then add/subtract numerators only.

• **Forgetting to simplify final answers**: Leaving 4/8 instead of 1/2, or 0.500 instead of 0.5. *Fix*: Make simplification a reflex — cancel HCF for fractions, drop trailing zeros for decimals.

• **Dividing fractions incorrectly**: Trying to divide numerators and denominators separately instead of using reciprocal. *Fix*: Remember the rule: division = multiply by reciprocal. Write it out every time until automatic.

• **Confusing terminating and non-terminating decimals**: Writing 1/3 = 0.33 (should be 0.333... or 0.3̄). *Fix*: Recognise fractions with denominators having only factors 2 and 5 give terminating decimals; others repeat.

Quick Reference

• Fraction to decimal: divide top by bottom.
• Decimal to fraction: write over power of 10, simplify.
• Add/subtract fractions: common denominator first.
• Multiply fractions: straight across, simplify.
• Divide fractions: multiply by reciprocal.
• Decimal operations: align points (add/sub); count places (multiply); shift decimal to make divisor whole (divide).
• Memorise: 1/2=0.5, 1/4=0.25, 1/5=0.2, 1/8=0.125, 1/3=0.333...