Fractions and Decimals Conversion
Proper fraction: numerator < denominator (3/5). Improper: numerator >= denominator (7/4). Mixed number: whole + fraction (1 3/4). To convert fraction to decimal, divide numerator by denominator. To convert a terminating decimal to a fraction, write digits over the place value (0.75 = 75/100 = 3/4). For comparing fractions, cross-multiply or make denominators equal. To add/subtract fractions, take LCM of denominators. To multiply, multiply across; to divide, multiply by the reciprocal. Memory aid for recurring decimals: 0.333... = 1/3, 0.666... = 2/3, 0.142857... = 1/7. Ascending order trick: smaller decimal value = smaller fraction. Always reduce final answers to lowest terms.
Surds and Square Roots
A surd is an irrational root like root2, root3, root5. Rules: root(a) x root(b) = root(ab); root(a)/root(b) = root(a/b); root(a^2 x b) = a root(b). Useful values to memorise: root2 = 1.414, root3 = 1.732, root5 = 2.236, root7 = 2.646. Rationalising the denominator: multiply numerator and denominator by the surd, e.g. 1/root2 = root2/2. For square roots of perfect squares, learn squares up to 30 (e.g. 25^2 = 625, 30^2 = 900). To find square root by factorisation, pair the prime factors: root324 = root(2^2 x 3^4) = 2 x 9 = 18. These shortcuts save crucial seconds in the exam.
Worked Example: Fraction Operations
Simplify 2/3 + 3/4 - 1/6. Step 1: LCM of 3, 4 and 6 is 12. Convert: 2/3 = 8/12, 3/4 = 9/12, 1/6 = 2/12. Step 2: 8/12 + 9/12 - 2/12 = (8 + 9 - 2)/12 = 15/12. Step 3: reduce 15/12 = 5/4 = 1 1/4. Answer = 5/4. For arranging in ascending order, e.g. 2/3, 3/5, 4/7: convert to decimals - 0.667, 0.600, 0.571 - so ascending order is 4/7 < 3/5 < 2/3. Decimal conversion is the fastest, error-free way to compare unlike fractions under exam time pressure.
Fractions, Decimals & Surds — Flashcards
Cover the answer, recall, then check. 12 cards on comparing, converting and simplifying.
Q1. Compare 3/7 and 4/9 by cross-multiplication.
A1. 3×9 = 27, 4×7 = 28. Since 27 < 28, 3/7 < 4/9.
Q2. Convert 0.375 to a fraction.
A2. 375/1000 = 3/8.
Q3. Convert 5/8 to a decimal.
A3. 0.625.
Q4. Rationalise 1/√2.
A4. Multiply by √2/√2 → √2/2 ≈ 0.707.
Q5. Simplify √50.
A5. √(25×2) = 5√2.
Q6. Approximate values of √2, √3 and √5?
A6. 1.414, 1.732 and 2.236.
Q7. Arrange in ascending order: 1/2, 2/3, 3/5, 5/8.
A7. As decimals 0.5, 0.667, 0.6, 0.625 → 1/2 < 3/5 < 5/8 < 2/3.
Q8. Rationalise 1/(√3 − 1).
A8. Multiply by (√3+1): (√3+1)/((√3)²−1²) = (√3+1)/2.
Q9. Write 0.4545… as a fraction.
A9. 45/99 = 5/11.
Q10. Simplify 2/5 + 3/10 − 1/4.
A10. LCM 20: 8/20 + 6/20 − 5/20 = 9/20 = 0.45.
Q11. (√7 + √3)(√7 − √3) = ?
A11. 7 − 3 = 4.
Q12. Fast rule for dividing by 0.25, 0.5, 0.125?
A12. ÷0.25 = ×4, ÷0.5 = ×2, ÷0.125 = ×8. e.g. 36 ÷ 0.25 = 144.
Fractions, Decimals & Surds — Summary
Fractions, decimals and surds are the plumbing of quantitative aptitude — rarely a big question block on their own in RPF (1–2 direct), but the fraction↔percentage↔decimal fluency they build powers percentage, ratio, DI and simplification. Weak here means slow everywhere, so this is a high-return foundation.
Core skills
- Fraction ⇄ decimal ⇄ percentage conversion at sight.
- Comparing fractions by cross-multiplication.
- Simplifying and rationalising surds.
- Converting recurring decimals to fractions.
Must-know values
| Fraction | Decimal | % |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 1/8 | 0.125 | 12.5% |
| 3/8 | 0.375 | 37.5% |
| 5/8 | 0.625 | 62.5% |
| 1/11 | 0.0909… | 9.09% |
Surds: √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236, √7 ≈ 2.646.
Exam Tricks & Tips
- 🎯 Compare fractions by cross-multiplication: for 3/7 vs 4/9, 3×9 = 27 < 4×7 = 28, so 3/7 is smaller — no LCM needed.
- 🎯 Dividing by 0.25/0.5/0.125 = multiplying by 4/2/8; use it to kill decimals.
- 🎯 Rationalise a√b denominators by multiplying top and bottom by the conjugate: 1/(√3−1) → (√3+1)/2.
- 🎯 Recurring decimal → fraction: put the repeating digits over as many 9s: 0.4545…=45/99=5/11.
- 🎯 Simplify surds by pulling out perfect squares: √50 = 5√2, √72 = 6√2.
- ❌ Don't leave a surd in the denominator when comparing or estimating — rationalise first or you'll misjudge size.
Expected exam pattern
"Arrange in ascending order", "which is the largest fraction", "simplify the √ expression", "convert 0.xy… to a fraction". Quick single-concept solves once conversions are automatic.
Quick recap
Burn the fraction–decimal–% table into memory, compare fractions by cross-multiplying, simplify/rationalise surds, and turn recurring decimals into (digits)/(9s). Fluency here speeds up the entire paper.