Fractions and Decimals

Fractions and Decimals

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

A fraction a/b (with b ≠ 0) represents a part of a whole, where a is the numerator and b the denominator. A decimal is the same quantity written in base-10 positional form, obtained by extending the division algorithm past the units digit.

Must-know rules:

• Equivalence: a/b = (a×k)/(b×k) for any non-zero k.
• Common denominator is required for + and −; not for × and ÷.
• Division by a fraction equals multiplication by its reciprocal: a/b ÷ c/d = ad/(bc).
• Terminating vs. repeating: a/b in lowest terms terminates iff b has no prime factors other than 2 or 5; otherwise it repeats.
• Recurring → fraction: 0.1̄6̄ = 15/99 = 5/33 (the 9s-and-0s shortcut).

For NAT-I: 3% weight, MCQ format, expect simplification, four-operation, and fraction-to-decimal conversion in under 90 seconds per question.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Definitions and Types

A fraction a/b is proper when |a| < |b| (e.g. 3/7), improper when |a| ≥ |b| (e.g. 11/4), and a mixed number when written as an integer plus a proper fraction (e.g. 2¾). Converting: divide a by b; the quotient is the integer part, the remainder over b is the fraction. Back-conversion: multiply the integer by the denominator and add the numerator.

Four Operations

• Addition/Subtraction: convert to a common denominator (the LCM of the original denominators is the smallest valid choice), then a/b ± c/d = (ad ± bc)/(bd).
• Multiplication: a/b × c/d = ac/(bd) — no common denominator needed. Cancel common factors across numerators and denominators before multiplying to keep numbers small.
• Division: a/b ÷ c/d = a/b × d/c = ad/(bc), valid when c and d are non-zero.

Fraction ↔ Decimal Conversion

Performing a ÷ b by long division yields the decimal. The decimal terminates if and only if, after reducing to lowest terms, the denominator’s prime factors are only 2 and/or 5 (since 10 = 2·5). Otherwise, the digits repeat with a period whose length divides φ(b), Euler’s totient.

Converting Recurring Decimals to Fractions

• Pure repeat 0.d̄₁d₂…dₖ: write the repeating block as the numerator and k nines as the denominator. Example: 0.1̄6̄ = 16/99 = 5/33.
• Mixed repeat 0.n₁…nₘd̄₁…dₖ: numerator = (all digits up to end of first repeat) − (non-repeating part); denominator = k nines followed by m zeros.

Comparing Fractions

Use cross-multiplication: a/b > c/d iff ad > bc (when b, d > 0). Alternatively, convert both to decimals and compare digit by digit from the left.

Form	Example	Decimal	Repeats?
1/4	denominator 4 = 2²	0.25	No
1/8	denominator 8 = 2³	0.125	No
1/3	denominator 3	0.333…	Yes, period 1
1/7	denominator 7	0.142857 142857…	Yes, period 6
1/6	denominator 6 = 2·3	0.1666…	Yes, period 1

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Place Value and Precision

Each digit after the decimal point represents a power of 10⁻¹: tenths, hundredths, thousandths, ten-thousandths, etc. Misplacing the decimal point by one position changes a value by a factor of 10 — the single most frequent error in NAT-I MCQs involving 0.25 vs 0.025. The number of significant decimal digits a calculator displays can mask the exact rational value; always reduce to lowest terms before judging termination.

Why the 2-and-5 Rule Works

Every terminating decimal x can be written as m/10ⁿ for some integers m, n. Reducing m/10ⁿ cancels factors of 2 and 5 only, because 10ⁿ = 2ⁿ·5ⁿ and no other primes appear in 10. Conversely, if a reduced denominator b contains any prime p ∉ {2, 5}, then a/b cannot equal m/10ⁿ for any n, and the long division cycles with period dividing φ(b). For b = 7, φ(7) = 6, predicting a period of 1, 2, 3, or 6 — and indeed 1/7 has period 6.

Common Traps in NAT-I

• Reciprocal slip: computing 3/4 ÷ 5/6 as 15/24 instead of 18/20 = 9/10.
• Mixed-number confusion: treating 2½ × 4 as 2·(½·4) = 4 instead of (5/2)·4 = 10.
• One-sided cancellation: cancelling across addition (e.g. simplifying 3/4 + 5/4 to 3+5/4 by writing it as 3/4 + 5/4 = 8/4 only after combining numerators over a common denominator, never 3+5).
• Sign mishandling: −3/4 × 2/5 = −6/20 = −3/10, not +3/10; double negatives flip signs in division too.
• Repeating-bar misread: 0.16̄ (1 repeats) = 16/99 ≈ 0.1616…; 0.1̄6̄ (both repeat) = 16/99 as well, but 0.166 (terminating) = 166/1000 = 83/500 — three different values from visually similar strings.

Worked Example

Evaluate: 2¼ + 5/6 × 3/5 ÷ (1/2).

1. Convert: 2¼ = 9/4.
2. Multiply then divide: 5/6 × 3/5 = 15/30 = 1/2; 1/2 ÷ 1/2 = 1.
3. Add: 9/4 + 1 = 13/4 = 3.25.

Practice Prompts

1. Reduce 84/126 to lowest terms, then state whether the decimal terminates and, if so, write its first four decimal digits. (Answer: 2/3, non-terminating repeating, 0.6666…)
2. Express 0.2̄7̄ as a fraction in lowest terms. (Answer: 27/99 = 3/11.)

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