Fractions, Decimals and Percentages

Fractions, Decimals and Percentages

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

A fraction a/b names a part of a whole — a is the numerator, b (≠ 0) is the denominator. A decimal is the same quantity written in base-10 positional form (0.75), and a percentage is a fraction with denominator 100, written with a % sign (75%). The three forms are interconvertible: divide a by b for decimals, multiply by 100 for percent. Memorise Percentage of Q = (p/100) × Q, New = Original × (1 ± p/100), and SP = CP × (1 ± P%/100). HAT-UG tests these conversions in 2–3 MCQs of the Quantitative Reasoning section, almost always disguised inside a word problem on profit, discount, or successive change.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Conversions and Simplest Form

Two fractions are equivalent when numerator and denominator are multiplied or divided by the same non-zero integer (3/4 = 6/8 = 9/12). The simplest form is reached when gcd(numerator, denominator) = 1. Converting among the three notations:

• Fraction → Decimal: long-divide a by b.
• Decimal → Percent: shift the decimal point two places right, append %.
• Percent → Fraction: write the number over 100, then cancel common factors (40% = 40/100 = 2/5).

Terminating vs Recurring Decimals

A fraction in lowest terms terminates (e.g. 3/8 = 0.375) iff its denominator’s only prime factors are 2 and 5. Any other prime factor forces a recurring decimal (1/3 = 0.333…). The length of the repeating block is at most b − 1.

Percentage Change and Successive Change

Single-step change: New = Original × (1 ± p/100). For two successive changes of p₁% and p₂%, the net factor is (1 + p₁/100)(1 + p₂/100), so the combined change is that product minus 1 — not p₁ + p₂. A +20% rise followed by a +20% rise gives 1.20 × 1.20 − 1 = +44%, a classic HAT-UG trap.

Comparing Fractions

Cross-multiply: a/b > c/d iff ad > bc (when b, d > 0). Equivalently, raise both to a common denominator and compare numerators. Decimal form gives the fastest comparison (0.625 vs 0.58).

Typical HAT-UG Question Patterns

Profit/Loss: P% = (Profit/CP) × 100; SP = CP(1 + P/100). Discount: D% = (Discount/MP) × 100, with SP = MP(1 − D/100). Expect one question testing successive change on price or population, and one on converting between fraction and percent inside a ratio problem.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Edge Cases in Conversion

A percentage above 100 is legal (250% = 2.5 = 5/2). A “percentage point” is not a percentage: going from 8% to 10% is a 2-percentage-point rise but a 25% relative increase ((10−8)/8). Examiners exploit this distinction.

Profit, Loss, Discount and Successive Discount

Two successive discounts of d₁% and d₂% on marked price M give effective SP = M × (1 − d₁/100)(1 − d₂/100), never M × (1 − (d₁+d₂)/100). For a chain of n equal discounts d%, the single equivalent discount is 1 − (1 − d/100)ⁿ, useful for verifying 10%–10%–10% ≈ 27.1%, not 30%.

Worked Solution

A shirt is marked at Rs 2400. A shopkeeper offers 10% discount, then applies 5% sales tax on the discounted price.

• Discounted price = 2400 × (1 − 0.10) = Rs 2160.
• Final price = 2160 × (1 + 0.05) = Rs 2268.
• Net effective change vs marked price = (2268 − 2400)/2400 = −5.5%.

Common Mistakes

• Adding fractions without a common denominator (½ + ⅓ ≠ 2/5).
• Forgetting to invert when dividing by a fraction.
• Confusing CP and MP in profit/discount chains.
• Treating +p% then −p% as net zero — it actually yields −p²/100% (e.g. +20% then −20% = −4%).

Practice Prompts

1. A price rises 25%, then falls 20%. Express the net change as a percentage.
2. An article is sold at Rs 1140 with 5% profit. Find the cost price, then compute the selling price needed for a 12% profit.

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