Basic Arithmetic & Number Operations

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Percentages, ratios, and proportions form the backbone of quantitative reasoning in the UI entrance exam. A percentage is simply a fraction with denominator 100, so 35% = 35/100 = 7/20. To convert between forms: divide percentage by 100 to get decimal, multiply decimal by 100 to get percentage. For ratio a:b, the value is a/b; the ratio 3:5 equals 3/5 = 0.6. A proportion states that two ratios are equal: a/b = c/d, which cross-multiplies to ad = bc.

Essential formulas:

• Percentage change = (new − old)/old × 100%
• Percentage of a value = (percentage/100) × value
• Ratio simplification: divide both terms by their HCF
• Proportion: if a/b = c/d, then a × d = b × c

⚡ Exam tip: When a question says “increased by 20% then decreased by 20%”, the final value is NOT the original. For example, 100 → 120 → 96. The order matters. Always apply each step sequentially, not by netting the percentages.

⚡ Exam tip: UI questions often embed percentages in word problems about population, profit, or tax. Identify the base quantity first — it’s usually the value BEFORE any change occurred.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Percentages — Deeper Understanding

Percentages appear in three main题型 in the UI exam: direct percentage calculation, successive percentage changes, and percentage reverse-engineering (finding the original given the final value).

For successive percentage changes, the multiplicative approach is faster than applying each change sequentially. If a value x undergoes changes of p%, q%, and r%, the final value is x × (1 + p/100) × (1 + q/100) × (1 + r/100). A value increasing by 10%, then 20%, then decreasing by 5% from 50,000 gives: 50,000 × 1.10 × 1.20 × 0.95 = 62,700.

For reverse problems: if 120 is 15% of a number, the number = 120 × 100/15 = 800. If the price after a 25% discount is Rp 187,500, the original price = 187,500 / 0.75 = Rp 250,000.

Ratios and Proportions

A ratio a:b compares two quantities. The ratio 8:12 simplifies to 2:3 by dividing by 4 (the HCF). When scaling recipes or maps, the ratio must stay constant.

Direct proportion: as one quantity increases, the other increases at the same rate (y = kx). Inverse proportion: as one increases, the other decreases (y = k/x).

Example — direct proportion: If 4 workers complete a job in 12 days, 6 workers complete it in 8 days (since 4 × 12 = 6 × 8 = 48 worker-days).

Example — inverse proportion: If speed increases from 60 km/h to 90 km/h, travel time for a 180 km journey drops from 3 hours to 2 hours.

Common Mistakes to Avoid:

Mistake	Correct approach
Conflating percentage points with percentage	5% + 3% = 8 percentage points, NOT 8% (unless calculating compound)
Assuming percentage increase then decrease cancels	Apply sequentially: 100 + 20% = 120, then 120 − 20% = 96
Forgetting to invert the divisor in reverse percentage	Divide by (1 − percentage/100) when given the reduced value
Misidentifying which quantity is the base	The original/before value is always the base for percentage change

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

The Mathematics of Successive Percentage Changes

The compound nature of percentage changes is a favourite UI trap. Consider: a town’s population of 80,000 grows by 12% in Year 1, then declines by 8% in Year 2. The naive student subtracts 12% − 8% = 4% and answers 80,000 × 1.04 = 83,200. The correct answer is 80,000 × 1.12 × 0.92 = 82,432. The discrepancy of 768 represents the compounding error.

This principle extends to profit and loss calculations. A shop selling an item at a 20% profit, then offering a 15% discount on the marked price, requires careful handling. If cost price = Rp 100,000, marked price with 20% profit = Rp 120,000. After 15% discount: Rp 120,000 × 0.85 = Rp 102,000. Net profit = Rp 2,000 or 2% (not 5% as some might initially assume).

Ratio Mathematics and Its Applications

Ratios extend into more complex scenarios including three-part ratios (a:b:c), where the total parts equal the sum. If a:b = 3:4 and b:c = 5:6, find a:b:c by making b equal. LCM of 4 and 5 = 20, so a:b = 15:20 and b:c = 20:24, giving a:b:c = 15:20:24.

The concept of continued proportion is also tested: three quantities a, b, c are in continued proportion if a/b = b/c. This means b² = ac. For example, if a = 8 and b = 12 are in continued proportion with c, then c = b²/a = 144/8 = 18.

Historical Context

The concept of percentage dates to Roman taxation — the Latin “per centum” means “by the hundred.” Roman Emperor Augustus levied a 1/100 tax on goods sold at auction, essentially the first percentage. The % symbol evolved from the Italian “c.per 0” (cento per 0) used in 15th-century merchant manuscripts.

Exam Pattern Analysis

UI quantitative sections typically include 2-3 questions on arithmetic operations per test paper. Common question types include:

1. “A car travels 240 km using 18 litres. How far can it travel using 25 litres?” (direct proportion)
2. “The population increased from 50,000 to 64,000 in two years at the same rate. Find the annual rate.” (compound percentage)
3. “If 35% of students are female and there are 520 male students, how many students total?” (reverse percentage)

Advanced Problem-Solving

For mixed-operations problems: “A shopkeeper uses a false weight of 900g instead of 1kg. He professes to sell at cost price but actually gains 20%. What is his actual profit percentage?” Solution: Cost per true kg = Rp C. He sells 900g at price of 1000g = Rp C (cost price). So he receives Rp C for 900g, meaning per true kg he receives Rp (C/900) × 1000 = Rp (10/9)C. Gain = (10/9 − 1)C = C/9, which is (1/9) × 100% ≈ 11.11%.

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