Ratio and Proportion
🟢 Lite — Quick Review (1h–1d)
Rapid summary for last-minute revision before your exam.
A ratio compares two quantities of the same kind. It is written as a:b and is equivalent to the fraction a/b. For example, if a class has 12 girls and 18 boys, the ratio of girls to boys is 12:18, which simplifies to 2:3 (divide both by their HCF = 6). A proportion states that two ratios are equal: a:b = c:d, which can be written as a/b = c/d.
Essential Formulas:
- If a:b = c:d, then a × d = b × c (cross-multiplication, or “product of means equals product of extremes”)
- Division in a given ratio: to share an amount N in the ratio a:b, total parts = a + b; A’s share = N × a/(a+b); B’s share = N × b/(a+b)
- Direct proportion: y ∝ x means y = kx for some constant k
- Inverse proportion: y ∝ 1/x means y = k/x
- If a:b = b:c, then b is the mean proportional between a and c, and b² = ac
Key Facts:
- Ratios must be expressed in the same units before comparing — 2 hours : 30 minutes = 120 min : 30 min = 4:1
- When sharing in a ratio, the total must equal the original amount
- In direct proportion, as one quantity increases, the other increases at the same rate
- In inverse proportion, as one quantity increases, the other decreases
⚡ Exam Tip: For sharing problems in the UI entrance test, always check that your shares add up to the original total. If they don’t, something has gone wrong. For proportion problems, identify whether it’s direct (both increase together) or inverse (one increases, other decreases) before setting up the equation.
🟡 Standard — Regular Study (2d–2mo)
Standard content for students with a few days to months.
Understanding Ratios
A ratio a:b means “a parts to b parts.” It tells us the relative size of two quantities.
Example: The ratio of copper to zinc in an alloy is 3:7 by mass. If the alloy contains 15 kg of copper, how much zinc is there? 3 parts = 15 kg → 1 part = 5 kg. Zinc = 7 parts = 7 × 5 = 35 kg.
Simplifying Ratios
Divide both parts by their HCF. Example: 24:36 = 2:3 (divide by 12). 1.5:2.5 = 15:25 = 3:5 (multiply by 10 first to clear decimals, then simplify).
Sharing in a Given Ratio
Example: Rp 2,400,000 is divided between Ahmad and Budi in the ratio 5:3. Find each share. Total parts = 5 + 3 = 8. Ahmad = (5/8) × 2,400,000 = 1,500,000. Budi = (3/8) × 2,400,000 = 900,000. Check: 1,500,000 + 900,000 = 2,400,000. ✓
Combining Ratios
Example: The ratio of males to females in a college is 3:2. After 100 males join, the ratio becomes 5:3. Find the original numbers. Let original males = 3k, females = 2k. After 100 males join: males = 3k + 100, females = 2k. New ratio: (3k + 100)/2k = 5/3 → 3(3k + 100) = 5(2k) → 9k + 300 = 10k → k = 300. Original: males = 900, females = 600. Check: after 100 males → 1000:600 = 5:3 ✓.
Direct Proportion
Two quantities are in direct proportion if they increase or decrease at the same rate: y ∝ x, so y = kx.
Example: If 4 identical pipes fill a tank in 6 hours, how long would 8 pipes take (assuming same flow rate)? More pipes → less time → inverse proportion? Actually in this context, doubling pipes doubles the rate, so time halves. 8 pipes = 4 × 2 pipes, so time = 6/2 = 3 hours. More formally: 4 pipes → 6 hours, 8 pipes → h hours. Rate per pipe = 4/6 tanks per hour. For 8 pipes: 8 × (4/6) = 32/6 = 16/3 tanks per hour → time = 1 / (16/3) = 3/8 tanks per hour? No — let me redo.
Rate approach: 4 pipes fill 1 tank in 6 hours → combined rate = 1/6 tank per hour → each pipe rate = 1/24 tank per hour. 8 pipes: rate = 8/24 = 1/3 tank per hour → time = 1 / (1/3) = 3 hours. ✓
Inverse (Indirect) Proportion
Two quantities are in inverse proportion if y ∝ 1/x, so y = k/x. As one increases, the other decreases.
Example: If 6 workers can complete a job in 20 days, how long would 15 workers take? More workers → less time → inverse proportion. 6 workers × 20 days = 15 workers × d days → d = (6 × 20) / 15 = 120/15 = 8 days.
Compound Proportion
Some problems involve multiple proportions at once.
Example: If 3 machines can produce 200 items in 5 hours, how many items can 6 machines produce in 8 hours? Items ∝ machines (direct) and Items ∝ time (direct). So items = k × machines × time. 200 = k × 3 × 5 → k = 200/15 = 40/3. For 6 machines, 8 hours: items = (40/3) × 6 × 8 = 40 × 16 = 640 items.
Proportional Parts — Dividing a Quantity
When dividing a quantity in a given ratio with more than two parts: for ratio a:b:c, total parts = a+b+c. Each share = N × (part/total).
Example: Rp 5,600,000 is divided among three people A, B, C in the ratio 2:3:7. Find each share. Total parts = 2+3+7 = 12. A = (2/12) × 5,600,000 = 933,333.33… ≈ 933,333. B = (3/12) × 5,600,000 = 1,400,000. C = (7/12) × 5,600,000 = 3,266,666.67… ≈ 3,266,667. Check sum: 933,333 + 1,400,000 + 3,266,667 = 5,600,000. ✓
Scale and Maps
A map scale of 1:50,000 means 1 unit on the map = 50,000 units in reality. If two cities are 4.5 cm apart on a map, actual distance = 4.5 × 50,000 = 225,000 cm = 2.25 km.
Problem-Solving Strategies:
- When a ratio changes after an addition or subtraction, set up an equation using the original ratio
- For “sharing” problems, always verify that shares sum to the original amount
- In proportion questions, first determine if the relationship is direct or inverse
- For compound proportion, set up the constant of proportionality and solve
Common Mistakes:
- Adding ratio parts incorrectly — a:b = 3:4 does NOT mean a = 3 and b = 4 unless the total is 7 parts
- Confusing direct with inverse proportion — the key question is: when one quantity increases, does the other increase or decrease?
- Forgetting to convert to the same units before forming or simplifying a ratio (e.g., mixing cm and m)
- In ratio change problems, not properly tracking the before and after states
🔴 Extended — Deep Study (3mo+)
Comprehensive coverage for students on a longer study timeline.
Ratios with More Than Two Quantities
Extended ratio a:b:c means a/b = b/c = c/a? No — it means the three quantities are in the ratio a:b:c. In a:b:c, the first is to the second as a is to b, and the second is to the third as b is to c.
Example: If three numbers are in the ratio 3:5:7 and their sum is 75, find the numbers. Total parts = 3+5+7 = 15. Each part = 75/15 = 5. Numbers: 3×5 = 15, 5×5 = 25, 7×5 = 35. Check: 15+25+35 = 75. ✓
Mean Proportional
If a:b = b:c, then b is the mean proportional between a and c, and b² = ac. Example: Find the mean proportional between 4 and 9. b² = 4 × 9 = 36 → b = 6. So 4:6 = 6:9 (ratio simplifies to 2:3 = 2:3 ✓).
Continued Ratio
Three or more quantities can be compared using a continued ratio. Example: The ratio of salt to water to acid in a solution is 2:7:3. If there are 14 kg of water, how much salt and acid are there? 2 parts salt : 7 parts water : 3 parts acid. 1 part = 14/7 = 2 kg. Salt = 2 × 2 = 4 kg. Acid = 3 × 2 = 6 kg.
Rates
A rate is a ratio comparing two quantities with different units. Examples: km/h (speed), Rp/kg (price), students/teacher (ratio). Unitary method: find the value of one unit, then multiply.
Example: A car travels 180 km in 4 hours. What is its speed? 180/4 = 45 km/h. How far in 7 hours at the same speed? 45 × 7 = 315 km.
Best Buy Problems
Compare prices using the unit rate (price per unit).
Example: Brand A: 500g for Rp 12,000. Brand B: 750g for Rp 17,250. Brand A price per gram: 12,000/500 = Rp 24/g. Brand B price per gram: 17,250/750 = Rp 23/g. Brand B is cheaper per gram.
Alligation
Alligation is used to find the ratio in which two ingredients of different prices must be mixed to produce a mixture at a given price.
Example: Tea costing Rp 30,000/kg and Rp 50,000/kg are mixed to produce a blend costing Rp 40,000/kg. Find the ratio. Using alligation medial: (50,000 − 40,000) : (40,000 − 30,000) = 10,000 : 10,000 = 1:1. So mix equal quantities of each.
Ratios in Data Presentation
Ratios are used to describe population demographics, market share, and other comparative data. Example: In a class of 40 students, 15 study Physics, 10 study Chemistry, and 15 study Biology. Express as a ratio: Physics:Chemistry:Biology = 15:10:15 = 3:2:3. The total of ratio parts is 8 (3+2+3). This doesn’t correspond to the total 40 — it means for every 8 parts, 3 are Physics, 2 Chemistry, 3 Biology. Scale factor = 40/8 = 5. So actual numbers are 15, 10, 15 ✓.
UI Entrance Exam Patterns
Ratio and proportion questions typically include:
- Simplifying ratios (including those with decimals and fractions)
- Sharing quantities in given ratios
- Ratio change problems (after addition/subtraction)
- Direct and inverse proportion word problems
- Best buy/unit rate comparisons
- Scale and map calculations
⚡ Exam Strategy: When a ratio changes after some event (people joining/leaving, ingredients added), always set up the “before” and “after” ratios carefully. If the question says “the ratio becomes X:Y,” use that directly. If it says “100 more of X are added and the ratio becomes,” substitute the new amounts accordingly.
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