Every time you cook rice and remember "one cup of rice needs one and a half cups of water," you are using a ratio. Every time you split a bill with friends so each pays a fair share, you are using ratio and proportion. This is one of the most practical, down-to-earth topics in all of mathematics — and one of the most heavily tested on the Civil Service Exam. The good news is that it runs on just two ideas and a single reliable method. Let's make them second nature.
A ratio compares two (or more) quantities of the same kind. A proportion is a statement that two ratios are equal. Together they answer questions like "if the recipe is for 4 people, how much do I need for 10?" or "if we split the winnings 3 to 5, how much does each get?"
On the Civil Service Exam, ratio and proportion appear frequently in Numerical Reasoning, and — just as importantly — the "shares" and "cross-multiply" methods you learn here reappear inside mixtures, partnerships, map scales, and recipe-scaling problems. Counting the disguised cases, ratio reasoning touches 5 to 9 items on a typical exam.
The difficulty is beginner to intermediate. The arithmetic is light; the exam is really testing whether you can (1) set up the ratio in the correct order and (2) tell a direct proportion from an inverse one. Master those two decisions and this topic becomes a source of fast, reliable marks.
After completing this lesson you will be able to:
You should already be comfortable with:
Quick refresher on the one idea we lean on hardest: simplifying. The ratio 12:18 can be reduced by dividing both numbers by their common factor 6, giving 2:3. A ratio, like a fraction, is usually written in its simplest whole-number form. Hold that in mind — we simplify ratios constantly.
Ratios are the math of "fair shares" and "keeping proportions," which appears everywhere in Filipino daily life:
Any time something must stay in balance while its size changes, ratio and proportion are the tools — which is exactly why the exam tests them.
Let's build the topic one idea at a time.
A ratio compares quantities of the same kind. "The ratio of boys to girls is 3:5" (read "3 to 5") means for every 3 boys there are 5 girls. We can write it as 3:5, as the fraction 3/5, or in words.
Analogy: A ratio is a recipe, not a total. "3:5" doesn't say there are exactly 3 boys and 5 girls — it says they come in batches of 3 boys per 5 girls. There could be 6 and 10, or 30 and 50; all share the same ratio 3:5.
Order matters absolutely. "Boys to girls, 3:5" is completely different from "girls to boys, 3:5." Always write the ratio in the exact order the problem names the quantities. Reversing the order is the single most common ratio mistake.
Because a ratio is about proportion, you can multiply or divide all its parts by the same number without changing it. 12:18 = 2:3 (divide both by 6). 2:3 = 8:12 (multiply both by 4). This freedom is what makes ratios so useful — you scale the recipe up or down while keeping the balance.
A proportion states that two ratios are equal, like 3/5 = 12/20. The most powerful tool for proportions is cross-multiplication: if a/b = c/d, then a × d = b × c.
3/5 = x/20 → 3 × 20 = 5 × x → 60 = 5x → x = 12.
Why does cross-multiplication work? It's just the balance rule from algebra: multiply both sides of a/b = c/d by b and by d, and the b's and d's cancel to leave a·d = b·c. It's a shortcut for "clear the denominators."
This is the workhorse technique. To split something in a ratio, add the parts to get the total number of shares, find the value of one share, then multiply.
Divide ₱4,000 between two people in the ratio 3:5. Total shares = 3 + 5 = 8. One share = 4,000 ÷ 8 = ₱500. First person: 3 × 500 = ₱1,500. Second: 5 × 500 = ₱2,500.
Check: 1,500 + 2,500 = 4,000. ✓ The same method extends to three or more parts: a ratio of 2:3:5 means 2 + 3 + 5 = 10 shares.
This is where marks are won or lost. Ask: when one quantity goes up, does the other go up or down?
Direct example: If 5 kg of rice costs ₱250, what do 8 kg cost? More rice, more cost → direct. 5/250 = 8/x → 5x = 2,000 → x = ₱400. (Or: ₱50 per kg × 8 = ₱400.)
Inverse example: If 4 workers finish a task in 12 days, how long for 6 workers at the same rate? More workers, fewer days → inverse. Workers × days is constant: 4 × 12 = 48. So 48 ÷ 6 = 8 days.
Memory tip: Direct → divide across (keep the ratio). Inverse → the product stays the same (workers × days = constant). If picturing it helps: "more men, less time" screams inverse.
Sometimes you're told A:B and B:C separately and need A:B:C. Make the shared term (B) match by scaling.
A:B = 2:3 and B:C = 4:5. B is 3 in the first and 4 in the second; the LCM is 12. Scale the first by 4 (→ 8:12) and the second by 3 (→ 12:15). Now A:B:C = 8:12:15.
Sometimes an outcome depends on two quantities at the same time — like workers and days both affecting how much work gets done. This is compound proportion. The reliable method: start from the known answer and multiply by each ratio, deciding for each factor whether it's direct or inverse.
If 6 men build a wall in 12 days, how many days will 8 men take to build a wall of the same size? Only the number of men changed. More men → fewer days (inverse), so multiply 12 by 6/8: 12 × 6/8 = 9 days.
When more than one factor changes, handle each in turn — put the ratio "right-side up" for a direct factor and "upside down" for an inverse factor — then multiply them all onto the starting value. We'll use exactly this reasoning again in Time & Work.
A ratio of 3:5 means the first part is 3 out of 8 total, i.e. 3/8 = 37.5%, and the second is 5/8 = 62.5%. Converting a ratio to "fraction of the total" is often the fastest way to check an answer or to compare two differently-worded questions.
A very common exam use of ratios is describing a mixture — milk and water, two blends of coffee, an alloy of metals. The ratio tells the proportion of each ingredient, and the "shares" method finds the actual amounts.
A 40-liter mixture of milk and water is in the ratio 5:3. How much milk does it contain? Total shares = 5 + 3 = 8. One share = 40 ÷ 8 = 5 liters. Milk = 5 × 5 = 25 liters; water = 15 liters.
The interesting twist is when you add one ingredient: the other stays fixed. If we add water to change the ratio, the 25 liters of milk doesn't move, and we solve for the new water amount around it. This "one thing stays constant" idea is exactly what powers the Mixtures & Alligation topic later.
A scale is just a ratio between drawing distance and real distance. A map scale of 1:50,000 means 1 unit on the map equals 50,000 of the same units in reality.
On a 1:50,000 map, two barangays are 8 cm apart. How far apart are they in real life? Real distance = 8 × 50,000 = 400,000 cm = 4 kilometers (since 100,000 cm = 1 km).
Scale problems are direct proportions in disguise: bigger map distance means bigger real distance, in a fixed ratio.
With the concepts in place, let's picture them, list the formulas, and drill examples.
| Formula / Rule | Meaning | Example |
|---|---|---|
| a : b (simplify by common factor) | A ratio in lowest terms | 12:18 = 2:3 |
| a/b = c/d → a·d = b·c | Cross-multiplication for a proportion | 3/5 = x/20 → x = 12 |
| one share = total ÷ (sum of parts) | Dividing in a ratio | ₱4,000 in 3:5 → share ₱500 |
| direct: a/b = c/d | Quantities rise/fall together | 5 kg→₱250, 8 kg→₱400 |
| inverse: x₁·y₁ = x₂·y₂ | One up, the other down | 4×12 = 6×8 days |
| mean proportional = √(a·b) | The x in a : x = x : b | between 4 and 9 → 6 |
| part fraction = part ÷ (sum of parts) | Ratio to fraction/percent | 3:5 → 3/8 = 37.5% |
Mean proportional note: if a : x = x : b, then x² = a·b, so x = √(a·b). For 4 and 9, x = √36 = 6. This shows up occasionally and is worth recognizing.
| When the problem says… | Do this… |
|---|---|
| "the ratio of A to B is…" | write A:B in that exact order |
| "divide / share / split in the ratio…" | shares method (sum the parts) |
| "more X gives more Y" (cost, distance) | direct proportion, cross-multiply |
| "more X gives less Y" (workers/days, speed/time) | inverse proportion, product constant |
| "for every… there are…" | a ratio |
| A:B and B:C both given | combine by matching B |
| "8 is added to each part" | set up a proportion with (part + 8) |
Step 1 — Identify the quantities and their order. Write the ratio exactly as named. ↓ Step 2 — Decide the type. Is it a straight proportion, a "divide in ratio," or a direct/inverse relationship? Ask "more means more, or more means less?" ↓ Step 3 — Set up. Cross-multiply (direct), equate products (inverse), or sum the shares (division). ↓ Step 4 — Solve for the unknown. ↓ Step 5 — Answer what's asked, and check the parts add back to the total (for division problems) or that the relationship makes sense (bigger/smaller as expected).
Why Step 2 matters most: choosing direct when it's really inverse (or vice versa) is the defining error of this topic. Always pause on "more means more, or more means less?"
Example 1. Simplify the ratio 20:35. Solution: Both divide by 5 → 4:7. Difficulty: ★☆☆☆☆
Example 2. Solve 4/9 = x/36. Solution: Cross-multiply: 4 × 36 = 9x → 144 = 9x → x = 16. Check: 16/36 simplifies to 4/9. ✓ Difficulty: ★☆☆☆☆
Example 3 (three-part division). Divide ₱6,000 among three people in the ratio 1:2:3. Solution: Shares = 1 + 2 + 3 = 6. One share = 6,000 ÷ 6 = ₱1,000. Amounts: ₱1,000, ₱2,000, ₱3,000. Check: 1,000 + 2,000 + 3,000 = 6,000. ✓ Difficulty: ★★☆☆☆
Example 4 (direct proportion). If 5 notebooks cost ₱175, how much do 8 identical notebooks cost? Thinking: More notebooks, more cost → direct. Solution: 5/175 = 8/x → 5x = 1,400 → x = ₱280. (Or ₱35 each × 8 = ₱280.) Difficulty: ★★☆☆☆
Example 5 (inverse proportion). 15 workers build a wall in 20 days. How many workers are needed to build it in 12 days? Thinking: Fewer days needed → more workers → inverse. Solution: workers × days is constant: 15 × 20 = 300. Needed workers = 300 ÷ 12 = 25 workers. Difficulty: ★★★☆☆
Example 6 (combine ratios). A:B = 3:4 and B:C = 6:7. Find A:B:C. Solution: B is 4 then 6; LCM = 12. Scale first by 3 (→ 9:12), second by 2 (→ 12:14). A:B:C = 9:12:14. Difficulty: ★★★☆☆
Example 7 (coins). A jar holds ₱1 and ₱5 coins in the ratio 3:2 (by count). If the coins are worth ₱65 in total, how many ₱5 coins are there? Thinking: Let the counts be 3x and 2x. Value = 1(3x) + 5(2x). Solution: 3x + 10x = 13x = 65 → x = 5. ₱5 coins = 2x = 10 (and fifteen ₱1 coins). Check: value = 15 + 50 = ₱65. ✓ Difficulty: ★★★★☆
Example 8 (add-to-each ratio). Two numbers are in the ratio 3:4. If 8 is added to each, the new ratio is 4:5. Find the original numbers. Thinking: Let them be 3x and 4x. Set up a proportion after adding 8. Solution: (3x + 8)/(4x + 8) = 4/5 → 5(3x + 8) = 4(4x + 8) → 15x + 40 = 16x + 32 → x = 8. Numbers: 24 and 32. Check: (24 + 8)/(32 + 8) = 32/40 = 4/5. ✓ Difficulty: ★★★★☆
Example 9 (ages). The present ages of Mona and Nita are in the ratio 4:5. In 5 years the ratio will be 5:6. Find their present ages. Solution: Let ages be 4x and 5x. (4x + 5)/(5x + 5) = 5/6 → 6(4x + 5) = 5(5x + 5) → 24x + 30 = 25x + 25 → x = 5. Ages: 20 and 25. Check: in 5 years, 25/30 = 5/6. ✓ Difficulty: ★★★★☆
Example 10 (partnership / profit sharing). Aling Rosa invests ₱20,000 and Mang Tonyo invests ₱30,000 in a sari-sari store. They agree to split profit in the ratio of their investments. If the profit is ₱25,000, how much does each receive? Thinking: Investment ratio 20,000:30,000 = 2:3 → 5 shares. Solution: One share = 25,000 ÷ 5 = ₱5,000. Rosa: 2 × 5,000 = ₱10,000. Tonyo: 3 × 5,000 = ₱15,000. Check: 10,000 + 15,000 = 25,000. ✓ Difficulty: ★★★★☆
Example 11 (mixture). A 60-liter mixture of milk and water is in the ratio 7:5. How many liters of water must be added to make the ratio 7:6? Thinking: Adding water leaves the milk unchanged. Find the milk, then solve for the new water amount. Solution: Milk = 7/12 × 60 = 35 L; water = 25 L. For the new ratio 7:6, milk (35) is 7 parts, so one part = 5 L and water must be 6 × 5 = 30 L. Water to add = 30 − 25 = 5 liters. Check: new mix is 35:30 = 7:6. ✓ Difficulty: ★★★★☆
Example 12 (map scale). On a map with scale 1:250,000, two towns are 6 cm apart. What is the actual distance in kilometers? Solution: Real distance = 6 × 250,000 = 1,500,000 cm = 15 km (100,000 cm per km). Difficulty: ★★★☆☆
Example 13 (speed–time inverse). Two buses travel the same route. Their speeds are in the ratio 3:4. What is the ratio of the times they take? Thinking: Same distance → speed and time are inversely proportional, so the time ratio is the reverse of the speed ratio. Solution: Time ratio = 4:3. Why: the faster bus (4) takes less time (3). Distance = speed × time is constant, so a higher speed forces a lower time. Difficulty: ★★★☆☆
Example 14 (chained relationship). A sum of ₱1,000 is divided among A, B, and C so that A gets twice what B gets, and B gets three times what C gets. How much does each receive? Thinking: Express everyone in terms of C. B = 3C, A = 2B = 6C, so A:B:C = 6:3:1. Solution: Shares = 6 + 3 + 1 = 10. One share = 1,000 ÷ 10 = ₱100. A = ₱600, B = ₱300, C = ₱100. Check: 600 + 300 + 100 = 1,000; A is twice B, B is thrice C. ✓ Difficulty: ★★★★☆
A ratio is a recipe comparing quantities in a fixed order; a proportion says two ratios are equal. Simplify ratios like fractions, and remember that scaling all parts by the same number leaves a ratio unchanged. Solve proportions by cross-multiplication, and divide quantities using the shares method (sum the parts, find one share, multiply). The decisive skill is telling direct proportion (more–more, ratio constant, cross-multiply) from inverse proportion (more–less, product constant). Combine ratios by matching the shared term, and convert freely between ratios, fractions, and percentages to check your work. Get the order right and the direct-vs-inverse call right, and this topic is easy, dependable points.
| Topic | Key point |
|---|---|
| Order | Write A:B exactly as stated |
| Simplify | Divide all parts by a common factor |
| Proportion | a/b = c/d → a·d = b·c |
| Divide in ratio | one share = total ÷ (sum of parts) |
| Direct | more–more; cross-multiply |
| Inverse | more–less; x₁y₁ = x₂y₂ |
| Combine A:B, B:C | match B to its LCM, then merge |
| Ratio → % | part ÷ (sum of parts) × 100 |
| Mean proportional | √(a·b) |
How do I know if a problem is direct or inverse? Ask: "If the first quantity increases, does the second increase or decrease?" Increase together = direct; opposite = inverse. Workers/days, speed/time, and taps/time to fill are the classic inverse pairs.
Do the units have to match in a ratio? For a pure ratio comparing two amounts of the same kind, yes — convert to the same unit first (e.g., both in grams) before writing the ratio.
What's the difference between a ratio and a fraction? They're closely related. The ratio 3:5 corresponds to the fractions 3/8 and 5/8 of the total. The ratio compares the two parts to each other; the fraction compares one part to the whole.
When would I use the mean proportional? When a value sits "between" two others in a continued proportion a : x = x : b. Then x = √(a·b). It's less common but easy points when you spot it.
Can a ratio have more than two parts? Yes — 2:3:5 compares three quantities. The shares method still works; just sum all the parts.
Tick every box and you've mastered a topic that also underpins mixtures, partnerships, and scaling problems across the whole exam.
Put it to the test with 1,952 practice questions on this topic.