Percentage: The Language of Parts per Hundred

Welcome to one of the most important lessons you will study for the Civil Service Examination. If you master percentages, you make the entire Numerical Reasoning section easier — because percentages quietly appear inside almost every other topic: profit and loss, interest, discounts, data interpretation, mixtures, even the "passing rate" of the exam you are about to take. Take your time with this chapter. By the end, "percent" will stop feeling like a math word and start feeling like plain language you already speak.

1. Lesson Overview

Percentages are everywhere. When a store in Divisoria posts "50% OFF," when a news report says the price of rice rose "by 8 percent," when your payslip shows a "12% SSS/Pag-IBIG deduction," or when the Civil Service Commission announces a "passing rate of 12.6%," they are all speaking the same language — the language of parts per hundred.

In the Civil Service Exam (both Professional and Sub-Professional levels), percentage questions are a guaranteed, high-frequency part of the Numerical Reasoning section. You can reliably expect several direct percentage questions, and many more that hide a percentage step inside a larger problem about discounts, interest, taxes, population, or reading a chart. It is not unusual for percentage-related reasoning to touch 8 to 12 of the numerical items on a typical exam once you count the topics that depend on it.

Here is the good news: percentages are considered a beginner-to-intermediate difficulty topic. There is really only one core idea, and once you understand it deeply, the "hard" questions are just the same idea wearing a costume. This lesson will take you from that single idea all the way to the trickiest exam-level problems, one careful step at a time.

2. Learning Objectives

After completing this lesson you will be able to:

• Define what a percentage really is, from first principles, and explain it in your own words.
• Convert fluently between percentages, decimals, and fractions in both directions.
• Calculate the three fundamental percentage quantities: the part, the percent, and the whole.
• Solve percentage-increase and percentage-decrease problems, including price mark-ups, discounts, taxes, and population changes.
• Handle successive (back-to-back) percentage changes without falling for the "just add them" trap.
• Work backwards from a final value to an original value (reverse-percentage problems).
• Distinguish between percentage points and percent change — a favorite exam trap.
• Recognize the problem type instantly from the wording of the question.
• Apply mental-math shortcuts to solve common percentage questions in seconds.
• Avoid the specific mistakes that cause examinees to lose easy marks.

3. Prerequisites

To get the most out of this lesson, you should already be comfortable with:

• Basic multiplication and division, including multiplying and dividing by decimals.
• Fractions — what they mean and how to simplify them (for example, knowing that 30/60 is the same as 1/2).
• Decimals — reading them and multiplying with them (for example, 0.25 × 40).

If any of these feel shaky, don't worry — we will review the exact pieces you need as we go. The most important prerequisite is one you already have: the willingness to read slowly and think about why, not just how.

Let's quickly refresh one prerequisite that percentages lean on constantly — the relationship between fractions, decimals, and division. The fraction 3/4 is not just a shape cut into pieces; it is also an instruction that says "3 divided by 4," which equals 0.75. Hold on to that idea: a fraction is a division waiting to happen. Percentages are built directly on top of it.

4. Why This Topic Matters

Percentages are arguably the single most useful math topic in everyday Filipino life — far more useful than, say, trigonometry for most people. Consider how often they appear:

• Shopping and sales. A ₱1,200 pair of shoes at an SM sale marked "25% off" — how much do you actually pay? A "buy one, take one" is really a 50% discount in disguise.
• Government and taxes. The 12% VAT added to most purchases, the withholding tax on your salary, the SSS, PhilHealth, and Pag-IBIG contributions deducted every payday — all are percentages.
• Salaries and business. A "10% commission" for a real-estate agent, a "20% markup" on goods sold in a sari-sari store, a "15% service charge" at a restaurant.
• Banking and loans. Interest rates on a Pag-IBIG housing loan or a credit card are quoted as percentages per year.
• News and the economy. "Inflation reached 6.1%," "the peso weakened by 3%," "unemployment fell to 4.5%" — understanding these headlines requires understanding percentages.
• Health and elections. "70% of Filipinos are fully vaccinated," "the candidate won with 58% of the vote," "the exam had a passing rate of 12.6%."

In other words, learning percentages is not just about passing the Civil Service Exam. It is a life skill that protects your money and helps you understand the world. The exam simply tests something you will use for the rest of your life.

5. Core Concepts

We will build your understanding brick by brick. Do not rush past any brick — each one holds up the next.

The meaning of "percent"

The word percent comes from the Latin per centum, which means "per hundred" or "for every hundred." The symbol % is just a shorthand for "/100" — literally "divided by one hundred."

So when we say 25%, we are really saying 25 out of every 100, or 25/100, or 0.25. These are three different costumes worn by the exact same number.

Analogy: Imagine every quantity in the world has been sliced into 100 equal pieces, like a pizza cut into 100 tiny slices. A percentage simply tells you how many of those 100 slices you are talking about. "40%" means "grab 40 of the 100 slices." It doesn't matter whether the whole pizza is big or small — percent always assumes it was cut into exactly 100 parts.

This is why percentages are so powerful: they put everything on the same scale of 100, so you can compare fairly. A student who got 45 out of 60 on one quiz and 70 out of 100 on another — which did better? Convert both to percent (75% and 70%) and the comparison becomes obvious. Percent is the great equalizer.

Think about how impossible comparison is without percentages. If Maria answered 38 of 50 items and Jose answered 52 of 70, who did better? The raw numbers (38 and 52) can't be compared because the totals differ. But 38 ÷ 50 = 76% and 52 ÷ 70 ≈ 74.3%, so Maria edged ahead. This is the everyday superpower of percentages: they let you compare things that started from different sizes. Keep this in mind — many exam questions are secretly asking "which is bigger?" and the fastest route is to convert everything to percent first.

Converting between percent, decimal, and fraction

Because a percentage is just "a number divided by 100," converting is easy once you internalize the pattern.

Percent → Decimal: divide by 100, which means move the decimal point two places to the left.

• 25% = 0.25
• 8% = 0.08
• 150% = 1.50
• 0.5% = 0.005

Decimal → Percent: multiply by 100, which means move the decimal point two places to the right, then add the % sign.

• 0.30 = 30%
• 1.2 = 120%
• 0.075 = 7.5%

Percent → Fraction: write it over 100, then simplify.

• 25% = 25/100 = 1/4
• 40% = 40/100 = 2/5
• 12.5% = 12.5/100 = 1/8

Fraction → Percent: divide the top by the bottom, then multiply by 100 (or just find an equivalent fraction with 100 on the bottom).

• 3/4 = 0.75 = 75%
• 1/5 = 0.20 = 20%
• 7/8 = 0.875 = 87.5%

Memory tip: Percent-to-decimal moves the point Left (percent is the Larger-looking number, so shrink it). Decimal-to-percent moves the point Right. Some students remember "% sign means the decimal is hiding two places to the right of where it should be."

The one relationship that runs everything

Here is the heart of the entire chapter. Almost every percentage problem — no matter how complicated it looks — is built from a single relationship:

part = percent × whole

In plain words: a part of something equals what fraction (the percent) you take, multiplied by the whole thing. The "whole" is the total or the base — the thing the percentage is of. The "part" is the piece you end up with.

Consider "20% of 150 is 30." Here:

• The whole (the base, the thing after "of") is 150.
• The percent (as a decimal) is 0.20.
• The part (the result) is 30, because 0.20 × 150 = 30.

Every basic percentage question gives you two of these three numbers and hides the third. Your only job is to figure out which one is missing and rearrange the relationship to find it. That's it. That is the whole game.

The three faces of the core relationship

Because part = percent × whole, we can rearrange it three ways depending on what is missing:

Face 1 — Find the part (percent and whole are known): part = percent × whole. Example: What is 20% of 150? → 0.20 × 150 = 30.

Face 2 — Find the percent (part and whole are known): percent = part ÷ whole. Example: 30 is what percent of 150? → 30 ÷ 150 = 0.20 = 20%.

Face 3 — Find the whole (part and percent are known): whole = part ÷ percent. Example: 30 is 20% of what number? → 30 ÷ 0.20 = 150.

Illustration Suggestion: Draw a triangle divided into three sections. Put PART in the top section, and PERCENT and WHOLE in the two bottom sections. To find any one quantity, cover it with your finger: covering PART shows "percent × whole" (side by side = multiply); covering WHOLE shows "part ÷ percent" (one on top of the other = divide); covering PERCENT shows "part ÷ whole." This is the same trick used for the speed-distance-time triangle.

Notice the pattern: whenever you are looking for the part, you multiply. Whenever you are looking for the percent or the whole, you divide — and the part is always on top of the division. Lock this in and you have conquered the basics.

A common misconception to clear up now

Many learners believe percentages can never go above 100. This is false. Percentages can be greater than 100%. If a sari-sari store's sales grew from ₱10,000 to ₱25,000, that is a 150% increase — perfectly valid. 100% simply means "the whole thing, once"; 200% means "twice the whole"; 150% means "one and a half times the whole." Whenever something more than doubles, you are in the land of percentages above 100. Do not let a big percent scare you.

Expressing one quantity as a percentage of another

A very common exam task is to express one number as a percentage of another. The rule is exactly Face 2: percent = (part ÷ whole) × 100 — but the trap is deciding which number is the whole. The whole is always the number you are comparing to, the number that comes after the words "as a percentage of."

Example: "Express ₱150 as a percentage of ₱600." Here ₱600 is the whole (it comes after "of"), so 150 ÷ 600 × 100 = 25%.

Be careful with part-to-part versus part-to-whole comparisons. Suppose a class has 12 boys and 18 girls. "What percent of the class are boys?" uses the whole class (30) as the base: 12 ÷ 30 × 100 = 40%. But "the number of boys is what percent of the number of girls?" uses girls (18) as the base: 12 ÷ 18 × 100 ≈ 66.7%. Same 12 boys, completely different answers — because the base changed. Always find the base first.

Percentage of a percentage

Sometimes a percentage is taken of another percentage. The word "of" still means multiply, so you simply multiply the decimals together. "60% of 40% of 500" means 0.60 × 0.40 × 500 = 120. It does not mean (60 + 40)% of 500, and it does not mean 60% of 40. This pattern hides inside many real problems — for instance, "40% of the workers are women, and 60% of those women are married," which means 0.60 × 0.40 = 24% of all workers are married women.

Percentage increase and decrease — think in multipliers

Most real-world percentage problems involve a change: a price goes up, a discount comes off, a population grows. The slow way is to find the change amount and then add or subtract it. The fast, exam-smart way is to think in multipliers.

To increase a value by r%, multiply it by (1 + r/100). To decrease it by r%, multiply by (1 − r/100).

A ₱500 item marked up 20%: new price = 500 × 1.20 = ₱600. A ₱500 item discounted 20%: new price = 500 × 0.80 = ₱400.

Why does this work? A 20% increase means the new price is "the whole original (100%) plus another 20%," which is 120% of the original, i.e. ×1.20. A 20% discount means you keep "100% minus 20%," which is 80%, i.e. ×0.80. Thinking in "what fraction of the original remains" is faster and less error-prone than adding and subtracting separately. When you shop, don't compute what you save — compute what you pay. A 30%-off item costs you 70% of the tag price in one multiplication.

Here is a fact that surprises nearly everyone: a 20% increase followed by a 20% decrease does not bring you back to the start. Watch: 500 × 1.20 = 600, then 600 × 0.80 = 480. You end at ₱480, which is 4% below ₱500. The reason is that the 20% decrease is taken from ₱600 (the bigger number), so it removes more than the 20% you originally added. Increasing then decreasing by the same percent always lands you lower than where you started.

Successive percentage changes — the fast way

When two percentage changes happen back-to-back, their combined effect is not simply the sum of the two percents. For two successive changes of a% and b%, the true combined change is:

combined % = a + b + (a × b ÷ 100)

A price rises 10%, then rises another 20%. Combined = 10 + 20 + (10 × 20 ÷ 100) = 30 + 2 = 32%, not 30%.

For a decrease, plug in a negative number for that change — the formula handles increases and decreases together. That extra "(a × b ÷ 100)" term is the second change landing on top of the first change; it's small but it's exactly what the exam checks. If you ever see "two discounts" or "rose then rose again," resist the urge to add — either apply the multipliers one after another, or use this formula.

Reverse percentage — working backwards

One of the most-tested — and most-failed — patterns gives you the value after a change and asks for the value before. The instinct is to apply the percentage the other way, but that over-corrects. The correct move is to divide by the multiplier.

After a 15% discount, an item costs ₱850. What was the original price? Original = 850 ÷ 0.85 = ₱1,000 (not 850 × 1.15, which gives the wrong ₱977.50).

Think of it this way: the discounted price of ₱850 already is 85% of the original. To get back to 100%, you undo the ×0.85 by dividing by 0.85. The same logic reverses an increase: if a price including 12% VAT is ₱3,360, the pre-VAT price is 3,360 ÷ 1.12 = ₱3,000. To go forward you multiply; to go backward you divide.

Percentage points versus percent change

These two sound identical and mean completely different things — and the exam loves the confusion. Suppose an interest rate rises from 5% to 8%.

• The change in percentage points is simply 8 − 5 = 3 percentage points (a plain subtraction of the two percents).
• The percent increase in the rate itself is 3 ÷ 5 = 0.60 = 60% (the change measured relative to the original rate).

Both statements describe the same event, but "3 percentage points" and "a 60% increase" are both correct depending on what is asked. Read the question carefully: if it says "by how many percentage points," subtract; if it says "by what percent did the rate increase," divide the change by the original. Getting these two mixed up is one of the most common ways examinees lose an otherwise-easy mark.

Now that the foundation is solid, let's give these ideas the visual and formula support they deserve.

6. Visual Learning Suggestions

Percentages become much easier when you can see them. Here are mental pictures worth drawing in your reviewer notebook:

• [Illustration Suggestion] The 100-grid. Draw a 10×10 grid of squares (100 squares total). Shade 25 of them to see what 25% looks like. This makes "parts per hundred" concrete. Shade 50 and you see exactly half; shade 5 and you see how small 5% really is.
• [Illustration Suggestion] The percentage bar. Draw a horizontal bar and label 0% on the far left and 100% on the far right. Mark the middle as 50%. Now any percentage has a place on the bar. A "30% discount" is a little less than a third of the way in. This helps you estimate before you compute.
• [Illustration Suggestion] The before-and-after arrows. For increase/decrease problems, draw the original amount as a box, then a taller box (for an increase) or a shorter box (for a decrease) beside it, with an arrow showing the change. Label the change amount. Seeing "the change compared to the original box" prevents the #1 mistake of comparing against the wrong base.
• [Illustration Suggestion] The pie slice. For "what percent of the total" questions (common in Data Interpretation), picture a pie chart. 25% is a quarter slice; 50% is half the pie; 10% is a thin slice. The whole pie is always 100%.

Keep these pictures in mind as we move to the formulas — the formulas are just these pictures written in symbols.

7. Formula Library

Below is every formula you need for percentages, organized so you can find them fast. Notation note: we write all math in plain symbols (×, ÷, ², etc.) so it reads the same on any device.

Let's unpack the two formulas students most often get wrong.

Formula 4 — Percentage change

Formula: % change = (new − old) ÷ old × 100

Meaning: How big is the change compared to where you started? The single most important word is old — the denominator is always the original value, never the new one.

Variable definitions: "old" is the starting/original amount; "new" is the amount after the change.

Why it works: A change of ₱10 feels huge on a ₱20 item (that's 50%) but tiny on a ₱2,000 item (that's 0.5%). To judge a change fairly, we measure it relative to the starting amount. That is exactly what dividing by "old" does.

Memory trick: "Change over Original." If you divide by the new value by mistake, you'll get a slightly wrong answer that the exam writers have deliberately included as a wrong choice.

Worked example: A jeepney fare rises from ₱12 to ₱15. % change = (15 − 12) ÷ 12 × 100 = 3 ÷ 12 × 100 = 25% increase.

Common mistake: Dividing by 15 (the new fare) to get 20%. Wrong — always divide by the original 12.

Formula 9 — Successive percentage changes

Formula: combined % = a + b + (a × b ÷ 100)

Meaning: When two percentage changes happen one after the other, their combined effect is not simply a + b. There is an extra piece because the second change acts on an already-changed amount.

Variable definitions: a is the first % change, b is the second. Use a negative number for a decrease.

Why it works: Multiplying by (1 + a/100) then (1 + b/100) expands to 1 + a/100 + b/100 + (a×b)/10000. Convert back to a percentage and you get a + b + (a×b ÷ 100). The last term is the "interaction" — the part of the second change that lands on top of the first change.

Memory trick: "Add them, then add their product-over-100."

Worked example: A price rises 10%, then rises another 20%. Combined = 10 + 20 + (10 × 20 ÷ 100) = 30 + 2 = 32%, not 30%.

Special case — increase then equal decrease: A 20% increase followed by a 20% decrease gives 20 + (−20) + (20 × −20 ÷ 100) = 0 − 4 = −4%. You end up 4% lower, never back at the start. This surprises almost everyone the first time.

8. Pattern Recognition

Half the battle is knowing which face of the formula a question is asking for. Train your eyes to catch these signal words:

Key habit: Before touching any numbers, underline the signal word and decide which pattern you're in. Nine out of ten "hard" percentage errors are really pattern-recognition errors — the student computes correctly but answered the wrong question.

9. Problem-Solving Framework

Use this repeatable six-step routine for every percentage problem. It feels slow at first and becomes automatic with practice.

Step 1 — Read the whole question carefully. ↓ Step 2 — Identify the WHOLE (the base). Find the number that comes right after the word "of," or the original amount before any change. This is your anchor. ↓ Step 3 — Identify what is KNOWN and what is UNKNOWN. Label each number as part, percent, or whole. ↓ Step 4 — Choose the right formula / face. Are you finding the part, the percent, the whole, a change, or a reversal? ↓ Step 5 — Solve carefully. Convert percents to decimals before multiplying. Keep your units (₱, students, votes) attached. ↓ Step 6 — Verify. Ask "does this answer make sense?" A discount should give a smaller number. A percentage of a quantity should be smaller than the whole (unless the percent is over 100). Estimate to check.

Why each step matters: Step 2 (finding the whole) prevents the most common error — using the wrong base. Step 4 prevents pattern mistakes. Step 6 catches arithmetic slips before they cost you a mark. Never skip Step 6 on the real exam; it takes three seconds and saves points.

10. Worked Examples

Study these slowly. Cover the solution, try it yourself, then check. The thinking process matters more than the answer.

Beginner

Example 1. What is 25% of 80? Thinking: Signal word "of" → find the part → multiply. Solution: 0.25 × 80 = 20. Shortcut: 25% is a quarter, so just divide 80 by 4 → 20. Difficulty: ★☆☆☆☆

Example 2. 15 is what percent of 60? Thinking: "is what percent of" → find the percent → divide part by whole. Solution: 15 ÷ 60 = 0.25 = 25%. Common mistake: Dividing 60 ÷ 15 = 4 and answering "4%" or "400%." Always put the part on top. Difficulty: ★☆☆☆☆

Intermediate

Example 3. A student answered 45 out of 60 items correctly. What is the percentage score? Thinking: part = 45, whole = 60, find percent. Solution: 45 ÷ 60 = 0.75 = 75%. Alternative: 45/60 simplifies to 3/4, and 3/4 = 75%. Difficulty: ★★☆☆☆

Example 4. A gadget costs ₱2,000. A 12% VAT is added. What is the final price? Thinking: "added" → increase → multiply by (1 + r). Solution: 2000 × 1.12 = ₱2,240. Alternative: VAT = 0.12 × 2000 = 240; final = 2000 + 240 = 2,240. Difficulty: ★★☆☆☆

Example 5 (reverse). After a 15% discount, a bag costs ₱850. What was the original price? Thinking: We're given the price after a decrease → reverse-percentage → divide by (1 − r). Solution: 850 ÷ 0.85 = ₱1,000. Verify: 1000 × 0.85 = 850. ✓ Common mistake: Computing 850 × 1.15 = 977.50. This over-corrects and is wrong — you must divide, not multiply back. Difficulty: ★★★☆☆

Advanced

Example 6 (successive discounts). A jacket priced at ₱1,500 gets a 20% discount, then an additional 10% off at the counter. What is the final price? Thinking: Two changes in a row → apply one after the other (never add to 30%). Solution: 1500 × 0.80 = 1,200; then 1,200 × 0.90 = ₱1,080. Shortcut (single equivalent discount): 0.80 × 0.90 = 0.72, so you pay 72% and the true discount is 28%, not 30%. 1500 × 0.72 = 1,080. ✓ Common mistake: Adding 20% + 10% = 30% and computing 1500 × 0.70 = 1,050. Wrong by ₱30. Difficulty: ★★★☆☆

Example 7 (repeated growth). A barangay's population is 12,000 and grows 5% each year. What will it be after 2 years? Thinking: Same 5% increase applied twice → multiply by 1.05 twice. Solution: 12,000 × 1.05 × 1.05 = 12,000 × 1.1025 = 13,230. Common mistake: Adding "5% + 5% = 10%" to get 13,200. That misses the second year's growth on the first year's increase (the missing 30 people). Difficulty: ★★★☆☆

Example 8 (concentration). A tank holds 40 liters of a solution that is 15% salt. How much water must evaporate so the solution becomes 20% salt? Thinking: Evaporating water removes water but leaves the salt unchanged. Find the fixed amount of salt, then find the new total that makes it 20%. Solution: Salt = 0.15 × 40 = 6 liters (this never changes). We need 6 liters to be 20% of the new volume: new volume = 6 ÷ 0.20 = 30 liters. Water evaporated = 40 − 30 = 10 liters. Difficulty: ★★★★☆

Civil Service Exam Level

Example 9 (marks and passing). In an exam, a candidate who scores 30% fails by 20 marks. Another who scores 45% gets 25 marks more than the passing mark. What are the maximum (total) marks? Thinking: Let the maximum marks be M and the passing mark be P. "Fails by 20" means their score is 20 below passing; "25 more than passing" means 25 above. Solution: 0.30M = P − 20 0.45M = P + 25 Subtract the first from the second: 0.15M = 45, so M = 45 ÷ 0.15 = 300. Verify: Passing mark = 0.30 × 300 + 20 = 110. Check: 0.45 × 300 = 135 = 110 + 25. ✓ Difficulty: ★★★★★

Example 10 (elections). In a two-candidate barangay election, the winner received 60% of the valid votes and won by 900 votes. How many valid votes were cast? Thinking: Winner 60%, loser 40%, so the margin is 60% − 40% = 20% of the total, and that margin equals 900. Solution: 20% of total = 900 → total = 900 ÷ 0.20 = 4,500 valid votes. Verify: 60% of 4,500 = 2,700; 40% = 1,800; difference = 900. ✓ Difficulty: ★★★★☆

Example 11 (the reversal trap). Ben's salary is 20% more than Ana's. By what percent is Ana's salary less than Ben's? Thinking: The answer is NOT 20%, because the base changes. Pick an easy number for Ana. Solution: Let Ana = ₱100. Then Ben = 100 × 1.20 = ₱120. Ana is less than Ben by 20 out of 120 = 20 ÷ 120 = 0.1667 = 16.67% (about 16⅔%). Why it differs: "20% more" is measured against Ana (base 100); "how much less" is measured against Ben (base 120). Different base, different percent. Difficulty: ★★★★☆

Example 12 (price and consumption). The price of rice increases by 25%. By what percent must a family reduce its rice consumption to keep its total rice spending unchanged? Thinking: Spending = price × quantity. If price ×1.25, quantity must ×(1/1.25) to keep the product the same. Solution: required decrease = 25 ÷ (100 + 25) × 100 = 25 ÷ 125 × 100 = 20%. Verify: Say price ₱50, buy 100 units → ₱5,000. New price ₱62.50; to still spend ₱5,000, buy 5000 ÷ 62.50 = 80 units. Drop from 100 to 80 = 20% less. ✓ Difficulty: ★★★★★

Example 13 (reading a budget — data interpretation). A barangay's annual budget of ₱2,500,000 is allocated as follows: 32% to infrastructure, 28% to health services, 15% to education, and the remainder to administration. How much is allocated to administration? Thinking: The four parts must total 100%. Find administration's percent first, then take that percent of the whole. Solution: Administration = 100% − (32% + 28% + 15%) = 100% − 75% = 25%. Amount = 0.25 × 2,500,000 = ₱625,000. Shortcut: 25% is a quarter, so just divide the budget by 4. Common mistake: Adding the three given percents wrong, or forgetting that the four slices must sum to 100%. Difficulty: ★★★☆☆

Example 14 (profit as a percentage — a preview of Profit & Loss). A fruit vendor in Baclaran buys mangoes at ₱80 per kilo and sells them at ₱100 per kilo. What is the percentage profit? Thinking: Profit percent is always measured against the cost (the original outlay), so cost is the whole. Solution: Profit = 100 − 80 = ₱20. Percent profit = 20 ÷ 80 × 100 = 25%. Common mistake: Dividing by the selling price (20 ÷ 100 = 20%). Profit and loss are measured on cost unless the problem says otherwise. This is the exact same "which is the base?" habit from percentage change — profit questions are just percentage-change questions in a business costume. Difficulty: ★★★☆☆

11. Exam Tricks

Exam writers rarely test whether you can multiply. They test whether you'll fall for a setup. Here are their favorite traps and how to see through them:

• The wrong-base trap. "A shirt's price rose 20%, then fell 20%. Is it back to the original?" Your instinct screams "yes." The correct answer is "no — it's 4% lower," because the 20% decrease is taken from the higher price. Whenever you see increase-then-decrease by the same percent, the answer is always slightly lower than the start.
• The "20% more" vs "20% less" trap (see Example 11). "A is 25% more than B" does not mean "B is 25% less than A." The bases differ, so the percents differ. Always pick a concrete number to check.
• The percentage-points trap. "Interest rose from 5% to 8%." That is a rise of 3 percentage points but a 60% increase in the rate (3 ÷ 5). If the question asks "by what percent did the rate increase," the answer is 60%, not 3%. Read which one they want.
• The successive-discount trap. "30% off, then 20% off" is not 50% off. It equals 0.70 × 0.80 = 0.56, i.e. 44% off. Two discounts are always less generous than their sum.
• The "of a percent" trap. "60% of 40% of 500" means multiply all three: 0.60 × 0.40 × 500 = 120. It is not (60 + 40)% of 500, and not 60% of 40.
• Elimination by estimation. If a ₱980 item is "35% off," the discount is a bit more than a third — roughly ₱340 — so the price is roughly ₱640. Any choice near ₱637 must be right; you can eliminate ₱340 (that's the discount, not the price) and ₱1,320 (that's an increase) instantly.

12. Common Mistakes

These are the specific errors that cost real examinees real marks. Read each one, understand why it happens, and consciously guard against it.

• Comparing a change against the wrong base. Using the new value instead of the original in a percentage-change problem. Why it happens: the new value is the last number you read, so it's fresh in mind. Fix: always ask "change compared to what I started with." Divide by the old value.
• Assuming a 20% increase then 20% decrease returns to the start. Why: the two 20%s look symmetric. Fix: remember they act on different bases; the result is always lower (specifically 4% lower for 20%).
• Adding successive percentages. Treating "20% off then 10% off" as "30% off." Why: addition is easier than multiplication. Fix: apply changes one at a time, or use the a + b + ab/100 formula.
• Confusing percentage points with percent change. Saying a rate "from 5% to 8%" rose "3%." Why: the word "percent" appears in both. Fix: "percentage points" = subtract; "percent increase" = divide the change by the original.
• Reversing a discount by multiplying instead of dividing. Computing final × (1 + r) to undo a discount. Why: it feels like "adding back" what was removed. Fix: to undo a change, divide by the multiplier — original = final ÷ (1 − r).
• Multiplying "60% of 40% of 500" as (60 + 40)%. Why: the word "of" gets read as "and." Fix: every "of" means multiply.
• Forgetting to convert percent to a decimal. Computing 20 × 150 = 3,000 instead of 0.20 × 150 = 30. Why: rushing. Fix: convert the percent first, every time.

13. Shortcuts

These legal shortcuts will save you precious seconds on the exam. Practice them until they're automatic.

• 10% is just moving the decimal one place left. 10% of 350 = 35. 10% of 42 = 4.2. This is your building block for everything else.
• 1% moves the decimal two places left. 1% of 350 = 3.5.
• 5% is half of 10%. 5% of 350 = half of 35 = 17.5.
• 15% = 10% + 5%. 15% of 200 = 20 + 10 = 30. (Handy for restaurant tips and service charges.)
• 20% = double 10%. 20% of 45 = 2 × 4.5 = 9.
• 25% = divide by 4; 50% = divide by 2; 75% = half plus a quarter. 75% of 80 = 40 + 20 = 60.
• The swap trick: X% of Y = Y% of X. They always give the same answer, so flip to whichever is easier. 16% of 25 is annoying, but 25% of 16 = 4 is instant. 8% of 50 = 50% of 8 = 4.
• Use fraction equivalents for "ugly" percents. 12.5% = 1/8, 37.5% = 3/8, 62.5% = 5/8, 87.5% = 7/8, 33⅓% = 1/3, 66⅔% = 2/3, 16⅔% = 1/6. So 37.5% of 800 = 3/8 × 800 = 300 — no decimals needed.
• To increase by r%, multiply by (1 + r/100) in one step. A 12% VAT on ₱2,000 is just 2000 × 1.12 = 2,240, faster than finding 240 and adding.
• Back-solving. On a multiple-choice item, you can test the answer choices. If "the original price after a 20% discount is ₱1,080" and one choice is ₱1,350, check: 1350 × 0.80 = 1,080. ✓ Done — no algebra needed.

14. Memory Techniques

• "Percent = per hundred = ÷ 100." Say it out loud until it's reflex. The % sign literally is the "/100."
• "IS over OF." In "what number is X% of Y," the value near "is" is the part (top of the fraction) and the value near "of" is the whole (bottom). percent = IS ÷ OF.
• "Of means multiply." Whenever you read "of" in a percentage problem, your hand should reach for the × sign.
• "Change over Original." For percentage change, the denominator is always the original. Chant "C-O-O: Change Over Original."
• The discount ladder. Memorize the pay-fractions: 10% off = pay 0.90; 20% off = pay 0.80; 25% off = pay 0.75; 50% off = pay 0.50. Thinking in "what you pay" instead of "what you save" removes a subtraction step.
• Fraction–percent flashcards. Write 1/2, 1/3, 1/4, 1/5, 1/6, 1/8, 3/4, 2/3, 3/8 on cards with their percents on the back. Five minutes a day for a week and these become instant recall — a huge time-saver on exam day.

15. Real Civil Service Exam Strategy

• How much time to spend. A direct percentage item ("what is 15% of 240?") should take under 30 seconds. A multi-step word problem might take 60–90 seconds. If you're past two minutes on one item, mark it and move on — every question is worth the same one point.
• When to estimate. If the answer choices are far apart, estimate instead of computing exactly. "48% of 610" is very close to "half of 610" ≈ 305 — usually enough to pick the right choice.
• When to skip. If a problem has heavy successive-percentage algebra and you're low on time, skip it, answer the quick items first, then return. Never let one hard percentage question eat the time of five easy ones.
• How to eliminate choices. Cross out choices that don't make sense: a discounted price can't be higher than the original; a percentage of a number (with percent under 100) can't be bigger than the number. This often removes two of four choices instantly.
• How to check answers. Plug your answer back into the situation. If you found an "original price," apply the discount to it and see if you get the given final price. Reversibility is the fastest self-check.
• How to stay calm. Percentages reward the calm reader who identifies the base first. If you feel rushed, take one breath and ask the single question that unlocks everything: "What is the whole here?"

16. Practice Questions

Try these on your own before reading the answers. Time yourself.

Easy

1. What is 30% of 200?
2. Convert 3/5 to a percentage.
3. 18 is what percent of 90?

Medium

4. A ₱1,500 phone is on sale at 20% off. What is the sale price?
5. A worker's daily wage rises from ₱610 to ₱671. What is the percentage increase?
6. 40% of a number is 52. What is the number?

Hard

7. A laptop is discounted 25%, then a further 20% off for a member. If the original price is ₱24,000, what is the final price?
8. After adding 12% VAT, a purchase totals ₱3,360. What was the price before VAT?

Challenge

9. In a class, 60% are girls. If there are 12 more girls than boys, how many students are in the class?
10. The price of cooking gas rose by 25%. By what percent must a household cut its usage to keep the same total spending?
11. A ₱4,500 appliance's price is first increased by 10%, then decreased by 10%. What is the final price, and is it higher or lower than ₱4,500?
12. Out of 2,500 examinees, 72% passed the Civil Service Exam. How many failed?

Answers and Explanations

1. 60. 0.30 × 200 = 60.
2. 60%. 3 ÷ 5 = 0.60 = 60%.
3. 20%. 18 ÷ 90 = 0.20 = 20%.
4. ₱1,200. 1500 × 0.80 = 1,200 (you pay 80%).
5. 10%. (671 − 610) ÷ 610 × 100 = 61 ÷ 610 × 100 = 10%.
6. 130. whole = 52 ÷ 0.40 = 130. Check: 0.40 × 130 = 52. ✓
7. ₱14,400. 24,000 × 0.75 = 18,000; 18,000 × 0.80 = 14,400. (Equivalent single discount: 0.75 × 0.80 = 0.60, so 40% off.)
8. ₱3,000. original = 3,360 ÷ 1.12 = 3,000. Check: 3,000 × 1.12 = 3,360. ✓
9. 60 students. Girls 60%, boys 40%, so girls exceed boys by 20% of the class, and that equals 12. Class = 12 ÷ 0.20 = 60. Check: girls = 36, boys = 24, difference = 12. ✓
10. 20%. decrease = 25 ÷ (100 + 25) × 100 = 25 ÷ 125 × 100 = 20%.
11. ₱4,455 — lower. 4,500 × 1.10 = 4,950; 4,950 × 0.90 = 4,455. Increase-then-equal-decrease always ends lower: here 1% lower (because 1.10 × 0.90 = 0.99).
12. 700. Failed = 100% − 72% = 28% of 2,500 = 0.28 × 2,500 = 700. (Or 2,500 − 0.72 × 2,500 = 2,500 − 1,800 = 700.)

17. Summary

Let's gather everything into a clear picture. A percentage is simply a number expressed as parts per hundred — the % sign means "÷ 100." Every basic problem runs on one relationship, part = percent × whole, rearranged three ways: multiply to find the part, and divide (with the part on top) to find the percent or the whole.

For changes, measure the change over the original value, and use multiplier thinking: ×(1 + r) to increase, ×(1 − r) to decrease. To reverse a change, divide by that same multiplier rather than multiplying back. When two changes stack, they don't simply add — apply them one after another, or use a + b + ab/100. And always keep percentage points (a plain subtraction) separate from percent change (a change divided by the original).

Above all, the exam mostly tests whether you can identify the whole (the base) and recognize the pattern from the wording. Nail those two habits and percentages become some of the easiest, fastest points on the entire test.

18. Cheat Sheet

Fraction ↔ percent quick table

Speed shortcuts: 10% = move decimal left once • 5% = half of 10% • 15% = 10% + 5% • 25% = ÷4 • 50% = ÷2 • X% of Y = Y% of X.

Traps to remember: wrong base • +20% then −20% ≠ original (it's −4%) • successive discounts aren't additive • percentage points ≠ percent change • "of" means multiply • reverse by dividing.

19. Frequently Asked Questions

Can a percentage be more than 100%? Yes. 100% is the whole thing once; anything that more than doubles goes above 100%. A sales jump from ₱10,000 to ₱30,000 is a 200% increase.

Why isn't a 20% increase cancelled by a 20% decrease? Because the decrease is taken from a bigger number. Twenty percent of the higher amount is more than twenty percent of the original, so you end up below where you started (4% below, for 20%).

Do I convert the percent to a decimal or a fraction — which is better? Whichever is faster for that problem. Decimals are great for calculator-style multiplying (0.12 × 2,000); fractions are great for "ugly" percents (37.5% = 3/8). Learn both and choose.

What's the difference between "percentage points" and "percent"? Percentage points measure the plain gap between two percentages (5% to 8% is 3 points). Percent change measures that gap relative to the starting percent (3 ÷ 5 = 60%). Exam questions deliberately test which one you'll report.

How do I "reverse" a discount to find the original price? Divide the discounted price by (1 − discount). If a 15% discount leaves ₱850, the original is 850 ÷ 0.85 = ₱1,000. Do not multiply by 1.15.

Is there a fast way to check my answer? Yes — reverse it. Apply the change to your answer and see if it reproduces the number the problem gave you. If it does, you're right.

20. Mastery Checklist

Before you move on, honestly check each box. If you can't tick it, revisit that section.

• ☐ I can explain what "percent" means in my own words (parts per hundred).
• ☐ I can convert quickly between percent, decimal, and fraction in both directions.
• ☐ I can find the part, the percent, and the whole using part = percent × whole.
• ☐ I always identify the whole/base before computing.
• ☐ I compute percentage change as change ÷ original × 100.
• ☐ I can increase and decrease using ×(1 + r) and ×(1 − r).
• ☐ I can reverse a change by dividing, not multiplying back.
• ☐ I know two successive changes aren't just added, and can use a + b + ab/100.
• ☐ I never confuse percentage points with percent change.
• ☐ I recognize the problem type from the wording before I calculate.
• ☐ I can solve a Civil Service–level percentage word problem and verify my answer.
• ☐ I feel calm and confident tackling percentage questions under time pressure.

When every box is ticked, you have genuinely mastered percentages — and you've built the foundation for profit and loss, interest, discounts, and data interpretation, which all stand on exactly what you just learned. Well done. On to the next topic.