8.2 Percentages

8.2 Percentages

Quick Answer: A percent is a special fraction with denominator 100. Convert to a decimal or fraction, then multiply by the whole to find the part—or rearrange the same relationship to find the whole or the percent.

Percentages appear throughout the USTET Mathematics subtest and also in Science and Mental Ability word problems. If you can translate “what percent,” “percent of,” and “percent change” into one equation, you will clear most items quickly.

Percent ↔ Decimal ↔ Fraction

Rules of thumb:

• Percent → decimal: divide by 100 (move the decimal point two places left).
• Decimal → percent: multiply by 100 (move two places right).
• Percent → fraction: write over 100 and simplify. 35% = 35/100 = 7/20.

Worked example: Write 62.5% as a fraction.

62.5/100 = 625/1000 = 5/8.

The Core Relationship

$$\text{Part} = \text{Percent (as decimal)} \times \text{Whole}$$

Equivalently: Part / Whole = Percent (as a decimal).

Finding the part

Worked example: What is 18% of 250?

0.18 × 250 = 45.

Finding the whole

Worked example: 36 is 15% of what number?

Let W be the whole. 0.15W = 36 → W = 36/0.15 = 240.

Finding the percent

Worked example: 42 is what percent of 70?

(42/70) × 100% = 0.6 × 100% = 60%.

Percent Increase and Decrease

$$\text{Percent change} = \frac{\text{new} - \text{original}}{\text{original}} \times 100\%$$

A positive result is an increase; a negative result is a decrease (often reported as a positive “percent decrease”).

Worked example — increase: A fee rises from PHP 500 to PHP 575.

Change = 75; (75/500)×100% = 15% increase.

Worked example — decrease: A price falls from PHP 800 to PHP 680.

Change = 120; (120/800)×100% = 15% decrease.

Notice both examples used a 15% change but different directions—always identify whether the problem asks for increase, decrease, or the new amount.

Finding a new amount after a percent change

• After a p% increase: new = original × (1 + p/100).
• After a p% decrease: new = original × (1 − p/100).

Worked example: A shirt costs PHP 1,200. After a 20% discount, what is the sale price?

Sale price = 1200 × (1 − 0.20) = 1200 × 0.80 = PHP 960.

Alternatively: discount amount = 0.20 × 1200 = 240; 1200 − 240 = 960.

Tax, Tip, Commission, and Markup

These are percent-of-whole problems with everyday labels.

Worked example — tax: An item costs PHP 2,500 before 12% VAT. Find the total with tax.

Total = 2500 × 1.12 = PHP 2,800.

Worked example — commission: A salesperson earns 8% of PHP 45,000 in sales.

Commission = 0.08 × 45000 = PHP 3,600.

Worked example — markup: A store buys an item for PHP 400 and marks it up 35%.

Selling price = 400 × 1.35 = PHP 540.

Successive Percent Changes

Applying 10% then another 10% is not a 20% total change on the original, because the second change acts on the new base.

Worked example: A value of 200 increases by 10%, then decreases by 10%. What is the final value?

After +10%: 200 × 1.10 = 220. After −10%: 220 × 0.90 = 198.

Final value 198 is 1% below the original 200—not a return to 200.

Worked example — successive discounts: An item priced at PHP 5,000 gets 20% off, then an additional 10% off the reduced price.

After 20%: 5000 × 0.80 = 4000. After 10%: 4000 × 0.90 = 3600.

Equivalent single factor: 0.80 × 0.90 = 0.72 → 28% total discount relative to the original, not 30%.

Reverse Percent Problems

Sometimes you know the result after a percent change and must recover the original.

Worked example: After a 25% increase, a score is 90. What was the original score?

90 = original × 1.25 → original = 90/1.25 = 72.

Worked example: A sale price of PHP 840 reflects a 30% discount. Find the original price.

840 = original × 0.70 → original = 840/0.70 = PHP 1,200.

Percent of a Percent and Compound Contexts

Sometimes a problem asks for a percent of a quantity that is already a percent of something else. Translate each clause in order.

Worked example: 40% of a class of 50 students passed a quiz. Of those who passed, 30% scored at least 90. How many scored at least 90?

Passed = 0.40 × 50 = 20 students. High scorers = 0.30 × 20 = 6 students.

Do not multiply 40% × 30% × 50 in one blur without knowing what each percent modifies—here the 30% applies only to the passers, which happens to equal 0.40 × 0.30 × 50 = 6, but only because the second percent is a subset of the first group.

Worked example — exam score: A student answered 48 of 60 items correctly. What percent score is that?

(48/60) × 100% = 80%.

If a later curve adds 5 percentage points to every score, the new reported score is 85%—that is an additive change to the percent, not a 5% increase of 80 (which would be 84%). Read the wording carefully: “5 percentage points” ≠ “5 percent of the score.”

Common Traps on Timed Items

Worked example — 200% increase: A population of 400 increases by 200%.

New = 400 × (1 + 2) = 1,200 (triple the original, not double).

Exam Strategy Checklist

1. Underline whether you need the part, the whole, or the percent.
2. Convert the percent to a decimal before multiplying.
3. For change problems, always divide by the original, not the new value (unless the problem explicitly asks for a different base).
4. For successive changes, multiply the factors; do not add the percents.
5. Sanity-check: 10% of a number is one-tenth; 5% is half of that—use these anchors to estimate.
6. Distinguish percentage points from percent of when scores or rates are adjusted.

Percent fluency also feeds Mental Ability speed items and Science data questions that quote percent composition or growth. Keep the part–whole equation written in the margin on harder stems so you do not lose the base mid-calculation.