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Percentages express a number as a fraction of 100. The word comes from the Latin "per centum" — per hundred. All percentage problems reduce to one of three basic types: finding a percentage of a number, finding what percentage one number is of another, and finding percentage change between two values.
Percentage Formulas
X% of Y =
Y × (X ÷ 100)
20% of 150 = 30
X is what % of Y =
(X ÷ Y) × 100
30 is 20% of 150
% Change =
((New − Old) ÷ Old) × 100
150→180 = +20%
Current salary: $65,000 | Raise offered: 8%
Raise amount = $65,000 × 0.08 = $5,200. New salary = $65,000 + $5,200 = $70,200. Alternatively: $65,000 × 1.08 = $70,200 directly (multiplying by 1 + decimal rate).
To find what percentage X is of Y: (X ÷ Y) × 100. Example: 45 is what percent of 180? (45 ÷ 180) × 100 = 25%. This is useful for calculating test scores (got 36 out of 40 questions right: 36/40 × 100 = 90%), discounts (saved $24 on a $120 item: 24/120 × 100 = 20% off), tip verification (tip of $15 on $75 bill: 15/75 × 100 = 20%), and market share calculations.
To find X% of Y: multiply Y by X, then divide by 100. Or equivalently: Y × (X/100). Example: 15% of $240 = 240 × 0.15 = $36. Mental math shortcut: to find 10%, move the decimal left one place ($240 → $24). For 15%, find 10% ($24) then add half of that ($12): $24 + $12 = $36. For 20%: double the 10% figure ($48). For 5%: halve the 10% figure ($12).
Percentage change = ((New Value – Old Value) ÷ Old Value) × 100. If the result is positive, it is an increase; negative means decrease. Example: price rose from $80 to $92: ((92 – 80) ÷ 80) × 100 = (12 ÷ 80) × 100 = 15% increase. Price dropped from $200 to $170: ((170 – 200) ÷ 200) × 100 = (–30 ÷ 200) × 100 = –15% decrease. Common uses: salary raises, stock price changes, population growth, inflation.
Percent difference compares two values without specifying a "before" and "after" — it treats both values equally. Formula: |Value1 – Value2| / ((Value1 + Value2) / 2) × 100. Example: comparing 80 and 100: |80 – 100| / ((80 + 100) / 2) × 100 = 20 / 90 × 100 = 22.2% difference. Unlike percentage change, percent difference is symmetric — swapping the two values gives the same result. Use percentage change when one value is a baseline; use percent difference when comparing two peers.
To find the original value before a percentage was applied: Original = Current Value ÷ (1 + %/100) for an increase, or ÷ (1 – %/100) for a decrease. Example: A price is $130 after a 30% markup — what was the original? $130 ÷ 1.30 = $100. A price is $68 after a 15% discount — what was the original? $68 ÷ 0.85 = $80. Common mistake: subtracting the same percentage back from the marked-up price — $130 – 30% of $130 = $91, not $100. Always divide, never subtract, to reverse a percentage.
To find the sale price after a discount: Sale Price = Original Price × (1 – Discount%/100). Example: 30% off $85 = $85 × 0.70 = $59.50. To find how much you save: Savings = Original × (Discount%/100) = $85 × 0.30 = $25.50. To find what discount percentage a sale represents: Discount% = (Original – Sale Price) ÷ Original × 100. Item was $120, now $90: (120 – 90) ÷ 120 × 100 = 25% off.
Percentage points (pp) measure the absolute arithmetic difference between two percentages. Percent measures the relative change. If unemployment rises from 4% to 6%: it rose by 2 percentage points (6 – 4 = 2 pp) but by 50% relative to the original (2/4 × 100 = 50%). Politicians and media often conflate these. "Interest rates rose 1%" could mean 1 percentage point (from 5% to 6%, a 20% relative increase) or 1% of 5% (from 5% to 5.05%, a 1% relative increase). Context is everything.
Percentages are everywhere: Sales tax (8% of purchase price added at register), tipping (15–20% of restaurant bill), discounts (30% off sticker price), investment returns (7% annual return means $1,000 becomes $1,070), mortgage rates (6.5% APR means 0.542%/month on the outstanding balance), body fat percentage (essential fat is ~3–5% for men, 10–13% for women), and test scores (80% = 80 out of 100 correct). Understanding percentages helps you quickly evaluate whether a "deal" is actually a deal and how financial products truly compare.
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