A percentage is just a ratio to 100 — but that single idea powers school grades, sale prices, bank interest, and business profit calculations everywhere.
This is the final topic in our Number Sense strand — and arguably the most useful in daily life. Percentages tie directly back to Ratio & Proportion, then branch out into averages, profit and loss, discounts, and interest — the exact math behind shopping, banking, and business.
A percentage is a ratio that always compares a part to a whole of exactly 100. The symbol % literally means "per hundred."
25% means 25 out of every 100 — the same amount as the fraction 25/100, or 1/4 in simplest form.
| Percentage | Fraction | Decimal |
|---|---|---|
| 25% | 1/4 | 0.25 |
| 50% | 1/2 | 0.5 |
| 60% | 3/5 | 0.6 |
| 75% | 3/4 | 0.75 |
To convert a decimal to a percentage, multiply by 100. To convert a percentage to a decimal, divide by 100.
Find 25% of 80. Convert to a decimal: 25% = 0.25. Multiply: 0.25 × 80 = 20.
45 out of 180 — what percentage is that? (45÷180) × 100 = 25%.
A price rises from $80 to $100. Increase = 100−80=20. Percentage increase = (20÷80)×100 = 25%.
A price falls from $100 to $80. Decrease = 100−80=20. Percentage decrease = (20÷100)×100 = 20%.
This feels intuitive, but percentage changes are always calculated from a different base value each time.
The fix: 100 increased by 20% becomes 120. But 120 decreased by 20% is 120 − 24 = 96, not back to 100. The second percentage is calculated on the new, larger value, not the original — so equal opposite percentage changes never fully cancel out.
After a 20% increase, a price is $120. What was the original price? $120 represents 120% of the original. Original = 120 ÷ 1.20 = $100.
The average (or mean) of a set of numbers is their sum divided by how many numbers there are. It's closely tied to percentages, since both describe a single representative value standing in for a whole group.
Find the average of 12, 15, 18, 21, 24. Sum = 12+15+18+21+24 = 90. Count = 5. Average = 90÷5 = 18.
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Every sale has a Cost Price (CP, what the seller paid) and a Selling Price (SP, what the buyer pays). If SP is higher, it's a profit; if lower, it's a loss — and both are always calculated as a percentage of the Cost Price, never the selling price.
CP=$200, SP=$250. Profit = 250−200=$50. Profit% = (50÷200)×100 = 25%.
CP=$300, SP=$270. Loss = 300−270=$30. Loss% = (30÷300)×100 = 10%.
Dividing by the selling price instead of the cost price gives a smaller, incorrect percentage.
The fix: profit and loss percentages are always calculated against the Cost Price, since that's the seller's original investment. For CP=$200, SP=$250: the correct profit% is 50÷200×100=25%, not 50÷250×100=20% (which incorrectly divides by SP).
A discount is a percentage reduction off a Marked Price (MP) — the original listed price before any sale.
MP=$500, discount=15%. Discount amount = 0.15×500=$75. Selling price = 500−75=$425.
Simple interest is calculated only on the original amount borrowed or invested — the Principal (P) — using the formula SI = (P × R × T) ÷ 100, where R is the annual rate and T is time in years.
Principal=$1,000, Rate=5% per year, Time=3 years. SI = (1000×5×3)÷100 = $150. Total amount owed = 1000+150=$1,150.
Unlike simple interest, compound interest is recalculated on the growing total each year — interest earns interest.
Principal=$1,000, Rate=5%, Time=2 years. Compound amount = 1000×(1.05)² = $1,102.50, giving compound interest of $102.50 — compared to just $100 in simple interest over the same 2 years. The extra $2.50 comes from year 2's interest being calculated on $1,050, not the original $1,000.
A store marks a jacket at $80, discounts it 25% for a sale ($60), then a shopper resells it at $75 — a profit of $15 on their $60 cost, or 25% profit. Meanwhile, $2,000 saved in a bank account at 4% simple interest for 3 years earns (2000×4×3)÷100=$240 in interest.
The percent symbol (%) evolved from the Italian "per cento" ("for a hundred"), gradually abbreviated over centuries from "P c°" into the two-circle-and-slash symbol used today.
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Percentages are the most consistently used piece of math in adult life — reading a sale discount, comparing loan interest rates, calculating a tip, understanding exam grades, or reading a company's profit margins. This topic — and the Number Sense strand it completes — hands you the tools to check every one of those numbers for yourself, rather than taking them on faith.
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