A percentage is just a ratio to 100, but that one idea powers exam marks, sale prices, bank interest and business profit. Learn the patterns behind every percentage problem.
Sale tags, exam results, bank interest, tax, tips and company profits all use percentages. Few topics in mathematics are used as often in adult life. A percentage is just a ratio to 100, and once you see that, the rest follows from a handful of patterns. This guide takes you from the basic idea to profit and loss, discounts and compound interest, with worked examples and real-life problems. It builds on Ratio & Proportion.
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. That is the same amount as the fraction 25/100, which is 1/4 in simplest form, and the decimal 0.25.
| Percentage | Fraction | Decimal |
|---|---|---|
| 1% | 1/100 | 0.01 |
| 5% | 1/20 | 0.05 |
| 10% | 1/10 | 0.1 |
| 12.5% | 1/8 | 0.125 |
| 20% | 1/5 | 0.2 |
| 25% | 1/4 | 0.25 |
| 50% | 1/2 | 0.5 |
| 60% | 3/5 | 0.6 |
| 75% | 3/4 | 0.75 |
Learn the first few rows by heart. They make mental calculation much faster, as the next section shows.
Turn the percentage into a decimal, then multiply.
Find 25% of 80. Convert: 25% = 0.25. Multiply: 0.25 × 80 = 20.
Mental shortcuts (build any percentage from easy ones):
A restaurant bill is Rs 3,500 and a 10% service charge is added. Charge = 0.10 × 3,500 = Rs 350, so the total is Rs 3,850. (Shortcut: 3,500 × 1.10 = 3,850.)
Write the part over the whole as a fraction, then multiply by 100: percentage = (part ÷ whole) × 100.
(a) 45 out of 180: (45 ÷ 180) × 100 = 25%.
(b) A student scores 68 out of 80 in a test: (68 ÷ 80) × 100 = 85%.
To find a percentage change, work out the change, divide by the original value, then multiply by 100. Percentage change = (change ÷ original) × 100.
A price rises from Rs 80 to Rs 100. Increase = 20. Percentage increase = (20 ÷ 80) × 100 = 25%.
A price falls from Rs 100 to Rs 80. Decrease = 20. Percentage decrease = (20 ÷ 100) × 100 = 20%.
Notice that the same Rs 20 gives a 25% rise one way and a 20% fall the other way. The base is different each time.
This is a fast way to increase or decrease an amount in one step. For an increase of r%, multiply by 1 + r/100. For a decrease of r%, multiply by 1 − r/100.
(a) Increase Rs 2,500 by 12%: multiplier 1.12, so 2,500 × 1.12 = Rs 2,800.
(b) Decrease Rs 4,000 by 35%: multiplier 0.65, so 4,000 × 0.65 = Rs 2,600.
This feels intuitive, but each percentage change is calculated from a different base value.
The fix: 100 increased by 20% becomes 120. But 120 decreased by 20% is 120 − 24 = 96, not 100. The second percentage is taken of the new, larger value, so equal and opposite percentage changes never cancel exactly. With multipliers: 1.20 × 0.80 = 0.96.
Sometimes you know the new value and the percentage change, and you must go back to the start. The new value stands for a percentage of the original: after a 20% increase it is 120% of the original, and after a 20% decrease it is 80% of the original. Divide by the multiplier.
(a) After a 20% increase, a price is Rs 120. Original = 120 ÷ 1.20 = Rs 100.
(b) A jacket is on sale at Rs 2,000 after a 20% discount. It is 80% of the marked price, so the original = 2,000 ÷ 0.80 = Rs 2,500.
(c) A price including 18% tax is Rs 5,900. The price before tax = 5,900 ÷ 1.18 = Rs 5,000.
This takes the percentage of the sale price, but the 20% was taken off the original price.
The fix: the discount was 20% of the original, so Rs 2,000 is 80% of the original. Divide by 0.80: 2,000 ÷ 0.80 = Rs 2,500. Check: 20% of 2,500 = 500, and 2,500 − 500 = 2,000. ✓
The average (or mean) of a set of numbers is their sum divided by how many numbers there are. Like a percentage, it describes a whole group with one number. (See Basic Data & Statistics for median, mode and range.)
Find the average of 12, 15, 18, 21 and 24. Sum = 90 and the count is 5, so the average = 90 ÷ 5 = 18.
A student scores 60%, 75% and 90% in three equal tests. Average = (60 + 75 + 90) ÷ 3 = 75%.
Now suppose the tests are different sizes: Test 1 is out of 20 (score 15, which is 75%) and Test 2 is out of 80 (score 56, which is 70%). You cannot just average 75% and 70%. Add the marks instead: (15 + 56) ÷ (20 + 80) = 71 ÷ 100 = 71%. The simple average of the two percentages would give 72.5%, which is wrong because the bigger test counts for more.
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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, there is a profit. If SP is lower, there is a loss. Both are always worked out as a percentage of the Cost Price, never the selling price.
CP = Rs 200 and SP = Rs 250. Profit = 250 − 200 = Rs 50. Profit% = (50 ÷ 200) × 100 = 25%.
CP = Rs 300 and SP = Rs 270. Loss = 300 − 270 = Rs 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 is the seller’s original investment. For CP = Rs 200 and SP = Rs 250, the correct profit% is 50 ÷ 200 × 100 = 25%, not 50 ÷ 250 × 100 = 20%.
(a) A trader buys a fan for Rs 800 and wants a 15% profit. SP = 800 × 1.15 = Rs 920.
(b) A shop sells a bag for Rs 540 at a 10% loss. SP is 90% of CP, so CP = 540 ÷ 0.90 = Rs 600.
A discount is a percentage reduction off a Marked Price (MP), the original listed price before any sale.
MP = Rs 500 and the discount is 15%. Discount = 0.15 × 500 = Rs 75. Selling price = 500 − 75 = Rs 425. (Or in one step: 500 × 0.85 = 425.)
Adding the percentages ignores that the second discount is taken from the already-reduced price.
The fix: multiply the multipliers. 0.80 × 0.90 = 0.72, so you pay 72% of the marked price, a total discount of 28%, not 30%. On Rs 1,000: after 20% off you pay Rs 800, then 10% off Rs 800 is Rs 80, leaving Rs 720.
Simple interest is calculated only on the original amount borrowed or invested, the Principal (P), using SI = (P × R × T) ÷ 100, where R is the yearly rate (%) and T is the time in years.
Principal = Rs 1,000, rate = 5% per year, time = 3 years. SI = (1,000 × 5 × 3) ÷ 100 = Rs 150. Total amount = 1,000 + 150 = Rs 1,150.
Rs 5,000 earns Rs 600 simple interest at 4% a year. For how long? From SI = PRT/100: T = (SI × 100) ÷ (P × R) = (600 × 100) ÷ (5,000 × 4) = 60,000 ÷ 20,000 = 3 years.
Unlike simple interest, compound interest is recalculated on the growing total each year, so interest earns interest. After n years: A = P × (1 + R/100)n.
Principal = Rs 1,000, rate = 5%, time = 2 years. Amount = 1,000 × 1.052 = Rs 1,102.50, so the compound interest is Rs 102.50. Simple interest over the same 2 years would be Rs 100. The extra Rs 2.50 comes from year 2’s interest being worked out on Rs 1,050, not on Rs 1,000.
Rs 10,000 at 8% for 3 years: simple interest gives 10,000 + 2,400 = Rs 12,400. Compound interest gives 10,000 × 1.083 = Rs 12,597.12. Compounding adds Rs 197.12 after only 3 years, and the gap keeps growing. (These are practice numbers, not real bank rates.)
The rule of 72: a handy estimate for how long money takes to double at a compound rate is 72 ÷ rate. At 8% that is about 72 ÷ 8 = 9 years. (Check: 1.089 ≈ 1.999, so the estimate is very close.)
A store marks a jacket at Rs 8,000 and discounts it 25%, so it sells for Rs 6,000. A shopper later resells it for Rs 7,500, a profit of Rs 1,500 on a cost of Rs 6,000, which is 25%. Meanwhile, Rs 200,000 saved at 4% simple interest for 3 years earns (200,000 × 4 × 3) ÷ 100 = Rs 24,000 in interest.
The percent sign is generally believed to have developed from the Italian phrase “per cento”, meaning “for a hundred”, which scribes gradually shortened over centuries into the two-circles-and-a-slash symbol we use today.
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1. 0.35 × 2,400 = Rs 840. (Shortcut: 10% = 240, so 30% = 720, and 5% = 120, giving 840.)
2. Increase = 6,000. (6,000 ÷ 40,000) × 100 = 15%.
3. The sale price is 85% of the marked price. 1,700 ÷ 0.85 = Rs 2,000.
1. 3 ÷ 8 = 0.375, so 37.5%.
2. Profit = 2,400. Profit% = (2,400 ÷ 12,000) × 100 = 20%.
3. 5,900 ÷ 1.18 = Rs 5,000.
4. Amount = 20,000 × 1.053 = 20,000 × 1.157625 = Rs 23,152.50. Interest = 23,152.50 − 20,000 = Rs 3,152.50.
Percentages become meaningful when learners use them on real numbers: prices on a shop shelf, their own exam marks, the interest in a savings plan. Start with the hundred square so the idea “out of 100” is visible. Teach the multiplier method early, because it makes reverse percentages and repeated changes much simpler. Ask “percent of what?” every time. This question prevents the most common errors (the wrong base) and also builds the habit of reading news and advertisements critically.
Percentages are among the most consistently used pieces of mathematics in adult life: reading a sale discount, comparing loan interest rates, working out a tip or a tax, understanding exam grades and reading a company’s profit margin. The skills above give you the tools to check every one of those numbers for yourself, instead of taking them on trust.
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