1. Chapter at a Glance
"50% off!", "GST 18%", "Pass percentage 94.5%", "Interest 7% per year" — percentages are everywhere. A percentage is simply a fraction in disguise: a fraction whose denominator is 100. This chapter shows how to convert between fractions, decimals and percentages, find percentages of quantities, calculate percentage increase and decrease, and solve problems on profit, loss, discount, tax and simple interest.
- The meaning of per cent and conversions among fractions, decimals and percentages.
- Finding a percentage of a quantity and the whole from a part.
- Percentage increase and decrease.
- Profit, loss, discount and GST.
- Simple interest.
2. What is a Percentage?
"Per cent" means "per hundred" and is written %. 27% means 27 out of 100 = 27/100 = 0.27.
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/8 | 0.125 | 12.5% |
| 1/3 | 0.333… | 33⅓% |
| 1/10 | 0.1 | 10% |
- Fraction → percentage: multiply by 100. 7/20 × 100 = 35%.
- Percentage → fraction: write over 100 and simplify. 45% = 45/100 = 9/20.
- Decimal ↔ percentage: move the point two places. 0.068 = 6.8%; 125% = 1.25.
3. Percentage of a Quantity and Finding the Whole
(a) Find 15% of ₹840. (b) 30% of a number is 54. Find the number. (c) What percent is 18 of 72?
Solution: (a) 0.15 × 840 = ₹126. (b) 54 × 100/30 = 180. (c) 18/72 × 100 = 25%.
4. Percentage Increase and Decrease
- Price rises from ₹400 to ₹460: increase = 60; 60/400 × 100 = 15% increase.
- To increase a quantity by r%, multiply by (1 + r/100). To decrease by r%, multiply by (1 − r/100).
- A 20% increase followed by a 20% decrease does not return to the start: 100 → 120 → 96 (a 4% net decrease).
5. Profit, Loss, Discount and GST
| Term | Meaning / formula |
|---|---|
| Cost price (CP) | Price at which an article is bought |
| Selling price (SP) | Price at which it is sold |
| Profit | SP − CP (when SP > CP); Profit % = Profit/CP × 100 |
| Loss | CP − SP (when CP > SP); Loss % = Loss/CP × 100 |
| Marked price (MP) | Price printed on the tag |
| Discount | MP − SP; Discount % = Discount/MP × 100 |
| GST | Tax added on the selling price: bill = SP × (1 + GST%/100) |
A shopkeeper buys a fan for ₹1,600 and sells it for ₹1,840. Find the profit percent.
Solution: Profit = 240. Profit % = 240/1600 × 100 = 15%.
A shirt is marked ₹1,200 with a 25% discount, and 12% GST is charged on the discounted price. Find the bill.
Solution: SP = 1200 × 0.75 = 900. Bill = 900 × 1.12 = ₹1,008.
6. Simple Interest
When money P (principal) is borrowed or deposited at R% per year for T years, the simple interest is:
Find the simple interest on ₹15,000 at 8% per year for 3 years.
Solution: SI = 15000 × 8 × 3 / 100 = ₹3,600; amount = ₹18,600.
7. Solved Problems (NCERT Style)
A village had 5,000 people. The population increased by 10% one year and 10% the next. Find the population after two years.
Solution: 5000 × 1.1 × 1.1 = 6,050 (not 6,000 — the second increase is on the larger number).
A cycle bought for ₹4,500 is sold at a loss of 12%. Find the selling price.
Solution: SP = 4500 × 0.88 = ₹3,960.
By selling a table for ₹2,760, a carpenter gains 15%. Find the cost price.
Solution: SP = 1.15 × CP → CP = 2760 ÷ 1.15 = ₹2,400.
8. Master Formula Table
| Quantity | Formula |
|---|---|
| x% of Q | xQ/100 |
| Whole from part | part × 100 / x |
| Percentage change | change / original × 100 |
| Increase by r% | × (1 + r/100) |
| Decrease by r% | × (1 − r/100) |
| Profit % / Loss % | on cost price |
| Discount % | on marked price |
| SP with gain g% | CP × (100 + g)/100 |
| Simple interest | PRT/100 |
9. Master Summary Table
| Topic | Key idea | Exam focus |
|---|---|---|
| Percentage | Fraction out of 100 | Conversions |
| Percent of quantity | Multiply by x/100 | Marks, money |
| Change | Base = original value | Increase/decrease |
| Commerce | Profit/loss on CP; discount on MP | Bills with GST |
| Interest | SI = PRT/100 | Amount after T years |
10. Mind Map
- Per hundred
- Fraction ↔ decimal ↔ %
- % of quantity
- Whole from part
- % change
- Profit, loss
- Discount, GST
- SI = PRT/100
- Amount
11. Tips and Tricks
- 10% is easy (divide by 10); build others from it: 15% = 10% + 5%.
- x% of y = y% of x: 8% of 50 = 50% of 8 = 4.
- Use multipliers for successive changes: +10% then −10% → 1.1 × 0.9 = 0.99 (1% loss).
- Calculating profit % on the selling price instead of the cost price.
- Calculating discount on the selling price instead of the marked price.
- Adding successive percentages directly.
12. Important One-liners
- Per cent means per hundred.
- Profit and loss percentages are calculated on the cost price.
- Discount is calculated on the marked price.
- GST is added to the selling price.
- Simple interest = PRT/100.
- Successive percentage changes multiply, they do not add.
- x% of y equals y% of x.
13. Exam-pattern Questions with Detailed Answers
Express 3/8 as a percentage.
- 37.5%
- 38%
- 3.75%
- 30%
Show answer
3/8 × 100 = 37.5%.
Answer: (A)
Find 12% of 250.
Show answer
0.12 × 250 = 30.
Answer: 30
A student scored 42 out of 60. What percentage is that?
Show answer
42/60 × 100 = 70%.
Answer: 70%
The price of milk rose from ₹52 to ₹60 per litre. Find the percentage increase.
Show answer
8/52 × 100 ≈ 15.38%.
Answer: About 15.4%
A TV marked ₹32,000 is sold at ₹28,800. Find the discount percent.
Show answer
Discount = 3200; 3200/32000 × 100 = 10%.
Answer: 10%
A price is increased by 25% and then decreased by 20%. The net change is:
- 5% increase
- No change
- 5% decrease
- 45% increase
Show answer
1.25 × 0.80 = 1.00 → no change.
Answer: (B)
Find the simple interest on ₹8,000 at 6.5% per annum for 4 years.
Show answer
8000 × 6.5 × 4 / 100 = 2080.
Answer: ₹2,080
In an election, a candidate got 62% of the votes and won by 8,640 votes over the only other candidate. Find the total votes.
Show answer
Difference = 62% − 38% = 24% of total = 8640 → total = 36,000.
Answer: 36,000
A shopkeeper marks goods 40% above cost and allows a 15% discount. Find his profit percent.
Show answer
Take CP = 100. MP = 140. SP = 140 × 0.85 = 119. Profit = 19%.
Answer: 19%
At what rate of simple interest will a sum double itself in 8 years?
Show answer
SI = P (since the amount is 2P). P = P × R × 8/100 → R = 12.5%.
Answer: 12.5% per annum
A man sells two phones for ₹9,900 each, gaining 10% on one and losing 10% on the other. Find his overall gain or loss.
Show answer
CP₁ = 9900/1.1 = 9000; CP₂ = 9900/0.9 = 11000. Total CP = 20,000; total SP = 19,800. Loss = ₹200 = 1%.
Answer: Loss of ₹200 (1%)
After a 12% GST, a bill comes to ₹5,600. What was the price before tax?
Show answer
Price × 1.12 = 5600 → price = ₹5,000.
Answer: ₹5,000
14. Quick Revision Summary
- % = per hundred; convert by × or ÷ 100.
- % change uses the original value as base; successive changes multiply.
- Profit/loss % on CP; discount % on MP; GST added to SP.
- SI = PRT/100; amount = P + SI.