Percentages: a quick recap
A percentage is a ratio out of 100. x% of a quantity = (x/100) × quantity. A percentage change is always worked out on the original value.
Marked price and discount
The marked price (MP) is the price printed on the tag. A discount is a reduction on the marked price, offered to attract customers.
Discount = marked price − selling price
Discount % = discount ÷ marked price × 100
Discount is always calculated on the marked price.
Example. A shirt marked ₹1200 is sold at a 15% discount. Discount = 15% of 1200 = ₹180, so the selling price = ₹1020.
Example. A bag marked ₹800 is sold for ₹680. Discount = ₹120, and discount % = 120/800 × 100 = 15%.
Profit and loss with overheads
A shopkeeper often spends extra money on repairs, transport or labour before selling. These overhead expenses are added to the cost price.
Example. Sunita buys an old scooter for ₹12,000, spends ₹1,500 on repairs and sells it for ₹15,120. Cost price = 12,000 + 1,500 = ₹13,500. Profit = ₹1,620, and profit % = 1620/13500 × 100 = 12%.
Profit and loss percentages are always calculated on the cost price.
Sales tax and GST
The government charges a tax when goods are sold. In India, most goods and services carry the Goods and Services Tax (GST), charged as a percentage of the selling price and added to the bill.
Example. A watch costs ₹2,000 and GST is 18%. Tax = 18% of 2,000 = ₹360, so the bill is ₹2,360.
Working backwards. If ₹1,180 includes 18% GST, the price before tax was 1180 ÷ 1.18 = ₹1,000. A common mistake is to take 18% of ₹1,180, which gives the wrong answer.
Simple interest versus compound interest
With simple interest, interest is paid only on the original principal every year: SI = P × R × T ÷ 100.
With compound interest, the interest earned each year is added to the principal, and next year's interest is calculated on this larger amount. You earn "interest on interest".
Example. ₹10,000 at 10% per year for 2 years.
Simple interest = 10,000 × 10 × 2 ÷ 100 = ₹2,000.
For compound interest, work year by year:
Year | Principal for the year | Interest at 10% | Amount at year end |
|---|---|---|---|
1 | ₹10,000 | ₹1,000 | ₹11,000 |
2 | ₹11,000 | ₹1,100 | ₹12,100 |
Compound interest = 12,100 − 10,000 = ₹2,100, which is ₹100 more than simple interest.
The compound interest formula
A = P (1 + R/100)ⁿ and CI = A − P
where A is the amount, P the principal, R the annual rate and n the number of years.
Example. ₹5,000 at 8% for 2 years: A = 5000 × (1.08)² = 5000 × 1.1664 = ₹5,832, so CI = ₹832.
Half-yearly compounding: use half the rate and twice the number of periods. ₹8,000 at 10% per year for 1 year, compounded half-yearly, gives A = 8000 × (1.05)² = ₹8,820.
Growth and depreciation
The same formula describes anything that grows by a fixed percentage each year, such as a population or bacteria. For something that loses value, such as a machine, use a minus sign:
Value after n years = P (1 − R/100)ⁿ
Example. A ₹50,000 machine depreciates by 10% a year. After 2 years it is worth 50,000 × (0.9)² = ₹40,500.
Summary
Discount is on the marked price; profit and loss are on the cost price including overheads.
GST is added to the selling price; to remove it, divide by (1 + rate/100).
Compound interest: A = P(1 + R/100)ⁿ; for half-yearly, halve R and double n.
Depreciation uses (1 − R/100)ⁿ.