Fractions in Disguise

When you see "25%," your brain might think "25 out of something" and get confused. Remember: 25% ALWAYS means "25 out of 100," regardless of what the total quantity is.

Wrong: 25% of 80 = 25 (thinking "25 out of something")

Right: 25% of 80 = (25/100) × 80 = 20

Now that you understand percentages, let's see how shopkeepers and business owners use them.

The Key Concept:

• Cost Price (CP): What the shopkeeper pays to buy something
• Selling Price (SP): What the customer pays to buy it
• Profit: SP - CP (when SP > CP)
• Loss: CP - SP (when CP > SP)

But how do we compare profits fairly?

If I make a profit of ₹100, is that good or bad? It depends on the cost price! A ₹100 profit on a ₹1000 item (10% profit) is very different from a ₹100 profit on a ₹100 item (100% profit).

Profit % Formula:

Profit % = (Profit / CP) × 100

Loss % = (Loss / CP) × 100

Example Problem: A shopkeeper buys apples for ₹50 per kg (CP = 50) and sells them for ₹60 per kg (SP = 60). Find the profit percentage.

Profit = SP - CP = 60 - 50 = ₹10

Profit % = (10 / 50) × 100 = 0.2 × 100 = 20%

The shopkeeper makes a 20% profit.

When you borrow money, the lender charges you interest — a percentage payment for using their money.

Key Vocabulary:

• Principal (P): The original amount borrowed
• Rate of Interest (R): The percentage charged per year
• Time (T): How long the money is borrowed (in years)
• Simple Interest (SI): Interest calculated only on the principal, not on accumulated interest

Why is it called "Simple" Interest?

Because the interest amount stays the same each year. If you borrow ₹100 at 10% per year, you pay ₹10 interest every year—not ₹10, then ₹11, then ₹12.1, etc.

Simple Interest Formula:

SI = (P × R × T) / 100

Amount = P + SI

Example Problem: You borrow ₹1000 from a bank at 5% per annum for 2 years. Find the simple interest and total amount to be paid.

P = 1000, R = 5%, T = 2 years

SI = (1000 × 5 × 2) / 100 = 10000 / 100 = ₹100

Total Amount = 1000 + 100 = ₹1100

If interest is 5% "per annum" (per year) but you only borrow for 6 months, you don't pay 5% interest! You pay half of 5% = 2.5%.

Convert months to years: 6 months = 6/12 = 0.5 years

SI = (P × 5 × 0.5) / 100

In simple interest, you always pay interest on the original principal. But banks often use compound interest, where each year's interest gets added to the principal, and next year's interest is calculated on this larger amount.

Understanding the Difference:

Simple Interest Example (5% per year on ₹1000):

• Year 1: Interest = ₹50, Total = ₹1050
• Year 2: Interest = ₹50 (still 5% of 1000), Total = ₹1100
• Year 3: Interest = ₹50, Total = ₹1150

Compound Interest Example (5% per year on ₹1000):

• Year 1: Interest = 5% of 1000 = ₹50, Total = ₹1050
• Year 2: Interest = 5% of 1050 = ₹52.50, Total = ₹1102.50
• Year 3: Interest = 5% of 1102.50 = ₹55.13, Total = ₹1157.63

See the difference? Compound interest gives you more money because you earn "interest on interest." That's why compound interest is better for savers but worse for borrowers!

Compound Interest Formula:

Amount = P × (1 + R/100)^T

CI = Amount - P

Example Problem: You deposit ₹1000 in a savings account at 10% per annum for 2 years (compounded annually).

P = 1000, R = 10%, T = 2 years

Amount = 1000 × (1 + 10/100)² = 1000 × (1.1)²

Amount = 1000 × 1.21 = ₹1210

CI = 1210 - 1000 = ₹210

Students often calculate: 1000 × (1 + 10/100) = 1000 × 1.1 = 1100 and forget to square it for 2 years!

Wrong: Amount = 1000 × 1.1 = 1100

Right: Amount = 1000 × (1.1)² = 1000 × 1.21 = 1210