🔢 Class 8 Mathematics — Comparing Quantities
Interactive Visual Revision Notes
Aligned with the Class 8 Mathematics syllabus. Independently created study material. Not affiliated with or endorsed by NCERT.
🌎 Real-World Connection: You see a “30% OFF” sale at a shop. You buy a shirt that was marked ₹800. The discount = 30% of 800 = ₹240. You pay ₹560. Later you deposit ₹5,000 in a bank at 6% interest. After one year you earn ₹300 as Simple Interest. When interest is added back and earns more interest the next year, that is Compound Interest. Comparing quantities — profit, loss, discounts, and interest — happen in every market and bank in the world!
🔢 Topic 1: What Is Percentage?
Percent means "out of 100."
The symbol for percent is %.
25% means 25 out of 100.
50% means half of something.
100% means the whole thing.
We use % to compare amounts easily.
🔢 Topic 2: Fractions to Percentages
To change a fraction to %, multiply by 100.
Example: 3/4 = 3/4 × 100 = 75%.
Example: 1/5 = 1/5 × 100 = 20%.
Example: 2/5 = 2/5 × 100 = 40%.
Always multiply the fraction by 100.
Then write the % sign after the number.
Percentage = (Fraction × 100) %
🔢 Topic 3: Percentages to Fractions
To change % to a fraction, divide by 100.
Example: 25% = 25/100 = 1/4.
Example: 60% = 60/100 = 3/5.
Example: 12.5% = 12.5/100 = 1/8.
Always simplify the fraction you get.
Remove the % sign and put 100 below.
Fraction = Percentage ÷ 100
🔢 Topic 4: Profit and Loss
Cost Price (CP) = what you pay to buy.
Selling Price (SP) = what you get when you sell.
If SP is more than CP, you earn profit.
Profit = SP − CP.
If SP is less than CP, you face loss.
Loss = CP − SP.
🔢 Topic 5: Profit and Loss Percentage
Profit % tells how much you gained per 100.
Loss % tells how much you lost per 100.
Profit % = (Profit ÷ CP) × 100.
Loss % = (Loss ÷ CP) × 100.
Always use CP in the bottom (denominator).
Never use SP to find profit % or loss %.
Profit % = (Profit ÷ CP) × 100
🔢 Topic 6: Discount
Marked Price (MRP) = price written on tag.
Discount = amount taken off from MRP.
Discount = MRP − SP.
Discount % = (Discount ÷ MRP) × 100.
SP = MRP − Discount.
Shops give discount to attract buyers.
SP = MRP − Discount
🔢 Topic 7: Sales Tax and GST
Tax is extra money the government takes.
Tax is added on top of selling price.
GST = Goods and Services Tax.
Bill amount = SP + Tax.
Tax = (Tax Rate ÷ 100) × SP.
You pay more than the price tag shows.
Bill = SP + (Tax% ÷ 100) × SP
🔢 Topic 8: Simple Interest
Principal (P) = money you borrow or save.
Rate (R) = percent charged per year.
Time (T) = number of years.
SI = P × R × T ÷ 100.
Interest stays the same each year.
Amount = P + SI.
SI = P × R × T ÷ 100
🔢 Topic 9: Compound Interest
In CI, interest is added to principal each year.
Next year, you earn interest on a bigger amount.
So the interest grows each year.
CI gives you more money than SI.
Banks use compound interest.
CI = Amount − Principal.
🔢 Topic 10: Compound Interest Formula
A = P × (1 + R/100)ⁿ.
A = total amount after n years.
P = principal (starting money).
R = rate of interest per year.
n = number of years.
CI = A − P.
A = P × (1 + R/100)ⁿ
🎨 Section 2 — Easy Illustrations
Picture 1: Percentage Bar
The bar has 100 equal parts. 40 shaded = 40%. The rest = 60%.
Picture 2: Profit and Loss Flowchart
Start with CP, then compare with SP. More = Profit. Less = Loss.
Picture 3: Discount Visual
MRP is the tag price. Discount is removed. You pay the lower SP.
Picture 4: Simple vs Compound Interest
SI stays the same each year. CI grows bigger because interest earns interest.
Picture 5: Compound Interest Growth Curve
SI grows in a straight line. CI curves upward because each year adds more interest.
✏️ Section 3 — Worked Examples
Example 1: Finding Profit Percentage
Question: A shopkeeper buys a bag for ₹600. He sells it for ₹750. Find the profit %.
1CP = ₹600 (cost price).
2SP = ₹750 (selling price).
3 SP > CP, so there is a profit.
4 Profit = SP − CP = 750 − 600 = ₹150.
5 Profit % = (Profit ÷ CP) × 100.
6 Profit % = (150 ÷ 600) × 100 = 25%.
Profit = ₹150 | Profit % = 25%
Always divide profit by CP, not by SP!
Example 2: Compound Interest Calculation
Question: Find compound interest on ₹8000 at 10% per year for 2 years.
1P = ₹8000, R = 10%, n = 2 years.
2 Use A = P × (1 + R/100)ⁿ.
3 A = 8000 × (1 + 10/100)².
4 A = 8000 × (110/100)².
5 A = 8000 × (11/10)² = 8000 × 121/100.
6 A = ₹9680.
7 CI = A − P = 9680 − 8000 = ₹1680.
Amount = ₹9680 | CI = ₹1680
Check: SI would be ₹1600. CI is ₹80 more!
Example 3: Discount on a Shirt
Question: A shirt is marked at ₹1200. The shop gives 15% discount. Find the SP and discount amount.
1MRP = ₹1200 (marked price).
2 Discount rate = 15%.
3 Discount = 15% of ₹1200.
4 Discount = (15/100) × 1200 = ₹180.
5 SP = MRP − Discount = 1200 − 180.
6 SP = ₹1020.
Discount = ₹180 | SP = ₹1020
SP = MRP − Discount. Always!
📊 Section 4 — Cheat Sheet
Term
What It Means
Key Formula
Percentage
Out of 100
(Value/Total) × 100
Fraction to %
Multiply by 100
3/4 × 100 = 75%
% to Fraction
Divide by 100
25% = 25/100 = 1/4
Profit
SP more than CP
Profit = SP − CP
Loss
SP less than CP
Loss = CP − SP
Profit %
Gain per 100 of CP
(Profit ÷ CP) × 100
Loss %
Loss per 100 of CP
(Loss ÷ CP) × 100
Discount
Price cut from MRP
MRP − SP
Sales Tax
Government charge on SP
SP + Tax on SP
SI
Interest on original only
P × R × T ÷ 100
CI
Interest on growing amount
A = P(1+R/100)ⁿ
CI Amount
Principal + CI
CI = A − P
📄 Section 5 — Take-Home Handout
📝 A. Fill in the Blanks
1. Percent means out of _________.
2. Profit = SP − _________.
3. Discount is given on the _________ price.
4. SI stands for _________ Interest.
5. In CI, interest is added to the _________ each year.
✓ B. True or False
1. 50% means half of something. _______
2. Profit % is found by dividing profit by SP. _______
3. Discount means price goes up. _______
4. CI gives more money than SI over time. _______
5. GST is a tax added on selling price. _______
✏️ C. Solve These
1. A toy costs ₹400. It is sold for ₹340. Find the loss and loss %.
Show your work here
2. Find the compound interest on ₹5000 at 8% for 2 years.
Show your work here
🔗 D. Match the Columns
Column A
Column B
1. SP − CP
a. Discount
2. MRP − SP
b. Simple Interest
3. P × R × T ÷ 100
c. Percentage
4. Out of 100
d. Profit
🎨 E. Draw and Compare
Draw a bar chart showing SI and CI on ₹2000 at 10% for 3 years. Which gives more money?
Draw your comparison here
✅ Detailed Solutions — Section 5 Handout
A. Fill in the Blanks — Answers
1. Percent means out of 100. (Per = for, cent = 100.)