Comparing Quantities — Class 8 Mathematics
1. Ratios and Percentages
A RATIO compares two quantities of the SAME unit. a:b read as 'a is to b.' Ratio has NO units. Percentage means 'per hundred' — a ratio with denominator 100. x% = x/100. To convert a ratio to percentage: (a/b)×100%. To convert percentage to ratio: x% = x:100 (simplify if possible).
Worked Examples
- Convert 3:4 to percentage: (3/4)×100 = 75%.
- 25% of 80 = (25/100)×80 = 20.
- Increase 200 by 15%: new value = 200×(115/100) = 230.
- Decrease 300 by 20%: new = 300×(80/100) = 240.
2. Profit and Loss
| Term | Formula |
|---|---|
| Profit | SP − CP (when SP > CP) |
| Loss | CP − SP (when CP > SP) |
| Profit % | (Profit/CP) × 100% |
| Loss % | (Loss/CP) × 100% |
| SP (given profit %) | CP × (100+P%)/100 |
| CP (given profit %) | SP × 100/(100+P%) |
Worked Examples
Example 1 — Basic: CP=₹600, SP=₹750. Profit = 150. Profit % = (150/600)×100 = 25%. Example 2 — Finding SP: CP=₹450, profit=12%. SP = 450×(112/100) = ₹504. Example 3 — Finding CP: SP=₹880, loss=12%. CP = 880×100/(100−12) = 880×100/88 = ₹1000.
3. Discount and Marked Price
Discount is always calculated on the MARKED PRICE (MP). Discount = MP × Discount% / 100. SP = MP − Discount = MP × (100−D%)/100. If discount% is given, net percentage is NOT simply (profit% − discount%). Calculate step by step.
Worked Example
An article is marked at ₹1200 and sold at a discount of 15%. Discount = 1200 × 0.15 = ₹180. SP = 1200 − 180 = ₹1020. If CP = ₹850, profit = 1020−850 = ₹170. Profit % = (170/850)×100 = 20%.
Successive Discounts
When MULTIPLE discounts are given (e.g., 20% + 10%): Do NOT simply add (20+10=30%). Each is applied sequentially on the REDUCED price. SP = MP × (100−D₁)/100 × (100−D₂)/100. For MP=₹1000, D₁=20%, D₂=10%: SP = 1000×0.80×0.90 = ₹720. Effective single discount = 28%, NOT 30%. 'Successive discounts are ALWAYS less than the sum of the individual discounts.'
4. Sales Tax, VAT, and GST
Tax is calculated as a PERCENTAGE of the SELLING PRICE (or value added). Final amount = SP × (100 + Tax%) / 100. In India, GST (Goods and Services Tax) replaced VAT and sales tax. Common GST slabs: 0%, 5%, 12%, 18%, 28%. 'GST is a consumption-based tax. It is charged at each stage of the supply chain, but the consumer bears the final burden.'
Worked Example
SP of a laptop = ₹45,000. GST = 18%. GST amount = 45,000×0.18 = ₹8,100. Total amount paid = 45,000+8,100 = ₹53,100.
5. Simple Interest (SI)
SI = (P × R × T) / 100. Amount A = P + SI = P(1 + RT/100). Where: P = Principal. R = Rate per annum (annual). T = Time in years.
Example: P=₹5000, R=8% p.a., T=3 years. SI = 5000×8×3/100 = ₹1200. Amount = ₹6200.
6. Compound Interest (CI)
In CI, interest is calculated on the PRINCIPAL PLUS accumulated interest. 'Interest earns interest — this is the EIGHTH WONDER of the world (attributed to Einstein).'
Formula
A = P(1 + R/100)^T. CI = A − P.
Example: P=₹8000, R=10% p.a., T=2 years. Year 1: Interest = ₹800. A = ₹8800. Year 2: Interest = ₹880. A = ₹9680. CI = 9680−8000 = ₹1680. Formula check: A = 8000(1.1)² = 8000×1.21 = ₹9680 ✓.
Compounding More Than Once a Year
Half-yearly: Rate becomes R/2 %. Time becomes 2T half-years. A = P(1 + R/200)^(2T). Quarterly: Rate = R/4 %. Time = 4T. A = P(1 + R/400)^(4T).
Example: P=₹10,000, R=8% p.a., T=1 year, compounded half-yearly. R/2 = 4%. T×2 = 2 half-years. A = 10000(1.04)² = 10000×1.0816 = ₹10,816. CI = ₹816.
Population Growth and Depreciation
Population growth: A = P(1 + R/100)^T (same formula as CI — compounding growth). Depreciation (machines, cars): Value after T years = P(1 − R/100)^T.
7. SI vs CI Comparison
| Aspect | Simple Interest | Compound Interest |
|---|---|---|
| Interest on | Principal ONLY | Principal + Accumulated Interest |
| Growth | Linear (same each year) | Exponential (grows faster) |
| Amount | P(1 + RT/100) | P(1 + R/100)^T |
| SI vs CI | CI > SI (for T > 1 year) | Difference = P(R/100)² for T=2 |
Example — Difference for 2 years: P=₹5000, R=6%. SI = 5000×6×2/100 = ₹600. CI = 5000[(1.06)²−1] = 5000×0.1236 = ₹618. Difference = ₹18. Formula: for 2 years, CI−SI = P(R/100)² = 5000(0.06)² = ₹18.
8. Common Mistakes
- Appying profit% on SP instead of CP: Profit% is ALWAYS on CP. '12% profit on SP of ₹560' means CP is different — use formula.
- Adding successive discount percentages directly: 20%+10% ≠ 30% effective. Calculate sequentially.
- Using the same formula for growth and depreciation: Growth: (1+R/100). Depreciation: (1−R/100). DON'T mix them.
9. AP Exam Focus
| Topic | Marks |
|---|---|
| Profit/Loss/Discount | 3-4 |
| Simple Interest | 2-3 |
| Compound Interest | 4-5 |
| Population growth / Depreciation | 3-4 |
Key Exam Tips
- For compound interest: WRITE the formula A = P(1+R/100)^T. Substitute. Calculate step by step. Show each year separately if T ≤ 3.
- 'Difference between CI and SI for 2 years' uses the shortcut: P(R/100)². State BOTH the full calculation AND the shortcut if applicable.
- For half-yearly/quarterly: clearly state how R and T change BEFORE substituting.
Quick Self-Test
- CP=₹400, SP=₹480. Profit%? (Answer: (80/400)×100 = 20%.)
- MP=₹2500, discount=12%. SP? (Answer: 2500×0.88 = ₹2200.)
- P=₹10000, R=10% p.a., T=2 yr, compounded annually. CI? (Answer: A=12100, CI=₹2100.)
- Machine worth ₹50000 depreciates at 20% p.a. Value after 2 years? (Answer: 50000×(0.8)² = ₹32000.)
- SI on ₹8000 at 5% for 4 years? (Answer: 8000×5×4/100 = ₹1600.)
