Comparing Quantities in Class 8 Mathematics teaches how to compare numerical values using ratios, percentages, profit and loss, discounts, GST, and compound interest. Unlike simple interest, compound interest reinvests accrued interest back into the principal amount periodically, accelerating growth according to \(A = P\left(1 + \frac{R}{100}\right)^n\).
High-Yield Chapter Key Takeaways
- Ratio to percentage conversion: (a / b) * 100%.
- Discount = Marked Price (MP) - Selling Price (SP); Discount% is always based on MP.
- Profit% and Loss% are always calculated strictly on the Cost Price (CP).
- GST is an indirect destination tax applied to the final discounted sale price.
- Compounded Annually: A = P(1 + R/100)^n; Compounded Half-Yearly: A = P(1 + (R/2)/100)^(2n).
1. Ratios and Percentages
A ratio is a mathematical comparison of two quantities of the same kind having the same units. A percentage is a special ratio where the denominator is always 100.
| Comparison Concept | Mathematical Notation | Applied Example |
|---|---|---|
| Ratio Representation | \(\frac{a}{b} = a : b\) | Speed ratio: 15 km/h to 30 km/h is \(15:30 = 1:2\) |
| Fraction to Percentage | \(\frac{a}{b} \times 100\%\) | \(\frac{4}{5} \times 100\% = 80\%\) |
| Percentage Change | \(\frac{\text{Change}}{\text{Original}} \times 100\%\) | Price increase from ₹200 to ₹250 = \(\frac{50}{200} \times 100 = 25\%\) |
2. Discounts, Profit, and Loss Calculations
In commercial transactions, retailers label goods with a Marked Price (MP). A reduction given on this price is known as Discount.
When Selling Price (SP) and Gain%/Loss% are given, directly compute Cost Price (CP) without quadratic confusion:
$$\text{CP} = \frac{\text{SP} \times 100}{100 + \text{Profit}\%} \quad \text{or} \quad \text{CP} = \frac{\text{SP} \times 100}{100 - \text{Loss}\%}$$3. Goods and Services Tax (GST)
GST is an indirect destination-based tax charged by the Government of India on supply of goods and services. Crucial rule: GST is calculated on the discounted selling price, never on the original marked price.
$$\text{GST Amount} = \frac{\text{GST}\%}{100} \times \text{Discounted Selling Price}$$ $$\text{Total Customer Invoice} = \text{Discounted Selling Price} + \text{GST Amount}$$4. Simple Interest vs. Compound Interest
While Simple Interest (\(SI = \frac{P \times R \times T}{100}\)) is earned only on the initial principal, Compound Interest adds the interest earned to the principal at each interval, yielding interest upon interest.
5. Applications: Population Growth and Depreciation
The compound interest framework models practical real-world exponential phenomena:
- Population Growth: If population grows at rate \(R\%\) annually, the population after \(n\) years is: $$P_n = P_0 \left(1 + \frac{R}{100}\right)^n$$
- Depreciation of Equipment & Vehicles: Value decreases over time at rate \(R\%\) per year: $$\text{Value after } n \text{ years} = P_0 \left(1 - \frac{R}{100}\right)^n$$
6. Step-by-Step Solved Exemplar Numericals
Problem: Calculate the amount and compound interest on ₹10,000 for \(1\frac{1}{2}\) years at \(10\%\) per annum, compounded semi-annually.
Step 1: Principal \(P = ₹10,000\).
Step 2: Semi-annual rate \(R' = \frac{10}{2} = 5\%\).
Step 3: Conversion periods \(n' = 1.5 \times 2 = 3\) half-years.
Step 4: Amount $$A = 10,000 \times \left(1 + \frac{5}{100}\right)^3 = 10,000 \times \left(\frac{21}{20}\right)^3 = 10,000 \times \frac{9261}{8000} = ₹11,576.25.$$
Step 5: Compound Interest \(CI = A - P = 11,576.25 - 10,000 = ₹1,576.25\).
7. Common Exam Pitfalls & Prevention
- Trap 1: Calculating discount percentage on the Selling Price (SP). Always compute discount on Marked Price (MP).
- Trap 2: Keeping the annual rate when interest is compounded half-yearly. Always divide \(R\) by 2 and multiply \(n\) by 2.
- Trap 3: Adding tax/GST to the marked price before applying discount. GST must be added strictly to the net discounted sale price.
