Comparing Quantities Introduction Percentage NCERT class8 Chapter 8 teachyousmart tys

Comparing Quantities: Percentages, Ratios, and Profit & Loss

The ability to compare quantities is essential in everyday life — comparing prices, calculating discounts, computing interest rates, and analysing data. This chapter in CBSE Class 8 Mathematics covers percentages, ratios, the concept of profit and loss, and the calculation of simple and compound interest. These concepts form the foundation of financial literacy and are applied in business, banking, and personal finance throughout life.

A percentage means "per hundred" and is a way of expressing a ratio as a fraction with a denominator of 100. The percentage symbol is %. For example, 45% means 45 out of 100, or 45/100, or 0.45. Converting between fractions, decimals, and percentages: to convert a fraction to a percentage, multiply by 100 (e.g., 3/5 = 60%); to convert a percentage to a fraction, divide by 100 and simplify (e.g., 75% = 3/4); to convert a decimal to a percentage, multiply by 100 (e.g., 0.8 = 80%). Percentages are used to compare quantities — 40% is greater than 25% regardless of the base amounts. When finding the percentage increase or decrease, the formula is: Percentage change = (Change / Original value) × 100. For example, if a shirt costs ₹400 after a 20% discount, the original price was ₹500, because 400 is 80% of 500 (100% − 20% = 80%). In profit and loss calculations, the cost price (CP) is what you paid, and the selling price (SP) is what you sell it for. Profit = SP − CP (when SP > CP); Loss = CP − SP (when CP > SP). Profit percentage = (Profit/CP) × 100; Loss percentage = (Loss/CP) × 100. Both are always calculated on the cost price, not the selling price.

Simple interest is calculated only on the original principal amount: SI = (P × R × T)/100, where P is the principal, R is the rate of interest per annum, and T is the time in years. The total amount after T years is A = P + SI. Compound interest, on the other hand, is calculated on the principal plus the accumulated interest from previous periods, making the money grow faster. The formula for compound interest compounded annually is A = P(1 + R/100)ⁿ, where n is the number of years. The compound interest is CI = A − P = P(1 + R/100)ⁿ − P. The amount grows exponentially with compound interest — this is why compound interest is called "the eighth wonder of the world" (often attributed to Einstein). For example, ₹10,000 at 10% per annum gives SI of ₹1,000 per year, but CI in the first year is also ₹1,000, in the second year it is ₹1,100 (10% of ₹11,000), and so on, accumulating to more than SI. Half-yearly compounding applies interest every 6 months at half the annual rate: A = P(1 + R/200)²ⁿ. Discount is a reduction on the marked price (MP) of an article: Discount% = (Discount/MP) × 100, and SP = MP − Discount. Marked price is often set higher than CP to allow for discounts while still making a profit.

• Percentage means "per hundred"; convert: fraction × 100 = %; % ÷ 100 = fraction; decimal × 100 = %.
• Profit = SP − CP; Loss = CP − SP; profit% and loss% are always calculated on cost price.
• Simple interest: SI = PRT/100; amount A = P + SI; interest is calculated only on the original principal.
• Compound interest: A = P(1 + R/100)ⁿ compounded annually; CI = A − P; grows faster than simple interest.
• Discount = MP − SP; Discount% = (Discount/MP) × 100; successive discounts are applied one after another, not added.