8.7 Going Deeper: Enrichment & Exam Preparation

Where Percentages Show Up in Real Life

Math in the Real World

  • Sale tags and discounts. A jacket priced ₹350 with “30% off” costs 0.70 \times 350 = ₹245. Every “flat X% off” banner is just the marked price multiplied by (1 - \tfrac{X}{100}).
  • GST and taxes. A ₹7500 phone with 18% GST rings up at 7500 \times 1.18 = ₹8850. The tax is a percentage added on top, so you multiply by (1 + \tfrac{X}{100}).
  • Bank interest. Savings accounts and fixed deposits quote a rate “per annum.” Left to compound, ₹25,000 at 10% for 2 years grows to 25000(1.1)^2 = ₹30{,}250, more than simple interest, because you earn interest on the interest.
  • Population growth. A town growing 4% a year multiplies by 1.04 each year; over 3 years that is (1.04)^3 \approx 1.1249, a 12.5% rise in all, not 12%.
  • Depreciation. A ₹32,000 laptop losing 8% of its value in a year keeps 0.92 \times 32000 = ₹29{,}440. Cars, phones and machinery all “depreciate” by multiplying by (1 - r) each period.
  • A bridge to later chapters. The same multiply-by-(1+r) idea returns as exponents (Chapter 2) and underlies the simple and compound interest problems you will meet again in higher classes.

Quick Reference: FDP and the Change Formulas

Knowing the common fraction–decimal–percent equivalents by heart turns slow conversions into instant recall, a real advantage under exam time pressure.

Fraction Decimal Percent Fraction Decimal Percent
\tfrac{1}{2} 0.5 50\% \tfrac{3}{4} 0.75 75\%
\tfrac{1}{3} 0.333\ldots 33.33\% \tfrac{1}{8} 0.125 12.5\%
\tfrac{2}{3} 0.666\ldots 66.67\% \tfrac{3}{8} 0.375 37.5\%
\tfrac{1}{4} 0.25 25\% \tfrac{5}{8} 0.625 62.5\%
\tfrac{1}{5} 0.2 20\% \tfrac{1}{10} 0.1 10\%
\tfrac{2}{5} 0.4 40\% \tfrac{1}{20} 0.05 5\%
\tfrac{3}{5} 0.6 60\% \tfrac{1}{25} 0.04 4\%
Quantity Formula
y\% of a value V \dfrac{y}{100}\times V
Percentage change \dfrac{\text{new} - \text{old}}{\text{old}}\times 100
Profit % \dfrac{\text{SP} - \text{CP}}{\text{CP}}\times 100
Loss % \dfrac{\text{CP} - \text{SP}}{\text{CP}}\times 100
Discount \text{MP}\times\Big(1 - \dfrac{d}{100}\Big)
Price after tax \text{price}\times\Big(1 + \dfrac{t}{100}\Big)
Simple growth (t years) A = p(1 + rt)
Compound growth A = p(1 + r)^{t}
Decline / depreciation A = p(1 - r)^{t}

Use the explorer to apply a percentage change, a discount-then-tax, or a profit/loss calculation to any starting amount:

Percentage-change & profit/loss explorer

Memory Tricks & One-Page Revision

Quick Revision Card

  • A percentage is a fraction over 100: x\% = \tfrac{x}{100}. To turn a fraction into a percent, multiply by 100; to find y\% of a value, multiply by \tfrac{y}{100}.
  • The 10% anchor: 10\% is the value with the decimal point moved one place left. Halve it for 5\%, double for 20\%, and build 15\%, 25\%, 30\% from these.
  • Percentage change is always over the original: \dfrac{\text{change}}{\text{old}}\times 100. Profit and loss are reckoned against the cost price, never the selling price.
  • Discount multiplies by (1-d); tax multiplies by (1+t). A 20% discount then 18% tax is \times 0.80 \times 1.18.
  • Growth: simple = p(1+rt); compound = p(1+r)^t; decline = p(1-r)^t.
  • Successive changes MULTIPLY, they don’t add. A 10% rise then a 10% fall is 1.1\times 0.9 = 0.99, a 1% loss, not break-even.
  • x\% of y always equals y\% of x, e.g. 8\% of 25 = 25\% of 8 = 2.

Spot the Mistake

Common Exam Mistakes

  • Adding successive percentages: treating “30% then 25% off” as a single 55% discount. It is really 0.70 \times 0.75 = 0.525, a 47.5% reduction.
  • Computing profit/loss on the selling price instead of the cost price. The base for profit and loss is always CP.
  • Thinking a 10% rise followed by a 10% fall returns you to the start. It does not, the fall acts on a bigger base, leaving a 1% net loss.
  • Confusing “is 120% of” with “is 20% more than”, both are correct here (\times 1.20), but “120% more” means \times 2.20.
  • Adding interest each year on the original principal when the question says compound, under compounding the base grows every year.
  • Forgetting that x\% of a quantity needs dividing by 100: writing “30% of 140 = 30 \times 140” instead of \tfrac{30}{100}\times 140 = 42.

Test Your Reflexes

A fast, friendly drill: convert a fraction to a percentage, or find a percentage of a number. Build a streak!

Percent reflex
Type the answer and press Check.
…
Score 0 · Streak 0

Exam Corner: CBSE-Style Practice

Mixed Practice (objective, short, long, HOTS)

Objective type (1 mark each)

  1. 35\% of 80 is ______.
  2. A price rises from ₹240 to ₹300. The percentage increase is ______.
  3. A coat marked ₹1200 is sold at a 25\% discount. The selling price is ______.

Short answer (2 marks each)

  1. A trader buys an item for ₹600 and sells it for ₹750. Find the profit percentage.
  2. A number increased by 20\% becomes 84. Find the original number.

Long answer (3 marks each)

  1. ₹15,000 is deposited at 8\% p.a., compounded annually, for 2 years. Find the final amount and the compound interest.
  2. A shopkeeper marks a shirt 25\% above its cost price of ₹800, then allows a 20\% discount on the marked price. Find the selling price and the profit or loss percentage.

HOTS (Higher Order Thinking)

  1. The price of a commodity rises 10\% one month and then falls 20\% the next. Find the overall percentage change, and say whether the final price is above or below the start.
  2. A village of 25{,}000 people grows at 4\% per year. Estimate the population after 3 years.

Assertion–Reason (Choose: (a) both true, R explains A; (b) both true, R does not explain A; (c) A true, R false; (d) A false, R true.)

  1. Assertion (A): A 30\% increase followed by another 30\% increase is a 69\% increase overall.   Reason (R): Successive percentage changes multiply rather than add.
  1. 35\% of 80 = \tfrac{35}{100}\times 80 = \mathbf{28}.
  2. Increase = 300 - 240 = 60 on a base of 240: \dfrac{60}{240}\times 100 = \mathbf{25\%}.
  3. 1200 \times (1 - 0.25) = 1200 \times 0.75 = \mathbf{₹900}.
  4. Profit = 750 - 600 = 150 on CP = 600: \dfrac{150}{600}\times 100 = \mathbf{25\%}.
  5. 1.20 \times n = 84 \Rightarrow n = \dfrac{84}{1.20} = \mathbf{70}.
  6. Amount = 15000(1.08)^2 = 15000 \times 1.1664 = \mathbf{₹17{,}496}. Compound interest = 17496 - 15000 = \mathbf{₹2496}.
  7. Marked price = 800 \times 1.25 = ₹1000; after 20\% discount, SP = 0.80 \times 1000 = ₹800. Since SP = CP = ₹800, there is no profit and no loss (0%).
  8. 1.10 \times 0.80 = 0.88, so the final price is 88\% of the start, a net decrease of 12\%, below the starting price. (The fall acts on a larger base than the rise did.)
  9. 25000 \times (1.04)^3 = 25000 \times 1.124864 \approx \mathbf{28{,}122} people.
  10. (a), Both true and R explains A: 1.30 \times 1.30 = 1.69, a 69\% increase, precisely because the changes multiply rather than add.

Connections to Other Chapters

How This Chapter Links Forward

  • Chapter 2 (Power Play): compound growth p(1+r)^t is an exponent in action, the rate (1+r) raised to the power of the number of periods.
  • Ratios and Proportion: every percentage is a ratio “out of 100,” so comparing proportions fairly is the same skill at heart.
  • Algebra: working backwards from a percentage to the whole (solving 1.20n = 84) is a one-step linear equation.
  • Data Handling: pie charts and bar graphs report shares as percentages, and reading them correctly depends on the FDP trio.

Glossary

Key Terms

  • Per cent (%): “out of a hundred”; x\% = \tfrac{x}{100}.
  • FDP trio: the three equivalent costumes of one proportion, fraction, decimal and percentage, e.g. \tfrac{1}{4} = 0.25 = 25\%.
  • Percentage change: the change divided by the original amount, times 100.
  • Cost price (CP), marked price (MP), selling price (SP): what the seller paid, the tag price, and the price actually paid.
  • Profit / loss %: profit or loss as a percentage of the cost price.
  • Discount: a percentage subtracted from the marked price; tax / GST: a percentage added to a price.
  • Simple growth: interest paid on the original principal only, p(1+rt). Compound growth: interest folded back in each period, p(1+r)^t.
  • Depreciation: a percentage decline each period, multiplying by (1-r) each time.