Fractions in Disguise

Chapter at a Glance

This chapter details the math and applications of percentages, presenting them as fractions with a constant denominator of $100$. It demonstrates how to convert fractions, decimals, and ratios to percentages and vice versa, using visual bar models and proportions. The chapter then applies percentages to compare proportions (such as test scores and food ingredients), calculate percentage increase or decrease, determine profit and loss margins in retail transactions, compute simple and compounded interest over time, handle asset depreciation, and analyze common retail calculations involving consecutive discounts and profit markups.

Key Definitions & Terminology

• Per cent: Derived from the Latin per centum, meaning "by the hundred" or "out of every hundred," represented by the symbol $\%$.
• Percentage: A fraction with a denominator of exactly $100$ (e.g., $x\% = \frac{x}{100}$).
• Cost Price (CP): The purchase price paid by a seller to acquire goods.
• Marked Price (MP) / MRP: The advertised or catalog price quoted by a seller.
• Selling Price (SP): The final price paid by a customer after discounts are applied.
• Discount: The reduction in price offered on the marked price of an item:
  $$\text{Discount} = \text{MP} - \text{SP}$$
• Profit: The positive difference when the selling price is greater than the cost price:
  $$\text{Profit} = \text{SP} - \text{CP}$$
• Loss: The negative difference when the selling price is less than the cost price:
  $$\text{Loss} = \text{CP} - \text{SP}$$
• Interest: The additional money paid on savings deposits or charged on borrowed loans.
• Principal ($P$): The initial sum of money deposited or borrowed.
• Rate of Interest ($r$): The percentage rate charged or paid per year, denoted as per annum (p.a.).
• Compounding: The process where interest earned is periodically added back to the principal, increasing the base on which future interest is calculated.
• Depreciation: The decrease in the financial value of an asset over time due to wear, use, and age.

Formulas, Rules & Properties

• Fraction to Percentage Conversion:
• To convert a fraction $\frac{a}{b}$ to a percentage, multiply by $100$:
  $$\text{Percentage} = \frac{a}{b} \times 100\%$$
• Percentage of a Quantity:
• To find $y\%$ of a value $x$:
  $$\text{Value} = \frac{y}{100} \times x$$
• Reversibility of Percentages:
• For any numbers $x$ and $y$, $x\%$ of $y$ is equal to $y\%$ of $x$:
  $$\frac{x}{100} \times y = \frac{y}{100} \times x$$
• Percentage Increase/Decrease:
• $$\text{Percentage Increase} = \frac{\text{Amount of Increase}}{\text{Original Amount (Base)}} \times 100\%$$
• $$\text{Percentage Decrease} = \frac{\text{Amount of Decrease}}{\text{Original Amount (Base)}} \times 100\%$$
• Profit and Loss Percentages:
• Formulated relative to the Cost Price (investment base) or Sales Revenue:
  $$\text{Profit } \% = \frac{\text{Profit}}{\text{CP}} \times 100\%$$
  $$\text{Loss } \% = \frac{\text{Loss}}{\text{CP}} \times 100\%$$
  $$\text{Net Profit } \% = \frac{\text{Net Profit}}{\text{Revenue}} \times 100\%$$
• Simple Interest (No Compounding):
• Interest: $I = P \times r \times t$.
• Total maturity amount:
  $$A = P + Prt = P(1 + rt)$$
• Compound Interest (Compounded Annually):
• Total maturity amount:
  $$A = P(1 + r)^t$$
• Depreciation Formula:
• Value after $t$ terms with depreciation rate $r$:
  $$A = P(1 - r)^t$$
• Consecutive Discounts:
• For consecutive discounts of $d_1\%$ and $d_2\%$, the final price is:
  $$\text{Final Price} = \text{Original Price} \times (1 - d_1)(1 - d_2)$$
  The total discount rate is $1 - (1 - d_1)(1 - d_2)$, which is strictly less than $d_1 + d_2$.

Core Concepts & Topics

• FDP Trio Equivalence:
• Percentages, fractions, and decimals represent equivalent quantities. E.g., $40\% = \frac{40}{100} = \frac{2}{5} = 0.40$.
• Mental Arithmetic Baselines:
• $10\%$ is found by dividing the number by $10$.
• $20\%$ is double $10\%$.
• $5\%$ is half of $10\%$.
• $25\%$ is a quarter of the value (divide by 4).
• $1\%$ is found by dividing by $100$.
• Linear vs. Exponential Growth:
• Simple Interest (No Compounding) is an example of linear growth because the interest earned remains constant every term since the principal remains unchanged.
• Compound Interest is an example of exponential growth because the interest is added back to the principal, causing the principal to grow each term, and the interest to increase proportionally.
• Tricky Retail Percentage Pitfalls:
• A retailer adding a $50\%$ markup and then offering a $50\%$ discount does not break even:
  $$\text{CP} = x \rightarrow \text{MP} = 1.5x \rightarrow \text{SP} = 1.5x \times (1 - 0.50) = 0.75x$$
  This results in a $25\%$ loss. To break even, the discount must be exactly $33.33\%$.
• Historical Context:
• India: Kautilya’s Arthaśhāstra (4th century BCE) states interest rates as panas per month per cent (e.g., $1.25$ panas standard, $5$ panas commercial, $10$ panas in forests, $20$ panas for sea traders).
• Rome: Ancient Romans collected taxes in fractions of $\frac{1}{100}$ and $\frac{1}{20}$ on auctions.

Worked Examples

• Marble Proportions (Page 3 Q2):
• Problem: Nandini has $25$ marbles, $15$ of which are white. What percentage are white?
• Solution: White fraction $= \frac{15}{25}$.
  • Percentage $= \frac{15}{25} \times 100 = 15 \times 4 = 60\%$.
• Comparing Score Proportions (Example 1, Page 14):
• Problem: Eesha scored $42/50$ in English and $70/80$ in Science. Compare her performance.
• Solution: Convert both to percentages:
  • English: $\frac{42}{50} \times 100 = 84\%$.
  • Science: $\frac{70}{80} \times 100 = 87.5\%$.
  • She did better in Science.
• Profit Margin (Page 19 Figure it Out Q1):
• Problem: Retailer buys a geometry box for ₹$75$ and sells it for ₹$110$. Find the profit margin.
• Solution: Profit $= 110 - 75 = $ ₹$35$.
  • Profit $\% = \frac{35}{75} \times 100 = \frac{35 \times 4}{3} \approx 46.67\%$.
• Simple vs. Compound Interest (Page 22 Figure it Out Q1):
• Problem: Deposit ₹$20,000$ for $2$ years at $10\%$ p.a. Compare maturity amounts with and without compounding.
• Solution:
  • Without compounding:
    $$A = P(1 + rt) = 20000(1 + 0.10 \times 2) = 20000(1.2) = \text{₹}24,000$$
  • With compounding:
    $$A = P(1 + r)^t = 20000(1 + 0.10)^2 = 20000(1.21) = \text{₹}24,200$$
  • Difference $= 24,200 - 24,000 = $ ₹$200$ more with compounding.
• Depreciation Calculation (Page 25 Example 11):
• Problem: A village population of $1250$ reduces by $10\%$ every decade. Find the population after $3$ decades.
• Solution:
  $$A = P(1 - r)^t = 1250 \times (1 - 0.10)^3 = 1250 \times (0.9)^3 = 1250 \times 0.729 = 911.25 \approx 911 \text{ people}$$
• Tricky Consecutive Discount (Page 26 Example 12):
• Problem: Cakely offers $30\% + 20\%$ discount, while Cakify offers a flat $50\%$ discount. Compare the cost of a ₹$200$ cake at both.
• Solution:
  • Cakify: $50\%$ discount $\rightarrow \text{Price} = 200 \times 0.5 = $ ₹$100$.
  • Cakely: Apply $30\%$ discount first: $200 \times 0.7 = $ ₹$140$. Apply $20\%$ discount on ₹$140$: $140 \times 0.8 = $ ₹$112$.
  • Cakify is cheaper.
• Left-Handed Room Riddle (Page 30 Q14):
• Problem: In a room of $100$ people, $99\%$ are left-handed. How many left-handed people must leave the room to bring that percentage down to $98\%$?
• Solution:
  • Initially, there are $99$ left-handed people and $1$ right-handed person.
  • If $x$ left-handed people leave, the number of left-handed people becomes $99-x$ and the total becomes $100-x$.
  • We want the right-handed person (who does not leave) to represent $100\% - 98\% = 2\%$ of the remaining population.
  • Thus:
    $$\frac{1}{100 - x} = \frac{2}{100} = \frac{1}{50} \rightarrow 100 - x = 50 \rightarrow x = 50$$
  • Answer: $50$ left-handed people must leave.

Practical Activities & Experiments

• Processed Food KYC Audit: Gather three food containers (e.g., cereal, snacks). Read the nutritional information to find the total serving size weight and sugar weight per serving. Calculate the sugar percentage of each product to determine which has the largest proportion of sugar.
• Discount Comparison Model: Create bar models representing consecutive discounts (like $20\% + 10\%$) vs. single flat discounts (like $30\%$) on graph paper. Present this model to show why consecutive discounts do not add up directly to a single sum.