Grade 11 Math: Percentage

Cheatsheet Content

### Introduction to Percentage - **Definition:** Percentage means "per hundred" or "out of 100". It is a way of expressing a number as a fraction of 100. - **Symbol:** The symbol for percentage is `%`. - **Conversion:** - To convert a fraction to a percentage: Multiply by 100%. - To convert a decimal to a percentage: Multiply by 100%. - To convert a percentage to a fraction: Divide by 100 and simplify. - To convert a percentage to a decimal: Divide by 100. #### Examples: - Fraction to percentage: $\frac{3}{4} = \frac{3}{4} \times 100\% = 75\%$ - Decimal to percentage: $0.25 = 0.25 \times 100\% = 25\%$ - Percentage to fraction: $60\% = \frac{60}{100} = \frac{3}{5}$ - Percentage to decimal: $15\% = \frac{15}{100} = 0.15$ ### Calculating Percentage of a Quantity - To find $x\%$ of a quantity $Q$: $$ \text{Percentage of Quantity} = \frac{x}{100} \times Q $$ #### Example: - Find $20\%$ of Rs. 800. $$ \frac{20}{100} \times 800 = 0.20 \times 800 = \text{Rs. } 160 $$ ### Percentage Increase and Decrease - **Percentage Increase:** $$ \text{Percentage Increase} = \frac{\text{Increase in Quantity}}{\text{Original Quantity}} \times 100\% $$ New Quantity = Original Quantity + Increase New Quantity = Original Quantity $\times (1 + \frac{\text{Percentage Increase}}{100})$ - **Percentage Decrease:** $$ \text{Percentage Decrease} = \frac{\text{Decrease in Quantity}}{\text{Original Quantity}} \times 100\% $$ New Quantity = Original Quantity - Decrease New Quantity = Original Quantity $\times (1 - \frac{\text{Percentage Decrease}}{100})$ #### Examples: - **Increase:** A price of Rs. 500 increases by $10\%$. Increase = $\frac{10}{100} \times 500 = \text{Rs. } 50$ New Price = $500 + 50 = \text{Rs. } 550$ Alternatively: New Price = $500 \times (1 + \frac{10}{100}) = 500 \times 1.10 = \text{Rs. } 550$ - **Decrease:** A price of Rs. 1200 decreases by $25\%$. Decrease = $\frac{25}{100} \times 1200 = \text{Rs. } 300$ New Price = $1200 - 300 = \text{Rs. } 900$ Alternatively: New Price = $1200 \times (1 - \frac{25}{100}) = 1200 \times 0.75 = \text{Rs. } 900$ ### Finding the Original Quantity - If a quantity $Q$ is increased by $x\%$ to become $Q_{new}$: $$ Q = \frac{Q_{new}}{1 + \frac{x}{100}} $$ - If a quantity $Q$ is decreased by $x\%$ to become $Q_{new}$: $$ Q = \frac{Q_{new}}{1 - \frac{x}{100}} $$ #### Example: - After a $15\%$ increase, the price of an item is Rs. 690. Find the original price. Let the original price be $P$. $P \times (1 + \frac{15}{100}) = 690$ $P \times 1.15 = 690$ $P = \frac{690}{1.15} = \text{Rs. } 600$ ### Profit and Loss Percentage - **Cost Price (CP):** The price at which an article is bought. - **Selling Price (SP):** The price at which an article is sold. - **Profit:** If SP > CP, then Profit = SP - CP $$ \text{Profit Percentage} = \frac{\text{Profit}}{\text{CP}} \times 100\% $$ - **Loss:** If CP > SP, then Loss = CP - SP $$ \text{Loss Percentage} = \frac{\text{Loss}}{\text{CP}} \times 100\% $$ #### Example: - A shopkeeper buys an item for Rs. 500 and sells it for Rs. 650. Profit = $650 - 500 = \text{Rs. } 150$ Profit Percentage = $\frac{150}{500} \times 100\% = 30\%$ - A shopkeeper buys an item for Rs. 800 and sells it for Rs. 700. Loss = $800 - 700 = \text{Rs. } 100$ Loss Percentage = $\frac{100}{800} \times 100\% = 12.5\%$ ### Discount Percentage - **Marked Price (MP):** The price listed on the article. - **Discount:** Reduction given on the Marked Price. - **Selling Price (SP):** MP - Discount $$ \text{Discount Percentage} = \frac{\text{Discount}}{\text{MP}} \times 100\% $$ #### Example: - An item is marked at Rs. 1500. A discount of Rs. 300 is given. Discount Percentage = $\frac{300}{1500} \times 100\% = 20\%$ Selling Price = $1500 - 300 = \text{Rs. } 1200$ ### Simple Interest - **Principal (P):** The initial amount of money. - **Rate (R):** The percentage of interest per annum. - **Time (T):** The period for which the money is borrowed or lent (in years). $$ \text{Simple Interest (I)} = \frac{\text{P} \times \text{R} \times \text{T}}{100} $$ $$ \text{Total Amount (A)} = \text{P} + \text{I} $$ #### Example: - Find the simple interest and total amount for Rs. 10,000 at $5\%$ per annum for 3 years. $P = 10,000$, $R = 5$, $T = 3$ $I = \frac{10000 \times 5 \times 3}{100} = \text{Rs. } 1500$ $A = 10000 + 1500 = \text{Rs. } 11500$ ### Tax and VAT (Value Added Tax) - **Tax:** A compulsory contribution to state revenue, levied by the government on workers' income and business profits, or added to the cost of some goods, services, and transactions. - **VAT:** A consumption tax placed on a product whenever value is added at each stage of the supply chain, from production to the point of sale. - Tax/VAT is usually calculated as a percentage of the price of the good or service. $$ \text{Tax Amount} = \frac{\text{Tax Rate}}{100} \times \text{Original Price} $$ $$ \text{Price Including Tax} = \text{Original Price} + \text{Tax Amount} $$ Or, $$ \text{Price Including Tax} = \text{Original Price} \times (1 + \frac{\text{Tax Rate}}{100}) $$ #### Example: - An item costs Rs. 2000 before VAT. If VAT is $8\%$. VAT Amount = $\frac{8}{100} \times 2000 = \text{Rs. } 160$ Price including VAT = $2000 + 160 = \text{Rs. } 2160$ Alternatively: Price including VAT = $2000 \times (1 + \frac{8}{100}) = 2000 \times 1.08 = \text{Rs. } 2160$