Top 10 Best Life Insurance Companies in the USA

Unit II: Percentage, Profit loss discount, Simple and Compound Interest

⭐PERCENTAGES

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1. What is Percentage?

• A percentage is a number or ratio expressed as a fraction of 100.
• It is denoted by the symbol %.
• Formula: $$\text{Percentage} = \left(\frac{\text{Value}}{\text{Total Value}}\right) \times 100$$
  

Example:
If a student scores 40 marks out of 50, the percentage of marks obtained is:

$$\left(\frac{40}{50}\right) \times 100 = 80\%$$

2. Calculation of Percentages

• Finding X% of a number:

  $$\left(\frac{X}{100} \times \text{Number} \right)$$
  

  Example:

  • 20% of 150 = (20/100) × 150 = 30

• Finding the number when percentage is given:

  $$\text{Number} = \left(\frac{\text{Given Percentage} \times \text{Total}}{100} \right)$$
  

  Example:

  • If 25% of a number is 50, then the number = (50 × 100) / 25 = 200

3. Percentage Comparison

• Used to compare two values in percentage terms.
• Formula: $$\text{Percentage Difference} = \left(\frac{\text{Difference of values}}{\text{Original Value}}\right) \times 100$$
  

Example:

• Price of an item increases from 200 to 250.
• Difference = 250 - 200 = 50
• Percentage increase = (50 / 200) × 100 = 25%

4. Percentage to Fraction Conversion

• To convert a percentage to a fraction, divide by 100 and simplify.

Fraction	Decimal	Percentage
1/2	0.5	50%
1/3	0.3333...	33.33%
1/4	0.25	25%
1/5	0.2	20%
1/6	0.1666...	16.67%
1/7	0.142857...	14.29%
1/8	0.125	12.5%
1/9	0.1111...	11.11%
1/10	0.1	10%
1/11	0.090909...	9.09%
1/12	0.0833...	8.33%
1/13	0.076923...	7.69%
1/14	0.071428...	7.14%
1/15	0.0666...	6.67%
1/16	0.0625	6.25%
1/17	0.058823...	5.88%
1/18	0.0555...	5.56%
1/19	0.052631...	5.26%
1/20	0.05	5%

Example:

• Convert 40% to a fraction $$40\% = \frac{40}{100} = \frac{2}{5}$$

5. Successive Percentage Change

• When a quantity changes by two or more percentages successively.
• Formula: $$\text{Final Percentage Change} = A + B + \frac{A \times B}{100}$$ where A and B are successive percentage changes.

Example:

• A price increases by 20% and then decreases by 10%. $$\text{Final Change} = 20 - 10 + \frac{20 \times (-10)}{100}$$ $$= 20 - 10 - 2 = 8\%$$ So, the net increase is 8%.

6. Problems Based on Percentage

(i) Problems on Marks

• Used to calculate marks obtained or percentage of marks.

Example:
A student scores 180 marks out of 300. Find the percentage of marks.

$$\left(\frac{180}{300}\right) \times 100 = 60\%$$

(ii) Election Problems

• Used to calculate votes received by candidates.

Example:
In an election, a candidate gets 60% of votes. If total votes = 10,000, find votes received.

$$\left(\frac{60}{100} \times 10000 \right) = 6000$$

(iii) Population Problems

• Used to calculate population growth or decrease.

Formula:

$$\text{Final Population} = \text{Initial Population} \times \left(1 + \frac{\text{Growth Rate}}{100}\right)^n$$

where n = number of years.

Example:
A town has 10,000 people. Population increases by 5% annually. Find the population after 2 years.

$$\text{Population after 1st year} = 10000 \times \left(1 + \frac{5}{100} \right) = 10000 \times 1.05 = 10500$$
$$\text{Population after 2nd year}=10500\times 1.05=11025$$

So, population after 2 years = 11,025.

(iv) Percentage Error

• Used to calculate errors in measurement.
• Formula: $$\text{Percentage Error} = \left(\frac{\text{Error}}{\text{Actual Value}}\right) \times 100$$

Example:
A measurement is recorded as 95 instead of 100. Find the percentage error.

$$\left(\frac{100 - 95}{100}\right) \times 100 = 5\%$$

⭐Profit, Loss, and Discount

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1. Basic Concepts

(i) Cost Price (C.P.)

• The price at which an article is bought.
• Example: If a shopkeeper buys a book for ₹200, then its cost price (C.P.) = ₹200.

(ii) Selling Price (S.P.)

• The price at which an article is sold.
• Example: If the shopkeeper sells the book for ₹250, then its selling price (S.P.) = ₹250.

(iii) Marked Price (M.P.)

• The original price of an article before any discount is given.
• It is also called list price.
• Example: A shopkeeper marks a bag at ₹1,000 but sells it at ₹800 after a discount. Here, M.P. = ₹1,000.

(iv) Profit (or Gain)

• If Selling Price (S.P.) > Cost Price (C.P.), there is a profit.
• Formula: $$\text{Profit} = \text{S.P.} - \text{C.P.}$$
  
• Example: A shopkeeper buys a phone for ₹5,000 and sells it for ₹6,000.
  • Profit = 6000 - 5000 = ₹1000

(v) Loss

• If Selling Price (S.P.) < Cost Price (C.P.), there is a loss.
• Formula: $$\text{Loss} = \text{C.P.} - \text{S.P.}$$
  
• Example: A shopkeeper buys a TV for ₹20,000 but sells it for ₹18,500.
  • Loss = 20000 - 18500 = ₹1500

2. Calculation of Profit and Loss Percentage

(i) Profit Percentage

$$\text{Profit \%} = \left(\frac{\text{Profit}}{\text{C.P.}}\right) \times 100$$

Example: If a shopkeeper buys a bag for ₹500 and sells it for ₹600:

• Profit = 600 - 500 = ₹100
• Profit % = (100 / 500) × 100 = 20%

(ii) Loss Percentage

$$\text{Loss \%} = \left(\frac{\text{Loss}}{\text{C.P.}}\right) \times 100$$

Example: If a shopkeeper buys a chair for ₹800 and sells it for ₹720:

• Loss = 800 - 720 = ₹80
• Loss % = (80 / 800) × 100 = 10%

3. Problems on Discount

(i) Concept of Discount

• A discount is a reduction in the marked price to attract customers.
• Formula: $$\text{Discount} = \text{Marked Price} - \text{Selling Price}$$
• Example: A shirt has an M.P. of ₹1,200, but a shopkeeper sells it for ₹1,000.
  • Discount = 1200 - 1000 = ₹200

(ii) Discount Percentage

$$\text{Discount \%} = \left(\frac{\text{Discount}}{\text{Marked Price}}\right) \times 100$$

Example: If M.P. = ₹1,500 and S.P. = ₹1,200, then:

• Discount = 1500 - 1200 = ₹300
• Discount % = (300 / 1500) × 100 = 20%

(iii) Successive Discounts

• When two or more discounts are given on a product successively.
• Formula: $$\text{Net Discount \%} = A + B - \left(\frac{A \times B}{100}\right)$$where A and B are two successive discount percentages.

Example: If a shop gives 10% and then 20% discount:

$$\text{Net Discount} = 10 + 20 - \left(\frac{10 \times 20}{100}\right) = 28\%$$

4. Faulty Weights

• Some shopkeepers use weights that are lighter than actual to show more quantity and gain extra profit.
• Formula for Gain % in Faulty Weights: $$\text{Gain \%} = \left(\frac{\text{Error in Weight}}{\text{Actual Weight} - \text{Error}}\right) \times 100$$
  

Example:

• A shopkeeper uses a 950g weight instead of 1kg (1000g).
• Error in weight = 50g
• Gain % = (50 / (1000 - 50)) × 100 = (50 / 950) × 100 ≈ 5.26%

5. Number of Articles

• Used to find the number of articles bought and sold when profit/loss is involved.

Formula:

$$\text{Total Cost}=\text{Number of Articles}\times \text{Cost Price per Article}$$

Example:

• A person buys 20 pens for ₹500. Find the C.P. of one pen.
  • C.P. per pen = 500 / 20 = ₹25
• If the person sells each pen for ₹30, find the profit per pen and total profit.
  • Profit per pen = 30 - 25 = ₹5
  • Total profit = 20 × 5 = ₹100

6. Common Selling Price

• When different products are bought at different prices and sold at a single selling price.

Formula:

$$\text{Common S.P.} = \frac{\text{Total Cost} + \text{Total Profit}}{\text{Total Articles}}$$

Example:
A shopkeeper buys:

• 5 shirts for ₹400 each
• 5 shirts for ₹500 each
• Sells all 10 shirts at ₹550 each

Find profit per shirt and total profit:

• Total C.P. = (5 × 400) + (5 × 500) = 2000 + 2500 = ₹4500
• Total S.P. = 10 × 550 = ₹5500
• Total Profit = 5500 - 4500 = ₹1000
• Profit per shirt = 1000 / 10 = ₹100

7. Summary of Important Formulas

Concept	Formula
Profit	S.P. - C.P.
Loss	C.P. - S.P.
Profit %	(Profit / C.P.) × 100
Loss %	(Loss / C.P.) × 100
Discount	M.P. - S.P.
Discount %	(Discount / M.P.) × 100
Selling Price (with profit)	S.P. = C.P. × (1 + Profit % / 100)
Selling Price (with loss)	S.P. = C.P. × (1 - Loss % / 100)
Common Selling Price	(Total C.P. + Total Profit) / Total Articles

⭐Simple and compound interest

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1. Interest and Its Types

(i) Interest

• Interest is the cost of borrowing money or the earnings on investments, usually expressed as a percentage of the principal amount.
• Principal (P): The initial amount of money invested or borrowed.
• Interest (I): The extra amount paid or earned over time, usually in percentage.

2. Simple Interest (SI)

(i) Concept of Simple Interest

• Simple Interest is calculated only on the initial principal throughout the period.
• It does not change over time.

(ii) Formula for Simple Interest

$$\text{Simple Interest (SI)} = \frac{P \times R \times T}{100}$$

Where:

• P = Principal amount (initial investment or loan)
• R = Rate of interest per year (%)
• T = Time period in years

(iii) Total Amount (A) with Simple Interest

The total amount after interest is added is calculated as:

$$A=P+SI$$

or,

$$A=P(1+\frac{R\times T}{100})$$

Where A is the total amount at the end of the time period.

(iv) Example of Simple Interest

Example:
A person invests ₹1,000 at an interest rate of 5% per year for 3 years.

• P = ₹1000, R = 5%, T = 3 years
• SI = (1000 × 5 × 3) / 100 = ₹150
• Total Amount (A) = 1000 + 150 = ₹1150

3. Compound Interest (CI)

(i) Concept of Compound Interest

• Compound Interest is calculated on the initial principal and also on the interest that accumulates over time.
• The interest gets added to the principal at regular intervals (such as annually, half-yearly, or quarterly), and future interest is calculated on the new total.

(ii) Formula for Compound Interest

The formula to calculate Compound Interest is:

$$A=P{(1+\frac{R}{100})}^T$$

Where:

• A = Total amount after time period
• P = Principal amount
• R = Rate of interest per year
• T = Time in years

To find Compound Interest (CI):

$$CI=A-P$$

(iii) Example of Compound Interest

Example:
A person invests ₹1,000 at an interest rate of 5% per year for 3 years, compounded annually.

• P = ₹1000, R = 5%, T = 3 years
• A = 1000 × (1 + 5/100)^3 = 1000 × (1.05)^3 = ₹1157.63
• CI = ₹1157.63 - ₹1000 = ₹157.63

4. Comparison of Simple and Compound Interest

(i) Simple Interest vs Compound Interest

• Interest Calculation:
  • SI is calculated on the original principal only.
  • CI is calculated on both the principal and the accumulated interest.
• Amount Earned:
  • Simple Interest: The interest amount is the same each year.
  • Compound Interest: The interest increases each year because it is calculated on the new amount (principal + accumulated interest).

(ii) When is CI better than SI?

• Compound Interest gives a higher amount compared to Simple Interest when the time period is long or the interest is compounded frequently (like quarterly or monthly).
• The longer the investment period, the more beneficial Compound Interest becomes.

(iii) Example Comparison

If ₹1,000 is invested for 3 years at 5% interest:

• With Simple Interest (SI):
  • SI = (1000 × 5 × 3) / 100 = ₹150
  • Total Amount = ₹1000 + ₹150 = ₹1150
• With Compound Interest (CI):
  • CI = 1000 × (1 + 5/100)^3 - 1000 = ₹157.63
  • Total Amount = ₹1000 + ₹157.63 = ₹1157.63

The Compound Interest (₹157.63) is higher than Simple Interest (₹150).

5. Problems on Depreciation and Growth

(i) Depreciation (Reduction in Value)

• Depreciation refers to the reduction in value of an asset over time due to factors like wear and tear, age, or obsolescence.
• Similar to compound interest, depreciation is often calculated on the reduced value each year.

Formula for Depreciation:

$$A=P{(1-\frac{R}{100})}^T$$

Where:

• A = Value of asset after time T
• P = Initial value of the asset
• R = Depreciation rate per year
• T = Time period in years

Example:
A machine worth ₹10,000 depreciates by 10% every year. Find its value after 2 years.

• P = ₹10,000, R = 10%, T = 2 years
• A = 10000 × (1 - 10/100)^2 = 10000 × (0.9)^2 = ₹8100
• The value of the machine after 2 years is ₹8100.

(ii) Growth (Increase in Value)

• Growth is the increase in value over time, often used in population growth, investments, etc.
• Similar to compound interest, growth is calculated on the increased value over time.

Formula for Growth (Similar to Compound Interest):

$$A=P{(1+\frac{R}{100})}^T$$

Example:
A population of 50,000 increases by 4% every year. Find its population after 3 years.

• P = 50,000, R = 4%, T = 3 years
• A = 50000 × (1 + 4/100)^3 = 50000 × (1.04)^3 = 50000 × 1.124864 = 56,243.20
• The population after 3 years is approximately 56,243.

6. Summary of Important Formulas

Concept	Formula
Simple Interest (SI)	$\text{SI} = \frac{P \times R \times T}{100}$
Total Amount (Simple Interest)	$A = P \left( 1 + \frac{R \times T}{100} \right)$
Compound Interest (CI)	$A = P \left( 1 + \frac{R}{100} \right)^T$
Compound Interest (CI)	$\text{CI} = A - P$
Depreciation (Value Reduction)	$A = P \left( 1 - \frac{R}{100} \right)^T$
Growth (Increase in Value)	$A = P \left( 1 + \frac{R}{100} \right)^T$

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