Unit II: Percentage, Profit loss discount, Simple and Compound Interest - PEA305 Analytical Skills-I | B.Tech CSE Notes PDF | FineNotes4U

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

Contents

1. ⭐PERCENTAGES
   1. 1. What is Percentage?
   2. 2. Calculation of Percentages
   3. 3. Percentage Comparison
   4. 4. Percentage to Fraction Conversion
   5. 5. Successive Percentage Change
   6. 6. Problems Based on Percentage
2. ⭐Profit, Loss, and Discount
   1. 1. Basic Concepts
   2. 2. Calculation of Profit and Loss Percentage
   3. 3. Problems on Discount
   4. 4. Faulty Weights
   5. 5. Number of Articles
   6. 6. Common Selling Price
   7. 7. Summary of Important Formulas
3. ⭐Simple and compound interest
   1. 1. Interest and Its Types
   2. 2. Simple Interest (SI)
   3. 3. Compound Interest (CI)
   4. 4. Comparison of Simple and Compound Interest
   5. 5. Problems on Depreciation and Growth
   6. 6. Summary of Important Formulas
4. 🚨Thanks for visiting finenotes4u✨

⭐PERCENTAGES

---

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

---

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

---

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

---

🚨Thanks for visiting finenotes4u✨

Welcome to a hub for 😇Nerds and knowledge seekers! Here, you'll find everything you need to stay updated on education, notes, books, and daily trends.

💗 Bookmark our site to stay connected and never miss an update!

💌 Have suggestions or need more content? Drop a comment below, and let us know what topics you'd like to see next! Your support means the world to us. 😍