Business Mathematics and Statistics

Unit I: Unitary Method, Number System, and Matrices

1. Unitary Method

• The unitary method is used to find the value of a single unit from the value of multiple units and then use it to calculate the value of any number of units.

• Steps:

  1. Find the value of 1 unit by dividing the total value by the number of units.
  2. Multiply the value of 1 unit by the required number of units to get the final result.

• Example:
  If 5 pens cost ₹100, find the cost of 8 pens.

  • Step 1: Cost of 1 pen = $\frac{100}{5} = ₹20$.
  • Step 2: Cost of 8 pens = $8 \times ₹20 = ₹160$.

• Application:
  It is used in everyday calculations like finding the price for a different quantity, time taken to complete work with different workers, etc.

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2. Concept of Integers

• Integers are whole numbers that can be positive, negative, or zero. They do not include fractions or decimals.

• Types:

  1. Positive Integers: Numbers greater than zero (e.g., 1, 2, 3).
  2. Negative Integers: Numbers less than zero (e.g., -1, -2, -3).
  3. Zero: Considered an integer but neither positive nor negative.

• Operations on Integers:

  1. Addition: Adding two integers. If both integers are of the same sign, add the numbers and keep the sign. If they are of different signs, subtract the smaller from the larger and keep the sign of the larger.
     • Example: $5 + (-3) = 2$.
  2. Subtraction: Subtracting one integer from another by adding its negative.
     • Example: $7 - (-2) = 7 + 2 = 9$.
  3. Multiplication and Division: Multiplying or dividing two integers gives a positive result if both are of the same sign, and a negative result if they are of different signs.
     • Example: $(-4) \times (-3) = 12$; $(-4) \times 3 = -12$.

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3. Exponential System

• An exponential system involves numbers raised to a power (exponent).

• Basic Form:
  $a^n$, where $a$ is the base and $n$ is the exponent. It means $a$ is multiplied by itself $n$ times.

• Rules of Exponents:

  1. $a^m \times a^n = a^{m+n}$ (Add powers when multiplying same base).
  2. $\frac{a^m}{a^n} = a^{m-n}$ (Subtract powers when dividing same base).
  3. $(a^m)^n = a^{m \times n}$ (Multiply powers when raising a power to another power).

• Example:
  $2^3 = 2 \times 2 \times 2 = 8$.

• Application:
  Exponents are used in scientific notation, calculating compound interest, and exponential growth or decay problems.

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4. Squares and Square Roots

• Square: The square of a number is that number multiplied by itself.

  • Example: $5^2 = 5 \times 5 = 25$.

• Square Root: The square root of a number is a value that, when multiplied by itself, gives the original number.

  • Example: $\sqrt{25} = 5$, because $5 \times 5 = 25$.

• Key Points:

  • The square of any positive number is positive.
  • The square of any negative number is also positive.
  • The square root of a negative number is not defined in real numbers (complex numbers are used).

• Application:
  Squares and square roots are widely used in geometry (e.g., finding the area of a square), physics, and algebra.

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5. Cubes and Cube Roots

• Cube: The cube of a number is that number multiplied by itself twice more.

  • Example: $3^3 = 3 \times 3 \times 3 = 27$.

• Cube Root: The cube root of a number is a value that, when multiplied by itself twice, gives the original number.

  • Example: $\sqrt[3]{27} = 3$, because $3 \times 3 \times 3 = 27$.

• Key Points:

  • The cube of a positive number is positive.
  • The cube of a negative number is negative.
  • Cube roots are easier to calculate for perfect cubes (numbers like 8, 27, 64, etc.).

• Application:
  Cubes and cube roots are used in volume calculations and many physics equations.

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6. Problems on Time and Work

• Concept: Time and Work problems deal with the relationship between the amount of work done, the time taken, and the rate at which work is done.

• Formula:
  $\text{Work Done} = \text{Rate of Work} \times \text{Time Taken}$.

  • If A can do a work in $n$ days, then the amount of work done by A in 1 day is $\frac{1}{n}$.

• Key Point: When multiple people work together, the total work done in a day is the sum of individual work rates.

• Example:
  If A can complete a task in 5 days and B can do it in 10 days, together they can complete it in: $\frac{1}{5} + \frac{1}{10} = \frac{3}{10}$ of the work in a day, so they’ll finish in $\frac{10}{3}$ days.

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7. Problems on Speed, Distance, and Time

• Concept: These problems involve three variables:

  • Speed: The rate at which an object moves.
  • Distance: The length of the path traveled.
  • Time: The duration of travel.

• Formulas:

  1. $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$.
  2. $\text{Distance} = \text{Speed} \times \text{Time}$.
  3. $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$.

• Example:
  If a car travels 120 km in 2 hours, its speed is $\frac{120}{2} = 60$ km/h.

• Application:
  These formulas are used in motion problems, planning travel schedules, and calculating fuel efficiency.

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8. Matrix

• A matrix is a rectangular array of numbers arranged in rows and columns.
  • A matrix with m rows and n columns is called an $m \times n$ matrix.
• Example:
  $\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$ is a 2x3 matrix.

9. Matrix Addition

• Definition: Matrix addition is the process of adding two matrices by adding their corresponding elements. Matrices must have the same dimensions (same number of rows and columns) for the addition to be valid.

• Formula:
  If $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$ and $B = \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix}$, then:

  $$A + B = \begin{pmatrix} a_{11} + b_{11} & a_{12} + b_{12} \\ a_{21} + b_{21} & a_{22} + b_{22} \end{pmatrix}$$

• Example:
  $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$.

  $$A + B = \begin{pmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{pmatrix} = \begin{pmatrix} 6 & 8 \\ 10 & 12 \end{pmatrix}$$

• Application: Matrix addition is used in areas such as computer graphics, economics (to add cost matrices), and data analysis.

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10. Matrix Multiplication

• Definition: To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. The element in the resulting matrix is found by taking the dot product of rows from the first matrix and columns from the second matrix.

• Formula:
  If $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$ and $B = \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix}$, then:

  $$AB = \begin{pmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \\ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{pmatrix}$$

• Example:
  $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$.

  $$AB = \begin{pmatrix} 1(5) + 2(7) & 1(6) + 2(8) \\ 3(5) + 4(7) & 3(6) + 4(8) \end{pmatrix} = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}$$

• Application: Matrix multiplication is used in physics (e.g., transformations in mechanics), economics (e.g., input-output analysis), and computer science (e.g., 3D transformations in graphics).

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11. Inverse of Matrix

• Definition: The inverse of a matrix $A$ is a matrix $A^{-1}$ such that $A \times A^{-1} = I$, where $I$ is the identity matrix. A matrix must be square (same number of rows and columns) and have a non-zero determinant to have an inverse.

• Formula (for a 2x2 matrix):
  If $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, then its inverse is:

  $$A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$

  where $ad - bc \neq 0$.

• Example:
  $A = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$.

  $$A^{-1} = \frac{1}{(2)(4) - (3)(1)} \begin{pmatrix} 4 & -3 \\ -1 & 2 \end{pmatrix} = \frac{1}{5} \begin{pmatrix} 4 & -3 \\ -1 & 2 \end{pmatrix}$$

  So, $A^{-1} = \begin{pmatrix} \frac{4}{5} & \frac{-3}{5} \\ \frac{-1}{5} & \frac{2}{5} \end{pmatrix}$.

• Application: Inverse matrices are used in solving systems of linear equations, cryptography, and computer graphics.

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12. Application of Matrix in the Business World

• Business Applications:
  • Economics: Input-output models in economics use matrices to analyze how different sectors of an economy interact.
  • Finance: Portfolio theory and risk assessment use covariance matrices to evaluate the performance of multiple investments.
  • Operations Research: Matrices are used in optimization problems, such as minimizing costs or maximizing profit.
  • Computer Science: Matrices are used in algorithms for data encryption, image processing, and 3D modeling.

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13. Concept of Profit & Loss

• Profit: When the selling price (SP) of an item is greater than the cost price (CP), the difference is called profit.

  • Formula:
    $\text{Profit} = \text{SP} - \text{CP}$.

• Loss: When the cost price (CP) is greater than the selling price (SP), the difference is called a loss.

  • Formula:
    $\text{Loss} = \text{CP} - \text{SP}$.

• Profit or Loss Percent:

  • Profit % = $\frac{\text{Profit}}{\text{CP}} \times 100$
  • Loss % = $\frac{\text{Loss}}{\text{CP}} \times 100$

• Example:
  If CP = ₹200 and SP = ₹250,
  Profit = ₹250 - ₹200 = ₹50.
  Profit % = $\frac{50}{200} \times 100 = 25\%$.

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14. Problems on Selling Price and Cost Price

• Selling Price (SP): The price at which an item is sold.

• Cost Price (CP): The price at which an item is bought.

• Formula for Profit and Loss:

  • If there is profit, SP = $\text{CP} + \text{Profit}$.
  • If there is a loss, SP = $\text{CP} - \text{Loss}$.

• Example:
  If CP = ₹150 and profit = ₹30,
  SP = ₹150 + ₹30 = ₹180.

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15. Concept of Percentages

• Definition: A percentage is a fraction with a denominator of 100. It represents a part of a whole.

• Formula:
  Percentage $P = \frac{\text{Part}}{\text{Whole}} \times 100$.

• Example:
  If a student scores 75 marks out of 100, the percentage is: $\frac{75}{100} \times 100 = 75\%$.

• Application: Percentages are widely used in calculating discounts, interest rates, profit and loss, population growth, etc.

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16. Problems on Percentages & Average

• Percentage Problems: These involve calculating how much one value is relative to another in percentage terms.

  • Example: If a product costs ₹500 and is sold at a 20% discount, the new price is: $500 - (20\% \times 500) = 500 - 100 = ₹400$.

• Average: The average of a set of numbers is the sum of all the numbers divided by the total number of values.

  • Formula:
    $\text{Average} = \frac{\text{Sum of Values}}{\text{Number of Values}}$.

• Example:
  Find the average of 5, 10, and 15.
  $\text{Average} = \frac{5 + 10 + 15}{3} = \frac{30}{3} = 10$.

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17. Data Interpretation using Percentage

• Definition: Data interpretation using percentage involves analyzing data (usually presented in tables, graphs, or charts) and calculating percentages to make comparisons or conclusions.

• Example:
  If a company’s revenue in 2020 was ₹100,000 and it increased by 25% in 2021, the new revenue is:
  $\text{New Revenue} = 100,000 + (25\% \times 100,000) = 100,000 + 25,000 = ₹125,000$.

18. Profit and Loss: Problems on Selling Price and Cost Price

• Understanding Selling Price (SP):

  • Selling Price is the final price at which an item is sold.
  • It can be affected by various factors such as discounts, offers, and market demand.

• Understanding Cost Price (CP):

  • Cost Price is the price at which a seller purchases an item before any markup.

Types of Problems:

1. Finding Selling Price:

   • If you know the Cost Price and the profit percentage:
   $$\text{SP} = \text{CP} + \left( \frac{\text{Profit \%}}{100} \times \text{CP} \right)$$
   • Example: If CP = ₹200 and Profit % = 20%: $$\text{SP} = 200 + \left( \frac{20}{100} \times 200 \right) = 200 + 40 = ₹240$$

2. Finding Cost Price:

   • If you know the Selling Price and profit percentage:
   $$\text{CP} = \text{SP} - \left( \frac{\text{Profit \%}}{100} \times \text{CP} \right)$$
   • Example: If SP = ₹300 and Profit % = 25%: $$\text{CP} = \text{SP} - \left( \frac{25}{100} \times \text{CP} \right)$$ Rearranging gives: $$CP = \frac{\text{SP}}{1 + \left( \frac{\text{Profit \%}}{100} \right)} = \frac{300}{1.25} = ₹240$$

3. Finding Profit/Loss:

   • Profit = SP - CP
   • Loss = CP - SP (if CP > SP)
   • Example: If CP = ₹150 and SP = ₹120:
     • Loss = 150 - 120 = ₹30

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19. Percentages: Problems on Percentages and Average

• Solving Percentage Problems:

  • Percentages can be tricky but follow a systematic approach:
  1. Identify the Total: Understand what the whole or total amount is.
  2. Determine the Part: Find out what portion of the total you are dealing with.
  3. Apply the Formula: Use $P = \frac{\text{Part}}{\text{Whole}} \times 100$.

• Average Calculations:

  • Step 1: Add all the values together.
  • Step 2: Divide by the total number of values.
  • Example: Find the average of 10, 15, and 20. $$\text{Average} = \frac{10 + 15 + 20}{3} = \frac{45}{3} = 15$$

Types of Average:

1. Simple Average: As calculated above.

2. Weighted Average: If some values contribute more than others.

   • Formula:
   $$\text{Weighted Average} = \frac{\sum (x_i \times w_i)}{\sum w_i}$$
   • Example: Grades in a course where some assignments weigh more.

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20. Data Interpretation using Percentage: Solving Problems

• Data Interpretation: It involves analyzing data (like charts or tables) to extract useful information.

Steps to Solve Problems:

1. Read the Data Carefully: Understand what the data represents.
2. Calculate Percentages: If data requires comparison, calculate percentages to find differences or similarities.
3. Draw Conclusions: Based on calculations, interpret what the data suggests.

• Example: A survey shows that 60% of people prefer product A over product B. If 1,000 people were surveyed, then:
  • Number of people preferring A = 60% of 1,000 = $0.60 \times 1,000 = 600$.

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21. Concept of Profit Management

• Profit Management: It involves strategizing and managing business operations to maximize profit.

Key Components:

1. Cost Control: Reducing unnecessary expenses.
2. Revenue Enhancement: Increasing sales through marketing and promotions.
3. Efficiency: Streamlining operations to reduce waste.

Formulas:

• Total Profit:
  $\text{Total Profit} = \text{Total Revenue} - \text{Total Costs}$

• Profit Margin:
  $\text{Profit Margin} = \left( \frac{\text{Total Profit}}{\text{Total Revenue}} \right) \times 100$

• Example: If Total Revenue = ₹500,000 and Total Costs = ₹300,000, then:

  • Total Profit = ₹500,000 - ₹300,000 = ₹200,000.
  • Profit Margin = $\left( \frac{200,000}{500,000} \right) \times 100 = 40\%$.

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22. Applications of Percentages in Business

• Financial Analysis: Assessing profitability, margins, and cost structures.

• Market Research: Understanding customer preferences and market trends.

• Pricing Strategies: Setting competitive pricing through percentage discounts or markups.

• Investment Decisions: Evaluating returns on investment using percentage gains/losses.