AIOU 1349 Introduction to Business Mathematics Solved Assignment 1 Spring 2025
AIOU 1349 Assignment 1
Q1 a). Ratio and Proportion: Define "ratio" and provide an example. How is it different from a rate?
What is a ratio? A ratio is a comparison of two quantities that shows the relative sizes of those quantities. It is often written as "a to b" (or a:b) and can also be expressed as a fraction. For example, if there are 3 apples and 5 oranges in a basket, the ratio of apples to oranges is 3:5 or 3/5.
What is a rate? A rate is a special type of ratio where the two quantities being compared have different units. It often involves time, distance, or price. For example, if a car travels 100 kilometers in 2 hours, the rate of speed is 50 kilometers per hour (100 km ÷ 2 hours).
Difference between ratio and rate: The key difference is that ratios compare similar units, while rates compare different units.
Q1 b). Percentage Application: Convert 22.5% into a fraction and decimal. If a student scores 84% in 150 exams, how many marks did they secure?
Convert 22.5% into a fraction and decimal:
Fraction: 22.5% means 22.5/100. Simplifying further:
22.5/100 = 225/1000 = 9/40
Decimal: 22.5% is equivalent to 0.225.
Calculate marks secured from 84% of 150 exams:
84% of 150 means:
(84/100) × 150
= 0.84 × 150 = 126
So, the student secured 126 marks out of 150.
Q1 c). Calculate the profit percentage if the cost price of an item is Rs. 200 and the selling price is Rs. 250.
Calculate the profit percentage if the cost price of an item is Rs. 200 and the selling price is Rs. 250.
To calculate the profit percentage, you can use the formula:
Profit Percentage = (Profit / Cost Price) × 100
First, find the profit:
Profit = Selling Price - Cost Price = 250 - 200 = 50
Now, calculate the profit percentage:
Profit Percentage = (50 / 200) × 100 = 25%
So, the profit percentage is 25%
Q2 a). Straight-Line Depreciation Question: A company purchases a computer for Rs. 30,000. The computer is expected to have a salvage value of Rs. 3,000 and a useful life of 5 years. Calculate the annual depreciation expense using the straight-line method.
Straight-Line Depreciation Question: A company purchases a computer for Rs. 30,000. The computer is expected to have a salvage value of Rs. 3,000 and a useful life of 5 years. Calculate the annual depreciation expense using the straight-line method.
Formula:
Annual Depreciation Expense = (Cost of Asset - Salvage Value) / Useful Life
Applying the formula:
Annual Depreciation Expense = (30,000 - 3,000) / 5 = 27,000 / 5 = 5,400
So, the annual depreciation expense for the computer is Rs. 5,400 per year. This means the company will record Rs. 5,400 as depreciation expense each year for 5 years.
Q2 b). Diminishing Balance Depreciation Question: A piece of machinery is bought for Rs. 80,000 with a depreciation rate of 15% per annum. Calculate the depreciation expense for the first year using the diminishing balance method.
Diminishing Balance Depreciation Question: A piece of machinery is bought for Rs. 80,000 with a depreciation rate of 15% per annum. Calculate the depreciation expense for the first year using the diminishing balance method.
The diminishing balance method calculates depreciation on the book value of the asset rather than its original cost. The formula is:
Depreciation = Book Value × Depreciation Rate
For Year 1:
- Initial cost of the machinery = Rs. 80,000
- Depreciation rate = 15% per annum
Applying the formula:
Depreciation Expense = 80,000 × 0.15
= 12,000
So, the depreciation expense for the first year is Rs. 12,000.
After this depreciation, the new book value of the machinery for the next year would be:
80,000 - 12,000 = 68,000
Q3 a). Define simple interest and provide the formula for its calculation.
Define simple interest and provide the formula for its calculation.
Simple interest is a method used to calculate the interest on a principal amount over a specific period. Unlike compound interest, simple interest does not consider the effect of previous interest earnings—it remains constant throughout the duration of the loan or investment.
Formula for Simple Interest:
SI = P × R × T
Where:
SI = Simple Interest
P = Principal amount (the initial sum of money)
R = Rate of interest (expressed as a percentage per year)
T = Time (in years)
To find the total amount after applying simple interest, you can use:
A = P + SI
Where A represents the final amount after the interest is added.
Q3 b). If a principal amount of Rs. 10,000 is borrowed at 5% p.a. for 3 years, what is the total interest paid?
Q: If a principal amount of Rs. 10,000 is borrowed at 5% p.a. for 3 years, what is the total interest paid?
You're dealing with simple interest, which is calculated using the formula:
Simple Interest = (P × R × T) / 100
where:
- P is the principal amount (Rs. 10,000),
- R is the annual interest rate (5%),
- T is the time in years (3).
Substituting the values:
Simple Interest = (10,000 × 5 × 3) / 100 = 1,500
So, the total interest paid over 3 years is Rs. 1,500.
Q4 a). What is an annuity, and how is it different from a perpetuity?
What is an annuity, and how is it different from a perpetuity?
An annuity is a financial product or investment where a series of payments are made at regular intervals over a specified period. These payments can be made monthly, quarterly, or annually, depending on the terms of the annuity. It’s often used for retirement planning, ensuring a steady income for a set number of years.
A perpetuity, on the other hand, is like an annuity with no end. It provides constant periodic payments indefinitely. Unlike an annuity, which has a finite duration, a perpetuity theoretically continues forever. A common example is certain preferred stock dividends that pay shareholders indefinitely.
In short:
Annuity → Fixed payments for a defined period
Perpetuity → Fixed payments forever
Q4 b). Calculate the future value of an ordinary annuity with annual payments of Rs. 1,000 for 5 years at 6% interest.
Calculate the future value of an ordinary annuity with annual payments of Rs. 1,000 for 5 years at 6% interest.
The future value (FV) of an ordinary annuity is calculated using the formula:
FV = P × ((1 + r)n - 1) / r
Where:
- P = annual payment (Rs. 1,000)
- r = annual interest rate (6% or 0.06)
- n = number of years (5)
Plugging in the values:
FV = 1000 × ((1.06)5 - 1) / 0.06
FV = 1000 × (1.3382 - 1) / 0.06
FV = 1000 × 0.3382 / 0.06
FV = 1000 × 5.6367
FV ≈ Rs. 5,637
So, after 5 years, the future value of the ordinary annuity would be approximately Rs. 5,637.
Q5 a). Solve the linear equations for x:
i). 3x + 5 = 20
We start with the equation:
3x + 5 = 20
Step 1: Isolate 3x
Subtract 5 from both sides:
3x = 15
Step 2: Solve for x
Divide both sides by 3:
x = 5
So, the solution is x = 5.
ii). 4x + 2(3x - 5) = 8
Solve the linear equation for x: 4x + 2(3x - 5) = 8
We start with the given equation:
4x + 2(3x - 5) = 8
Step 1: Expand the parentheses
Distribute the 2 into (3x - 5):
4x + 6x - 10 = 8
Step 2: Combine like terms
10x - 10 = 8
Step 3: Isolate x
Add 10 to both sides:
10x = 18
Step 4: Solve for x
Divide by 10:
x = 18/10
x = 9/5
So, the solution is x = 9/5 or x = 1.8.
Q5 b). Quadratic Equation: What is a quadratic equation? Provide the standard form and an example.
What is a quadratic equation?
A quadratic equation is a polynomial equation of degree 2, which means the highest power of the variable is 2.
The standard form of a quadratic equation is:
ax² + bx + c = 0
where:
- a, b, c are constants (with a ≠ 0).
- x is the unknown variable.
Example:
Consider the equation:
2x² + 3x - 5 = 0
Here, a = 2, b = 3, and c = -5. This equation represents a quadratic equation and can be solved using methods like factoring, the quadratic formula, or completing the square.
Solve x² + 2x - 3 = 0 using factorization.
Step 1: Find two numbers that multiply to -3 (the constant term) and add up to 2 (the coefficient of x).
The numbers 3 and -1 satisfy these conditions because:
3 × (-1) = -3
3 + (-1) = 2
Step 2: Rewrite the middle term using these numbers:
x² + 3x - x - 3 = 0
Step 3: Factor by grouping:
(x² + 3x) - (x + 3) = 0
Factor out common terms:
x(x + 3) - 1(x + 3) = 0
Factor (x + 3):
(x - 1)(x + 3) = 0
Step 4: Solve for x:
x - 1 = 0 → x = 1
x + 3 = 0 → x = -3
So, the solutions are x = 1 and x = -3.
Q5 c). Simultaneous Equations: Solve using matrices:
3x + 4y = 24
2x - 3y = 15
We are given the following system of linear equations:
1. Represent the system in matrix form:
The system can be represented as \(AX = B\), where:
\( A = \begin{bmatrix} 3 & 4 \\ 2 & -3 \end{bmatrix} \), \( X = \begin{bmatrix} x \\ y \end{bmatrix} \), and \( B = \begin{bmatrix} 24 \\ 15 \end{bmatrix} \)
So, the matrix equation is:
2. Find the determinant of the coefficient matrix (A):
The determinant of A is:
3. Find the inverse of the coefficient matrix (\(A^{-1}\)):
The inverse of A is:
4. Solve for the variables (X):
We use the formula \(X = A^{-1}B\):
Performing the matrix multiplication:
Therefore, the solution to the system of equations is:
\( x = \frac{132}{17} \) and \( y = \frac{3}{17} \)
AIOU 1349 Introduction to Business Mathematics Solved Assignment 2 Spring 2025
AIOU 1349 Assignment 2
Q1 a). Function Definition: Explain the vertical line test and why y² = x is not a function.
Vertical Line Test:
A function is a relation where each input (x-value) has exactly one output (y-value).
To check this visually, imagine drawing vertical lines (parallel to the y-axis) across the graph.
If any vertical line intersects the graph at more than one point, the relation is not a function.
Why y² = x is NOT a Function:
Rewriting y in terms of x, we get:
y = ±√x
This equation gives two possible values of y for each x—one positive and one negative (except when x = 0).
Graphing this equation shows that a vertical line passes through two points (one above the x-axis and one below).
Since this violates the vertical line test, y² = x is not a function.
Q1 b). Graph Analysis: Sketch the graph of f(x) = 2x + 1 and identify its x - intercept and y - intercept.
X-Intercept (Where f(x) = 0):
The x-intercept occurs when the function equals zero:
0 = 2x + 1
Solving for x:
x = -1/2
So the x-intercept is (-1/2, 0).
Y-Intercept (Where x = 0):
The y-intercept occurs when x = 0:
f(0) = 2(0) + 1 = 1
So the y-intercept is (0, 1).
Graph
Q2 a). Matrix Types: Differentiate between a scalar matrix and an identity matrix. Provide examples.
Matrix Types: Scalar Matrix vs. Identity Matrix
A scalar matrix and an identity matrix are both special types of square matrices, but they differ in the values of their diagonal elements. Here's a breakdown of their differences:
Scalar Matrix:
Definition: A scalar matrix is a square matrix in which all the principal diagonal elements are equal to some non-zero scalar value, and all the other elements are zero.
Form:
Key Feature: The diagonal entries are the same non-zero number.
Example of a Scalar Matrix:
A 3x3 scalar matrix with a scalar value of 7 would look like this:
Identity Matrix:
Definition: An identity matrix is a square matrix in which all the principal diagonal elements are equal to 1, and all the other elements are zero. It is a specific case of a scalar matrix where the scalar value is 1.
Form:
Key Feature: The diagonal entries are all 1.
Example of an Identity Matrix:
A 3x3 identity matrix is:
Summary of Key Differences:
| Feature | Scalar Matrix | Identity Matrix |
|---|---|---|
| Diagonal Elements | All equal to some non-zero scalar (\(k\)) | All equal to 1 |
| Other Elements | All equal to 0 | All equal to 0 |
| Relationship | A scalar matrix is a multiple of an identity matrix (\(kI\)) | An identity matrix is a special case of a scalar matrix (where \(k=1\)) |
In essence, an identity matrix is a scalar matrix where the scalar is specifically 1. Therefore, every identity matrix is a scalar matrix, but not every scalar matrix is an identity matrix (unless the scalar value is 1).
Q2 b). Determinants: Find the determinant of the below matrix. Is this matrix singular? $$ \begin{pmatrix} 2 & 3 \\ 1 & -5 \end{pmatrix} $$
Determinant and Singularity Check
Let's find the determinant of the given matrix:
The determinant of a 2 x 2 matrix see below example, is calculated as ad - bc. \begin{pmatrix} a & b \\ c & d \end{pmatrix}
So, for the given matrix, the determinant is:
$$\det(A) = (2)(-5) - (3)(1)$$
$$\det(A) = -10 - 3$$
$$\det(A) = -13$$
The determinant of the matrix is: -13
A matrix is considered singular if its determinant is equal to zero. Since the determinant of the given matrix is -13, which is not zero, the matrix is not singular.
Q3 a). Matrix Operations: Add \( \begin{bmatrix} 2 & 6 \\ 5 & 8 \end{bmatrix} \) and \( \begin{bmatrix} 7 & 0 \\ 4 & 3 \end{bmatrix} \)
Matrix Addition
To add two matrices, we add their corresponding elements. This is only possible if the matrices have the same dimensions. In this case, both matrices are 2x2, so we can add them.
Given Matrices:
Step-by-step Solution:
Step 1: Write the formula for matrix addition
The formula to add matrices A and B is:
Step 2: Add the corresponding elements
Substitute the values from matrices A and B into the formula:
Step 3: Calculate the sums
Perform the addition for each element:
Result:
Therefore, the sum of the two matrices A and B is:
Q3 b). Inverse Matrix: Find the inverse of \( \begin{bmatrix} 5 & 9 \\ 4 & 1 \end{bmatrix} \)
Inverse of Matrix A
Given Matrix:
\[ A = \begin{bmatrix} 5 & 9 \\ 4 & 1 \end{bmatrix} \]
Step 1: Compute the Determinant
\[ \text{det}(A) = (5 \times 1) - (9 \times 4) \]
\[ = 5 - 36 = -31 \]
Step 2: Check if the Inverse Exists
The determinant is nonzero (\(-31\)), so the inverse exists.
Step 3: Compute the Adjugate Matrix
The adjugate matrix is obtained by swapping \( a \) and \( d \), and negating \( b \) and \( c \):
\[ \text{Adj}(A) = \begin{bmatrix} 1 & -9 \\ -4 & 5 \end{bmatrix} \]
Step 4: Multiply by \( \frac{1}{\text{det}(A)} \)
\[ A^{-1} = \frac{1}{-31} \begin{bmatrix} 1 & -9 \\ -4 & 5 \end{bmatrix} \]
Step 5: Compute the Final Inverse Matrix
\[ A^{-1} = \begin{bmatrix} -\frac{1}{31} & \frac{9}{31} \\ \frac{4}{31} & -\frac{5}{31} \end{bmatrix} \]
Q4 a). Conversion: Convert 101101₂ to decimal.
Convert binary 101101₂ to decimal:
Each digit in a binary number represents a power of 2, starting from right to left:
101101₂ = (1 × 2⁵) + (0 × 2⁴) + (1 × 2³) + (1 × 2²) + (0 × 2¹) + (1 × 2⁰)
Breaking it down:
(1 × 32) + (0 × 16) + (1 × 8) + (1 × 4) + (0 × 2) + (1 × 1)
= 32 + 0 + 8 + 4 + 0 + 1
= 45
Final Answer: 101101₂ = 45₁₀
Q4 b). Binary Arithmetic: Add 1101₂ and 1011₂
Binary Addition: 1101₂ + 1011₂
Here's a step-by-step explanation of how to add the binary numbers 1101₂ and 1011₂:
Step 1: Set up the addition
Write the two binary numbers vertically, aligning the rightmost digits:
1101₂
+ 1011₂
-------
Step 2: Add the rightmost bits (2⁰ place)
Add the digits in the rightmost column: 1 + 1 = 10₂. Write down the 0 and carry-over the 1.
¹
1101₂
+ 1011₂
-------
0₂
Step 3: Add the next bits (2¹ place)
Add the digits in the next column, including the carry-over: 0 + 1 + 1 = 10₂. Write down the 0 and carry-over the 1.
¹¹
1101₂
+ 1011₂
-------
00₂
Step 4: Add the next bits (2² place)
Add the digits in the next column, including the carry-over: 1 + 0 + 1 = 10₂. Write down the 0 and carry-over the 1.
¹¹¹
1101₂
+ 1011₂
-------
000₂
Step 5: Add the leftmost bits (2³ place)
Add the digits in the leftmost column, including the carry-over: 1 + 1 + 1 = 11₂. Write down the 11.
¹¹¹
1101₂
+ 1011₂
-------
11000₂
Result
Therefore, the sum of 1101₂ and 1011₂ is: 11000₂
Q5 a). Application: Why is the binary system critical in computing? Explain with an example.
Why is the binary system critical in computing? Explain with an example.
The binary system is the foundation of computing because it simplifies complex operations into a language that electronic devices can easily understand. Computers process data using tiny electronic switches called transistors, which can either be ON or OFF. This aligns perfectly with binary, which consists of only two digits: 0 and 1.
Example:
Consider storing a simple text message: "Hi" on a computer. Each character in "Hi" is translated into binary using ASCII encoding:
- H → 01001000
- i → 01101001
These binary sequences are then stored and processed by the computer using electrical signals—0 represents OFF, and 1 represents ON. This system allows computers to perform calculations, store data, and execute commands efficiently.
Binary’s simplicity makes it ideal for reliable, high-speed computing. Even complex operations like graphics rendering and artificial intelligence ultimately rely on this fundamental system.
Q5 b). Discuss how the letter A is 01000001 in ASCII (a binary-based encoding).
Discuss how the letter A is 01000001 in ASCII (a binary-based encoding).
The letter A is represented by 01000001 in ASCII because ASCII (American Standard Code for Information Interchange) is a character encoding standard that assigns unique binary values to characters.
How It Works:
- ASCII assigns numbers to characters, with uppercase A being 65 in decimal.
- The binary equivalent of 65 is 01000001 (using 8-bit binary representation).
- Each bit in 01000001 corresponds to a power of 2, making up the decimal value 65:
0 × 128 + 1 × 64 + 0 × 32 + 0 × 16 + 0 × 8 + 0 × 4 + 0 × 2 + 1 × 1 = 65
Computers use binary-based encoding like ASCII to store and process text data efficiently. Since computers only understand 0s and 1s, this system allows seamless translation between human-readable text and machine instructions.
Q5 c). Adding Two Numbers in Binary.Suppose a computer adds two decimal numbers 3 and 2. Find the sum of these two numbers in Binary.
Adding Two Numbers in Binary
To add two decimal numbers (3 and 2) and find their sum in binary, follow these steps:
1. Convert Decimal to Binary:
3 in binary: 11₂
2 in binary: 10₂
2. Perform Binary Addition:
11 (3 in binary) + 10 (2 in binary) ------ 101₂ (Sum in binary)
Just like decimal addition, add from right to left:
1 + 0 = 1
1 + 1 = 10 (which means carry 1 to the next column)
Thus, 3 + 2 = 5, and in binary, 5 is represented as 101₂.
No comments:
Post a Comment