a. Find the present value of 350 000 which is due after 10 years if the interest rate is 8% per year
i. compounded annually, or (4 marks) ii. compounded continuously.

The Area of the rhombus with base 21 cm and height 17.3 cm is 363.3 cm².

What is area of Rhombus?

To calculate the base of the rhombus, we will follow the steps below;

write the formula for calculating the area of a rhombus

area of a rhombus = b × h

where h is the height of the rhombus and b is the base of the rhombus

Given:

Base = 21 cm

Height = 17.3 cm

We know area of Rhombus =  base x height

                                             = 21 x 17.3

                                             = 363.3 cm²

So, the area of Rhombus is 363.3 cm².

The summary table of the function obtained using differential calculus, and the graph of the function are presented as follows;

[tex]\begin{tabular}{|c|c|c|} \cline{1-3} Interval & Increasing/Decreasing &Concave Up/Down\\ \cline{1-3}(-\infty, 0)&Decreasing&Concave Up \\ \cline{1-3} (0, 3/2) & Decreasing & Concave Down \\ \cline{1-3} (3/2, \infty) & Increasing& Concave Up\\ \cline{1-3}\end{tabular}[/tex]

Please find attached, the graph of the function, created using MS Excel

Differential calculus is the part of calculus that is concerned with the rate of change of quantities.

The steps to determine the intervals where f(x) = x⁴ - 2·x³ is increasing or decreasing, or constant are presented as follows;

The derivative of the function is first found using calculus, differentiation as follows;

f'(x) = 4·x³ - 6·x²

The critical points at which the slope of the graph changes from increasing are found at the points where f'(x) = 0 as follows;

f'(x) = 0 = 4·x³ - 6·x²

4·x³ - 3·x² = 0

2·x²·(2·x - 3) = 0

Therefore, at the critical points, x = 0, and x = 3/2

The first derivative test can be used to determine the interval the function is increasing or decreasing as follows;

f(x) is increasing at x < a, if f'(x) > 0 for x < a and f(x) is decreasing at x > a if f(x) < 0 at x > a, where a is a critical point

f(x) is decreasing at x < a, if f'(x) < 0 for x < a and f(x) is increasing at x > a if f(x) > 0 at x > a,

The concavity of the function at x = a if f'(x) = 0 at x = a can be determined using the second derivative as follows;

f''(x) = 12·x² - 12·x

f''(x) > 0 at x = a if f(x) is concave up

f''(x) < 0 at x = a if f(x) is concave down

When a = 0, we get;

f''(0) = 12 × 0² - 12 × 0 = 0

f''(0.5) = 12 × 0.5² - 12 × 0.5 = -3 (Concave down)

f''(-0.5) = 12 × (-0.5)² - 12 × (-0.5) = 9 (Concave up)

f''(0) = 0 at x = 0, therefore, x = 0 is an inflection point

f'(-1) = 4·(-1)³ - 6·(-1)² = -10

f'(1) = 4·(1)³ - 6·(1)² = -2

Therefore, the graph changes from decreasing and is also decreasing at x > 0

Similarly we get;

f'(2) = 4·(2)³ - 6·(2)² = 8 (Increasing)

Therefore;

f'(1) = 4·(1)³ - 6·(1)² = -2

There

f''(3/2) = 12 × (3/2)² - 12 × (3/2) = 9

The second derivative at x = 3/2 = 9 > 0, therefore;

The function, f(x) is concave up at x = 3/2

Therefore, we get the following table;

Please find attached the graph of the function created using MS Excel

and the following points;

x   [tex]{}[/tex]      f(x)

-0.9    [tex]{}[/tex]2.1141

-0.8 [tex]{}[/tex]   1.4336

-0.1 [tex]{}[/tex]    0.0021

0 [tex]{}[/tex]       0

0.4 [tex]{}[/tex]    -0.1024

0.5 [tex]{}[/tex]    -0.1875

1.5 [tex]{}[/tex]     -1.6875

2 [tex]{}[/tex]        0

2.5 [tex]{}[/tex]     7.8125

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The experimental probability is0.33 or 33% to dock .

The probability calculated without performing any experiments is known as theoretical probability.  It can be characterised as the proportion of favourable results to all other possible outcomes.  No experiment is required to determine the theoretical probability.  As an alternative, we should be aware of the circumstances in order to determine the likelihood that an event will occur.

The actual outcome of an experiment is known as the experimental probability, which may differ from the theoretical probability.

In this instance, the marina 1 is home to 6 boats from Maine and 12 vessels from other states.

Hence,

6/(6+12) = 0.33 or 33%

the chance that the next boat to port comes from Maine.

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Answer:

1/3

Step-by-step explanation:

The result of the matrix multiplication for  F³  is determined as [[1,0],[-2,3]].

Matrix multiplication is a mathematical operation that takes two matrices and produces a new matrix.

Given two matrices A and B, the resulting matrix C is obtained by multiplying the elements of the rows of matrix A with the corresponding elements of the columns of matrix B, and summing the products.

To calculate F³, we need to multiply the matrix F by itself three times:

F² = F * F = [[1,0],[2,-3]] * [[1,0],[2,-3]] = [[11+02,10+0(-3)],[21+(-3)2,20+(-3)(-3)]] = [[1,0],[-4,9]]

F³ = F * F² = [[1,0],[2,-3]] * [[1,0],[-4,9]] = [[11+02,10+09],[21+(-3)(-4),2*0+(-3)*9]] = [[1,0],[-2,3]]

Therefore, F³ = [[1,0],[-2,3]].

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After answering the provided question, we can conclude that So the experimental probability that the next freshman will register for earth science is 23/66.

Probability is a measure of how likely an event is to occur. It is represented by a number between 0 and 1, with 0 representing an unlikely event and 1 representing an unavoidable event. Switching a fair coin and coin flips has a probability of 0.5 or 50% because there are two equally likely outcomes. (Heads or tails). Probability theory is a branch of mathematics that studies random events rather than their properties. It is applied in many fields, including statistics, finance, science, and engineering.

There are a total of 77 + 5 + 4 + 46 = 132 freshmen who have registered for a science class. Out of these, 46 have registered for earth science.

P(earth science) = number of freshmen who registered for earth science / total number of freshmen who registered for a science class

P(earth science) = 46 / 132

Simplifying the fraction, we get:

P(earth science) = 23 / 66

So the experimental probability that the next freshman will register for earth science is 23/66.

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According to the given rate the total profit made by the company in the first year is R240,000.

The problem statement does not provide enough information to determine the rate of return or interest rate. In the absence of this information, it is not possible to calculate the rate of return or interest rate earned on the capital contributions or on the profit made by the company.

According to the agreement, the profit made by the company should be divided among the partners in the ratio of their capital contributions, which in this case is 225,000:150,000:75,000 or 3:2:1.

Let's assume that the total profit made by the company in the first year is P. Dee's share of the profit is R40,000, which is 1/6 of the total profit, since her capital contribution is 1/6 of the total capital.

So, we can set up the equation:

1/6 * P = R40,000

Multiplying both sides by 6, we get:

P = R240,000

Therefore, according to the given rate the total profit made by the company in the first year is R240,000.

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The equation of the oscillating circle of the curve y = ¹/₄ x² +2 at point (2,1) is (x - 5/2)² + (y - 1)² = 1/4.

To find the equation of the oscillating circle of the curve y = ¹/₄ x² +2 at point (2,1), we need to follow these steps:

Find the first and second derivatives of the given curve:

y = ¹/₄ x² + 2

y' = ¹/₂ x

y'' = ¹/₂

Use the first and second derivatives to find the equation of the tangent line to the curve at point (2,1):

y - 1 = 1/2(x - 2)

y = 1/2 x + 0.5

Find the perpendicular bisector of the tangent line at point (2,1):

The slope of the tangent line is 1/2, so the slope of the perpendicular bisector is -2.

y - 1 = -2(x - 2)

y = -2x + 5

Find the intersection of the perpendicular bisector with the x-axis:

When y = 0, -2x + 5 = 0, so x = 5/2.

Find the center of the oscillating circle:

The center of the oscillating circle is (5/2,1).

Find the radius of the oscillating circle:

The radius of the oscillating circle is the distance between the center and the point of tangency, which is:

r = √[(2 - 5/2)² + (1 - 1)²]

r = [1/4 + 0] = 1/2.

Write the equation of the oscillating circle in standard form:

The equation of the oscillating circle is:

(x - 5/2)² + (y - 1)² = (1/2)².

Simplifying, we get:

(x - 5/2)² + (y - 1)² = 1/4.

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The complete question is below:

Find the equation of oscillating circle of curve y=1/4 x^2 +2 at (2,1)

The function notation  formula is given as g(x) = f(x/2)

Function notation is a way of representing a mathematical function using symbols and variables. It is a shorthand way of writing the rule or formula for a function, which specifies how the input values (usually denoted by x) are transformed into output values (usually denoted by y or f(x)).

In function notation, a function is typically represented by a letter (usually f, g, or h) followed by parentheses that enclose the input value or variable, such as f(x), g(t), or h(y). This notation indicates that the function takes an input value (x, t, or y) and produces a corresponding output value (f(x), g(t), or h(y)).

Assuming we want to horizontally stretch the graph of f(x) by a factor of 2 to obtain the graph of g(x), the function notation for g(x) can be written as:

g(x) = f(x/2)

In other words, we plug in x/2 instead of x in the formula for f(x) to get the corresponding value of g(x) that is horizontally stretched by a factor of 2. This is because when we substitute x/2 in f(x), it effectively multiplies the input (x) by 2, which horizontally stretches the graph by a factor of 2.

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Step-by-step explanation:

To solve the system of equations:

2y = -2x + 6 ----(1)

x - 5y = -15 ----(2)

We can use substitution method or elimination method. Here, we will use elimination method to eliminate y.

Multiplying equation (1) by -5, we get:

-10y = 10x - 30 ----(3)

Adding equations (2) and (3), we get:

-15y = -45

Dividing by -15 on both sides, we get:

y = 3

Substituting y = 3 in equation (2), we get:

x - 5(3) = -15

x - 15 = -15

Adding 15 on both sides, we get:

x = 0

Therefore, the solution of the system of equations is (x, y) = (0, 3).

To solve the system of equations:

5x + y = 2 ----(1)

20x + 3y = -4 ----(2)

We can use substitution method or elimination method. Here, we will use elimination method to eliminate y.

Multiplying equation (1) by -3, we get:

-15x - 3y = -6 ----(3)

Adding equations (2) and (3), we get:

5x = -10

Dividing by 5 on both sides, we get:

x = -2

Substituting x = -2 in equation (1), we get:

5(-2) + y = 2

-10 + y = 2

Adding 10 on both sides, we get:

y = 12

Therefore, the solution of the system of equations is (x, y) = (-2, 12).

Answer:

1)   (0, 3)

2)  (-2, 12)

Step-by-step explanation:

1)

2y = -2x + 6     →    y = -x + 3

x - 5y = -15

Substitute the simplified first equation into the second equation and solve for x.

x - 5( -x + 3 ) = -15

x + 5x - 15 = -15

6x = 0

x = 0

Substitute into first equation to get y.

y = -(0) + 3

y = 3

The solution to the system is the coordinate (0, 3).

2)

5x + y = 2

20x + 3y = -4

Solve the first equation for y.

y = -5x + 2

Substitute this into the second equation and solve for x.

20x + 3 ( -5x + 2 ) = -4

20x - 15x + 6 = -4

5x = -10

x = -2

Substitute this value on the first equation to get y.

y = -5( -2 ) + 2

y = 10 + 2

y = 12

The solution to the system is the coordinate (-2, 12).

The unit of labor required to produce one unit of output for the two countries are 2 and 1/3.

The world automobiles-to-rice relative supply curve slopes upward, indicating that as the world price ratio of rice to automobiles rises, so will the relative supply of automobiles to rice.

To draw the world automobiles-to-rice relative supply curve, calculate the relative supply of automobiles to rice for each country. The relative supply is the ratio of the quantity of automobiles produced to the quantity of rice produced, multiplied by the ratio of the world price of rice to the world price of automobiles. Express this mathematically as:

Relative supply of automobiles to rice = (QA/RA)/(QR/RR) x (PR/PA)

where QA and RA are the quantities of automobiles and rice produced by a country, QR and RR are the quantities of rice produced by a country, and PR and PA are the world prices of rice and automobiles, respectively.

Using the unit labor requirements given in the table, calculate the relative supply of automobiles to rice for China and India as follows:

Relative supply of automobiles to rice for China = (1/4)/(1/2) x (2/1) = 2

Relative supply of automobiles to rice for India = (1/12)/(1/2) x (2/1) = 1/3

We can then plot these points on a graph, where the x-axis represents the relative supply of automobiles to rice and the y-axis represents the world price ratio of rice to automobiles. Assume that the world price ratio of rice to automobiles is initially set at 1, which means that the price of rice is equal to the price of automobiles. We can then draw a line connecting the two points, which represents the world automobiles-to-rice relative supply curve.

The kink point of the relative supply curve occurs when the relative supply of automobiles to rice is equal to 2, which is the point where China switches from being a net exporter of automobiles to a net importer of automobiles. At this point, the world price ratio of rice to automobiles increases, which means that the price of rice increases relative to the price of automobiles. As a result, China reduces its production of automobiles and increases its production of rice, which reduces its relative supply of automobiles to rice.

The second segment of the relative supply curve occurs when the relative supply of automobiles to rice is less than 1/3, which is the point where India switches from being a net exporter of automobiles to a net importer of automobiles. At this point, the world price ratio of rice to automobiles decreases, which means that the price of automobiles increases relative to the price of rice. As a result, India reduces its production of rice and increases its production of automobiles, which increases its relative supply of automobiles to rice.

The world automobiles-to-rice relative supply curve is upward sloping, which means that as the world price ratio of rice to automobiles increases, the relative supply of automobiles to rice also increases. This reflects the fact that countries specialize in the production of goods for which they have a comparative advantage, and trade with other countries to obtain goods for which they do not have a comparative advantage.

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Answer:

the answer is : 2

Given the above conditions with regard to Rolle's Theorem, the answers are:

Therefore, the solution is:

F(0) = 0

f(4) = (-8) e^4

C = 1

To find F(0), we can simply substitute x=0 into the expression for f(x):

f(0) = (0^2 - 4(0)) e^0 = 0

To find f(4), we can substitute x=4:

f(4) = (4^2 - 4(4)) e^4 = (-8) e^4

Next, we need to find the critical point(s) of f(x) on the interval [0,4]. To do this, we find the derivative of f(x) and set it equal to zero:

f'(x) = (2x-4) e^x + (x^2-4x) e^x = (x^2-2x) e^x

Setting f'(x) equal to zero, we have:

(x^2-2x) e^x = 0

Since e^x is never zero, we can divide both sides by e^x to get:

x^2 - 2x = 0

Factoring out x, we have:

x(x-2) = 0

Therefore, the critical points on [0,4] are x=0 and x=2.

Since both critical points are in the open interval (0,4), Rolle's Theorem applies. This means there exists a number c in (0,4) such that f'(c) = 0. To find this value of c, we can simply take the average of the critical points:

c = (0 + 2)/2 = 1

Therefore, the solution is:

F(0) = 0

f(4) = (-8) e^4

C = 1

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The next two numbers in the sequence are 35 and 107.

The given number chain seems to be following the pattern of adding the previous number multiplied by a specific factor.

To find the next two numbers in the sequence, we will use the same pattern.

1 ► 3 ► 11 ► ?

To get the next number in the sequence, we multiply the previous number by 3 and add 2.

So, the next number in the sequence would be:

11 x 3 + 2 = 35

Therefore, the next number in the sequence is 35.

To get the number after 35, we apply the same pattern:

35 x 3 + 2 = 107

Therefore, the next number in the sequence is 107.

Hence, the next two numbers in the sequence are 35 and 107.

So the complete number sequence is: 1 ► 3 ► 11 ► 35 ► 107 ►

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Answer:

C. 6 inches

Step-by-step explanation:

Base = 25 inches

Area = 75 square inches

Using the formula

A = HbB/2

Solving for Hb

Hb = 2 A/b = 2 . 75/25 = 6 inches

Answer:

shorts/ and a t-shirt

Step-by-step explanation:

25° c is equal to 77° f

Answer: 22 sophomores were surveyed

Step-by-step explanation:

Answer:

The answer is (d) 200 m².

Step-by-step explanation:

Let's denote the width of the rectangle as w. Then the length of the rectangle is twice the width, which means the length is 2w.

The perimeter of a rectangle is given by the formula:

P = 2l + 2w

Substituting the values, we get:

60 = 2(2w) + 2w

Simplifying the equation, we get:

60 = 6w

Dividing both sides by 6, we get:

w = 10

So the width of the rectangle is 10 m and the length is twice the width, which is 20 m.

The area of a rectangle is given by the formula:

A = lw

Substituting the values, we get:

A = 10 x 20

Simplifying the equation, we get:

A = 200

Therefore, the area of the rectangle is 200 m².

Answer:23200cm^3

Step-by-step explanation:

we are going to watch this figure as 2 different prisms. 1st-parallelepiped

and 2nd-trapezoidal prism;

The volume of the parallelepiped is equal to a*b*c which is the product of its sides(width, height, and length) and in this case, it is equal to V1=a*b*c=10*12*80(because the trapezoidal prisms side is parallelogram and facing sides of parallelogram are equal so c=80cm)=9600cm^3;

V2=Area of base*h

area of base is equal to area of transpazoid which is

A=((a+b)/2)*h=(12+22)/2*10=34/2*10=17*10=170cm^2

and as i said earlier h=80 because its a parallelogram so,

V2=170+80=13600cm^3

V=V1+V2=9600+13600=23200cm^3

The total garden area obtained by adding the new area of the garden is obtained as 36  sq. ft.

Your polygon's four sides must be two congruent pairings in order to have two pairs with parallel sides.

The length of the left and right sides, as well as the base and top, will be equal. A four-sided polygon with two sets of parallel, congruent sides with four internal angles of 90° each is required to be a rectangle.

Given data:

current area of garden = 24 square feet.

Coordinates of other garden:

(−17, 15), (−20, 15), (−17, 11), and (−20, 11)

Length of new garden = -20 + 17 = 3 ft.

Width of new garden = 15 - 11 = 4 ft.

Area of new garden = length * width

Area of new garden = 3ft * 4 ft

Area of new garden = 12 sq. ft.

total area of the garden = current area +  new garden area

total area of the garden = 24 sq. ft. + 12 sq. ft.

total area of the garden = 36  sq. ft.

Thus, the total area of the garden obtained by adding the new area of the garden is obtained as 36  sq. ft.

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Answer:

4.85423

Step-by-step explanation:

D equals C/PI

Where D=Diameter and C=Circumference and PI=PI(3.14)

The coordinates of the final point after the reflections across the x-axis twice is (6, -4)

Reflecting a point across the x-axis means that we keep the x-coordinate the same, but change the sign of the y-coordinate.

So reflecting (6,-4) across the x-axis gives us the point (6,4).

Reflecting this result again across the x-axis means that we keep the x-coordinate the same, but change the sign of the y-coordinate again.

So reflecting (6,4) across the x-axis gives us the point (6,-4).

Therefore, the final point after both reflections is (6, -4).

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We can conclude after answering the provided question that Because each element in the domain is paired with exactly one element in the range, this mapping diagram represents a function.

The domain of a function is the range of possible values that it can accept. These numbers represent the x-values of a function such as f. (x). The domain of a function is the range of possible values that can be used with it. This set includes the value returned by the method after inserting the x value. A function with y as the dependent variable and x as the independent variable has the formula y = f. (x). When a single value of y can be produced successfully from a value of x, that value of x is said to be in the function's domain.

Each element in the domain (set of x values) is paired with exactly one element in the range (set of y values), as represented by arrows connecting the two sets in the mapping diagram. For example, the domain element 2 is paired with the range element 4, and the domain element -1 is paired with the range element 1.

Because each element in the domain is paired with exactly one element in the range, this mapping diagram represents a function. The function is not one-to-one because multiple domain elements are paired with the same range element (for example, both -2 and 3 are paired with 1).

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Since the limit is just 33 μg/ft³, it exceeded the given limit. Thus, with this condition, there would be strong odor and irritation of the eyes.

A cube's volume is defined as the total number of cubic units that it takes up in its entirety. To calculate the cube's volume, we need to know how long each cube side is. In order to compute volume, use the following equation: Side x Side x Side equals volume.

33 μg/ft³ is the maximum allowed formaldehyde concentration. Our foundation should be this. It is now assumed that wood panelling generates 3,365 μg of formaldehyde per square foot. The wood panelling measures 5 feet by 6 feet; therefore the area is equal to 30 ft². As a result, the formaldehyde emission is calculated as follows: Emitted formaldehyde = (3,365 μg/ft²) (30 ft²) = 100,950 μg

The size of the room is then measured:

V = 10 ft × 10 ft × 6 ft = 600 ft³

Let's compare the ratio to our basis:

Ratio = 100,950 μg/600 ft³ = 168.25 μg/ft³

Since the limit is just 33 μg/ft³, it exceeded the given limit. Thus, with this condition, there would be strong odor and irritation of the eyes.

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Answer:

IT IS B hope this helped

Step-by-step explanation:

The club made 20 pieces of pasta.

What is the equivalent expression?

Equivalent expressions are expressions that perform the same function despite their appearance. If two algebraic expressions are equivalent, they have the same value when we use the same variable value.

The cafeteria donated four pies, and each pie was cut into four pieces. Therefore, the total number of pieces donated is:

4 pies × 4 pieces/pie = 16 pieces

The total number of pieces to sell is given as 36. So the number of pieces the club made is:

Number of pieces made = Total number of pieces to sell - Number of pieces donated

Number of pieces made = 36 - 16

Number of pieces made = 20

Therefore, the club made 20 pieces of pasta.

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Answer:

Step-by-step explanation:

Solve from the inside outwards:

         [tex]g(-4)=\frac{-4}{2} =-2[/tex]

         [tex]f(g(-4))=f(-2)=10\times(-2)-4=-24[/tex]

Solution:  [tex]f(g(-4))=-24[/tex]

The volume of the solid whose base is a triangle with vertices (0,0), (0,3), and (5,0) is 25π/12 cubic units.

What is a triangle?

A triangle is a closed geometric shape that is formed by connecting three line segments. These line segments are called sides, and the points where they meet are called vertices. A triangle has three sides, three angles, and three vertices.

To start, let's graph the triangle to get a better understanding of the problem:

(0,3)   *

       |\

       | \

       |  \

       |   \

       |    \

(0,0)   *-----*-----> x

        (5,0)

The height of this slice is given by the line from the point (x,0) to the point (0,3), which has equation y = 3/5 * x + 0. The radius of the semicircle is half the height of the slice, which is given by the equation r = 3/10 * x.

The area of a semicircle is πr²/2, so the volume of this slice is:

V(x) = π * (3/10 * x)² / 2 * dx

To find the total volume of the solid, we need to integrate this expression over the range of x values that covers the entire base of the solid, which is from x=0 to x=5:

V = ∫₀₅ π * (3/10 * x)² / 2 dx

V = π/20 * ∫₀₅ x² dx

V = π/20 * [x³/3] from 0 to 5

V = π/20 * (5³/3)

V = π/4 * (25/3)

V = 25π/12

Therefore, the volume of the solid is 25π/12 cubic units.

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Step-by-step explanation:

रामले 3 वर्षको लागि 12% प्रतिवर्ष साधारण ब्याजदरले रु. 50000 लगानी गरेको थियो। यसको आधारमा उनले प्राप्त गरेको ब्याज हो:

Simple Interest (S.I.) = (P × R × T) / 100

यहाँ, P = रु. 50000 हो, R = 12% = 12/100 = 0.12 हो, र T = 3 वर्ष हो।

S.I. = (50000 × 0.12 × 3) / 100 = रु. 18000

त्यसैपछि, श्यामले चक्रीय ब्याजदरले रु. 50000 लगाएर 3 वर्षमा त्यति रकम लिन्छन्। चक्रीय ब्याजदरले ब्याज प्राप्त गर्नको लागि प्रत्येक वर्षको अन्तिम रकममा पूर्ववत ब्याज पनि जोडिन्छ। त्यसको आधारमा उनले प्राप्त गरेको ब्याज हो:

Compound Interest (C.I.) = P × [(1 + R/100)ᵀ - 1]

यहाँ, P = रु. 50000 हो, R = 10% = 10/100 = 0.10 हो, र T = 3 वर्ष हो।

C.I. = 50000 × [(1 + 0.10)³ - 1] = रु. 16532.10

त्यस्तै ब्याजदर र समयको आधारमा रामले पनि रकम लगाएर बराबर ब्याज प्राप्त गर्न चाहनुहुन्छ। यसलाई गणना गर्न प्रयोग गरिएको समान ब्याज तथा समय भएको सिस्टमको नाम "Compound Interest with annual installments" हो। यसक