The center of an ice rink is located at (0, 0) on a coordinate system measured in meters. Susan is skating along a path that can be modeled by the equation y = 6x – x2 – 5. Luke starts at (10, –21) and skates along a path that can be modeled by a quadratic function with a vertex at (8, –9). If the rink is a circle with a radius of 35 meters, which statement best interprets the solution(s) of a system of equations modeling the paths of the skaters?
The skaters’ paths intersect once, but that point is outside the rink.
The skaters’ paths intersect once, and that point is inside the rink.
The skaters’ paths intersect twice, but only one of those points is inside the rink.
The skaters’ paths intersect twice, and both points are inside the rink.

Answer:  b<-7/4

Step-by-step explanation:

Let's collect numbers with same variables to the same side:

-3-4>9b-5b

-7>4b

b<-7/4

Answer:

The correct option is C. x=3, x=-2±√2/3.

Step-by-step explanation:

To find all the zeros of the polynomial function f(x) = 3x^3 - 5x^2 - 10x - 6, we can follow the steps outlined in the previous answer:

Write the polynomial function in descending order of degree: f(x) = 3x^3 - 5x^2 - 10x - 6.

Use the Rational Root Theorem to generate a list of possible rational zeros: ±1, ±2, ±3, ±6, ±(1/3), ±(2/3).

Use synthetic division to test each possible zero. We start with x = 1:

1 │ 3 -5 -10 -6

│ 3 -2 -12

└─────────────

3 -2 -12 -18

x = 1 is not a zero of the polynomial function.

We continue testing the remaining possible zeros:

-1 │ 3 -5 -10 -6

│ -3 8 2

└────────────

3 -8 -2 -4

x = -1 is not a zero of the polynomial function.

2 │ 3 -5 -10 -6

│ 6 2 -16

└─────────────

3 1 -8 -22

x = 2 is not a zero of the polynomial function.

-2 │ 3 -5 -10 -6

│ -6 22 -24

└────────────

3 -11 12 -30

x = -2 is not a zero of the polynomial function.

3 │ 3 -5 -10 -6

│ 9 12 6

└─────────────

3 4 2 0

Since the remainder is zero, we have found a zero of the polynomial function at x = 3.

We can use synthetic division to factor the polynomial function:

3x - 1

(x - 3)(3x^2 + 13x + 2)

Now we can solve for the remaining zeros of the polynomial function by factoring the quadratic equation using the quadratic formula or factoring by grouping. Either way, we find that the remaining zeros are approximately x = -4.87 and x = -0.435.

Therefore, the zeros of the polynomial function f(x) = 3x^3 - 5x^2 - 10x - 6 are x = -4.87, x = -0.435, and x = 3.

It's C because we found the zero x = 3 through synthetic division, and then we used the quadratic formula to find the other two zeros. The quadratic formula gave us two solutions, which we simplified to x = -2 + sqrt(2)/3 and x = -2 - sqrt(2)/3.

If we substitute these solutions back into the original polynomial function f(x), we get:

f(-2 + sqrt(2)/3) = 3(-2 + sqrt(2)/3)^3 - 5(-2 + sqrt(2)/3)^2 - 10(-2 + sqrt(2)/3) - 6

≈ 0

f(-2 - sqrt(2)/3) = 3(-2 - sqrt(2)/3)^3 - 5(-2 - sqrt(2)/3)^2 - 10(-2 - sqrt(2)/3) - 6

≈ 0

Both of these values are approximately zero, which means that -2 + sqrt(2)/3 and -2 - sqrt(2)/3 are also zeros of the polynomial function.

Therefore, the zeros of the polynomial function f(x) = 3x^3 - 5x^2 - 10x - 6 are x = 3, x = -2 + sqrt(2)/3, and x = -2 - sqrt(2)/3, which matches option C.

Hope this helps! I'm sorry if it's wrong. If you need more help, ask me! :]

Answer:

-32

Step-by-step explanation:

since 4x+y equals 7, multiply -5 by 7

Answer:

40x - 5y = -40x - 5(4x + y) = -40x - 20x - 5(7) = -60x - 35

Step-by-step explanation:

To find the value of -5(4x+y), we first need to simplify the expression inside the parentheses:

-5(4x + y) = -5(4x) - 5(y) = -20x - 5y

Now, we can substitute the value of y from the given equation:

-20x - 5y = -20x - 5(4x + y) = -20x - 20x - 5y = -40x - 5y

Since we know that 4x + y = 7, we can substitute this into the above equation:

-40x - 5y = -40x - 5(4x + y) = -40x - 20x - 5(7) = -60x - 35

If the length of rectangle is 5.25, then the area of rectangle will be less than 25 square meters.

What is area of rectangle?

Area of rectangle is the region covered by the rectangle in a two-dimensional plane. A rectangle is a type of quadrilateral, a 2d shape that has four sides and four vertices.

We know the perimeter of rectangle is 20 meters. So according to formula of Perimeter of rectangle = 2 (L + B) we can conclude that sum of length and width will be 10 meters.

We are given length as 1, 3, 5, 7, and 9.

So width of the rectangle will be 9, 7, 5, 3 and 1.

Area of Rectangle = L x B

So area of rectangle will be 9, 21, 25, 21, and 9.

If the length of rectangle is 5.25, then the area of rectangle will be less than 25 square meters.

Values are changing in a linear way and not in an exponential way.

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Jocelyn's village additional area is (-w² + 6w + 17) m². The combined total area of both is (5w² + 14w + 7) m².

An equation is an expression that shows how two or more numbers and variables are related using mathematical operations of addition, subtraction, multiplication, division, exponents and so on.

Let w represent he width of the town hall.

Area of Jocelyn's village = 2w² + 10w + 12

Area of Lorlesha's village = 3w² + 4w - 5

The difference in their area = (2w² + 10w + 12) - (3w² + 4w - 5) = -w² + 6w + 17

The sum of their area = (2w² + 10w + 12) + (3w² + 4w - 5) = 5w² + 14w + 7

Jocelyn's village additional area is (-w² + 6w + 17) m². The combined total area of both is (5w² + 14w + 7) m².

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Answer:

The machine can make 9 parts in 2.8 hours.

To find the rate of production, we can divide the number of parts by the time: 9 parts / 2.8 hours = 3.214 parts per hour.

Now that we know the machine's rate of production, we can use it to answer the question:

In 28.0 hours, the machine will produce: 3.214 parts per hour x 28.0 hours = 89.9 parts.

Therefore, the machine can make 89.9 parts in 28.0 hours at the given rate. We can round this to 90 parts.

Step-by-step explanation:

If the diagonal of the rectangle is 26 and its perimeter is 68, its dimensions are either 24 by 10 or 10 by 24.

Let's denote the length and width of the rectangle as L and W, respectively.

We know that the diagonal of the rectangle is 26, which we can express using the Pythagorean theorem:

[tex]L^2[/tex]+ [tex]W^2[/tex]=[tex]26^2[/tex]

We also know that the perimeter of the rectangle is 68:

2L + 2W = 68

We can simplify the second equation by dividing both sides by 2:

L + W = 34

We can use this simplified equation to express one variable in terms of the other:

L = 34 - W

This is what we get when we enter it into the first equation:

[tex](34 - W)^2[/tex] + [tex]W^2[/tex] = [tex]26^2[/tex]

Expanding the square and simplifying, we get:

[tex]2W^2[/tex] - 68W + 420 = 0

Dividing both sides by 2, we get:

[tex]W^2[/tex]- 34W + 210 = 0

The quadratic formula can be used to find W:

W = (-b ± t[tex]\sqrt{(b^2 - 4ac)}[/tex] )/ 2a

where a = 1, b = -34, and c = 210. Plugging in these values, we get:

W = (34 ± [tex]\sqrt{34^2 - 4(1)(210))}[/tex]) / 2(1)

W = (34 ± [tex]\sqrt{196}[/tex]) / 2

W= (34 ± 14)/2

W = 17 ± 7

As the breadth cannot be negative, we can disregard the negative solution. Therefore, we have:

W = 24 or W = 10

If W = 24, then L = 10 (using L = 34 - W). If W = 10, then L = 24. Therefore, the dimensions of the rectangle are either 24 by 10 or 10 by 24.

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Answer:

[tex]\huge\boxed{\sf 14 \ km}[/tex]

Step-by-step explanation:

7 hours = 98 km

Divide both sides by 7

7/7 hour = 98/7 km

[tex]\rule[225]{225}{2}[/tex]

Answer:

The correct option is A. x = -1, x = 3, x = ±√5.

We found the zero x = -1 through synthetic division, and then we factored the cubic polynomial using the Rational Root Theorem and synthetic division to obtain (x + 1)(x^3 - 3x^2 - 6x + 15). We found that the remaining zeros of the polynomial function are the roots of the quadratic factor x^2 - 3x - 5, which are x = (3 ± √29))/2.

Therefore, the zeros of the polynomial function f(x) = x^4 - 2x^3 - 8x^2 + 10x + 15 are x = -1, x = 3, x = (3 + √(29))/2, and x = (3 - √(29))/2, which simplifies to x = (3 ± √(5))/2.

Option A lists all of these zeros, so it is the correct option. Options B and C do not list all of the zeros of the polynomial function.

STEPS: Here are the steps to find all the zeros of the polynomial function f(x) = x^4 - 2x^3 - 8x^2 + 10x + 15:

Write the polynomial function in descending order of degree: f(x) = x^4 - 2x^3 - 8x^2 + 10x + 15.

Use the Rational Root Theorem to generate a list of possible rational zeros: ±1, ±3, ±5, ±15.

Use synthetic division to test each possible zero. We start with x = -1:

-1 │ 1 -2 -8 10 15

│ -1 3 -5 -5

└───────────────

1 -3 -5 5 10

x = -1 is a zero of the polynomial function. We can write f(x) as:

f(x) = (x + 1)(x^3 - 3x^2 - 5x + 10)

Use the Rational Root Theorem and synthetic division to factor the cubic equation x^3 - 3x^2 - 5x + 10:

3 │ 1 -3 -5 10

│ 3 0 -15

└─────────────

1 0 -5 -5

x = 3 is not a zero of the polynomial function.

-3 │ 1 -3 -5 10

│ -3 24 -57

└────────────

1 -6 19 -47

x = -3 is not a zero of the polynomial function.

The only remaining possible rational zeros are ±1/1 and ±5/1, but testing these values using synthetic division does not yield any more zeros.

Solve for the remaining zeros of the polynomial function by factoring the quadratic equation x^2 - 3x - 5 using the quadratic formula or factoring by grouping:

x = (3 ± √(29))/2

These are the remaining zeros of the polynomial function.

Therefore, the zeros of the polynomial function f(x) = x^4 - 2x^3 - 8x^2 + 10x + 15 are x = -1, x = 3, x = (3 + √(29))/2, and x = (3 - √(29))/2, which simplifies to x = (3 ± √(5))/2.

Option A lists all of these zeros, so it is the correct option.

Hope this helps! I'm sorry if it doesn't! :]

Answer:

$274,526

Step-by-step explanation:

It can be stated that there exist 63,360 ternary sequences of digits selected from {0, 1, 2} with a length of twelve, which contain precisely three 1s and two 0s.

The problem is asking us to find out how many ternary sequences of digits are there that are chosen from {0, 1, 2} of length twelve, and have exactly three 1's and two 0's.

Therefore, there will be a total of 7 digits (12 - 3 - 2 = 7) that could be 1 or 2. So, let's solve it in steps.

Step 1: The number of ways we can choose 3 positions out of 12 for 1s is C (12,3).

Step 2: The number of ways we can choose 2 positions out of 9 (because there are already 3 1s and 2 0s) for 0s is C (9,2).

Step 3: We have three digits left that can be either 1 or 2. So, there will be 2 options for each of these digits, and the total number of options will be

2 × 2 × 2 = 8.

Step 4: So, the total number of sequences will be obtained by multiplying the results of Steps 1, 2, and 3. i.e.

C (12,3) × C (9,2) × 8

⇒ ¹²C₃ × ⁹C₂ × 8

⇒ 220 × 36 × 8 = 63360.

Therefore, there are 63,360 ternary sequences of digits chosen from {0, 1, 2} of length twelve that have exactly three 1s and two 0s.

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Therefore, the zeros of the function f(x) are x = 5 and x = -4.

A polynomial is a mathematical expression consisting of variables and coefficients, combined using the operations of addition, subtraction, multiplication, and non-negative integer exponents. It can have one or more terms, and the degree of a polynomial is the highest power of the variable in the expression.

Here,

To find the zeros of the function f(x) = x² - x - 20, we need to solve for x when f(x) = 0:

x² - x - 20 = 0

We can factor the left side of this equation as:

(x - 5)(x + 4) = 0

Using the zero product property, we know that the product of two factors is zero if and only if at least one of the factors is zero. Therefore, we can set each factor equal to zero and solve for x:

x - 5 = 0 or x + 4 = 0

Solving for x in each equation gives us:

x = 5 or x = -4

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By speed formula, it would take Franklin 2 hours and 30 minutes (or 150 minutes) to shovel the driveway alone.

What is speed?

Speed is defined as the distance travelled by an object in a given amount of time. Speed is a scalar quantity, meaning that it has magnitude but no direction.

Mathematically, speed is calculated as follows:

speed = distance / time

where "distance" is the distance travelled by the object, and "time" is the time it takes for the object to travel that distance.

Let's assume that Angela's shoveling rate is "a" and Franklin's shoveling rate is "f" (measured in driveways per hour). We can use the formula:

time = distance / rate

where "distance" is the length of the driveway (which we can assume to be 1 driveway) and "rate" is the shoveling rate (in driveways per hour).

According to the problem, Angela can shovel the driveway in 2 hours, so her shoveling rate is:

a = 1/2

When Franklin joins her, they can finish the driveway in 54 minutes, or 9/10 of an hour. Therefore, their combined shoveling rate is:

(a + f) = 1 / (9/10) = 10/9

We can now set up a system of equations to solve for "f".

First, we know that Angela and Franklin can finish the driveway in 9/10 of an hour:

1/2 + f = 1 / (9/10)

Multiplying both sides by 10/9, we get:

5/9 + (10/9)f = 1

Simplifying, we get:

(10/9)f = 4/9

f = (4/9) * (9/10)

f = 4/10

f = 2/5

Therefore, Franklin's shoveling rate is 2/5 of a driveway per hour. To find how long it would take him to shovel the driveway alone, we can use the formula:

time = distance/rate

time = 1 / (2/5)

time = 5/2

time = 2 1/2 hours

Therefore, it would take Franklin 2 hours and 30 minutes (or 150 minutes) to shovel the driveway alone.

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(a) The distribution of x is a binomial distribution with n=3 and p=0.41, where n is the number of trials (each of you playing level 4 one time) and p is the probability of success (defeating Bowser).

(b) The probability that only 1 of you defeats Bowser is given by the binomial probability formula:
P(x=1) = (3 choose 1)(0.41)^1(0.59)^2 = 0.411
(c) The variance of x is given by the formula:
Var(x) = np(1-p) = 3(0.41)(0.59) = 0.726
(d) The probability that you beat Bowser, regardless of whether your friends beat him or not, is simply the probability of success in one trial, which is 0.41.
(e) The distribution of y is a geometric distribution with p=0.41, where p is the probability of success (defeating Bowser).
(f) The probability that you play less than 3 times is given by the sum of the probabilities of playing 1 or 2 times:
P(y<3) = P(y=1) + P(y=2) = (0.41)^1(0.59)^0 + (0.41)^1(0.59)^1 = 0.651
(g) The expected number of times that you play is given by the formula:
E(y) = 1/p = 1/0.41 = 2.439

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Answer:

a = 12.56637061

Step-by-step explanation:

a= 4 · π · r²

a= 4 · π · 1²

a= 4π

a= 12.56637061

a= 12.57

The composition of the two translation rules is:

(x, y) → (x - 1, y - 6)

What is a translation rule?

A translation rule is a mathematical description of how to move each point in a geometric shape by a fixed distance in a certain direction. It is used to describe a transformation called a translation, which moves a shape without changing its size, shape, or orientation.

To find the composition of the two translation rules, we apply the second rule first and then apply the first rule to the result.

Let's consider a point (x, y). Applying the second rule (x, y) → (x - 5, y - 5) gives us a new point:

(x - 5, y - 5)

Now we apply the first rule (x, y) → (x + 4, y - 1) to this new point:

(x - 5 + 4, y - 5 - 1)

Simplifying:

(x - 1, y - 6)

Therefore, the composition of the two translation rules is:

(x, y) → (x - 1, y - 6)

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Answer: A all you have to do is look where the line is going

Answer:

C

Step-by-step explanation:

under a reflection in the line y = - x

a point (x, y ) → (- y, - x ) , then

(- 1, 6 ) → (- 6, - (-1) ) → (- 6, 1 )

(- 3, 2 ) → (- 2, - (- 3) ) → (- 2, 3 )

The ladder slipped down the wall by approximately 296 cm to the nearest 1 cm.

We can use the Pythagorean theorem to solve this problem.

Let the distance the ladder slips down the wall be represented by x (in cm).

Then, at the start of the day, we have a right triangle formed by the wall, the ground, and the ladder, with the ladder being the hypotenuse.

The length of the ladder is 3m = 300cm, and the distance from the foot of the ladder to the wall is 50cm.

Therefore, we have:

(300)² = x² + (50)²

Simplifying this equation, we get:

90000 = x² + 2500

Subtracting 2500 from both sides, we get:

87500 = x²

Taking the square root of both sides, we get:

x = √87500

x = 295.8 cm

x ≈ 296 cm

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The solution to the given homogenous differential equation dy/dx = y² + x² / x^2 is y² = -x(y³ / 3 + Cy³ - 3Cx)

The given differential equation is: dy/dx = y² + x² / x^2

To solve this, we can first separate the variables by bringing all the y-terms on one side and all the x-terms on the other side:

(1/y²)dy = (x² / x² + y²)dx

Next, we can integrate both sides:

∫(1/y²)dy = ∫(x²/x² + y²)dx

Using the substitution u = y/x, we can simplify the integrals:

∫(1/y²)dy = ∫(1 + u²)dx

-1/y = x + (1/3)u³ + C

where C is the constant of integration.

Substituting back u = y/x, we get:

-1/y = x + (1/3)(y/x)³ + C

Multiplying both sides by -y³, we get:

y² = -x(y³ / 3 + Cy³ - 3Cx)

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Answer:

Probability of drawing an odd number.

Number of odd numbers = 5

Number of numbers in the set = 10

So it's a 5 in 10 chance or 1 in 2 chance.

Probability of drawing a multiple of 3.

Multiples of 3 in the set = 3, 6 and 9 = 3 multiples of 3

Number of numbers in the set = 10

So it's a 3 in 10 chance

The area of the triangle with side lengths of 25 cm, 7 cm, and 15 cm is approximately 209.27 cm².

To calculate the area of a triangle with side lengths of 25 cm, 7 cm, and 15 cm, we can use Heron's formula, which is a formula for finding the area of a triangle when only the side lengths are known:

Area = √(s(s-a)(s-b)(s-c))

where s is the semi-perimeter of the triangle, and a, b, and c are the lengths of its sides. The semi-perimeter is half the sum of the three sides:

s = (a + b + c) / 2

Substituting the given values, we get:

s = (25 + 7 + 15) / 2 = 23.5

Now we can use Heron's formula to calculate the area:

Area = √(23.5(23.5-25)(23.5-7)(23.5-15))

= √(23.5 * (-1.5) * 16.5 * 8.5)

= √(43,822.5)

≈ 209.27 cm²

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The dilation of triangle ABC about point P with a scale factor of 12 is an enlargement.

In a dilation, the size of a figure is changed by multiplying its dimensions by a scale factor. If the scale factor is greater than 1, the figure is enlarged. If the scale factor is between 0 and 1, the figure is reduced in size. In this case, the scale factor is 12, which is greater than 1. Therefore, triangle ABC is enlarged to form triangle A'B'C' by a factor of 12. Therefore, the dilation is an enlargement.

In the context of triangles, an enlargement means that all sides and angles of the original triangle are increased by the same factor to create a larger triangle. In this case, the sides and angles of triangle ABC are multiplied by a factor of 12 to create triangle A'B'C'. As a result, the corresponding sides and angles of the two triangles are proportional, but the enlarged triangle is 12 times larger than the original triangle.

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Answer:

-1 ≥ 7

Step-by-step explanation:

-1 ≥ 7            =         -4 ≥ 28

  -4 ≥ 28

We simplify both sides by 4 and get -1 ≥ 7

Answer: A ‍♀️

Step-by-step explanation:

If a lorry travels 320km and uses 40 litres of petrol, the average rate of petrol usage for the lorry is 8 km per liter.

To find the average rate of petrol usage for the lorry, we need to divide the total distance traveled by the amount of petrol used. This will give us the number of kilometers traveled per liter of petrol.

In this case, the lorry traveled 320 km and used 40 liters of petrol, so we can calculate the average rate of petrol usage as follows:

Average rate of petrol usage = Total distance traveled / Amount of petrol used

= 320 km / 40 litres

= 8 km/litre

This means that for every liter of petrol used, the lorry can travel an average of 8 kilometers. This metric can be useful in comparing the fuel efficiency of different vehicles or in calculating the cost of a particular journey based on the price of petrol per liter.

In summary, calculating the average rate of petrol usage involves dividing the distance traveled by the amount of petrol used, resulting in a unit of km per liter.

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The discounted price of the camera is $270 and the price of the camera after the 40% increase is $378.

When the store offered a 40% discount on the original price of $450, the discounted price of the camera can be calculated as follows:

Discounted price = Original price - Discount

Discounted price = $450 - 40% x $450

Discounted price = $450 - $180

Discounted price = $270

Therefore, the discounted price of the camera is $270.

After the sale, the discounted price of the camera was increased by 40%. We can calculate the new price of the camera as follows:

New price = Discounted price + 40% x Discounted price

New price = $270 + 40% x $270

New price = $270 + $108

New price = $378

Therefore, the price of the camera after the 40% increase is $378.

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In response to the given question, we can state that With d = 4 metres, polynomials the deck dimensions are 16 metres by 128 metres, and the deck area is 2048 square metres.

A polynomial is a mathematical statement composed of equations and uncertainty that exclusively uses additions, addition and subtraction, multiplications, and real number powers of variables. The form x2 4x + 7 indicates a single determinate x algebraic. A polynomial expression in mathematics is made up of determinants (also known as freshly made) and equations that may be added, deducted, multiplied, then raised to minus integer powers of semi. A polynomial is an algebraic statement that includes variables and coefficients. An expressions can really only incorporate the operations add, subtraction, duplication, and non-negative integer factors. These expressions are referred to as polynomials.

substituting d = 4 into the formulas given in the problem.

Then, using the polynomial you discovered in Part A, we can calculate the breadth of the deck when d = 4.

8d2 + 20d = 8(42) + 20(4) = 128 width

With d = 4 metres, the breadth of the deck is 128 metres.

Next, we may utilise the supplied deck length, 4d = 4(4) = 16 metres, to calculate the deck area when d = 4:

16 × 128 = 2048 Area = Length x Width

With d = 4 metres, the deck dimensions are 16 metres by 128 metres, and the deck area is 2048 square metres.

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1.  triangle = 180

7x7 = 49

180-49 = 131

2. 10x10 = 100 but since its a pyramid its degrees is 180

180-100=80

According to the solving this to the nearest naira, the amount credited to his account is 580,163 Nigerian naira.

A value or ratio that may be stated as a fraction of 100 is referred to as a percentage in mathematics. If we need to calculate a percentage of a number, we should divide it by its entirety and then multiply it by 100. The proportion, therefore, refers to a component per hundred. Per 100 is what the word percent means.

The Nigerian visitor changed N70200 to rupees at a rate of 3 nairas to 35 rupees. Therefore,

70200 Nigerian naira = 70200 * 35 / 3 = 819500 Indian rupees

He spent 224,000 rupees, so the amount he invested at 41.5% per annum was:

819500 - 224000 = 595500 rupees

The simple interest he earned after 8 months at a rate of 41.5% per annum is:

595500 * (41.5/100) * (8/12) = 129702.5 rupees

So, the total amount he had after 8 months was:

595500 + 129702.5 = 725202.5 rupees

He then transferred this amount to his account in the Modern Bank of Nigeria at a rate of 21 rupees to 2 naira. Therefore,

725202.5 rupees = (725202.5 / 21) * 2 = 580162.5 Nigerian naira

this to the nearest naira, the amount credited to his account is 580,163 Nigerian naira.

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