rationalize the denominator of $\displaystyle \frac{1}{\sqrt{2} \sqrt{3} \sqrt{7}}$, and write your answer in the form\[ \frac{a\sqrt{2} b\sqrt{3} c\sqrt{7} d\sqrt{e}}{f}, \]where everything is in simplest radical form and the fraction is in lowest terms, and $f$ is positive. what is $a b c d e f$?

The average rate of change of f(x) from x = 1 to x = 3 is 0.5

The average rate of change of a function over an interval is the change in the function value (y) divided by the change in the input value (x) over that interval.

To find the average rate of change of f(x) = [tex]x^{3}[/tex] - [tex]4x^{2}[/tex] + 5 from x = 1 to x = 3, we can use the following formula:

(f(3) - f(1)) / (3 - 1)

substituting the function value, we get

([tex]3^{3}[/tex] - 4[tex](3)^{2}[/tex] + 5 - ([tex]1^{3}[/tex] - 4[tex](1)^{2}[/tex] + 5)) / (3 - 1)

= (27 - 36 + 5 - (1 - 4 + 5)) / 2

= (1) / 2

So the average rate of change of f(x) from x = 1 to x = 3 is 0.5

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1. Total 30 4th year children travel.

2. Total 40 children from 5th year travel.

3. Total 10 adults travel.

4. Total 40 students from 6th year travel.

5. The fraction of the total number of passengers together represent the students of 4th and 5th years who traveled is 7/12.

6. The fraction of the total does not have students of 4th or 5th is 5/12.

7. The fraction of the total number of travelers are adults is 1/12.

Total passenger = 120

4th year student = 1/4 × 120

4th year student = 30

5th year student = 1/3 × 120

5th year student = 40

Now the remaining passenger

Remaining passenger = Total passenger - (4th year student + 5th year student)

Remaining passenger = 120 - (30 + 40)

Remaining passenger = 120 - 70

Remaining passenger = 50

Adults = 1/5 × 50

Adults = 10

Now the 40 passengers are left that are 6th year students.

Now we determine fraction of the total number of passengers together represent the students of 4th and 5th years who traveled.

Fraction = (4th year student + 5th year student)/Total passenger

Fraction = (30 + 40)/120

Fraction = 70/120

Fraction = 7/12

Now we determine fraction of the total does not have students of 4th or 5th.

Fraction = (Adults + 6th year passenger)/Total Passenger

Fraction = (10 + 40)/120

Fraction = 50/120

Fraction = 5/12

Now we determine fraction of the total number of travelers are adults.

Fraction = Adults/Total Passenger

Fraction = 10/120

Fraction = 1/12

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The complete question is:

In a school excursion 120 passengers travel: 1/4 of the total: 4th year students, 1/3 of the total: 5th year students and the rest, 1/5 are adults.

1. How many 4th year children travel?

2. How many children from 5th year travel?

3. How many adults travel?

4. How many students from 6th year travel?

5. What fraction of the total number of passengers together represent the students of 4th and 5th years who traveled?

6. Which fraction of the total does not have students of 4th or 5th?

7. What fraction of the total number of travelers are adults?

The equation of the line passing through M and perpendicular to PR would be y = -x + 6.

What is the equation of the line?

An equation of a line is a mathematical expression that describes the relationship between the variables x and y that defines the position of a point on the line. There are several ways to write the equation of a line, including slope-intercept form (y = mx + b), point-slope form (y - y1 = m(x - x1)), and standard form (ax + by = c).

The midpoint of a line segment is the point that is exactly halfway between the two endpoints of the line. To find the midpoint of a line segment, you need to take the average of the x-coordinates and the y-coordinates of the two endpoints.

Given the line segment Q(-6, 2) and R(10, 6), the midpoint is:

((-6+10)/2 , (2+6)/2) = (2,4)

So the midpoint of the line segment Q(-6, 2) and R(10, 6) is (2,4).

Now find the slope of PR = 6-(-4)/10 = 10/10=1

Hence, the slope of the line passing through M and perpendicular to PR would be = -1.

Now we can form the equation of the line:

y - 4 = -1(x - 2)

y - 4 = -x + 2

y = -x + 6

Hence, the equation of the line passing through M and perpendicular to PR would be y = -x + 6.

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[tex]\huge\text{Hey there!}[/tex]

[tex]\huge\textbf{Looking for improper or mixed fraction:}[/tex]

[tex]\mathsf{310\% \rightarrow \dfrac{310}{100} \rightarrow 3.1}[/tex]

[tex]\mathsf{\rightarrow 3.1 \times \dfrac{10}{10}}[/tex]

[tex]\mathsf{\rightarrow \dfrac{3.1}{1}\times\dfrac{10}{10}}[/tex]

[tex]\mathsf{\rightarrow \dfrac{3.1\times10}{1\times10}}[/tex]

[tex]\mathsf{\rightarrow \dfrac{31}{10}}[/tex]

[tex]\mathsf{\rightarrow 3 \dfrac{1}{10}}[/tex]

[tex]\huge\text{Therefore, your answers should be:}[/tex]

[tex]\huge\boxed{\mathsf{Fraction \rightarrow \dfrac{310}{100}\ \& \ Mixed\ Number \ \rightarrow 3\dfrac{1}{10}}}\huge\checkmark[/tex]

[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]

The following algebraic expressions are factorized using their highest common factor HCF:

a) x² - 2x = x(x - 2)

b) 6x² + 12x = 6x(x + 2)

c) 3x³ - 9x = 3x(x² - 3)

d) 4x² + 28x³ = 4x²(1 + 7x)

The H.C.F. defines the highest common factor present in between given two or more numbers or mathematical expression.

x² and 2x have HCF as x such that;

x² - 2x = x × x - 2 × x

x² - 2x = x(x - 2)

6x² and 12x have HCF as 6x such that;

6x² + 12x = x × 6x + 2 × 6x

6x² + 12x = 6x(x + 2)

3x³ and 9x have HCF as 3x such that;

3x³ - 9x = x² × 3x - 3 × 3x

3x³ - 9x = 3x(x² + 3)

4x² + 28x³ have HCF as 4x² such that;

4x² + 28x³ = 1 × 4x² + 7x × 4x²

4x² + 28x³ = 4x²(1 + 7x)

Therefore, the algebraic expressions are factorized using their highest common factor HCF to be:

a) x² - 2x = x(x - 2)

b) 6x² + 12x = 6x(x + 2)

c) 3x³ - 9x = 3x(x² - 3)

d) 4x² + 28x³ = 4x²(1 + 7x)

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The measure of angle m∠EDC = 25°, round to the nearest tenth.

For the given question

As the both diagonal of the rectangle are equal.

ΔAED is isosceles triangle.

Then,

m∠ADE = m∠DAE = 50°

As all 4 angles of the triangle are equal.

m∠EDC = 90 - m∠ADE

             = 90 - 65

m∠EDC = 25°

Thus, the measure of the angle m∠EDC = 25°, round to the nearest tenth.

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The diagram for the question is attached.

Answer:

-7/2

Step-by-step explanation:

The slopes of perpendicular lines are negative reciprocals of each other.

To find the reciprocal of a fraction, flip the fraction.

For the negative part, change the sign.

Slope of line: 2/7

Slope of perpendicular line: -7/2

Answer:

  zero

Step-by-step explanation:

You want the rate of change between the points (3, 1) and (4, 1).

The "rate of change" is also known as the "slope" of the line between the points. That is given by the formula ...

  m = (y2 -y1)/(x2 -x1)

For your points, this is ...

  m = (1 -1)/(4 -3) = 0/1 = 0

The rate of change is zero.

__

Additional comment

The y-value is constant. That means the equation of the line is y=1, a horizontal line. It does not rise or fall. Its rate of change is zero.

(a) the sample mean is 0.0494 and the sample variance is 0.000016, (b) the upper quartile is 0.04775, and the lower quartile is 0.0510, (c) the sample median is 0.04975, (d) boxplot is attached, and (e) the 5th and 95th percentiles of the inside diameter are 0.03974 and 0.056995 respectively.

(a) The mean = sum of all values divided by the number of values

μ = (x1 + x2 + ..... + xn)/n

n = 20

μ = (0.0395 + 0.0443+ 0.0450 + ... + 0.0550 + 0.0571)/20

μ = 0.9878/20

μ = 0.0494

(b) Variance = sum of squared deviations from the mean divided by n-1

s² = {(x1-μ)² + (x2-μ)² + .... (xn - μ)²)/(n-1)

s² = {(0.0395-0.0494)² + (0.0443-0.0494)² + .... +(0.0571-0.0494)²}/19

s² = 0.000016

(b) The minimum is 0.0395 and the maximum is 0.0571.

since the number of data is even, the median will be the average of two middle values.

M = Q2 = (0.0496+0.0499)/2 = 0.04975

Now, the first quartile is the median of the data values below the median

so Q1 = (0.0470+0.0485)/2 = 0.04775

And third quartile will be the median of the data values above the median

Q3 = (0.0504+0.0516)/2 = 0.0510

(c) Since we know that the number of data values is even, the median will be the average of the two middle values of the data set

so M = (0.0496+0.0499)/2

or M = 0.04975

(d) The boxplot is at maximum and minimum values. It will start in Q1 and end in Q3 and has a vertical line at the median or Q2.

The boxplot is attached.

(e) The 5th percentile means 0.05(n+1)th data value

or = 0.05(20+1) = 1.05th data value

5th percentile = 0.0550 + 0.05(0.0443-0.0395) = 0.03974

similarly,

95th percentile = 0.0550 + 0.95(0.0571-0.0550) = 0.056995

Therefore, (a) the sample mean is 0.0494 and the sample variance is 0.000016, (b) the upper quartile is 0.04775, and the lower quartile is 0.0510, (c) the sample median is 0.04975, and (e) the 5th and 95th percentiles of the inside diameter are 0.03974 and 0.056995 respectively.

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I began my job as a student on 08/21/2019 and worked for a total of 905 days until my graduation on 05/20/2022. This is calculated by taking the total number of days between the start and end date - 3 years - which is 1095, and then subtracting the weekends and holidays - 190 days - during that period. This demonstrates my commitment to my job as a student.

08/21/2019 - 05/20/2022

= 3 years

= 1095 days

- 190 days (weekends and holidays)

= 905 days

Starting on 08/21/2019, I have been working at my job as a student for 905 days as of 05/20/2022. This is determined by taking the total number of days - 3 years - between the start date and end date, which is 1095 days, and then subtracting 190 days for the weekends and holidays during that period. This leaves me with 905 days of work as a student.

I began my job as a student on 08/21/2019, and my graduation was on 05/20/2022. After calculating the total number of days between the two dates - 3 years - which is 1095, and then subtracting the weekends and holidays during that period - 190 days - I have worked as a student for 905 days up until my graduation date. This demonstrates the commitment and dedication I had to my job as a student.

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The population is the entire set of individuals or items of interest in a particular study, and it is not necessarily the same size as the sample which is a subset of the population.

Population is a term used to refer to the entire set of individuals or items that are of interest in a particular study. Population size is not necessarily the same as sample size. Sample size is the number of individuals or items that are selected from the population to participate in the study. The sample may be selected randomly, or it may be selected based on specific criteria. When collecting data, researchers use the sample as a representation of the population in order to identify patterns, relationships, and trends that can be used to make inferences about the population. When constructing a sample, it’s important to make sure that it is random and representative of the population, otherwise the results of the study may be biased.

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The numbers is greater than 50 and less than 54 are 51, 52 and 53.

Now, According to the question:

A number is an arithmetic value used for representing the quantity and used in making calculations. A written symbol like “3” which represents a number is known as numerals. A number system is a writing system for denoting numbers using digits or symbols in a logical manner.

Adam writes a number.

The number is greater than 50 and less than 54.

Based on the given condition:

A number is > and  <  .

Then,

51, 52, and 53 are all greater than 50, but less than 54

Hence, The numbers is greater than 50 and less than 54 are 51, 52 and 53.

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Answer: 50%

Step-by-step explanation:

If fourteen are men, and there are twenty-eight in the class, then the fraction that are men are -

14/28.

14/28 is 1/2 which is equal to 0.5

0.5 converted to percent is 50%

For the trigonometric function lim x→(π/2) tan (x), the RHL and LHL are not equal to f(a), hence the limit does not exist.

What is trigonometric function?

Simply put, trigonometric functions also referred to as circular functions are the functions of a triangle's angle. This means that these trig functions provide the relationship between the angles and sides of a triangle.

The given trigonometric function is -

lim x→(π/2) tan (x)

To find the value of trigonometric function use x = π/2.

First deduce the right hand limit (RHL).

Right Hand Limit = lim h→0^+ tan [(π/2) + h]

= lim h→0^+ -cot h

= lim h→0^+ [(-cot h)/h] × h

= -1 × 0 = 0

Now deduce the left hand limit (LHL).

Left Hand Limit = lim h→0^- tan [(π/2) - h]

= lim h→0^- cot h

= lim h→0^- [(cot h)/h] × h

= 1 × 0 = 0

Thus, LHL = RHL.

Left hand limit = right hand limit = f(a)

If the above condition is satisfied then limit exists.

f(π/2) = lim x→(π/2) tan (x)

= tan (π/2)

= ∞

Left hand limit = right hand limit ≠ f(a)

Therefore, the limit does not exist.

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The number of square centimeters of the cup covered by the sleeve is 141.42 square cm

From the question, we have the following parameters that can be used in our computation:

Height, h = 5 cm

Diameter, d = 9 cm

This means that the radius, r is

r = 9/2

r = 4.5 cm

Since the sleeve is a cylinder, we can use the formula for the lateral area of a cylinder to calculate the area of the sleeve:

Lateral area = 2 * pi * r * h

So in this case, the lateral area of the sleeve is:

2 * pi * (4.5) * 5 = 141.42 square cm

This is the area of the cup covered by the sleeve.

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Rigid motions and dilations are transformations in Euclidean space. They are both types of transformations that preserve distances and angles.

Similarities between Rigid motions and Dilations:

Both preserve the length of lines and distances between points.

Both preserve angles between lines.

Differences between Rigid motions and Dilations:

Rigid motions are a combination of rotations, translations, and reflections, while dilations are just scaling transformations.

Rigid motions preserve orientation, while dilations change it.

Rigid motions preserve the shape of figures, while dilations change the size of figures but preserve the ratio of lengths of corresponding sides.

Properties and the transformations that preserve them:

Length of lines: Rigid motions and dilations preserve the length of lines.

Distances between points: Rigid motions and dilations preserve distances between points.

Angles between lines: Rigid motions and dilations preserve angles between lines.

Orientation: Rigid motions preserve orientation, while dilations change it.

Shape of figures: Rigid motions preserve the shape of figures, while dilations change the size of figures but preserve the ratio of lengths of corresponding sides.

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His sister will be 47 years old.

Now, According to the question:

To find Joshua's sisters age when Joshua is 23 years old. We are given that Joshua is currently 12 years old and that his sister is three times as old as he is.

3 × 12 = 36

We have that his sister's current age is 36 years old.

We want to know how old she will be when Joshua is 23 years old. Since Joshua is currently 12 years old,

23 - 12 = 11

We have that in 11 years, Joshua will be 23 years old.

So we want to know what his sister's age will be in 11 years.

To find this, we simply add 11 to her current age of 36.

36 + 11 = 47

We get that in 11 years, his sister will be 47 years old, so when Joshua is 23 years old, his sister will be 47 years old.

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A dependent variable is the variable that being check in a scientific experiment.

The dependent variable is "dependent" on the other independent variable. As the experimenter alter the independent variable, the change in the dependent variable is observed and recorded. When you take data in an experiment, the dependent variable is the one being advised.

A scientist is testing the effect of light and dark on the efforts of moths by turning a light on and off. The independent variable is the amount of light and the moth's reply is the dependent variable.

A change in the independent variable (total amount of light) straight source a change in the dependent variable (moth behavior).

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The solution for the system of inequalities is (-0.83,-1.34).

What is an inequality?

In algebra, the inequality symbol is used to show how two expressions are related in an inequality assertion. Uneven expressions can be seen on both sides of a sign of inequality.

We are given a graph on which two inequalities are plotted

So, first, we will have to frame the equations from the points.

For the blue line, we have points (0,-3) and (-1.5,0)

So, using slope-intercept form, we get the equation as

y + 2x ≥ -3

For the red line, we have points (-1,-2) and (0,2)

So, using slope-intercept form, we get the equation as

y - 4x ≥ 2

For finding the solution, we will subtract both the equations

On subtracting, we get

⇒(y+2x) - (y-4x) = -3 - 2

⇒y+2x - y + 4x = -5

⇒6x = -5

⇒x = -0.83

Substituting this value in the equation, we get

⇒y+2(-0.83) = -3

⇒y-1.66 = -3

⇒y = -1.34

Hence, the solution for the system of inequalities is (-0.83,-1.34).

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Answer: c. a = 2y and b = -3

Step-by-step explanation:

To find the values of a and b in this division problem, we need to perform polynomial division. Starting with the dividend, 8y^2 – 12y + 4, we divide the leading term of the dividend, 8y^2, by the leading term of the divisor, 4y. This gives a quotient of 2y.

Next, we multiply the divisor, 4y, by 2y and subtract it from the dividend, 8y^2 – 12y + 4. This gives a remainder of -3.

So, in this division problem, the values of a and b are a = 2y and b = -3.

Allan would need to sell 427 boxes to make 1600 dollars in total sales. The number line representation of the number of boxes sold is 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ..............., 1600].

A pair of coordinates’ vertical value. How high or low the point is. In an ordered pair of coordinates (x,y), the Y Coordinate is always written second, as in (12,5). In this case, the Y Coordinate is “5”. Also known as “Ordinate” Pick x and solve the equation for y to locate points on the line y = mx + b, or choose y and solve for x.

When you know the slope of the line to be investigated and the point supplied is also the y intercept, you may utilize the slope intercept formula y = mx + b. (0, b). The y value of the y intercept point is represented by b in the formula. Exemplification 2: Determine the equation.

In the given question,

a. The point on the graph that represents 1600 dollars in total sales is (427, 1600).

b. Allan would have to sell 427 boxes to make 1600 dollars in total sales.

c. The number line representation of the number of boxes sold when the total sales is equal to 1600 dollars would be:

[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ................., 1600].

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Answer:

Step-by-step explanation:

a couple slices of pizza

First, we need to calculate the remaining balance after the down payment of $1200. The remaining balance is: $6995 - $1200 = $5795

To calculate the total interest for 36 months, we can use the formula:

I = P * r * t

Where:

I = total interest

P = remaining balance ($5795)

r = interest rate (10.4%/12)

t = number of months (36)

So the total interest for 36 months would be:

$5795 * (0.104/12) * 36 = $2,199.40

To calculate the total interest for 48 months, we can use the same formula but with a different value for t (48 months):

$5795 * (0.104/12) * 48 = $2,932.80

To find out how much Nancy will save by repaying the loan in 36 months instead of 48 months, we can subtract the total interest for 36 months from the total interest for 48 months:

$2,932.80 - $2,199.40 = $733.40

So, Nancy will save $733.40 in interest if she repays the loan in 36 months instead of 48 months.

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The minimum blood sugar reading that James measured was 92 mg/dl. This is based on the boxplot shown above, which shows the range of blood sugar readings that James recorded over the course of a week.

The boxplot shown above provides a graphical representation of the range of blood sugar readings that James measured over the course of a week. A boxplot is composed of five different components: the minimum, the first quartile (Q1), the median, the third quartile (Q3), and the maximum. The minimum is the lowest value in the dataset, and in this case, it is represented by the lowermost point of the boxplot. This point indicates that the minimum blood sugar reading that James measured was 92 mg/dl. This value is lower than the normal blood sugar range for someone without diabetes, which is less than 140 mg/dl, two hours after eating. This indicates that James's blood sugar levels are within a healthy range.

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Mary's speed, given the kilometers she rode in 7 hours, can be found to be 12. 8 km / hr

To find the speed that Mary was going in order to have been able to travel 89.6 kilometers in 7 hours, you can use the speed formula which is:

= Distance / Time

Distance =  89. 6 kilometers

Time = 7 hours

The speed that Mary was going at is therefore :

= 89. 6 km / 7 hours

= 12. 8 km / hr

In conclusion, the speed that Mary was going at was 12. 8 km / h.

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The angle of rotation that will move the clockwise position of 5 to 8 is 225 degrees.

The circular movement of an item about a center is referred to as rotation. Different forms can be rotated by an angle about their central point. A rotation is a map in mathematics. The rotation group of a certain space is made up of all the rotations that revolve around a fixed point and form a group beneath a structure.

The given nut is in the shape of an octagon.

The total angle of a polygon is 360 degrees.

An octagon has 8 sides, so each rotation of the nut will result in an angle of:

Rotation = 360 / 8

Rotation = 45 degrees.

To move the position of 5 to 8, a total of 5 rotations are required.

Total rotation = (45)(5)

Total rotation = 225 degrees.

Hence, the angle of rotation that will move the position of 5 to 8 is 225 degrees.

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The angle of elevation from the tip of a building's shadow to the top of the building is 63⁰. The height of the building is 432 ft.

The tip of the building's shadow, the top of the building, and the base of the building form a right triangle.

Angle of elevation is the angle formed an object is placed above the observer.  It is the angle between the  the line of sight and the horizontal line.

If we know the angle of elevation and the distance, we can compute the height of the building using trigonometric function.

tan α = y/x

Since it is a right triangle, using the trigonometric formula

height of the building / distance = tan 63⁰

height of the building / 220 = 1.96

height of the building = 220 x 1.96 = 432 feet.

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1. At least 4 ordered pairs that comply with the conditions are (0, 25), (6, 20.5), (12, 16) and (16, 13)

2. Graph is drawn according to the points and equation which is given below.

Alegria have total money $20.

The cost of one pound rice = $0.60

The cost of one pound sugar = $0.80

Let Algeria buy x pound rice and y pound sugar.

So the total cost of x pound rice is 0.60x.

The total cost of y pound sugar is 0.80y.

Now the required equation is:

0.60x + 0.80y = 20

Now we find the values buy putting the values as x = 1, 2, 3, ...

x       0         6         12          16

y      25      20.5      16          13

The graph of the values are given below:

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The complete question is:

Alegria goes to the store to buy rice and sugar, but only has $ 20. If each pound of rice costs $ 0.60 and sugar costs $ 0.80, how many pounds does each pound cost to buy?

a. Write at least 4 ordered pairs that comply with the condition

b. Realize the graphics.

In option B Charlie get more money. So he picks the option B that will give him $700 more after 10 years and deposits $10,000.

A bank offers two interest-rate options.

First we check which option can give more amount after 10 years by solving the options. Then we determine how much more money Charlie will have after 10 years.

Option A gives 6% annual interest compounded quarterly.

There are 4 quarters in a year. So n = 4

So r = 6/4 = 3/2%

The formula of compound interest is

A = P(1 + r/100)^n

From the question P = 10,000, Now putting the value

A = 10000(1 + 3/200)^4

A = 10000(203/200)^4

A = 10000(1.015)^4

A = 10000(1.0614)

A = $10,614

Option B gives 7% annual interest compounded yearly.

r = 7%, P = $10000, n = 1

The formula of compound interest is

A = P(1 + r/100)^n

Now put the values.

A = 10000(1 + 7/100)^1

A = 10000(107/100)

A = 10000 × 1.07

A = $10,700

Now we can see that in option B Charlie get more money.

More Money after 10 years = 10,700 - 10,000

More Money after 10 years = $700

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