3rd Grade Math Question

The previous year's total revenue for the small business was approximately $25,130.43.

The total revenue of the small business the previous year can be calculated by finding the value that is 15% more than the revenue of $28,900 this year.

To determine the previous year's revenue, we can set up the equation:

Previous Year's Revenue + 15% of Previous Year's Revenue = $28,900

Let's solve the equation to find the previous year's revenue:

Let x be the previous year's revenue.

x + 0.15x = $28,900

Combining like terms, we have:

1.15x = $28,900

Dividing both sides by 1.15, we get:

x = $28,900 / 1.15 ≈ $25,130.43

Therefore, the previous year's total revenue for the small business was approximately $25,130.43.

We set up the equation by adding 15% of the previous year's revenue to the previous year's revenue. This is done to find the total revenue that is 15% more than the previous year's revenue. By solving the equation, we find that the previous year's revenue was approximately $25,130.43.

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The given pair of equations is y = 4x + 2 and y = x + 5, and we are to determine which of the given graphs represents their solution. The first equation is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. Comparing this with the given equation, we see that its slope is 4 and y-intercept is 2.

The second equation is also in slope-intercept form, y = mx + b. Comparing it with the given equation, we see that its slope is 1 and y-intercept is 5.Since we have two lines, we need to find their point of intersection. Substituting y = 4x + 2 into y = x + 5, we have4x + 2 = x + 5Simplifying the equation, we get3x = 3, which gives x = 1.

Substituting this value of x into either of the equations, say y = 4x + 2, we have y = 4(1) + 2 = 6. Hence, the point of intersection is (1, 6). Now, let's examine the given graphs and see which one has (1, 6) as a point of intersection:

Graph 1: The line y = 4x + 2 passes through (0, 2) and has a slope of 4, which means it will be steeper than the line y = x + 5. The line y = x + 5 passes through (0, 5) and has a slope of 1, which means it will be flatter than the line y = 4x + 2. Hence, these two lines cannot intersect at (1, 6). Graph 1 is not the solution.

Graph 2: The line y = 4x + 2 passes through (-1, -2) and has a slope of 4, which means it will be steeper than the line y = x + 5. The line y = x + 5 passes through (0, 5) and has a slope of 1, which means it will be flatter than the line y = 4x + 2. Hence, these two lines cannot intersect at (1, 6). Graph 2 is not the solution.

Graph 3: The line y = 4x + 2 passes through (-1, -2) and has a slope of 4, which means it will be steeper than the line y = x + 5. The line y = x + 5 passes through (0, 5) and has a slope of 1, which means it will be flatter than the line y = 4x + 2.

Hence, these two lines cannot intersect at (1, 6). Graph 3 is not the solution.Graph 4: The line y = 4x + 2 passes through (0, 2) and has a slope of 4, which means it will be steeper than the line y = x + 5. The line y = x + 5 passes through (0, 5) and has a slope of 1, which means it will be flatter than the line y = 4x + 2. Hence, these two lines cannot intersect at (1, 6). Graph 4 is not the solution.

Therefore, none of the given graphs represents the solution to the pair of equations.

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The correct answer is "Oy-axis" as it represents the line of reflection for triangle DEF to triangle D'EF.

To determine the line of reflection that maps triangle DEF to triangle D'EF, we can examine the coordinates of the corresponding vertices.

Given that E' is located at (10, 4) and the corresponding vertex E in triangle DEF is at (-10, 4), we can observe that the x-coordinate of E' is the negation of the x-coordinate of E.

This suggests that the line of reflection is the y-axis. A reflection across the y-axis results in a transformation where the x-coordinate of each point is negated while the y-coordinate remains the same.

Since "Oy-axis" indicates the line of reflection from triangle DEF to triangle D'EF, it is the appropriate response.

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Answer:

362880

Step-by-step explanation:

There are 9 ways to place the first book

There are 8 ways to place the second book given the first book

...

There is 1 way to place the ninth book given the eighth.

This can be represented by 9! (9 factorial), which is 9*8*7*6*5*4*3*2*1 = 362880.

The best use for a sign chart when graphing rational functions is Option D: To check whether F(x) is positive or negative for values of x.

A sign chart, also known as an interval chart, is a tool used to determine the sign (positive or negative) of a function over different intervals. When graphing rational functions, it is essential to identify the intervals where the function is positive or negative.

By constructing a sign chart for the rational function, you can analyze the behavior of the function and determine its sign over different intervals. This information is crucial for sketching an accurate graph of the rational function.

The sign chart helps identify intervals where the function is positive (above the x-axis) or negative (below the x-axis), which directly impacts the shape and position of the graph. It provides information about the regions where the function is increasing or decreasing.

The best use for a sign chart when graphing rational functions is Option D: To check whether F(x) is positive or negative for values of x. Options A, B, and C are not the best uses for a sign chart when graphing rational functions. Checking whether the function is a straight or curved line (Option A), the slope (Option B).

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Answer:

1/4 I think

You haven't said which number line, so please provide some context.

Answer:

A

Step-by-step explanation:

subtract 2 from both sides and then divide by -2 on both sides.

when dividing by a negative sign, remember to switch the inequality sign (in this case from > to <)

(a) The annual interest rate is approximately 14.79%.

(b) It would take approximately 6.24 years for the investment to be worth at least $40,000.

(a) To find the value of r, we can use the compound interest formula:

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

Where:

A = Final amount

P = Principal amount (initial investment)

r = Annual interest rate (in decimal form)

n = Number of times interest is compounded per year

t = Number of years

In this case, the principal amount P is $6000, the final amount A is $16000, the number of years t is 10, and the interest is compounded annually (n = 1). We need to solve for r.

16000 = 6000(1 + r/1)^(1*10)

Dividing both sides by 6000:

16000/6000 = (1 + r)^10

Simplifying the left side:

2.6667 = (1 + r)^10

Taking the 10th root of both sides:

(1 + r) = (2.6667)^(1/10)

(1 + r) ≈ 1.1479

Subtracting 1 from both sides:

r ≈ 1.1479 - 1

r ≈ 0.1479

Therefore, the annual interest rate is approximately 14.79%.

(b) To find the time taken for the investment to be worth at least $40,000, we can again use the compound interest formula and solve for t:

40000 = 6000(1 + 0.1479/1)^(1*t)

Dividing both sides by 6000:

40000/6000 = (1 + 0.1479)^t

6.6667 = (1.1479)^t

Taking the logarithm (base 1.1479) of both sides:

log base 1.1479 (6.6667) = t

t ≈ 6.24

Therefore, it would take approximately 6.24 years for the investment to be worth at least $40,000.

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Jane has 6 - 14 - 12 = -20 boxes of beads left.

Jane initially had 6 boxes of beads.

She used 14 of those boxes to make face mask holders and gave 12 boxes to her sister.

To find out how many boxes of beads she has left, we need to subtract the boxes used and given away from the initial number.

First, let's calculate the total number of boxes used and given away:

Total boxes used and given away = Boxes used for face mask holders + Boxes given to sister

Total boxes used and given away = 14 + 12

Total boxes used and given away = 26

Next, we can subtract the total boxes used and given away from the initial number of boxes Jane had:

Boxes left = Initial number of boxes - Total boxes used and given away

Boxes left = 6 - 26

Boxes left = -20

The result is -20, which implies that Jane has a deficit of 20 boxes of beads.

This means she doesn't have any boxes of beads left; she has a shortage of 20 boxes based on the activities she performed.

It's important to note that having a negative value indicates that Jane doesn't have enough boxes to fulfill her activities.

If the result were positive, it would represent the number of boxes remaining.

However, in this case, Jane has used and given away more boxes than she initially had, resulting in a negative value.

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The amount of money increases by $125,000 due to the decrease in the reserve rate. The correct answer is B. It increases the amount by $125,000.

When the Federal Reserve decreases the reserve rate, it allows banks to hold a smaller portion of their deposits as reserves and lend out a larger portion. This change stimulates lending and increases the money supply in the economy through the process of fractional-reserve banking.

In this scenario, the initial deposit of $15,000 can be multiplied by the money multiplier, which is the reciprocal of the reserve rate. Initially, with a reserve rate of 6%, the money multiplier is 1/0.06 = 16.67. Therefore, the initial deposit can lead to a maximum increase in the money supply of $15,000 x 16.67 = $250,050.

When the reserve rate is decreased to 4%, the money multiplier becomes 1/0.04 = 25. Therefore, the same initial deposit of $15,000 can now result in a maximum increase in the money supply of $15,000 x 25 = $375,000.

The difference between the two scenarios is $375,000 - $250,050 = $124,950, which is approximately $125,000. Hence, the amount of money increases by $125,000 due to the decrease in the reserve rate.

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The true statement in the augmented matrix is that  (d) the system is inconsistent

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

The augmented matrix

In the augmented matrix, we have the number of rows and columns to be

Rows = 4 and Columns = 5

This means that

There are 4 equations in the system and there are 5 variables

The general rule is that

The number of equations must be at least the number of variables

Hence, there is no solution in the system

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z1 = 3 + 3i and z2 = 7(cos(5π/9) + i sin(5π/9)). z1z2, we first need to convert z2 into Rectangular form using the trigonometric form of complex numbers.z1z2 = 42 √2 cos(13π/36) + i 42 √2 sin(13π/36).Thus, the answer is z1z2 = 42 √2 cos(13π/36) + i 42 √2 sin(13π/36).

Given that, z1 = 3 + 3i and z2 = 7(cos(5π/9) + i sin(5π/9)).To find z1z2, we first need to convert z2 into rectangular form using the trigonometric form of complex numbers.

As we know, cosθ = Re^(iθ), where Re is the real part and i is the imaginary part.

Therefore, cos(5π/9) = Re^(i5π/9) = Re^(iπ/9) * Re^(i2π/3) = cos(π/9) + i sin(π/9) * (-1/2 + i √3/2) = (7/2) cos(π/9) - (7/2) sin(π/9) i √3 and sinθ = Im^(iθ),

where Im is the imaginary part and i is the imaginary unit.

Therefore, sin(5π/9) = Im^(i5π/9) = Im^(iπ/9) * Im^(i2π/3) = sin(π/9) + i cos(π/9) * (-1/2 + i √3/2) = (7/2) sin(π/9) + (7/2) cos(π/9) i √3So, z2 = 7(cos(5π/9) + i sin(5π/9)) = 7[(7/2) cos(π/9) - (7/2) sin(π/9) i √3 + (7/2) sin(π/9) + (7/2) cos(π/9) i √3]= 7(2 cos(π/9) + i 2 sin(π/9))= 14 cos(π/9) + i 14 sin(π/9)Now, z1z2= (3 + 3i)(14 cos(π/9) + i 14 sin(π/9))= 42 cos(π/9) + i 42 sin(π/9) + 42i cos(π/9) - 42 sin(π/9)= 42(cos(π/9) + i sin(π/9)) (1 + i)= 42(cos(π/9) + i sin(π/9)) √2 cis(π/4)= 42 √2 cos(π/9 + π/4) + i 42 √2 sin(π/9 + π/4)

Therefore, z1z2 = 42 √2 cos(13π/36) + i 42 √2 sin(13π/36).Thus, the answer is z1z2 = 42 √2 cos(13π/36) + i 42 √2 sin(13π/36).

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Answer:

[tex]21\sqrt{2} (cos\frac{29\pi }{36} +i sin\frac{29\pi }{36} )[/tex]

Step-by-step explanation:

Convert z1 to trig form, multiply the modulus and add the angles together with a common denominator of 36

The total value of the loan with quarterly compounded interest is approximately $45,288.38, while the total value of the loan with monthly compounded interest is approximately $45,634.84. The difference in total interest accrued is approximately $346.46.

Part A: To determine the total value of the loan with quarterly compounded interest, we can use the formula for compound interest:

A = P(1 + r/n)^(nt),

where:

A is the total value of the loan,

P is the principal amount (initial loan amount),

r is the interest rate (in decimal form),

n is the number of times interest is compounded per year,

and t is the number of years.

Given:

P = $20,000,

r = 9% or 0.09,

n = 4 (quarterly compounding),

t = 10 years.

Substituting the values into the formula, we have:

A = 20000(1 + 0.09/4)^(4*10).

Calculating this value, we find:

A ≈ $45,288.38.

Therefore, the total value of the loan with quarterly compounded interest is approximately $45,288.38.

Part B: To determine the total value of the loan with monthly compounded interest, we follow the same formula but with a different value for n:

n = 12 (monthly compounding).

Substituting the values into the formula, we have:

A = 20000(1 + 0.09/12)^(12*10).

Calculating this value, we find:

A ≈ $45,634.84.

Therefore, the total value of the loan with monthly compounded interest is approximately $45,634.84.

Part C: The difference between the total interest accrued on each loan can be calculated by subtracting the principal amount from the total value of each loan.

For the loan with quarterly compounding:

Total interest = Total value - Principal

Total interest = $45,288.38 - $20,000

Total interest ≈ $25,288.38.

For the loan with monthly compounding:

Total interest = Total value - Principal

Total interest = $45,634.84 - $20,000

Total interest ≈ $25,634.84.

The difference between the total interest accrued on each loan is approximately $346.46.

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The final rule for g(x) is g(x) = 3|3x| + 12.

To stretch the graph of f(x) = −|3x| − 4 vertically by a factor of 3, we multiply the function by 3. This will result in a vertical stretching of the graph.

So, the rule for g(x) is g(x) = 3f(x).

Now, let's find the expression for g(x) using the given function f(x) = −|3x| − 4:

g(x) = 3f(x)

g(x) = 3(-|3x| - 4)

g(x) = -3|3x| - 12

This is the expression for g(x), which represents the transformed graph.

To reflect the graph of g(x) across the x-axis, we change the sign of the function. This means that the negative sign in front of the absolute value will become positive, and the positive sign in front of the constant term will become negative.

Therefore, the final rule for g(x) is g(x) = 3|3x| + 12.

Now, let's consider the graph of g(x). The graph will have the same shape as f(x), but it will be stretched vertically by a factor of 3 and reflected across the x-axis.

The original graph of f(x) = −|3x| − 4 is a V-shaped graph that opens downward and passes through the point (0, -4). The transformed graph of g(x) will have a steeper V-shape, opening downward, and passing through the point (0, 12) instead of (0, -4).

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The surface area of the square pyramid with height of 12 cm, side of 10 cm, and slanting height of 13 cm is 360 cm².

A square pyramid has a square base, so to find the surface area of the pyramid, we need to find the area of each face and add them together. The pyramid has four triangular faces and a square base, so:

Surface area = area of base + sum of areas of all faces

The base of the pyramid is a square with side length of 10 cm, so its area is:

Area of base = side length^2 = 10 cm x 10 cm = 100 cm^2

To find the area of each triangular face, we can use the formula for the area of a triangle:

Area of triangular face = (1/2) x base x height

where the base is equal to the side length of the square base and the height is equal to the slanting height of the pyramid.

So, we have:

Area of triangular face = (1/2) x 10 cm x 13 cm = 65 cm^2

Therefore, the total area of all four triangular faces is:

Total area of triangular faces = 4 x Area of one triangular face

= 4 x 65 cm^2

= 260 cm^2

Now we can add the area of the base and the total area of the four triangular faces to get the total surface area of the pyramid:

Surface area = Area of base + Total area of triangular faces

= 100 cm^2 + 260 cm^2

= 360 cm^2

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The value of the function g(-7) when evaluated is 48

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

g(x) = 4(5 - x)

Also, we have

g(-7)

The above means that the value of x is

x = -7

Substitute the known values in the above equation, so, we have the following representation

g(-7) = 4(5 + 7)

Evaluate

g(-7) = 48

Hence, the function when evaluated is 48

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Answer:

a) See below.

b) $911 billion

c) See below.

Step-by-step explanation:

Linear regression is a statistical technique used to model the relationship between a dependent variable and one or more independent variables by fitting a linear equation to the observed data.

It estimates the slope and y-intercept of a straight line that minimizes the overall distance between the observed data points and the predicted values. The linear regression equation is y = ax + b.

The association between year and amount of disposable income is indicated by the linear regression equation y = ax + b.

The value of a is the slope of the linear regression line, and represents the average rate of change in disposable income per year. As a = 14.0545, it means that the disposable income increases by approximately $14.0545 billion dollars each year.  

As the value of r (correlation coefficient) is very close to +1, it indicates a very strong positive linear correlation between the year and disposable income. This suggests that as the years progress, the disposable income tends to increase.

Linear regression equation:

[tex]\boxed{y=14.05454545x-27198}[/tex]

To predict the amount of disposable income for the year 2000, we can substitute x = 2000 into the linear regression equation:

[tex]y = 14.05454545 \cdot 2000 - 27198[/tex]

[tex]y=28109.0909...-27198[/tex]

[tex]y=911.0909...[/tex]

[tex]y=911[/tex]

Therefore, the predicted amount of disposable income for the year 2000 is approximately $911 billion.

The predicted value of $911 billion for the year 2000 is significantly lower than the actual value of $8128 billion. This implies that the model is not accurately capturing the increasing trend in disposable income over time, leading to an underestimation of the income level in 2000. This suggests that the model may have limitations or inaccuracies when extrapolating beyond the range of the provided data. It indicates the need for caution and further analysis when using the model to make predictions outside of the given timeframe.

We can conclude that point A is located at the origin (0, 0).

There is only one possible location for point A is at the origin.

Let's revisit the given information to determine the possible location of point A.

Point C has the same y-coordinate as point B.

This means that the y-coordinate of point C is equal to the y-coordinate of point B.

The distance between point B and point C is equal to the distance between point C and the y-axis.

Let's assume the distance between point B and point C is represented by "d".

According to the information given, the distance between point C and the y-axis is also "d".

Point A has the same x-coordinate as point C.

This implies that the x-coordinate of point A is equal to the x-coordinate of point C.

The distance between point A and point C is twice the distance between point B and point C.

Let's assume the distance between point B and point C is represented by "d".

According to the information given, the distance between point A and point C is 2d.

Based on this information, we can analyze the relationships between the points:

Since the distance between point B and point C is equal to the distance between point C and the y-axis, we can infer that point B lies on the y-axis.

The x-coordinate of point B is 0.

As point C has the same y-coordinate as point B, the y-coordinate of point C is also determined to be the same as the y-coordinate of point B.

Since point A has the same x-coordinate as point C, the x-coordinate of point A will also be 0.

The distance between point A and point C is twice the distance between point B and point C.

As the distance between point B and point C is "d", the distance between point A and point C is 2d.

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The probability that it will rain on exactly three of the six days they are there is 0.242.

The given information can be represented as follows:P(Rain on any given day) = 4/9

The hiking club is camping for 6 days and the probability of rain on any given day is given as 4/9.

We are to find the probability that it will rain exactly three days out of the six days they are camping.

The number of ways that it can rain exactly three days out of the six days is given by:C(6,3) = 6!/(3!3!) = 20The probability of rain for any one of the 20 ways of getting exactly three days of rain is:

P(Rain on three days and no rain on three days) = (4/9)³(5/9)³

The probability of getting exactly three days of rain out of six is given by: P(exactly three days of rain) = 20(4/9)³(5/9)³ = 0.242, rounded to the nearest thousandth.

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The measure of angle LOP is given as follows:

m < LOP = 52º.

The segment OP bisects the angle O of the rhombus into two smaller angles, which are NOP and LOP.

.

A bisection means that the larger angle is divided into two smaller angles of equal measure.

The measure of angle NOP is given as follows:

m < NOP = 52º.

Hence the measure of angle LOP is given as follows:

m < LOP = 52º.

The final sentence is to find the measure of angle LOP.

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A. The distribution of the time spent is X ~ N(41, 19²)

B. The distribution of the sample mean is x-bar ~ N(41, (19²)/17)

C. The distribution of the total study time is X_total ~ N(17 × 41, 17 × 19²)

D. The probability is 0.3168

E. The probability is 0.6837

F. The probability is 0.6368

G. The least total time that a group of 17 students can study and still receive a sticker for "Great dedication" is approximately 350.3618 minutes.

To answer these questions, use the properties of the normal distribution. Given the mean and standard deviation provided, calculate the required probabilities and values.

a) The distribution of the time spent studying by a single student is given by:

X ~ N(41, 19²)

b) The distribution of the sample mean for a sample size of 17 students is given by:

x-bar ~ N(41, (19²)/17)

c) The distribution of the total study time for 17 students is given by:

X_total ~ N(17 × 41, 17 × 19²)

d) To find the probability that a randomly selected student's time is between 37 and 43 minutes, use the normal distribution:

P(37 ≤ X ≤ 43) = P((37 - 41) / 19 ≤ Z ≤ (43 - 41) / 19)

Using standard normal distribution tables or a calculator, find the corresponding probabilities for Z = -0.2105 and Z = 0.1053:

P(37 ≤ X ≤ 43) ≈ P(-0.2105 ≤ Z ≤ 0.1053) ≈ 0.3168

e) To find the probability that the average time studying for the 17 students is between 37 and 43 minutes, use the distribution of the sample mean:

P(37 ≤ x-bar ≤ 43) = P((37 - 41) / (19 / √(17)) ≤ Z ≤ (43 - 41) / (19 / √(17)))

Again, using standard normal distribution tables or a calculator, find the corresponding probabilities for Z = -0.8292 and Z = 0.8292:

P(37 ≤ x-bar ≤ 43) ≈ P(-0.8292 ≤ Z ≤ 0.8292) ≈ 0.6837

f) To find the probability that the randomly selected 17 students will have a total study time less than 731 minutes, use the distribution of the total study time:

P(X_total < 731) = P((X_total - (17 × 41)) / (√(17) × 19) < (731 - (17 * 41)) / (√(17) × 19))

Again, using standard normal distribution tables or a calculator, find the corresponding probability for Z ≈ 0.3503:

P(X_total < 731) ≈ P(Z < 0.3503) ≈ 0.6368

g) Yes, the assumption of normality is necessary for parts e) and f) since we are using the properties of the normal distribution.

h) To find the least total time that a group of 17 students can study and still receive a sticker for "Great dedication," find the cutoff point where the top 15% of the distribution lies.

Using the inverse normal distribution (also known as the Z-score or percent-point function), we can find the Z-score corresponding to the top 15% of the distribution. The Z-score for the top 15% is approximately 1.0364.

Now, solve for the least total time using the formula for the total study time:

X_total = (Z × √(17) × 19) + (17 × 41)

Substituting Z = 1.0364, we can calculate:

X_total ≈ (1.0364 × √(17) × 19) + (17 × 41) ≈ 350.3618

Therefore, the least total time that a group of 17 students can study and still receive a sticker for "Great dedication" is approximately 350.3618 minutes.

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The next number that continues the sequence: 4, 36, 18, 6, 45, 15, 8, 44 is 11.

Let us identify each of the numbers position:

n₁ = 4,

n₂ = 36,

n₃ = 18,

n₄ = 6,

n₅ = 45,

n₆ = 15,

n₇ = 8,

n₈ = 44

n₁, n₄, n₇ formed a sequence such that the common difference is:

d = n₄ - n₁ = n₇ - n₄ = 2

The next value (n₁₀) in this sequence will be

n₁₀ = n₇ + d

n₁₀ = 8 + 2 = 10

But we need to calculate n₉:

Consider the following,

n₂ and n₃

n₂/n₃ = 36/18 = 2

n₅ and n₆

n₅/n₆ = 45/15 = 3

n₈ and n₉

n₈/n₉ = 44/n₉ = 4

n₉ = 44/4 = 11

Therefore the 9th and 10th positions of the sequence are 11 and 10 respectively.

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The rectangular form of the complex number 12(cos(π) + isin(π)) is -12 + 0i.

To convert a complex number from polar form to rectangular form, we can use Euler's formula, which states that e^(ix) = cos(x) + isin(x). In this case, we have 12(cos(π) + isin(π)).

Since cos(π) = -1 and sin(π) = 0, we can substitute these values into the expression:

12(cos(π) + isin(π)) = 12(-1 + 0i) = -12 + 0i

In the rectangular form, a complex number is expressed as a combination of a real part (the coefficient of the real unit, represented as "a") and an imaginary part (the coefficient of the imaginary unit, represented as "bi").

In this case, the real part is -12 and the imaginary part is 0. The imaginary part is multiplied by the imaginary unit "i," which results in 0i. Therefore, the rectangular form of 12(cos(π) + isin(π)) is -12 + 0i.

In simpler terms, the rectangular form represents the complex number on the real number line (horizontal axis) and the imaginary number line (vertical axis). In this case, the complex number is purely real, as the imaginary part is 0, so it lies entirely on the real axis at -12.

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2.1. Number sense refers to a person's intuitive understanding and flexibility with numbers.

2.2 Number sense is of utmost importance in elementary mathematics education as it serves as the foundation for more advanced mathematical concepts and skills.

2.1 It involves the ability to comprehend and manipulate numerical quantities, make reasonable estimates, recognize patterns and relationships, and use numbers in practical and meaningful ways. Number sense goes beyond mere memorization of numerical facts and procedures; it encompasses a deep understanding of the structure and properties of numbers.

2.2 Number sense is of utmost importance in elementary mathematics education as it serves as the foundation for more advanced mathematical concepts and skills.

In summary, number sense is vital in energizing mathematics because it fosters conceptual understanding, supports problem-solving abilities, enhances mathematical fluency, promotes mathematical reasoning, and facilitates real-world applications of mathematics.

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The solution to the color puzzle in decimal is 10.5

Blue:

8z - 15 = 17

Add 15 to both sides

8z = 17 + 15

8z = 32

divide both sides by 8

z = 32/8

z = 4

Purple:

22 = 4y - 2

22 + 2 = 4y

24 = 4y

divide both sides by 4

y = 24/4

y = 6

Green:

3(-7) + 25

= -21 + 25

= 4

Red:

16 - 6 ÷ 2

= 16 - 3

= 13

Hence,

Red + (purple ÷ blue) - green

= 13 + (6 ÷ 4) - 4

= 13 + 1.5 - 4

= 14.5 - 4

= 10.5

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Apart from division or a fraction, the symbol '/' can also represent a ratio or indicate a relationship between two quantities.

In addition to representing division or a fraction, the symbol '/' can have different meanings depending on the context in which it is used. Here are a few common interpretations of the '/' symbol:

Ratio: The '/' symbol can indicate a ratio between two quantities. For example, if we say the ratio of boys to girls is 3/2, it means there are three boys for every two girls.

Per: The '/' symbol can be used to express "per" or "each." For instance, when we see a speed limit sign that says "60 km/h," it means "60 kilometers per hour."

Similarly, in sports statistics, we might see "10 goals/season," which means an average of 10 goals per season.

Probability: In probability theory, the '/' symbol can represent the probability of an event.

For example, if we say the probability of rolling a six on a fair die is 1/6, it means there is a one in six chance of rolling a six.

Quotient: While '/' is commonly used to represent division, it specifically represents the quotient in mathematical expressions.

For instance, in the expression 6/3, '/' indicates that we are finding the quotient of 6 divided by 3, which is equal to 2.

Symbolic Representation: In certain cases, the '/' symbol can be used as a shorthand or symbol to represent a concept or operation in specific fields or contexts.

For instance, in computer programming, '/' is often used to indicate file path or directory structure.

It is important to consider the context in which '/' is used to determine its meaning, as it can vary across different disciplines and applications.

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a) 4.516 multiplied by 215 equals 971.54.

b) (29.774 + 4.5) minus 8,451 equals -8,416.726.

c) 486.71 minus (29.774 + 4.5) plus 8,451 plus 15 equals 8,903.451.

a) To calculate 4.516 multiplied by 215, we multiply the two numbers:

4.516 x 215 = 971.54

Therefore, the result is 971.54.

b) To evaluate the expression (29.774 + 4.5) - 8,451, we perform the addition and then the subtraction:

29.774 + 4.5 = 34.274

34.274 - 8,451 = -8,416.726

Therefore, the result is -8,416.726.

c) To calculate 486.71 minus (29.774 + 4.5) plus 8,451 plus 15, we first perform the addition within parentheses, then the subtraction, and finally the addition:

(29.774 + 4.5) = 34.274

486.71 - 34.274 = 452.436

452.436 + 8,451 = 8,903.436

8,903.436 + 15 = 8,903.451

Therefore, the result is 8,903.451.

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The width of the walkway around the community garden is 19.5 feet.

To find the width of the walkway around the community garden, we can subtract the area of the garden from the total area of the garden and walkway combined.

Let's denote the width of the walkway as 'w'.

The area of the garden is given as 12 feet wide by 15 feet long, so its area is:

Area of the garden = 12 feet * 15 feet = 180 square feet

The total area of the garden and walkway combined is given as 648 square feet.

Area of the garden and walkway = 648 square feet

Now, we can set up an equation to find the area of the walkway:

Area of the walkway = Area of the garden and walkway - Area of the garden

Area of the walkway = 648 square feet - 180 square feet

Area of the walkway = 468 square feet

Since the walkway surrounds the garden on all sides, we can calculate its width by dividing the area of the walkway by the length of the garden plus twice the width of the walkway:

Width of the walkway = Area of the walkway / (Length of the garden + 2 * Width of the walkway)

Width of the walkway = 468 square feet / (15 feet + 2w)

Now we can substitute the given values and solve for 'w':

468 square feet / (15 feet + 2w) = w

Multiply both sides by (15 feet + 2w):

468 square feet = w(15 feet + 2w)

Rearrange the equation:

2w^2 + 15w - 468 = 0

Now, we can solve this quadratic equation for 'w' using factoring, completing the square, or the quadratic formula.

In this case, factoring is the most convenient method:

(2w - 39)(w + 12) = 0

Setting each factor to zero gives us two possible solutions:

2w - 39 = 0 or w + 12 = 0

Solving each equation for 'w' gives us:

2w = 39 or w = -12

w = 39/2 = 19.5

Since the width cannot be negative, the width of the walkway is 19.5 feet.

Therefore, the width of the walkway around the community garden is 19.5 feet.

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