Of the people who attended the school play, 5/12 were students and 1/8 were teachers. What fraction of the total audience were students or teachers

The quotient of the expression (7a³ + ab + 3) / (7a³ + ab - 3) divided by 4ab is (7/4a²) + (1/4b) + (3/4ab).

By dividing each term of the numerator by 4ab:

(7a³ + ab + 3) / (7a³ + ab - 3) ÷ 4ab

= (7a³ / 4ab) + (ab / 4ab) + (3 / 4ab)

Simplifying each term individually:

= (7/4a²) + (1/4b) + (3/4ab)

Thus, the quotient of the expression (7a³ + ab + 3) / (7a³ + ab - 3) divided by 4ab is (7/4a²) + (1/4b) + (3/4ab).

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The equation can be used to determine how many points Patrick earned is ;5 + 6 + y = 14.

What is an equation?

An equation is an expression that shows the relationship between two or more numbers and variables.

A mathematical equation is a statement with two equal sides and an equal sign in between.  4 + 6 and 10 can be seen on the left and right sides of the equal sign, respectively.

Given that Billy, Kip, and Patrick earned a combined 14 points during their team's game. Billy earned 5 points and Kip earned 6 points.

We can determine the direct answer by taking Billy and Kip's points and subtracting it from the total combined points.

That will give us 11, which means Patrick earned 3 points.

The equation we can use to represent this is ;5 + 6 + y = 14.

The probability that an antigen test comes back positive is approximately 0.05225, or about 5.225%.

We have,

To find the probability that an antigen test comes back positive, we need to consider both the true positive rate (probability of a positive test given that the person is infected) and the false positive rate.

Now,

Prevalence of coronavirus in California: 3 out of 1000

False positive rate of the antigen test: 0.05 (5 out of 100)

Let's calculate the probability of a positive test result.

The true positive rate can be calculated as 1 minus the false negative rate (probability of a negative test given that the person is infected):

True positive rate = 1 - 0.2 = 0.8 (or 80 out of 100)

The probability of a positive test result can be calculated using Bayes' theorem:

P(Positive test) = P(Positive test | Infected) x P(Infected) + P(Positive test | Not Infected) x P(Not Infected)

P(Positive test) = (0.8 x 3/1000) + (0.05 x 997/1000)

P(Positive test) = 0.0024 + 0.04985

P(Positive test) = 0.05225

Therefore,

The probability that an antigen test comes back positive is approximately 0.05225, or about 5.225%.

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The prove of the statement is described below.

First of all ,

Labeling the given figure

And draw  a diagonal,

In the quadrilateral PQRS,

The side PQ is equal to the side RS.

Therefore,

PQ = RS

And also, the side PQ is parallel to the side RS.

Then,

PQ || RS.

Now, draw the diagonal PR.

We know that a parallelogram's diagonal divides the parallelogram into two congruent triangles.

By the rule of  Side-Angle-Side,

we can show that ∆ PQR ≅ ∆ RSP.

Therefore,

The side QR is parallel to PS

Then,

QR || PS.

Hence,

PQRS is a parallelogram.

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The translation used to create the image of the second polygon is A, 7 units to the right.

To determine the translation used to create the image of the second polygon (A' B' C' D') from the original polygon (ABCD), compare the corresponding vertices.

Original polygon (ABCD):

A (-2, -1)

B (0, -4)

C (4, -4)

D (2, -1)

Image polygon (A' B' C' D'):

A' (5, -1)

B' (7, -4)

C' (11, -4)

D' (9, -1)

To find the translation, calculate the horizontal and vertical differences between the corresponding vertices.

Horizontal difference:

For point A: A'x - Ax = 5 - (-2) = 7

For point B: B'x - Bx = 7 - 0 = 7

For point C: C'x - Cx = 11 - 4 = 7

For point D: D'x - Dx = 9 - 2 = 7

Vertical difference:

For point A: A'y - Ay = -1 - (-1) = 0

For point B: B'y - By = -4 - (-4) = 0

For point C: C'y - Cy = -4 - (-4) = 0

For point D: D'y - Dy = -1 - (-1) = 0

From the calculations, see that the horizontal difference is 7 units for all the corresponding vertices, while the vertical difference is 0 units for all the corresponding vertices.

Therefore, the translation used to create the image of the second polygon is 7 units to the right.

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Mathematical form of the question:

Use the graph to answer the question.

Graph of polygon ABCD with vertices at A (-2, -1), B (0, -4), C (4, -4), D (2, -1). A second polygon A' (5, -1), B' (7, -4), C' (11, -4), D' (9, -1).

Determine the translation used to create the image.

7 units to the right

3 units to the right

7 units to the left

3 units to the left

Answer:

  b. 5.9 mL

Step-by-step explanation:

You want the value that represents the 99th percentile of a normal distribution with a mean of 5 mL and a standard deviation of 0.4 mL.

The inverse of the normal CDF function is used to find the X value for a given area. Here, we want X such that P(x < X) = 0.99 for μ = 5 and σ = 0.4.

The attached calculator output shows that is about X = 5.9305. Rounded to 1 decimal place, the volume is ...

  5.9 mL

<95141404393>

The scatter plot for the above data and it's line of best fit is attached accordingly. You may achieve this result using Microsoft Excel.

Enter your data into two columns in an Excel worksheet. For example, in columns A and B, enter the years and corresponding life expectancies from your given data.

Select the data range by clicking and dragging over the cells containing the data.

Go to the "Insert" tab in the Excel ribbon.

In the "Charts" group, click on the "Scatter" chart type. Choose the scatter plot style that suits your preference (e.g., "Scatter with Straight Lines").

A scatter plot will be created on your worksheet.

Right-  click on any data point on the scatter plot and choose "Add Trendline" from the context menu.

In the  "Format Trendline" pane  that appears on the right side of the screen, select the "Linear"  trendline option.

Check the box for  "Display Equation on Chart" if you want to show the equation of the line of best fit on the  chart.

You can further customize the appearance of the scatter plot and the trendline using various chart formatting options in Excel.

Your scatter plot with the line of best fit is now plotted on Excel.

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The triangles are similar triangles. Then the value of 'x' will be 27 cm.

The polygonal shape of a triangle has a number of sides and three independent variables. Angles in the triangle add up to 180°.

The ratio of the matching sides will remain constant if two triangles are comparable to one another.

From the above theorem, the equation is given as,

(x/15)² = 178.2 / 55

Simplify the equation, then we have

(x/15)² = 3.24

x/15 = 1.8

x = 27 cm

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Answer:

Graph coloring is a widely used concept in real-world situations, particularly in computer science and engineering. It involves assigning colors to the vertices of a graph, with no adjacent vertices having the same color. One real-world application of graph coloring is in scheduling problems, where tasks must be assigned to resources such as workers or machines, with the constraint that no two tasks requiring the same resource can be scheduled at the same time. Another application is in map coloring, where regions on a map are assigned colors in such a way that no two adjacent regions have the same color.

Challenges and limitations associated with using graph coloring algorithms include the exponential increase in computation time as the size of the graph increases, making it difficult to apply the algorithm to very large graphs. Additionally, some graph coloring problems may be NP-hard, meaning that no polynomial-time algorithm is known that can solve them for all possible inputs. This can make it challenging to find an optimal solution, and instead approximate solutions may need to be used.

For example, in telecommunication networks, channel assignment is a common problem of graph coloring. In mobile networks, different channels must be assigned to different cells to avoid interference and ensure efficient use of resources. The channel assignment problem can be formulated as a graph coloring problem, where the vertices represent cells and the edges represent interference.

Another example is in scheduling of sports leagues. In this case, teams are assigned to play on different days and times in such a way that no two teams play against each other more than once and no team plays two games on the same day. This can be formulated as a graph coloring problem, with vertices representing teams and edges representing games between teams.

In conclusion, graph coloring algorithms have numerous real-world applications. While there are challenges and limitations associated with these algorithms, they continue to be an important tool for solving optimization problems in a wide range of fields.

Step-by-step explanation:

The pattern in the table should be completed as follows;

Number of                      1          3        9       27        81

students with pets

Total number                  3         9        27     81        243

of students

A ratio that compares the number of blue squares to the total number of squares is 3 : 4.

In Mathematics and Geometry, a proportional relationship can be modeled or represented by the following mathematical equation:

y = kx

Where:

Next, we would find the constant of proportionality (k) by using various data points as follows:

Constant of proportionality, k = y/x

Constant of proportionality, k = 3/1 = 9/3

Constant of proportionality, k = 3.

y = 3x

81 = 3x

x = 27

x = 9/3

x = 3

y = 3(81)

y = 243

For the ratio of blue squares to total squares, we have:

18/24 = 3/4 = 3 : 4

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Answer:

To determine the total number of different ways Ms. Edwards can redecorate her office, we need to multiply the number of choices for each item:

Number of paint color choices = 8

Number of curtain choices = 5

Number of carpet color choices = 6

To calculate the total number of ways, we multiply these choices together:

Total number of ways = Number of paint color choices × Number of curtain choices × Number of carpet color choices

= 8 × 5 × 6

= 240

Therefore, there are 240 different ways for Ms. Edwards to redecorate her office. So, the answer is 240 ways.

The calculated value of the element d₁₂ is -8

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

The matrix D given as

D = [-3 -8 6 -4]

The element d₁₂ represents the element located at the first row and the second column

In this case, the element located at the first row and the second column is -8

Using the above as a guide, we have the following:

d₁₂ = -8

Hence, the value of the element d₁₂ is -8

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The total interest charges to the Roth Family, after 15 years, given the monthly payment would be $ 118, 000

First, find the total amount paid over the 15 years by the Roth Family to be :

= Number of years x Monthly payment x Number of months per year

= 15 x 1, 850 x 12

= $ 333, 000

The total interest charges that the Roth Family paid would be :

= Total amount paid after 15 years - Total amount borrowed as mortgage

= 333, 000 - 215, 000

= $ 118, 000

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The value of x for each item is given as follows:

28. x = 5.

29. x = 3.44.

For item 28, we apply the crossing chord theorem, which states that the products of the parts of the chords are equal, hence the value of x is obtained as follows:

16x = 10 x 8

16x = 80

x = 5.

For item 29, we apply the two secant theorem, hence the value of x is obtained as follows:

10(x + 10) = 12(12 + 25)

10x + 100 = 444

10x = 344

x = 3.44.

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The identification of the polygon that are given in the diagram above would be =

Polygon 1 = Isosceles trapezoid

polygon 2 = parallelogram

polygon 3 = Rectangle

polygon 4 = Rhombus

polygon 5 = Square

polygon 6 = trapezoid

Isosceles trapezoid : This is the type that of polygon in which the the top and base are not equal in length, but they are parallel to each other.

Parallelogram : This is defined as the type of polygon that has two pairs of parallel sides.

Rectangle : This has two pairs of sides that are equal and opposite.

Rhombus: All the sides are equal in length with its opposite angles equal .

Square: All sides are equal in length but and all angles are equal to 90°.

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The difference in mean growth per week is 3.7cm/week

From the graph given :

Growth per week for Basil = 2, 6, 9, 10, 14, 16

Growth per week for Mint = 1, 2, 4, 7, 9, 12

Mean growth per week = (2+6+9+10+14+16) / 6 = 9.5 cm/week

Mean growth per week (Mint) = (1+2+4+7+9+12)/6 = 5.83 cm/week

The difference cannbe obtained by subtracting the mean Values thus ;

Difference in average growth = 9.5 - 5.83 = 3.67 cm/week

Therefore, the difference in mean growth per week is 3.7cm/week

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No, the inference is not necessarily valid. The fact that a comedian received the most votes in an online poll does not guarantee that they are the funniest comedian in the nation.

The poll's results are influenced by various factors such as the sample size, the demographics of the voters and potential biases in the voting process.

Also, humor is subjective and what one person finds funny may not be the same for someone else. Therefore, determining the funniest comedian based solely on the number of votes in an online poll is an oversimplified and flawed approach.

To improve the poll, several steps can be taken which are:

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Answer:

-12a-1/4a2

Step-by-step explanation:

1. The value of angle NSR is 110°

2. The value of r 12

If two secants or chords intersect inside a circle, then the measure of the angle formed is equal to half the sum of the measures of the intercepted arcs.

Therefore;

S = 1/2( 170+40)

S = 1/2 ( 210)

S = 110

Therefore the value of angle NSR is 110°

2. There is a theorem that states that the angle between a radius and tangent is 90°

Using Pythagorean theorem

(25+r)² = r² +35²

= 625 + 50r + r² = r² +35²

625 +50r = r² - r² +35²

50r = 1125-625

50r = 600

r = 600/50

r = 12

therefore the value of r is 12

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Determine Joy's annual income tax liability. If the amount is significantly different from the R4000 monthly deduction, she may need to consult a tax advisor to ensure accurate tax planning.

To assess whether the monthly income tax deduction of R4000 is too much for Joy, we need to calculate her annual income tax liability and compare it to the deduction.

First, let's calculate the amount Joy contributes to her pension fund annually:

Pension fund contribution = 7% of annual salary

Pension fund contribution = 7/100 * R286,500

Pension fund contribution = R20,055

Next, we need to determine Joy's taxable income by subtracting her pension fund contribution from her annual salary:

Taxable income = Annual salary - Pension fund contribution

Taxable income = R286,500 - R20,055

Taxable income = R266,445

Now, we can calculate the income tax liability based on the South African income tax brackets for the 2023 tax year. Please note that tax brackets are subject to change, so it's essential to verify with the latest tax regulations.

According to the current tax brackets, the tax rates for the 2023 tax year are as follows:

18% on the first R216,200 of taxable income

26% on the portion of taxable income between R216,201 and R337,800

31% on the portion of taxable income between R337,801 and R467,500

36% on the portion of taxable income between R467,501 and R613,600

39% on the portion of taxable income above R613,601

Using these tax rates, we can calculate the income tax liability:

Tax liability = (18% * amount within first bracket) + (26% * amount within second bracket) + (31% * amount within third bracket) + (36% * amount within fourth bracket) + (39% * amount within fifth bracket)

First, let's calculate the amounts within each tax bracket:

Amount within first bracket = min(Taxable income, R216,200)

Amount within second bracket = min(max(Taxable income - R216,200, 0), R337,800 - R216,201)

Amount within third bracket = min(max(Taxable income - R337,800, 0), R467,500 - R337,801)

Amount within fourth bracket = min(max(Taxable income - R467,500, 0), R613,600 - R467,501)

Amount within fifth bracket = max(Taxable income - R613,600, 0)

Now, let's substitute the values and calculate the income tax liability:

Tax liability = (18% * min(Taxable income, R216,200)) + (26% * min(max(Taxable income - R216,200, 0), R337,800 - R216,201)) + (31% * min(max(Taxable income - R337,800, 0), R467,500 - R337,801)) + (36% * min(max(Taxable income - R467,500, 0), R613,600 - R467,501)) + (39% * max(Taxable income - R613,600, 0))

By calculating this equation, you can determine Joy's annual income tax liability. If the amount is significantly different from the R4000 monthly deduction, she may need to consult a tax advisor to ensure accurate tax planning.

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Answer:

Equation of the triangle:  (x + 74) + 73 + (37 + x) = 180°

Solve for x:  x = -2

Find the measure of angle A:  A = 35°

Step-by-step explanation:

Step 1:  We know that the 73° angle and the angle inside the triangle that lies opposite it are vertical angles ,so this angle is also 73° .

Step 2:  Similarly, the (x + 74)° angle and the angle inside the triangle that lies opposite it are also vertical angles, so this angle is also represented by the expressions (x + 74)°

Equation of the triangle:  The triangle sum theorem states that the sum of the interior angles of a triangle equals 180°.  Thus, the equation of the triangle is (x + 74) + 73 + (37 + x) = 180

Solving for x:  

Step 1:  Rearrange and combine like terms:

(x + 74) + 73 + (37 + x) = 180

(x + x) + (74 + 73 + 37) = 180

2x + 184 = 180

Step 1:  Subtract 184 from both sides:

(2x + 184 = 180) - 184

2x = -4

Step 3:  Divide both sides by 2 to solve for x:

(2x = -4) / 2

x = -2

Finding the measure of angle A:  Now we can plug in -2 for x in the expression representing the measure of angle A to find the actual measure:

A = 37 - 2

A = 35

Thus, the measure of angle A is 35°.

Optional Step:  We can check that our equation of the triangle, the value we've found for x, and the measure of angle A is correct by simply plugging in -2 for x in the equation of the triangle and checking that we get 180 on both sides:

(-2 + 74) + 73 + (37 - 2) = 180

72 + 73 + 35 = 180

145 + 35 = 180

180 = 180

Thus, all of our answers are correct.

Answer:

108 sq ft

Step-by-step explanation:

The formulas are given in the bottom right.

For the triangles*, the base = 8 ft. The height = 6 feet.

Area = (bh)/2.

Area = (8*6)/2 = (48)/2 = 24 sq ft.

There are two of those identical triangles. 24+24 = 48.

(*Another way to look at this is to say that the base = 8+8 = 16. Area = (bh)/2 = (16*6)/2 = 96/2 = 48. It doesn't matter, mathematically. I can't tell if they have tried to show this as 2 triangles or one.)

For the bottom rectangle, area = length x width = 6*10 = 60 sq ft.

Add them up! 24 + 24 + 60 = 108 sq ft.

The population of the country will be 796 million approximately in the year 2022.

To find when the population of the country will be 796 million, we can set the equation equal to 796 and solve for t:

[tex]A = 433.4e^{(0.032t)}\\796 = 433.4e^{(0.032t)}[/tex]

Divide both sides by 433.4:

[tex]e^{(0.032t)} = 1.836[/tex]

Take the natural logarithm of both sides:

[tex]ln(e^{(0.032t)}) = ln(1.836)[/tex]

0.032t = 0.605

Divide both sides by 0.032:

t = 18.91

Therefore, the population of the country will be 796 million approximately 19 years after 2003. To find the year, we add 19 to 2003:

2003 + 19 = 2022

Therefore, the population of the country will be 796 million approximately in the year 2022.

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It would take 21 weeks for the population to surpass 16,000.

The insect population P in a certain area fluctuates with the seasons and is estimated by the function P = 15,000 + 2500 sin(πt/52), where t is given in weeks.

For the population to surpass 16,000, we can set up the following equation:

15,000 + 2500 sin(πt/52) = 16,000

Subtracting 15,000 from both sides, we get:

2500 sin(πt/52) = 1000

Dividing both sides by 2500, we get:

sin(πt/52) = 0.4

We know that sin(πt/52) is positive when t is between 0 and 52 and between 104 and 156 since sine is positive in the first and second quadrants. Therefore, we can write:

πt/52 = sin⁻¹(0.4)

Multiplying both sides by 52/π, we get:

t = (52/π) sin⁻¹(0.4)

Using a calculator, we can evaluate sin⁻¹(0.4) to be approximately 0.4115 radians.

t = (52/π) (0.4115)

t = 21.02

Therefore, it would take 21 weeks for the population to surpass 16,000.

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Answer:

23p -6k - 42p + 30k - 5p

23p - 42p - 5p + 30k - 6k

23p - 47p + 24k

-24p + 24k

24(- p + k)

24( k - p)

Answer:

The answer is

60p + 24k

Step-by-step explanation:

1: Expand

=23p - 6k + 42p + 30k - 5p

step2: collect like terms

=42p + 23p - 5p - 6k + 30k

step 3: add/subtract them to get your answer

=60p + 24k

your answer☝

The value of x is 14

Fractions are described as expression that shows the part of a whole number, a whole variable. a whole element.

The different types of fractions are listed as;

From the information given, we have that;

1/5 ( x - 10 ) = 4/5

This is written as;

(x -10)/5 = 4/5

cross multiply the values, we have;

5(x-10) = 20

expand the bracket

5x - 50 = 20

Collect like terms

5x = 70

Divide the values

x =14

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The measure of Angle 2 is 17 degrees.

The measure of angle 2,  the relationships between the given angles formed by the parallel lines and transversals.

1. The angles formed by lines s, t, and y, clockwise from top left, are:

  - Blank

  - Blank

  - (10x + 5) degrees

  - Blank

  - (4x - 7) degrees

  - Blank

2. The angles formed by s and z are:

  - 65 degrees

  - 1

  - Blank

  - Blank

3. The angles formed by z and t are:

  - 2

  - Blank

  - Blank

  - Blank

From the given information, we can deduce the following:

- Angle 1 is congruent to the angle formed by z and t (corresponding angles).

- Angle (10x + 5) degrees is congruent to angle 65 degrees (alternate interior angles).

- Angle (4x - 7) degrees is congruent to angle 2 (alternate interior angles).

Therefore, we can set up the following equations:

65 = 10x + 5     (Equation 1)

2 = 4x - 7       (Equation 2)

Solving Equation 1:

65 - 5 = 10x

60 = 10x

x = 6

Substituting x = 6 into Equation 2:

2 = 4(6) - 7

2 = 24 - 7

2 = 17

Hence, the measure of angle 2 is 17 degrees.

Therefore, the correct answer is:Angle 2 = 17 degrees.

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Answer:

78

Step-by-step explanation:

Let the number of hamburgers sold = x.

Then, the number of cheeseburgers sold = 2x.

Total number of sales: x + 2x = 3x

Total number of sales: 234

3x = 234

x = 78

Answer: 78

The value of x, considering the vertical angles in this problem, is given as follows:

x = 11.

Vertical angles are angles that are opposite by the same vertex on crossing segments, hence they share a common vertex, thus being congruent, meaning that they end up having the same angle measure.

The vertical angles for this problem are given as follows:

As the vertical angles are congruent, the value of x is obtained as follows:

2x - 4 = x + 7

x = 11.

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