S.
5.
You may be surprised to learn that prime
numbers are used for sending information
securely over the internet. The internet uses
computers, so they do this by multiplying two
huge prime numbers. It is hard work. Here is
a challenge for you. The number 51 is the
product of two prime numbers. What are the
two prime numbers?

The largest area you can enclose with 20 meters of fencing against a long, straight wall is 50 square meters. Itcan be found using optimization methods.

Step 1: Define the variables
Let x be the length of the fencing parallel to the wall and y be the length of the fencing perpendicular to the wall.
Step 2: Set up the constraint equation
Since you have 20 meters of fencing, the sum of x and 2y (both sides perpendicular to the wall) should equal 20. The constraint equation is:x + 2y = 20
Step 3: Express one variable in terms of the other
Solve the constraint equation for one variable. In this case, solve for x: x = 20 - 2y

Step 4: Write the area function
The area of the rectangle can be expressed as A = xy. Substitute x from the previous step into this equation: A(y) = (20 - 2y)y.

Step 5: Find the critical points
Differentiate the area function with respect to y and set it to zero to find the critical points: dA/dy = 20 - 4y = 0, Solve for y:4y = 20, y = 5
Step 6: Find the corresponding value for x
Plug the value of y back into the equation for x: x = 20 - 2(5), x = 10

Step 7: Check for maximum area
The critical point we found (x=10, y=5) is indeed a maximum since the second derivative of the area function is negative.Step 8: State the largest area
The largest area you can enclose with 20 meters of fencing against a long, straight wall is A = xy = 10 * 5 = 50 square meters.

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Conclusion
Therefore, for some natural number N, n ≥ N and n ∈ N implies |(p - q)/n| < ε.

To show that for some natural number N, n ≥ N and n ∈ N implies |p/n - q/n| < ε, we'll use the Archimedean property of real numbers. The Archimedean property states that for any positive real numbers a and b, there exists a natural number n such that n*a > b.

Let's consider the inequality we want to prove: [tex]|p/n - q/n| < ε.[/tex]

Step 1: Rewrite the inequality
First, we can rewrite the inequality as |(p - q)/n| < ε, since we are allowed to combine the fractions.

Step 2: Apply the Archimedean property
By the Archimedean property, we know that for any ε > 0, there exists a natural number N such that N > (p - q)/ε.

Step 3: Rearrange the inequality
We can rearrange the inequality from step 2 to get[tex] N*ε > p - q. [/tex]

Step 4: Divide by N
Now, divide both sides of the inequality by N to get [tex]ε > (p - q)/N.[/tex]

Step 5: Relate this to our original inequality
We want to show that |(p - q)/n| < ε for some n ∈ N, where n ≥ N. Since n ≥ N, and ε > (p - q)/N, we have ε > (p - q)/n for n ∈ N and n ≥ N

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The average of the other two numbers is 16.

This is because the average of 8 numbers is 14, and the average of 6 of those numbers is 12. Therefore, to find the average of the remaining two numbers, you need to find the difference between the average of 8 and the average of 6.

The difference between 14 and 12 is 2. Therefore, if the average of 6 is 12, the average of the remaining two numbers must be 12 + 2 = 16.

To solve this question, you need to use the fact that the average of 8 numbers is 14, and the average of 6 of those numbers is 12. The average of 8 numbers is the sum of all 8 numbers divided by 8. Therefore, if the average of 8 numbers is 14, the sum of all 8 numbers must be 14x8 = 112.

Similarly, the average of 6 numbers is 12, so the sum of those 6 numbers must be 12x6 = 72. The difference between these two sums is 112-72 = 40. This is the sum of the remaining two numbers.

Therefore, if the sum of the remaining two numbers is 40, then the average of the remaining two numbers must be 40/2 = 20.

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Answer:

(3x-4)(x+1)

Step-by-step explanation:

Workout is provided in the attachment.

Answer:

48 square centimeters

Step-by-step explanation:

[tex] \frac{1}{2} (4)(8)(3) = 48[/tex]

Answer:

Step-by-step explanation:

c

The closest option, 24 cm, is the length of the mode, or the length that appears the most frequently in the data set, but none of the other possibilities fit this value's exact range.

The mean, which is used to indicate the average value of a group of values in statistics, is a measure of central tendency. It is determined by adding up all of the data set's values and dividing the result by the number of values. The mean is frequently chosen as a data set's representative value because it can provide light on the typical value or average performance of a population or sample.

given

The following formula can be used to determine the total number of roses multiplied by each length of roses using the information in the table:

(2 * 22) + (4 * 23) + (5 * 24) + (3 * 25) + (1 * 26) = 222

There are: roses in all.

2 + 4 + 5 + 3 + 1 = 15

As a result, the data set's mean is:

222 / 15 = 14.8

Thus, the data set's roses' average length is 14.8 centimetres.

The closest option, 24 cm, is the length of the mode, or the length that appears the most frequently in the data set, but none of the other possibilities fit this value's exact range.

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If the area is 384 square feet, then the length of the altitude is 36 feet, and the length of the base is 44 feet.

Let's start by using the formula for the area of a triangle in terms of its base and altitude:

[tex]Area = (1/2) * base * altitude[/tex]

We know that the area of the triangle is 384 square feet:

[tex]384 = (1/2) * base * altitude[/tex]

Next, we're given that the base is 8 feet longer than the altitude:

[tex]base = altitude + 8[/tex]

We can substitute this expression for the base into the area formula:

[tex]384 = (1/2) * (altitude + 8) * altitude[/tex]

Simplifying and multiplying out the right-hand side:

[tex]384 = (1/2) * (altitude^2 + 8altitude)\\768 = altitude^2 + 8altitude[/tex]

Now we have a quadratic equation in terms of the altitude. We can solve for the altitude by first rearranging the equation:

[tex]altitude^2 + 8altitude - 768 = 0[/tex]

Then, we can solve for the altitude using the quadratic formula:

[tex]altitude = (-8 ± sqrt(8^2 - 4(1)(-768))) / (2(1))\\altitude = (-8 ± sqrt(6400)) / 2\\altitude = (-8 ± 80) / 2[/tex]

We get two possible values for the altitude:

altitude = -44 or altitude = 36

Since the altitude of a triangle must be positive, we discard the negative solution and choose altitude = 36 feet.

We can then use the expression for the base in terms of the altitude to find the length of the base:

[tex]base = altitude + 8 = 36 + 8 = 44 feet[/tex]

Therefore, the length of the altitude is 36 feet and the length of the base is 44 feet.

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Correct question:

Find the lengths of the triangles altitude and base if its area is 384 squared feet. If its base is 8 feet longer than its altitude, find the length of the base and the altitude.

Answer:

The circumference of a circle with radius 2.8 is 17.59(*)

Step-by-step explanation:

Circumference of a cicle in terms of radius:Circumference = 2·π·r = 2·3.14·2.8 = 17.59(*)In terms of diameter:Circumference = π·d = 3.14·5.6 = 17.58(*)In terms of area:Circumference C = √4·π·A = √4·π·24.63 = 17.59(*)(*) 17.59 inches exactly or limited to de precision of this calculator (13 decimal places).Note: for simplicity, the operations above were rounded to 2 decimal places and π was rounded to 3.14.

The set of integers is a Pythagorean triple are option (A) 10, 24, 25 and (D) 6, 8, 10

A Pythagorean triple is a set of three positive integers a, b, and c, such that a^2 + b^2 = c^2, which represents the sides of a right triangle.

Let's check each set of integers

A. 10^2 + 24^2 = 100 + 576 = 676 = 25^2. Therefore, (10, 24, 25) is a Pythagorean triple.

B. 9^2 + 12^2 = 81 + 144 = 225 = 15^2. Therefore, (9, 12, 15) is a Pythagorean triple. However, the set given is (9, 12, 21), which is not a Pythagorean triple.

C. 8^2 + 15^2 = 64 + 225 = 289 = 17^2. Therefore, (8, 15, 17) is a Pythagorean triple. However, the set given is (8, 15, 23), which is not a Pythagorean triple.

D. 6^2 + 8^2 = 36 + 64 = 100 = 10^2. Therefore, the set of integers (6, 8, 10) is a Pythagorean triple.

Therefore, the correct options are (A) 10, 24, 25 and (D) 6, 8, 10

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The given question is incomplete, the complete question is:

Which set of integers is a Pythagorean triple?

A.

10, 24, 25

B.

9, 12, 21

C.

8, 15, 23

D.

6, 8, 10

Answer: Assuming the student walks along a rectangular grid and not diagonally, the possible locations the student could end up after walking 4 miles in any direction would be the points that are 4 units away from the starting point in a north, south, east, or west direction.

Using this information, we can determine that the possible locations the student could end up at are:

(-2, 10)

(-6, 14)

(-10, 10)

(-6, 6)

Therefore, the answer is:

All of the above locations are possible endpoints.

Step-by-step explanation:

If the student walks 4 miles north, south, east, or west,  the location at the end of the walk are respectively, (-6, 14). (-6, 6), (-2,10). (-10, 10).

The Cartesian plane, named after the mathematician René Descartes (1596 - 1650), is a plane with a rectangular coordinate system that associates each point in the plane with a pair of numbers.

Given that,

A student starts a walk at (-6, 10)

If the student walks 4 miles north, south, east, or west,

The location at the end of the walk =

Plotting dotted diagram,

                  East

                     |

North --------- |-------------- South

                     |

                  West

Suppose, he goes to north  direction

Then, 4 will be added to y coordinate,

It can be written as,

(-6, 14)

Similarly, for South, (-6, 6),  

East, (-2,10).

West, (-10, -10)

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Option C, The greatest common factor of −20x³y − 8[tex]xy^4[/tex] + 4xy³ is 4xy by simplifying expressions and solving equations.

To find the greatest common factor (GCF) of the given terms, we need to factor out any common terms that they share.

First, we can factor out 4xy from each term:

−20x³y − 8[tex]xy^4[/tex] + 4xy³ = 4xy(-5x² - 2y³ + y)

Now we need to check if any further common factors can be factored out from the remaining expression (-5x² - 2y³ + y). However, there are no other factors that can be factored out from this expression.

As it enables us to factor out frequent terms and reduce the expression to its most basic form, the GCF is beneficial for simplifying expressions and solving equations.

Therefore, the GCF of the original terms is 4xy, option (c).

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Answer:

4xy

Step-by-step explanation:

I did the test and got it right !

It is the greatest common factor of 20, 8, and 4. And we add the “xy”.

Answer:

3a + 2b

Step-by-step explanation:

Let the unknown side have length X.

X + X + 5a - b + 5a - b = 16a + 2b

2X + 10a - 2b = 16a + 2b

2X = 6a + 4b

X = 3a + 2b

Answer: 3a + 2b

Probability of drawing a red or white ball.

One ball is drawn at random from a box containing 4 red balls, 7 white balls, and 5 blue balls.

Determine the probability that the ball drawn is red or white.

The total number of balls in the box is 4+7+5=16 balls. The probability of drawing a red ball is 4/16. The probability of drawing a white ball is 7/16.The probability of drawing either a red or white ball is given by the sum of the individual probabilities. Hence, the probability of drawing a red or white ball is:(4/16) + (7/16) = (11/16)

Therefore, the probability that the ball drawn is red or white is 11/16.

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The LCD of 2/5, 1/2, and 3/4 is equal to 20.

In Mathematics, LCD is an abbreviation for least common denominator or lowest common denominator and it can be defined as the smallest number that can act as a common denominator for a given set of fractions.

Next, we would determine the factors of the denominators for the given fraction 5, 2, and 4 as follows;

5 = 5 × 1

2 = 2 × 1

4 = 2 × 2 × 1

Therefore, the least common denominator (LCD) would be calculated as follows:

Least common denominator (LCD) = 5 × 2 × 2 × 1

Least common denominator (LCD) = 20.

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The new length of the rectangle is approximately 2.29 units. We use the formula for the area of a rectangle to solve for the new length, given the new width.

If the width and length of a rectangle are 3 and 8, respectively, and the width is increased to 10.5, we can calculate the new length of the rectangle using the formula for the area of a rectangle, which is length multiplied by width.

The original area of the rectangle is 3 x 8 = 24 square units. If we increase the width to 10.5, the new area of the rectangle becomes: 10.5 x length = 24 Solving for the length, we get: length = 24/10.5 = 2.29 (rounded to two decimal places)

It's important to note that changing one dimension of a rectangle can affect the other dimension, especially if we want to maintain the same area.

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The quotient of 3.013 × [tex]10^{8}[/tex] and 6.55 × [tex]10^{5}[/tex] expressed in scientific notation is equal to 0.46 × [tex]10^{3}[/tex].

In Mathematics, a quotient is a mathematical expression that is simply used to represent the division of a number (numerator) by another number (denominator).

Generally speaking, any numerical data (number) can be written in scientific notation as shown below:

a × [tex]b^n[/tex]

Where

Next, we would rewrite the given numerical data (number) in scientific notation by dividing by 6.55 × [tex]10^{5}[/tex] as follows:

Scientific notation = 3.013 × [tex]10^{8}[/tex]/(6.55 × [tex]10^{5}[/tex])

Scientific notation = 0.46 × [tex]10^{8-5}[/tex]

Scientific notation = 0.46 × [tex]10^{3}[/tex].

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The length οf the hypοtenuse AC is 10 units.

In mathematics, a graph is a visual representatiοn οf data οr infοrmatiοn that cοnsists οf pοints, called vertices οr nοdes, and lines, called edges οr arcs, that cοnnect them.

Graphs are οften used tο represent relatiοnships between οbjects, events, οr quantities.

They can be used tο mοdel and sοlve prοblems in variοus fields, such as cοmputer science, engineering, ecοnοmics, and sοcial sciences. There are many types οf graphs, including bar graphs, line graphs, scatter plοts, pie charts, and netwοrk graphs, amοng οthers. Each type οf graph is suited fοr visualizing different types οf data οr relatiοnships.

Based οn the given diagram, we can use the Pythagοrean theοrem tο find the length οf the hypοtenuse AC οf the right triangle ABC:

[tex]AC^2 = AB^2 + BC^2[/tex]

Substituting the given side lengths, we get:

[tex]AC^2 = 6^2 + 8^2[/tex]

[tex]AC^2 = 36 + 64[/tex]

[tex]AC^2 = 100[/tex]

Taking the square rοοt οf bοth sides, we get:

AC = 10

Therefοre, the length οf the hypοtenuse AC is 10 units.

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ANSWER: dA/dt=0.17A A(0)=950

The correct differential equation and initial condition for the continuous compounding of a sum of 950$ at 17% interest are given by dA/dt = 0.17A and A(0) = 950. This represents the rate of change of the amount A(t) with respect to time and the initial investment value, respectively.

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Part A: The system of equations are:

j + g = 23

j = 2g + (- 4)

PART B: The solution of the system is:

Jodi:  14 games

Gerry: 9 games

Part A:

The first equation comes from the fact that they bought a total of 23 games:

j + g = 23 ----- (1)

The second equation comes from the fact that "Jodi buys 4 less than twice the number of games Gerry buys":

j = 2g + (- 4)

j = 2g - 4  ----- (2)

Part B:

We can solve this system of equations by substitution. We can solve the second equation for j, and then substitute that expression for j in the first equation. That is:

j + g = 23

2g-4 + g = 23

3g - 4 = 23

3g = 23 + 4

3g = 27

g = 27/3

g = 9

Put g = 9 in (1):

j + g = 23

j + 9 = 23

j = 23 - 9

j = 14

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To find the probability of rolling six standard, six-sided dice and getting six distinct numbers, we need to calculate the total number of possible outcomes and the number of outcomes that satisfy the given condition.

Total Number of Outcomes:

The total number of possible outcomes when rolling six dice is 6^6, which is 46,656. Each die has six possible outcomes (1, 2, 3, 4, 5, or 6), and there are six dice being rolled.

Number of Outcomes with Six Distinct Numbers:

To get six distinct numbers when rolling six dice, each number must be different from the others. We can choose any six numbers from 1 to 6, and there are 6! (6 factorial) ways to arrange them on the six dice.

6! = 6 x 5 x 4 x 3 x 2 x 1 = 720

Probability of Getting Six Distinct Numbers:

To find the probability of rolling six standard, six-sided dice and getting six distinct numbers, we divide the number of outcomes with six distinct numbers by the total number of possible outcomes:

P(six distinct numbers) = number of outcomes with six distinct numbers / total number of outcomes

P(six distinct numbers) = 720 / 46,656

P(six distinct numbers) = 5 / 324

Therefore, the value of the probability is 5/324 or approximately 0.0154.

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The required length of YZ is [tex]\frac{145}{16}$[/tex]  units.

To find the length of YZ, we can use the fact that the length of XZ is the sum of the lengths of XY and YZ. We are given that XZ is [tex]12 \frac{11}{16}$[/tex]units and XY is [tex]3 \frac{5}{8}$[/tex] units. Let's convert these mixed numbers to improper fractions so that we can add them:

[tex]$XZ = 12 \frac{11}{16} = \frac{12 \times 16 + 11}{16} = \frac{203}{16}$[/tex]

[tex]$XY = 3 \frac{5}{8} = \frac{3 \times 8 + 5}{8} = \frac{29}{8}$[/tex]

Now we can use the formula:

XZ = XY + YZ

to solve for YZ. Rearranging the equation, we get:

YZ = XZ - XY

Plugging in the values we calculated, we get:

[tex]$YZ = \frac{203}{16} - \frac{29}{8}$[/tex]

To subtract these fractions, we need to find a common denominator. The smallest number that both 16 and 8 divide into evenly is [tex]16 \times 2 = 32$[/tex], so we can rewrite the fractions with a denominator of 32:

[tex]$YZ = \frac{203}{16} \times \frac{2}{2} - \frac{29}{8} \times \frac{4}{4} = \frac{406}{32} - \frac{116}{32}$[/tex]

Now we can subtract:

[tex]$YZ = \frac{406 - 116}{32} = \frac{290}{32}$[/tex]

We can simplify this fraction by dividing both the numerator and denominator by the greatest common factor, which is 2:

[tex]$YZ = \frac{145}{16}$[/tex]

Therefore, the length of YZ is [tex]\frac{145}{16}$[/tex]  units.

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Complete question:

XY is a line, Z is a point on this line such that  XZ is 12 11/16 units XY is 3 5/8 units find YZ​.

Answer:

To calculate the time it will take Tom to complete the 1-mile kayak trip, we need to take into account the speed of the current and the speed at which Tom can paddle relative to the current.

Let's first consider the time it would take for Tom to go upstream against the current. In this case, the speed of the current is subtracted from Tom's paddling speed, so his effective speed is:

Effective speed upstream = 4 miles per hour - 3 miles per hour = 1 mile per hour

To travel 1 mile at 1 mile per hour, Tom would take:

Time upstream = Distance / Effective speed upstream = 1 mile / 1 mile per hour = 1 hour

Now let's consider the time it would take for Tom to go downstream with the current. In this case, the speed of the current is added to Tom's paddling speed, so his effective speed is:

Effective speed downstream = 4 miles per hour + 3 miles per hour = 7 miles per hour

To travel 1 mile at 7 miles per hour, Tom would take:

Time downstream = Distance / Effective speed downstream = 1 mile / 7 miles per hour ≈ 0.14 hours or about 8 minutes

Therefore, the total time it would take Tom to complete the 1-mile kayak trip, including going upstream and downstream, would be:

Total time = Time upstream + Time downstream = 1 hour + 0.14 hours ≈ 1.14 hours or about 68 minutes.

An expression for the number of amoebas after 24 hours is 1 × 2²⁴ = 16,777,216

Single-celled creatures knοwn as amοebas reprοduce asexually. An amοeba begins tο reprοduce when its genetic material dοubles, twο nuclei are fοrmed, and it begins tο alter shape by develοping a thin "waist" in the centre. Typically, this prοcedure gοes οn until the cells are finally divided intο twο.

Amοeba reprοduce typically asexually thrοugh a prοcess called binary fissiοn. The cell splits intο twο daughter cells οf equal size fοllοwing the replicatiοn οf its genetic material by mitοtic divisiοn.

a. It is given that an amοeba divides tο fοrm twο amοebas after οne hοur sο there are 2 amοebas after 1 hοur.

b. It is given that οne hοur later, each οf the twο amοebas divides tο fοrm twο mοre amοebas sο there are 2×2=4 amοebas after 2 hοurs.

c. The number οf amοebas is dοubling after each hοur since each amοeba divides intο twο amοebas every hοur. This means that the number οf amοebas after 6 hοurs can be fοund by multiplying the οriginal number οf amοebas, which was 1, by 2 six times. The number οf amοebas after 6 hοurs is then 1 × 2⁶ = 2⁶= 64 amοebas.

d. Using the same pattern frοm part (c), the number οf amοebas after 24 hοurs is 1 × 2²⁴ = 2²⁴ = 16,777,216 amοebas.

Expression = 1 × 2²⁴ = 16,777,216

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I think it equals 18 tens

Answer:

3 x 60 x 30 = 5400

Step-by-step explanation:

If john paul is 6 years older than Angelina and in 9 years the sum of their ages will be 86, then John Paul is 35 years old now.

Let's assume Angelina's current age to be x years old. Then, according to the given information, John Paul's current age would be (x + 6) years old.

In 9 years, the sum of their ages will be 86, which means that (x + 9) + (x + 6 + 9) = 86.

Simplifying the equation, we get 2x + 24 = 86.

Solving for x, we get x = 31.

Therefore, John Paul's current age would be x + 6 = 31 + 6 = 35 years old.

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To dilate a point S by a scale factor of 1/2, we need to multiply the coordinates of the point by the scale factor.So the dilated point S' will have coordinates (2, 3).

A scale factor is a number that scales or multiplies another quantity by a certain amount, either making it larger or smaller. In geometry, scale factor is used to describe the ratio of the corresponding side lengths of two similar figures.

If the coordinates of point S are (x,y), then the coordinates of the dilated point S' will be:

(x', y') = (1/2 * x, 1/2 * y)

In other words, the x-coordinate of the dilated point is half of the x-coordinate of the original point, and the y-coordinate of the dilated point is half of the y-coordinate of the original point.

For example, if point S has coordinates (4, 6), then the coordinates of the dilated point S' will be:

(x', y') = (1/2 * 4, 1/2 * 6) = (2, 3)

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The sweets Candy Bars 1 through 3 and Each individual will get a one - third of a candy bar as  each candy bar is cut into three equal pieces .

A fraction is a percentage or ratio between two numbers that is expressed numerically. Typically, it is expressed as a/b, where a stands for the numerator and b for the denominator. The denominator is the total number of equal parts that make up the whole, while the numerator is the number of equal parts that are being taken into account. Three out of four equal portions, or three-fourths of the entire, are represented by the fraction 3/4, for instance. Mathematicians frequently use fractions, particularly in the areas of algebra, geometry, and arithmetic.

given

Each individual will get a one - third of a candy bar.

A visual representation of this might be the following, where each candy bar is cut into three equal pieces and presented to a different friend:

[tex]| 1/3 | | 1/3 | | 1/3 |[/tex]

The sweets Candy Bars 1 through 3 and Each individual will get a one - third of a candy bar as  each candy bar is cut into three equal pieces .

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The answer is 31.10.

Based on the given conditions, formulate: 648 * 24 / 500 Cross out the common factor:dfrac162 * 2412.

The theoretical probability of not drawing an "o" is 3/4.

We have,

Each of the letters of the word Colorado are written on a piece of paper and then put into a bag.

Here, The word "Colorado" has 8 letters, including 2 "o"s.

Therefore, the probability of drawing an "o" is,

P = 2/8

P = 1/4

Hence, The probability of not drawing an "o" is the complement of this probability is,

1 - 1/4 = 3/4.

Therefore, the theoretical probability of not drawing an "o" is 3/4.

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