Male and female students were surveyed about dancing and playing sports. They had the following preferences:


Do you prefer dancing or playing sports?
Playing sports Dancing Row totals
Male students 0.25 0.27 0.52
Female students 0.17 0.31 0.48
Column totals 0.42 0.58 1


Which of the following is a two-way conditional relative frequency table for gender?

Do you prefer dancing or playing sports?
Playing sports Dancing
Male students 0.60 0.47
Female students 0.40 0.53
Column totals 1 1

Do you prefer dancing or playing sports?
Playing sports Dancing Row totals
Male students 0.48 0.52 1
Female students 0.35 0.65 1

Do you prefer dancing or playing sports?
Playing sports Dancing Row totals
Male students 25% 27% 52%
Female students 17% 31% 48%
Column totals 42% 58% 100%

Do you prefer dancing or playing sports?
Playing sports Dancing Row totals
Male students 50 54 104
Female students 34 62 96
Column totals 84 116 200

The expected recorded score of the given data of short quiz is,

Expected recorded score is 8.5.

And maximum of the two expected recorded score is 12.45.

Expected recorded score e(x + y),

e(x + y) = ΣΣ (x + y)  p(x, y)

Simplify this expression by rearranging the sum,

e(x + y) = ΣΣ x p(x, y) + ΣΣ y  p(x, y)

Using the table given,

ΣΣ x p(x, y)

= (0×0.02) + (5×0.04) + (10×0.01) + (0×0.06) + (5×0.13) + (15×0.15) + (0×0.02) + (5×0.20) + (10×0.16)

= 4.6

ΣΣ y p(x, y)

= (0×0.02) + (0×0.06) + (0×0.02) + (5×0.10) + (5×0.13) + (5×0.01) + (10×0.20) + (10×0.15) + (10×0.16)

= 3.9

Expected recorded score is,

e(x + y)

= 4.6 + 3.9

= 8.5

The maximum of the two scores is recorded,

Then the recorded score will be either x or y, whichever is larger.

Probability distribution of the maximum score,

P(max(x, y) = 0)

= P(x=0, y=0)

= 0.02

P(max(x, y) = 5)

= P(x=5, y=0) + P(x=0, y=5) + P(x=5, y=5)

= 0.06 + 0.04 + 0.13

= 0.23

P(max(x ,y) = 10)

= P(x=10, y=0) + P(x=0, y=10) + P(x=10, y=5) + P(x=5, y=10) + P(x=10, y=10)

= 0.01 + 0.15 + 0.20 + 0.10 + 0.16

= 0.62

P(max(x, y) = 15)

= P(x=15, y=0) + P(x=0, y=15) + P(x=15, y=5) + P(x=5, y=15) + P(x=15, y=10) + P(x=10, y=15) + P(x=15, y=15)

= 0.01 + 0.10 + 0.10 + 0.10 + 0.01 + 0.01 + 0.01

= 0.34

Expected recorded score can be calculated as the weighted sum of the possible maximum scores,

E(max(x, y)) = Σ max(x, y) × P(max(x, y))

Substituting the probabilities, we get,

E(max(x ,y))

= 0×0.02 + 5×0.23 + 10×0.62 + 15×0.34

= 0+ 1.15 + 6.2 + 5.1

= 12.45

Expected recorded score is 12.45 (upto two decimal places).

Therefore, expected recorded score is 8.5 and maximum of the expected recorded score is 12.45.

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The above question is incomplete, the complete question is:

An instructor has given a short quiz consisting of two parts. for a randomly selected student, let x = the number of points earned on the first part and y = the number of points earned on the second part. suppose that the joint pmf of x and y is given in the accompanying table.

                                                 y

     p(x, y)                     0               5                  10                               15

    0                           0.02          0.06              0.02                          0.10

x   5                           0.04           0.13              0.20                          0.10

   10                           0.01            0.15              0.16                          0.01

(a) if the score recorded in the grade book is the total number of points earned on the two parts, what is the expected recorded score e(x + y)? (enter your answer to one decimal place.)

(b) if the maximum of the two scores is recorded, what is the expected recorded score? (enter your answer to two decimal places.)

Answer:

-2299

Step-by-step explanation:

12(-72) = -864

(-3 - 32) = -35

41(-35) = -1435

-864 + -1435 = -2299

The thirty-fourth term of the sequence is 1282.

First, we can find the second term of the ap series as follows:

a₂ = a₁ + 36 = a₄₃ + 36

Similarly, we can find the third term of the sequence as:

a₃ = a₂ + 36 = a₄₄ + 36

Continuing this pattern, we can find the thirty-fourth term of the sequence as:

a₃₄ = a₃₃ + 36 = (a₃₂ + 36) + 36 = ((a₃₁ + 36) + 36) + 36 = so on = a + 33(36)

Substituting the given value of a = 146, we get:

a₃₄ = 146 + 33(36) = 1282

a₃₄ = 1282

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Ian's score as a percentage is 73.3%, rounded to one decimal place. This means that he obtained 73.3% of the total marks possible in the test.

The percentages are a useful way to express scores or quantities relative to the whole. In this case, Ian's percentage score indicates how well he performed on the test compared to the maximum score possible.

To calculate Ian's score as a percentage, we can use the formula:

percentage score = (marks obtained / total marks) x 100

In this case, Ian scored 44 out of a total of 60 marks. Plugging these values into the formula, we get:

percentage score = (44 / 60) x 100 = 73.3%

Therefore, Ian's score as a percentage is 73.3%, rounded to one decimal place. This means that he obtained 73.3% of the total marks possible in the test.

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Answer:

To fine mean (average) you add up all the numbers and divide it by the amount of numbers. there are 8 scores

Step-by-step explanation:

55+60+65+70+75+80+85+90

=580

580/8=72.5

the mean is 72.5

x = 18 is given by:

f(x) = (x - 18)²
      -------------

             x

Or, if you have choices it is CHOICE A :) :P

For the first student with a score of 1910, the z-score is: 1.34 ,For the second student with a score of 1220, the z-score is: 0.81 , For the third student with a score of 2160, the z-score is:2.14 , For the fourth student with a score of 1370, the z-score is: -0.34 and , out of the four students, only the one with the test score of 2160 is considered an unusual score based on the z-score.

To find the z-score for a given data point, we use the formula:

z = (x - μ) / σ

where x is the data point, μ is the mean, and σ is the standard deviation.

For the first student with a score of 1910, the z-score is:

z = (1910 - 1479) / 317 = 1.34

For the second student with a score of 1220, the z-score is:

z = (1220 - 1479) / 317 = -0.81

For the third student with a score of 2160, the z-score is:

z = (2160 - 1479) / 317 = 2.14

For the fourth student with a score of 1370, the z-score is:

z = (1370 - 1479) / 317 = -0.34

Now, we need to determine whether any of these z-scores are unusual. One way to do this is to use the rule of thumb for normal distributions, which states that about 68% of the data falls within one standard deviation of the mean, about 95% of the data falls within two standard deviations of the mean, and about 99.7% of the data falls within three standard deviations of the mean.

Using this rule, we can say that a z-score greater than 2 or less than -2 is unusual, since it corresponds to data that falls more than two standard deviations from the mean.

From the z-scores we calculated, only the third student with a z-score of 2.14 falls outside of this range and can be considered an unusual score.

Therefore, out of the four students, only the one with the test score of 2160 is considered an unusual score based on the z-score.

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Francesca salary in the first year is $169684.14

Let us assume that P be the first salary of Francesca.

The salary increases by 2% every year.

Let A be the amount she earns after n years at a rate of r.

Using the formula for the compound interest is:

A = P (1 + r)ⁿ

Here, A = $187345

n = 5 years

R = 2%

So, r = 0.02

We need to find the value of P

substituting these values in above equation we get,

187345 = P (1.02)⁵

187345 = 1.10408 × P

P = $169684.14

Therefore,her salary in the first year is $169684.14

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The complete question is:

Each year, Francesca earns a salary that is 2 percent higher than her previous year's salary. In her first 5 years at this job, she earned a total of $187,345. What was Francesca's salary in her 1st year at this job?

Step-by-step explanation:

I answered this on the other question

The center is (1,-4)

The new center is

(-4,-6)

The equation of the center is

[tex](x + 4) {}^{2} + (y + 6) {}^{2} = 16[/tex]

The venn diagram is given below and A - (AnBnC) is {a, b, d}.

What is venn diagram?

A Venn diagram consists of one or more circles, each representing a set. The items that belong to each set are placed inside the appropriate circle, and the overlapping areas between the circles represent items that belong to both sets.

(a) Here's a Venn diagram that shows the relation of U, A, B, and C:

The circle labeled "A" represents the set A, the circle labeled "B" represents the set B, and the circle labeled "C" represents the set C. The regions where the circles overlap represent the elements that are in more than one set.

(b) A - (AnBnC) means the elements that are in set A but not in the intersection of sets A, B, and C. To find these elements, we first need to find the intersection of sets A, B, and C:

AnBnC = {f}

Then, we can subtract that set from set A:

A - (AnBnC) = {a, b, d}

Therefore, A - (AnBnC) is {a, b, d}.

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Answer:

2/3

Step-by-step explanation:

1) find the gcf of 16 and 24.

Method 1: By Listing Factors

List the factors of each number.

Factors of 16 : 1, 2, 4, 8, 16

Factors of 24 : 1, 2, 3, 4, 6, 8, 12, 24

Find the largest number that is shared by all rows above. This is the GCF.

GCF = 8

Method 2: By Prime Factors

List the prime factors of each number.

Prime Factors of 16 : 2, 2, 2, 2

Prime Factors of 24 : 2, 2, 2, 3

Find the intersection of these primes.

2,2,2

Multiply these numbers: 2x2x2=8. this is the gcf.

gcf = 8

2) Divide both the numerator and the denominator by the GCF.

16/8 / 24/8

3)simplify

2/3

The requried radius of the cylinder is 4 inches.

We are given the volume of the cylinder as 48π cubic inches and the height as 3 inches. So we can use the formula V = πr²h to solve for the radius.

48π = πr²(3)

16 = r²

r = ±4

Since the radius of a cylinder cannot be negative, we take the positive value of r, which is 4.

Therefore, the radius of the cylinder is 4 inches.

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Consider the events , , . if a player rolls this weighted die and wins $1 if the outcome is in event , wins $ 2 if the outcome is in event , and loses $3 if the outcome is in event .

The expected value of this game is 0.70. The player should expect to win 0.70.

To find out the expected value of this game, we will use the formula

E(X)= ∑(xP(x))

where X represents the possible outcomes and P(x) represents the probabilities of those outcomes.

Let's determine the probabilities of each event:

Event 1: P(1) = 0.3

Event 2: P(2) = 0.5

Event 3: P(3) = 0.2

Using the information given in the question, we know that a player wins 1 if the outcome is in event 1, wins 2 if the outcome is in event 2, and loses $3 if the outcome is in event 3.

Therefore, the values of each event are:

Event 1: 1

Event 2: 2

Event 3: -3

Now, we can use the formula to calculate the expected value:

E(X)= ∑(xP(x))E(X)

      = (1 x 0.3) + (2 x 0.5) + (-3 x 0.2)

E(X) = 0.3 + 1 - 0.6E(X)

       = 0.7

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Step-by-step explanation:

Arrange each of the equations into y = mx+ b to compare....m is the slope and b is the y -axis intercept

- 8x - y = 3     ======>    y = -8x -3

8x + y = - 3    ======>    y = - 8x -3

These are equations of the SAME LINE with   slope = -8    intercept = -3

Answer:

The equations have the same slopes and the same y-intercepts.

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{6.4 cm}\underline{Slope-intercept form of a linear equation}\\\\$y=mx+b$\\\\where:\\ \phantom{ww}$\bullet$ $m$ is the slope. \\ \phantom{ww}$\bullet$ $b$ is the $y$-intercept.\\\end{minipage}}[/tex]

Rearrange both equations so that they are in slope-intercept form.

[tex]\underline{\sf Equation\;1}\\\\\begin{aligned}-8x-y&=3\\-8x-y+8x&=8x+3\\-y&=8x+3\\y&=-8x-3\end{aligned}[/tex]

[tex]\underline{\sf Equation\;2}\\\\\begin{aligned}8x+y&=-3\\8x+y-8x&=-8x-3\\y&=-8x-3\end{aligned}[/tex]

Therefore, the equations have the same slopes and the same y-intercepts.

The correct table is:

A 2-column table with 4 rows. Column 1 is labeled cups with entries 1, 2, 4, 8. Column 2 is labeled tablespoons with entries 16, 32, 64, 128 which is option first.

This is because we know that there are 16 tablespoons in one cup. So, if we multiply the number of cups by 16, we get the corresponding number of tablespoons forming a geometric sequence. For example, 2 cups is equal to 2 x 16 = 32 tablespoons. This relationship is correctly shown in the first table where each entry in the second column is obtained by multiplying the corresponding entry in the first column by 16.

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The correct table that relates the number of cups to the number of tablespoons is:

A 2-column table with 4 rows.

- Column 1 is labeled cups with entries 1, 2, 4, 8.

- Column 2 is labeled tablespoons with entries 16, 32, 64, 128.

In this table, the first row tells us that there is 1 cup which is equivalent to 16 tablespoons. The second row shows that 2 cups is equal to 32 tablespoons. The third row indicates that 4 cups is equal to 64 tablespoons. And finally, the fourth row states that 8 cups is equal to 128 tablespoons.

This table follows the relationship where for every 1 cup, there are 16 tablespoons. So, to find the number of tablespoons for a given number of cups, you can use this table as a reference.

For example, if you have 3 cups, you can find the number of tablespoons by looking at the table. Since there is no row for 3 cups in the given table, you can approximate it by knowing that 2 cups is equal to 32 tablespoons, and 4 cups is equal to 64 tablespoons. Therefore, 3 cups would be somewhere between 32 and 64 tablespoons.

option (C) is the box plot that best represents the given data set.

Based on the given data set, the box plot that best represents the data is (C). Here are the reasons:

The median of the data set is 30, and this is represented by the line inside the box in all four options.

The box in option (C) ranges from 22.5 to 33.5, which includes the interquartile range (IQR) of the data set, which is from 16 to 37. This means that 50% of the data falls within this range, and the box in option (C) represents this well.

The whiskers in option (C) extend to the minimum and maximum values in the data set, which are 9 and 39 respectively. This means that none of the data points fall outside the whiskers.

The range of the number line in option (C) (7 to 46) and the tick marks (every one unit) also match the range and increments of the given data set.

Therefore, option (C) is the box plot that best represents the given data set.    

Here is the visual representation of option (C):

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[tex]\textit{equation of a circle}\\\\ (x- h)^2+(y- k)^2= r^2 \hspace{5em}\stackrel{center}{(\underset{-10}{h}~~,~~\underset{4}{k})}\qquad \stackrel{radius}{\underset{2}{r}} \\\\[-0.35em] ~\dotfill\\\\ ( ~~ x - (-10) ~~ )^2 ~~ + ~~ ( ~~ y-4 ~~ )^2~~ = ~~2^2\implies (x+10)^2+(y-4)^2=4[/tex]

Each of the expressions has been simplified by using properties of exponents as shown below.

In Mathematics, an exponent refers to a mathematical operation that is typically used in conjunction with an algebraic expression in order to raise a quantity to the power of another.

This ultimately implies that, an exponent is represented by the following mathematical expression;

bⁿ

Where:

By applying the multiplication and division law of exponents for powers to each of the expressions, we have the following:

(1.25 × 10⁷) + 63,000,000 = (1.25 × 10⁷) + (6.3 × 10⁷) = (1.25 + 6.3) × 10⁷ = 7.55 × 10⁷

12,000 + 7 × 10⁴ = 1.2 × 10⁴ + 7 × 10⁴ = (1.2 + 7) × 10⁴ = 8.2 × 10⁴

5.88 × 10⁵ - 3.44 × 10⁵ = (5.88 - 3.44) × 10⁵ = 2.44 × 10⁵

6 × 10⁸ ÷ 120 = 6 × 10⁸ ÷ 1.2 × 10² = (6 ÷ 1.2) × 10⁸⁻² = 5 × 10⁶

9 × 10¹² ÷ 4.5 × 10⁴ = (9 ÷ 4.5) × 10¹²⁻⁴ = 2 × 10⁸

1.3 × 10³ · 2,200 = 1.3 × 10³ × 2.2 × 10³ = (1.3 × 2.2) × 10³⁺³ = 2.86 × 10⁶

(6.3 × 10⁴) + 1.3 × 10⁴ = (6.3 + 1.3) × 10⁴ = 7.6 × 10⁴

3,400 · 2 × 10⁴ = 3.4 × 10³ × 2 × 10⁴ = (3.4 × 2) × 10³⁺⁴ = 6.8 × 10⁷

9.78 × 10⁵ - 732,000 = (9.78 - 7.32) × 10⁵ = 2.46 × 10⁵

3.5 × 10² · 2 × 10⁴ = (3.5 × 2) × 10²⁺⁴ = 7 × 10⁶

1.1 × 10⁸ ÷ 22,000 = 1.1 × 10⁸ ÷ 2.2 × 10⁴ = (1.1 ÷ 2.2) × 10⁸⁻⁴ = 0.5 × 10⁴ = 5 × 10³.

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If Pancake Paradise makes the best Pineapple Pancakes. Each batch makes 3 whole pancakes.

To find the number of whole pancakes in each batch, we need to first calculate the area of one pancake.

The diameter of each pancake is 2.5 inches, which means the radius is 1.25 inches (diameter = 2 x radius). The area of each pancake is:

A = πr^2 = π(1.25 in)^2 ≈ 4.91 in^2

Next, we need to determine how many pancakes can be made from a batch of batter that covers an area of 15 inches.

Number of pancakes = Total area / Area of one pancake

Number of pancakes = 15 in^2 / 4.91 in^2 ≈ 3.05

Since we can't make a fraction of a pancake, we need to round down to the nearest whole number. Therefore, each batch makes 3 whole pancakes.

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a.  At the moment when the batter is halfway to first base, his distance from second base is not changing.

b. His distance from third base increasing at the same moment at rate 6√2 ft/s.

Let A be the starting point of the batter, B be second base and C be third base, and $P$ be the position of the batter when he is halfway to first base.

Note that AP = AB/2 = 45 ft, B = AB - AP = 45 ft, and PC = AC - AP = 45√2 ft.

(a) We want to find [tex]$\frac{d}{dt} PB$[/tex]  when AP = 45 ft and [tex]$\frac{dAP}{dt} = 24$[/tex] ft/s.

By the Pythagorean theorem, we have

PB² = AB² - AP² = 90² - 45² = 4050, so PB = √4050 ft.

To find [tex]$\frac{dPB}{dt}$[/tex] , we take the derivative of both sides of the equation PB² = 4050 with respect to time t:

[tex]2PB\frac{dPB}{dt} &= 0 \\\\\\\frac{dPB}{dt} &= 0[/tex]

Therefore, at the moment when the batter is halfway to first base, his distance from second base is not changing.

(b) We want to find [tex]$\frac{d}{dt} PC$[/tex] when AP = 45 ft and [tex]$\frac{dAP}{dt}[/tex] = 24 ft/s.

By the Pythagorean theorem, we have

PC² = AC² - AP² = (90\√2})² - 45² = 5850,

so PC = √5850 ft.

To find [tex]$\frac{dPC}{dt}$[/tex] we take the derivative of both sides of the equation PC² = 5850 with respect to time t:

[tex]2PC\frac{dPC}{dt} &= 2\frac{dAP}{dt}(AC - AP) \\\\\frac{dPC}{dt} &= \frac{dAP}{dt} \frac{AC - AP}{PC} \&= \frac{24(90\sqrt{2} - 45)}{\sqrt{5850}} \&= 16\sqrt{2} \approx 22.63 \text{ ft/s}.\end{align*}[/tex]

Therefore, at the moment when the batter is halfway to first base, his distance from third base is increasing at a rate of 16√2 ft/s.

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A set of information displaying string lengths in inches. The values in the data are as follows: 6, 5, 8, 4, 9, 7, 5, 7, 4, 8, 7, 6. This data set's mean is 6.

To find the mean of the data set, we need to add up all the values and then divide by the total number of values.

Adding up all the values:

[tex]6 + 5 + 8 + 4 + 9 + 7 + 5 + 7 + 4 + 8 + 7 + 6 = 76[/tex]

There are 12 values in the data set, so we divide the sum by 12:

76/12 = 6.33 (rounded to two decimal places)

Therefore, the mean of the data set is 6.33.

Therefore the correct answer is thus 6.

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If person is standing at the end of a pier 12 ft above the water, then the velocity of the boat when it is 16 ft from the pier is 7.5 ft/min.

The distance between the person and the water (y) = 12 ft,

The distance between the boat and the pier is (x) = 16 ft,

So, p = √(12² + 16²) = 20 ft,

From the triangle given below,

⇒ x² + y² = p²,

taking "derivative" with respect to "x",

We get,

⇒ 2x(dx/dt) + 2y(dy/dt) = 2p(dp/dt);

⇒ dy/dt = 0 ..because height is constant,

The water is pulling on a rope at speed of (dp/dt) = 6 ft/min,

⇒ 2x(dx/dt) = 2p(dp/dt),

⇒ 2×16×(dx/dt) = 2×20×6,

⇒ dx/dt = 7.5 ft/min.

Therefore, the velocity of the boat is 7.5ft/min.

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The given question is incomplete, the complete question is

A person is standing at the end of a pier 12 ft above the water and is pulling on a rope attached to a rowboat at the waterline at a rate of 6 ft per minute. What is the velocity of the boat when it is 16 ft from the pier?

The numbers in scientific notation 3.26E21 and 5.95E-27 are 3.26E21 = 3.26 * 10^21 and 5.95E-27 = 5.95 * 10^-27

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

3.26E21 and 5.95E-27

A number represented using aEb can be represented as

aEb = a * 10^b

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

3.26E21 = 3.26 * 10^21

5.95E-27 = 5.95 * 10^-27

Hence, the numbers in scientific notation are 3.26E21 = 3.26 * 10^21 and 5.95E-27 = 5.95 * 10^-27

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The probability that the first person who walks in wearing an N95 mask is the third person is 0.128 or 12.8%. The probability that the first person who walks in wearing an N95 mask is the third person can be found using the geometric probability distribution formula.

The formula for the probability of the first success occurring on the kth trial is:

P(k) = (1 - p)^(k-1) * p

Where p is the probability of success (in this case, 20% or 0.2), and k is the number of trials.

In this case, we want to find the probability that the third person who walks in is wearing an N95 mask, so k = 3.

Substituting the values, we get:

P(3) = (1 - 0.2)^(3-1) * 0.2

P(3) = 0.64 * 0.2

P(3) = 0.128

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Answer:

1and 2/3

Step-by-step explanation:

using the factorization method

the product of the equation is positive 6(3×2)

then we find the factors of 6 which give us the product 6 and the sum of-5

the factors are-3 and -2

we then substitute negative 5 with these factors

it then be 3x^2-3x-2x+2=0

then we group factorise

we will have 3x(x-1)-2(x-1)=0

we find the common factors of the 2 groups

we will then have (x-1)(3x-2)=0

then we equate the both brackets to 0

(x-1)=0 and(3x-2)=0

for the first bracket add 1 both sidesand the second add 2 both sides and divide by 3

A city has a population of 28,000 people. Suppose that each year the population grows by 4.5%. What will the population be in 13 years?

Answer: Approximately 496,21 people

This is an exponential growth question, specifically population growth. We will use the following equation to help us solve this situation:

[tex]P(t) = a(1+r)^t[/tex]

[tex]P(t) = \text{total population} \ (??)[/tex]

[tex]\text{a = initial population = 28,000}[/tex]

[tex]\text{r = growth factor (the percent as a decimal) = 0.045}[/tex]

[tex]1 + r = 1 + 0.045 = 1.045[/tex]

[tex]\text{t = time in years = 13}[/tex]

So, we can substitute all our information and solve:

[tex]P(13) = 28,000(1.045)^{13} = 496,21.49[/tex]

Rounding to the nearest whole person, 496,21 people in 13 years.

The radius of the cylinder is 4.5 cm.

What is a cylinder?

A cylinder is a three-dimensional solid in mathematics that maintains two parallel bases that are separated at a fixed distance by a curving surface. An axis joins the centres of each base, and the bases are often circular in shape.

We are given that the sum of the height and radius of a right circular cylinder is 9 cm.

This means that r + h = 9

We are also given that the total surface area is 81π cm square.

So,

⇒ 81π = 2πr (r + h)

⇒ 81 = 2 * r * 9

⇒ 81 = 18r

⇒ r = 4.5 cm

Hence, the radius of the cylinder is 4.5 cm.

Learn more about cylinder from the given link

brainly.com/question/27440983

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we can simply do that by multiplying each by the other's denominator.

[tex]\cfrac{19}{21}\hspace{9em}\cfrac{8}{9} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{19}{21}\cdot \cfrac{9}{9}\implies \boxed{\cfrac{171}{189}}\hspace{9em}\cfrac{8}{9}\cdot \cfrac{21}{21}\implies \boxed{\cfrac{168}{189}}[/tex]