Simplify the expression (Picture provided)

Answer:

a) 0.789, this is the complement of the probability of no repeat offenders; b) 0.411; c) 0.033; d) μ = 1.429; e) σ = 9.58

Step-by-step explanation:

For part a,

The probability that no parolees are repeat offenders is 0.211.  This means the probability of at least one is a repeat offender is the complement of this event.  To find this probability, subtract from 1:

1-0.211 = 0.789.

For part b,

To find the probability that 2 or more are repeat offenders, add together the probability that 2, 3, 4 or 5 parolees are repeat offenders:

0.216+0.162+0.032+0.001 = 0.411.

For part c,

To find the probability that 4 or more are repeat offenders, add together the probabilities that 4 or 5 parolees are repeat offenders:

0.032+0.001 = 0.033.

For part d,

To find the mean, we multiply each number of parolees by their probability and add them together:

0(0.211)+1(0.378)+2(0.216)+3(0.162)+4(0.032)+5(0.001)

= 0 + 0.378 + 0.432 + 0.486 + 0.128 + 0.005 = 1.429

For part e,

To find the mean, we first subtract each number of parolees and the mean to find the amount of deviation.  We then square it and multiply it by its probability.  Then we add these values together and find the square root.

First the differences between each value and the mean:

0-1.429 = -1.429;

1-1.429 = -0.429;

2-1.429 = 0.571;

3-1.429 = 1.571;

4-1.429 = 2.571;

5-1.429 = 3.571

Next the differences squared:

(-1.429)^2 = 2.0420

(-0.429)^2 = 0.1840

(0.571)^2 = 0.3260

(1.571)^2 = 2.4680

(2.571)^2 = 6.6100

(3.571)^2 = 12.7520

Next the squares multiplied by the probabilities:

0(2.0420) = 0

1(0.1840) = 0.1840

2(0.3260) = 0.652

3(2.4680) = 7.404

4(6.6100) = 26.44

5(12.7520) = 63.76

Next the sum of these products:

0+0.1840+0.652+0.7404+26.44+63.76 = 91.7764

Lastly the square root:

√(91.7764) = 9.58

Probabilities are used to determine the outcomes of events.

The table is given as:

[tex]\left[\begin{array}{ccccccc}x &0 &1 &2 &3 &4 &5 &P(x) &0.211 &0.378 &0.216& 0.162 &0.032 &0.001\end{array}\right][/tex]

(a) Probability that one or more are repeat offenders

This is represented as: [tex]P(x \ge 1)[/tex]

Using the complement rule, we have:

[tex]P(x \ge 1) = 1 - P(x = 0)[/tex]

So, we have:

[tex]P(x \ge 1) = 1 - 0.211[/tex]

[tex]P(x \ge 1) = 0.789[/tex]

The probability that one or more are repeat offenders is 0.789

(b) Probability that two or more are repeat offenders

This is represented as: [tex]P(x \ge 2)[/tex]

Using the complement rule, we have:

[tex]P(x \ge 2) = 1 - P(x = 0) - P(x = 1)[/tex]

So, we have:

[tex]P(x \ge 2) = 1 - 0.211 - 0.378[/tex]

[tex]P(x \ge 2) = 0.411[/tex]

The probability that two or more are repeat offenders is 0.411

(c) Probability that four or more are repeat offenders

This is represented as: [tex]P(x \ge 4)[/tex]

So, we have:

[tex]P(x \ge 4) = P(x = 4) + P(x = 5)[/tex]

[tex]P(x \ge 4) = 0.032 + 0.001[/tex]

[tex]P(x \ge 4) = 0.033[/tex]

The probability that four or more are repeat offenders is 0.033

(d) The expected number of repeat offenders

This is calculated as:

[tex]\mu = \sum x \times P(x)[/tex]

So, we have:

[tex]\mu = 0 \times 0.211+ 1\times 0.378 + 2 \times 0.216 + 3 \times 0.162 + 4 \times 0.032 + 5 \times 0.001[/tex]

[tex]\mu = 1.429[/tex]

The expected number of repeat offenders is 1.429

(e) The standard deviation

This is calculated as:

[tex]\sigma= \sqrt{\sum (x^2 \times P(x)) - \mu^2}[/tex]

[tex]\sum (x^2 \times P(x))[/tex] is calculated as:

[tex]\sum (x^2 \times P(x)) = 0^2 \times 0.211+ 1^2 \times 0.378 + 2^2 \times 0.216 + 3^2 \times 0.162 + 4^2 \times 0.032 + 5^2 \times 0.001[/tex]

[tex]\sum (x^2 \times P(x)) = 3.237[/tex]

So, we have:

[tex]\sigma= \sqrt{\sum (x^2 \times P(x)) - \mu^2}[/tex]

[tex]\sigma = \sqrt{3.237 - 1.429^2}[/tex]

[tex]\sigma = \sqrt{1.194959}[/tex]

[tex]\sigma = 1.093[/tex]

The standard deviation of repeat offenders is 1.093

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Answer:

Step-by-step explanation:

Answer:

75 "I think"

Step-by-step explanation:

Answer:

The area of the triangle is 43.5 cm².

Step-by-step explanation:

Since,  the area of a triangle is,

[tex]A=\frac{1}{2}\times s_1\times s_2\times sin \theta[/tex]

Where, [tex]s_1[/tex] and [tex]s_2[/tex] are adjacent sides and [tex]\theta[/tex] is the included angle of these sides,

Given,

[tex]s_1=12\text{ cm}[/tex]

[tex]s_2=8\text{ cm}[/tex]

[tex]\theta = 65^{\circ}[/tex]

Hence, the area of the given triangle is,

[tex]A=\frac{1}{2}\times 12\times 8\times sin 65^{\circ}[/tex]

[tex]=\frac{96\times 0.90630778703}{2}[/tex]

[tex]=\frac{87.0055475555}2}=43.5027737778\approx 43.5\text{ square cm}[/tex]

Answer:

18.15

Step-by-step explanation:

Percent means per hundred, so we can convert 4.6% and 3.65% to equivalent decimals.

4.6%= 4.6 divided by 100 = 0.046

3.65%= 3.65 divided by 100 = 0.0365

Since both sales tax rates apply to $220, we can add the two rates.

0.046 + 0.0365= 0.0825

0.0825 x 220 = 18.15

And so, Yuki pays$18.15 in sales tax for her handbag purchase.

Answer:

It would be $13.05

Step-by-step explanation:

Since you're dividing negative $65.25 with a negative number of 5 stores. The five stores are negative because he bought from those stores. So, when dividing the 2 negatives you get a positive $13.05.

If Tom has bought only one t-shirt  the change to Tom's account balance

would have been - $13.05.

A unitary method is a mathematical way of obtaining the value of a single unit and then deriving any no. of given units by multiplying it with the single unit.

Given, Tom bought 5 t-shirts from a store.

After he bought the t-shirts, his account balance showed a change of

- $65.25.

Assuming all 5 t-shirts cost the same so the cost of 1 t-shirt would be

= - (65.25/5).

= - 13.05 dollars.

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All you need to do is divide 1470 by 35, which is 42.

Answer:

Two parallel lines never intersect.

Step-by-step explanation:

The definition of a parallel line is: 2 lines that are exactly the same, but will not touch each other.

Answer with explanation:

~p means other line is Similar to p.

If there are two parallel lines , it means Perpendicular distance between the two lines is always constant, if measured by taking a point anywhere on the line.

⇒Two Parallel lines never intersect.

Option B

Is there answer choices?

Rolling a 5 in 300 rolls: Probability of 1/6 [tex]*[/tex] 300 ≈ 50 times.

Sure, let's break it down step by step:

1. Understand Experimental Probability : Experimental probability is based on actual outcomes from an experiment or trial. It's calculated by dividing the number of favorable outcomes by the total number of outcomes.

2. Identify the Probability of Rolling a 5 : Since a standard number cube has 6 faces numbered 1 through 6, each face has an equal probability of 1/6 of showing up in a single roll.

3. Calculate the Probability of Rolling a 5 : The probability of rolling a 5 is 1/6, since there's one favorable outcome (rolling a 5) out of the total 6 possible outcomes.

4. Use Probability to Predict Outcomes : To predict how many times you'll roll a 5 in 300 rolls, multiply the probability of rolling a 5 by the total number of rolls.

Let's calculate:

Probability of rolling a 5 = 1/6

Total number of rolls = 300

Number of times you'll roll a 5 = (Probability of rolling a 5) * (Total number of rolls)

= (1/6) [tex]*[/tex] 300

≈ 50

So, based on experimental probability, you can predict that you will roll a 5 approximately 50 times if you roll the number cube 300 times.

Answer:

C) 120 degrees

Step-by-step explanation:

m<A + m<B + m<C = 180

45 + m<B + 15 = 180

m<B + 60 = 180

m<B = 120

The answer is C) 120

Answer:

A

Step-by-step explanation:

Nature of roots is determined by using the Discriminant, D, which is:

[tex]D=b^2-4ac[/tex]

If

D > 0, there are 2 real unique solutions

D = 0, there are 2 real equal solutions

D < 0, there are no real solutions

Note: a is the coefficient of x^2, b is the coefficient of x, and c is the independent term (the constant)

Now, for our quadratic expression, a = 3, b = 12, & c = -3. Plugging it in the discriminant formula, we get:

[tex]D=b^2-4ac\\D=(12)^2-4(3)(-3)\\D=180[/tex]

Thus, D > 0 , which means there are 2 distinct real solutions. Answer choice A is right.

Answer:

300°

Step-by-step explanation:

This may seem really strange and counterintuitive to you, but when you turn 1° 300 times, you have turned 300°.

_____

If you don't want to add it up ...

1° + 1° + 1° ... + 1° . . . . . 300 times

you can use multiplication. It was invented as a shortcut to repeated addition:

1° × 300 = 300°

The gear has turned a total of 300 degrees.

To solve this problem, we need to consider the information given and apply basic multiplication. The gear turns in 1° sections, and it does so a total of 300 times. Since each section is 1°, we can simply multiply the number of sections by the size of each section to find the total degrees turned.

So, the calculation is as follows:

Number of sections = 300

Size of each section = 1°

Total degrees turned = Number of sections × Size of each section

Total degrees turned = 300 × 1°

Total degrees turned = 300°

Therefore, the gear has turned 300 degrees in total.

Answer:

0.6826

Step-by-step explanation:

To solve this, we find the z scores for both sample means.  We then us a z table to find the area under the curve to the left of (probability less than) each z score, and subtract them to find the area between them.

The formula we use, since we are using sample means, is

[tex]z=\frac{\bar{X}-\mu}{\sigma \div \sqrt{n}}[/tex]

Our x-bar will be 116 in the first z-score and 120 in the second; our mean, μ, is 118; our standard deviation, σ, is 6.3; and our sample size, n, is 10:

[tex]z=\frac{116-118}{6.3\div \sqrt{10}}=\frac{-2}{1.9922}\approx -1.0039\\\\z=\frac{120-118}{6.3\div \sqrt{10}}=\frac{2}{1.9922}\approx 1.0039[/tex]

Using a z table, we see that the area under the curve to the left of -1.00 is 0.1587.  The area under the curve to the left of 1.00 is 0.8413.  This makes the area between them

0.8413-0.1587 = 0.6826.

To solve this statistics problem, calculate the z-scores for the lengths 116 cm and 120 cm, using the given mean, standard deviation, and sample size. Then, use a z-table to find the probabilities associated with these z-scores. The difference between these probabilities will give the likelihood of the machine producing shells between these lengths.

This problem is essentially a question about probability in statistics, specifically relating to the normal distribution. Given that the machine is operating properly, and assuming that the lengths of the shells it produces are normally distributed with a mean of 118 cm and a standard deviation of 6.3 cm, we want to find the probability that the average length of 10 randomly selected shells is between 116 and 120 cm.

First, we need to find the z-scores corresponding to the lengths of 116 cm and 120 cm. The formula for the z-score is (X - μ)/σ, where X is the measurement (in this case, the average length of the shells), μ is the mean, and σ is the standard deviation. However, in this scenario, due to the application of the Central Limit Theorem, we must use σ/sqrt(N) as the standard deviation for the distribution of sample means, where N=10. Therefore, for 116 cm we get Z1 = (116 -118) / (6.3/ sqrt(10))  and for 120 cm we get Z2 = (120 - 118) / (6.3/sqrt(10)).

Once the Z-scores are calculated, we can use Z-tables (commonly found in statistics textbooks or online) to find the probabilities associated with these z-scores. Subtract the probability of Z1 from the probability of Z2 to get the desired range probability.

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Answer:

1. The marked answer is correct.

2. The only equivalent equation is the 2nd one: x^2 -12x +36 = 28

Step-by-step explanation:

The square of a binomial is ...

(x -a)^2 = x^2 -2ax +a^2

This tells you the constant term in the perfect square trinomial is the square of half the x-coefficient.

1. The x-coefficient is -8, so half that is -4. The square of -4 is 16. To complete the square, the added constant term must be 16, the marked answer.

___

2. The x-coefficient is -12, so the square of half that is (-6)^2 = 36. Adding 36 to the equation gives ...

x^2 -12x +36 = 28 . . . . . . matches the 2nd choice

_____

The rule that makes algebra work is this: whatever you do on one side of the equal sign must also be done on the other side.

If you add 36 on one side of the equation, you must add 36 on the other side. Of course addition of signed numbers is done in the usual way: 36 + (-8) = 28.

Answer: third option.

Step-by-step explanation:

By definition we know tht the geometric sequences has a common ratio, which is represented with r.

This ratio can be calculated by dividing a term by the previous term.

Therefore, keeping the information above on mind, you have that the ratio r  of the geometric sequence given in the problem, is the shown below:

[tex]r=\frac{-\frac{1}{2}}{\frac{1}{10}}\\\\r=-5[/tex]

Answer:

The ratio of the sequence is -5

Step-by-step explanation:

The given geometric series is ;

[tex]\frac{1}{10},-\frac{1}{2},\frac{5}{2},...[/tex]

The ratio of the geometric sequence is given by;

[tex]r=\frac{a_n}{a_{n-1}}[/tex]

[tex]r=\frac{\frac{5}{2}}{-\frac{1}{2}}[/tex]

We simplify to get;

[tex]r=-5[/tex]

To find the integer c that satisfies the given congruences, we can use the properties of modular arithmetic. For each congruence, we substitute the given values of a and b and simplify the congruences to solve for c. The possible values of c are then determined using the Chinese Remainder Theorem when appropriate.

To find the integer c that satisfies the given congruences, we can use the properties of modular arithmetic:

a) To find c such that a.c ≡ 13a (mod 19), we divide both sides of the congruence by a. This gives us c ≡ 13 (mod 19).

b) To find c such that b.c ≡ 8b (mod 19), we divide both sides of the congruence by b. This gives us c ≡ 8 (mod 19).

c) To find c such that c.c ≡ a - b (mod 19), we square both sides of the congruence. This gives us c^2 ≡ (a - b)^2 (mod 19). Since we know a ≡ 11 (mod 19) and b ≡ 3 (mod 19), we substitute these values and simplify the congruence to c^2 ≡ 8 (mod 19). To solve this quadratic congruence, we can use the Chinese Remainder Theorem to find the two square roots of 8 modulo 19, which are 7 and 12. Therefore, c ≡ 7 (mod 19) or c ≡ 12 (mod 19).

d) To find c such that c.c ≡ 7a + 3b (mod 19), we substitute the given congruences for a and b into the congruence and simplify to obtain c^2 ≡ 7(11) + 3(3) ≡ 4 (mod 19). Similar to part (c), we use the Chinese Remainder Theorem to find the square roots of 4 modulo 19, which are 2 and 17. Therefore, c ≡ 2 (mod 19) or c ≡ 17 (mod 19).

e) To find c such that c.c ≡ 2a^2 + 3b^2 (mod 19), we substitute the given congruences for a and b into the congruence and simplify to obtain c^2 ≡ 2(11^2) + 3(3^2) ≡ 2(121) + 3(9) ≡ 149 ≡ 2 (mod 19). Therefore, c ≡ ±4 (mod 19).

f) To find c such that c ≡ a^3 + 4b^3 (mod 19), we substitute the given congruences for a and b into the congruence and simplify to obtain c ≡ 11^3 + 4(3^3) ≡ 11^3 + 4(27) ≡ 11^3 + 4(8) ≡ 11 + 32 ≡ 17 (mod 19). Therefore, c ≡ 17 (mod 19).

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3x+3=6x-57

60=3x

3x=60

60÷3=20

x=20

ANSWER

[tex] P(G)= \frac{1}{3}[/tex]

EXPLANATION

The number of green marbles in the bag is

[tex]n(G) = 2[/tex]

The total number of marbles in the bag is

[tex]n(S)=1+2+3 = 6[/tex]

The probability of selecting a green marble from the bag without looking is

[tex]P(G)= \frac{n(G)}{n(S)} [/tex]

Substitute the values to get,

[tex]P(G)= \frac{2}{6} [/tex]

[tex]P(G)= \frac{1}{3} [/tex]

Answer:

The probability of drawing a green marble out of the bag without looking = 1/3

Step-by-step explanation:

It is given that, a bag contains 1 blue, 2 green, and 3 red marbles

Therefore total number of marble in the  bag = 1 + 2 + 3 = 6

To find the probability

Total number of marble = 6

Number of green marble  = 2

The probability of drawing a green marble = 2/6 = 1/3

Answer:

C

Step-by-step explanation:

A function in the form  [tex]y=a*b^x[/tex] is an exponential function. If

The given function can be written as  [tex]y=1*0.8^x[/tex], so a > 0 and 0 < b < 1, hence this is exponential decay function.

For end behavior, we take limits from -∞ and from ∞. If we do that we can see that C is the correct answer. Also, looking at the graph explains it. Attached is the graph.

From the graph, as we move towards negative infinity, the graph goes towards positive infinity and as we move towards positive infinity, the graph goes towards 0.

Answer:

c. [tex]\frac{5}{33}[/tex]

Step-by-step explanation:

The given decimal is  [tex]0.\bar {15}[/tex]

Let  [tex]y=0.\bar {15}...(1)[/tex]

Multiply equation (1) by 100.

[tex]100y=15.\bar {15}...(2)[/tex]

Subtract equation (1) from equation (2)

[tex]\Rightarrow 100y-y=15.\bar {15}-0.\bar {15}[/tex]

[tex]\Rightarrow 99y=15[/tex]

Divide both sides by 99.

[tex]\Rightarrow y=\frac{15}{99}[/tex]

Simplify;

[tex]\Rightarrow y=\frac{5}{33}[/tex]

The recurring decimal 0.15 can be converted to the fraction 5/33.

To convert the repeating decimal 0.15 to a fraction, follow these steps:

Let x be the repeating decimal: x = 0.151515...

Express this equation by multiplying both sides by 100 to shift the decimal point two places: 100x = 15.151515...

Next, subtract the original equation from this new equation: 100x - x = 15.151515... - 0.151515...

This simplifies to: 99x = 15

Solve for x by dividing both sides by 99: x = 15/99

Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor (GCD), which is 3: x = 5/33

Therefore, the repeating decimal 0.15 can be expressed as the fraction 5/33.

Answer:

100

Step-by-step explanation:

The total length of ten pens laid end-to-end is

10 pens × 130 mm/1 pen = 1300 mm

The number of tiles needed to stretch the same length is

1300 mm × 1 tile/13 mm = 100 tiles

Helen uses 100 tiles.

Answer:

A=158

Step-by-step explanation: SA = 2 × l × w  +  2 × l × h  +  2 × w × h

Answer: Option D

(D) 158 square feet

Step-by-step explanation:

A=2(wl+hl+hw)=2·(3·8+5·8+5·3)=158ft²

Answer:

P(A') = 0.75

Step-by-step explanation:

We are given that the probability P(A) = 0.25 and we are to determine the probability of the complement of A.

According to the Complement Rule of any probability, the sum of the probabilities of an event and its complement must be equal to 1.

So for for the event A,

P(A) + P(A') = 1

0.25 + P(A') = 1

P(A') = 1 - 0.25

P(A') = 0.75

The probability of an event and its complement add up to 1. Therefore, if P(A) = 0.25, the probability of the complement of A is 0.75.

In the field of probability theory, the probability of an event and its complement always add up to 1. Therefore, if P(A) = 0.25, then the probability of the complement of A is 1 - 0.25 = 0.75.

This is because the complement of A, denoted by A', includes all outcomes that are not in A. So, all probabilities in the sample space (which has a total probability of 1) must either be in A or in A'. Therefore, P(A) + P(A') = 1.

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Answer:

a) 20.16; b) 20.49 and 21.51

Step-by-step explanation:

We use z scores for each of these.  The formula for a z score is

[tex]z=\frac{X-\mu}{\sigma}[/tex].

For part a, we want the 20th percentile; this means we want 20% of the data to be lower than this. We find the value in the cells of the z table that are the closest to 0.20 as we can get; this is 0.2005, which corresponds with a z score of -0.84.

Using this, 21 as the mean and 1 as the standard deviation,

-0.84 = (X-21)/1

-0.84 = X-21

Add 21 to each side:

-0.84+21 = X-21+21

20.16 = X

For part b, we want the middle 39%.  This means we want 39/2 = 19.5% above the mean and 19.5% below the mean; this means we want

50-19.5 = 30.5% = 0.305 and

50+19.5 = 69.5% = 0.695.

Looking these values up in the cells of the z table, we find those exact values; 0.305 corresponds with z = -0.51 and 0.695 corresponds with z = 0.51:

-0.51 = (X-21)/1

-0.51 = X-21

Add 21 to each side:

-0.51+21 = X-21+21

20.49 = X

0.51 = (X-21)/1

0.51 = X-21

Add 21 to each side:

0.51+21 = X-21+21

21.51 = X

The 20th percentile for incubation times is approximately 20.16 days. The incubation times that make up the middle 39% of fertilized eggs fall between roughly 20.5 days and 21.5 days.

This question focuses on statistics and their application in a biological context, specifically about the incubation time of fertilized eggs. In statistics, the normal distribution is a common continuous probability distribution that is symmetric about the mean.

(a) The 20th percentile of the normal distribution can be found using the z-table or a calculator that has the capability. Using the formula Z = (X - μ) / σ, where Z is the Z-score, X is the value in the data set, μ is the mean, and σ is the standard deviation, a Z-score associated with the 20th percentile is approximately -0.84. So the incubation time in days for the 20th percentile is (0.84 * 1) + 21 = 20.16 days.

(b) Similarly, to find the incubation times that make up the middle 39% of the fertilized eggs, note that since this is symmetric, this implies the incubation times falls between the 30.5 percentile and the 69.5 percentile. Using the formula Z = (X - μ) / σ and the Z-table, the Z-scores associated with these percentiles are approximately -0.5 and 0.5 respectively. Hence the incubation time falls between (0.5 * 1) + 21 = 20.5 days and (0.5 * 1) + 21 = 21.5 days

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Answer:

[tex]\large\boxed{x\approx0.52}[/tex]

Step-by-step explanation:

[tex]2^{5x}=6\Rightarrow\log_22^{5x}=\log_26\qquad\text{use}\ \log_ab^n=n\log_ab\\\\5x\log_22=\log_26\qquad\text{use}\ \log_aa=1\\\\5x=\log_26\qquad\text{divide both sides by 5}\\\\x=\dfrac{\log_26}{5}\\\\\log_26\approx2.585\\\\x\approx\dfrac{2.585}{5}\to\boxed{x\approx0.52}[/tex]

Answer:

slope = - 5

Step-by-step explanation:

given the equation of a line in the form

y = mx ← m is the slope

y = - 5x is in this form with slope m = - 5

Answer:

The height of the building is approximately [tex]34\ ft[/tex]

Step-by-step explanation:

Let

h------> height of the building from the student's eye level

H----> height of the building from the ground (H=h+5 ft)

we know that

[tex]tan(30\°)=\frac{h}{50}[/tex]

[tex]h=tan(30\°)(50)=28.87\ ft[/tex]

Find the height of the building H

[tex]H=h+5=28.87+5=33.87\ ft[/tex]

The height of the building is approximately [tex]34\ ft[/tex]

Answer:

72 members

Step-by-step explanation:

Total surveyed members = 40

members voting yes = 12

probability of the members voting yes = 12/40 = 3/10

It is observed that, out of every 'n' members, n*(3/10) members are expected to vote yes:

Therefore,

number of members expecting to voye yes out of 240 are = 240 * 3/10

=> 24 * 3 = 72 members

If you multiply the numerator and the denominator by the same number, it will be equal to 1/3.

For example:

1/3 * 3/3 = 3/9

1/3 * 8/8 = 8/24

Answer:

2/6, 3/9, and 4/12

Step-by-step explanation:

Any fraction where you can divide both the numerator and the denominator by the same number to get 1/3 also equals 1/3.

For example: 2/6

When you divide both numbers by 2, you get 1/3. Which means they are equivalent fractions (or equal).