A spinner is spun twice. Find the probability of spinning a 5 then spinning an even number.

The value is 50

A fraction represents a part of a whole or, more generally, any number of equal parts.

Given:

p/4 + pq

As, p= 8, q= 6

We have,

8/4 + 8*6

= 2 + 48

= 50

Hence, p/4 + pq= 50

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The equation results to get 50.

To evaluate the expression p/4 + pq when p = 8 and q = 6, follow these steps:

Therefore, the value of the expression when p = 8 and q = 6 is 50.

The complete question is:

Evaluate p/4 +pq when p=8 and q=6

Answer:

0.22

Step-by-step explanation:

Sample given is 500, so use z-score for the critical value

Given 95% confidence interval;

∝=100-95 =5% =0.05

∝/2 = 0.05/2 =0.025 ----because you are interested with one tail area

1-0.025= 0.975 -----area to the left

proceed to z-table to read 0.975 = 1.96 as the critical value

standard deviation from the question is 2.5 but because this is a sample then;

standard error for the mean is= standard deviation/√sample= 2.5/√500

=0.1118

Margin of error=1.96*0.1118 =0.2191

The margin of error is approximately 0.219 inches. Therefore, the correct answer is approximately 0.22 inches. Option (c) is correct.

To find the margin of error for a 95% confidence interval, we first need to determine the critical value for a 95% confidence interval.

For a normal distribution, with a 95% confidence level, the critical value is approximately 1.96.

Then, we use the formula for the margin of error:

[tex]\[ \text{Margin of Error} = \text{Critical Value} \times \frac{\text{Standard Deviation}}{\sqrt{\text{Sample Size}}} \][/tex]

Given:

Let's calculate the margin of error:

[tex]\[ \text{Margin of Error} = 1.96 \times \frac{2.5}{\sqrt{500}} \][/tex]

[tex]\[ \text{Margin of Error} = 1.96 \times \frac{2.5}{\sqrt{500}} \][/tex]

[tex]\[ \text{Margin of Error} \approx1.96 \times \frac{2.5}{\sqrt{500}} \][/tex]

[tex]\[ \text{Margin of Error} \approx 1.96 \times \frac{2.5}{22.36} \][/tex]

[tex]\[ \text{Margin of Error} \approx 1.96 \times 0.1118 \][/tex]

[tex]\[ \text{Margin of Error} \approx 0.219 \][/tex]

So, the margin of error is approximately 0.219 inches. Therefore, the correct answer is approximately 0.22 inches.

Answer:

At first we should know that:

The properties of the square are:

The properties of Rhombus

So, Wade is incorrect because the quadrilateral may be Rhombus

And the quadrilateral to be a square, she needs to show that It has two pairs of perpendicular lines.

Answer:

5x-3

Step-by-step explanation:

x+4+7x-1-4x

5x-3

Answer:

7 points total

Step-by-step explanation:

So basically the x represents the total amount of times you landed on that color so since its one of each you just replace the x with one and solve normally. 1+4=5 points for green, 7(1)-1= 6 points for blue -4(1)= -4 points for red  

Answer:

63 intervals would be expected to contain the true population average.

Step-by-step explanation:

90% confidence interval:

Mean that we are 90% sure that the interval contains the true population mean. That is, 90% of the intervals are expected to contain the true population average.

Of these 70 intervals, how many of these intervals would you expect to contain the true population average?

Following the explained logic

0.9*70 = 63

63 intervals would be expected to contain the true population average.

Answer:A

Go from the dark blue to light green flips on y and down 1

Step-by-step explanation:

Answer:

2z^2+7z+6

because you multiply 2z with z and 2 then multiply 3 with z and 2 and combine like terms

Answer:

(2z + 3) (z +2)

combine like terms: (2z⋅z)+(2z⋅2)+3z+(3⋅2)

2z^2+7z+6

Answer:

using the calculator

Step-by-step explanation:

The mathematical value to 54-200 divided by 4 is -36.5

A subject of mathematics known as arithmetic operations deals with the study and use of numbers in all other branches of mathematics. Basic operations including addition, subtraction, multiplication, and division are included.

Given that we should find the value of 54-200 divided by 4, which can be expressed mathematically as ;

[tex]\frac{ 54-200}{4}[/tex]

Then we can find the value of the numerator as ;

[tex]54-200 = -146[/tex]

Then we have

[tex]\frac{-146}{4} \\\\= -36.5[/tex]

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Answer:

(a) The estimated proportion of defective in the population is 0.05.

(b) The 95% confidence interval for the proportion defective cups is (2.5%, 7.5%).

(c) Zack does not needs to return the lots.

Step-by-step explanation:

Let X = number of defective cups.

The random sample of cups selected is of size, n = 300.

The number of defective cps in the sample is, X = 15.

(a)

The proportion of the defective cups in the population can be estimated by the sample proportion because the sample selected is quite large.

The sample proportion of defective cups is:

[tex]\hat p=\frac{X}{n}=\frac{15}{300}=0.05[/tex]

Thus, the estimated proportion of defective in the population is 0.05.

(b)

The (1 - α)% confidence interval for population proportion is:

[tex]CI=\hat p \pm z_{\alpha/2}\times\sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]

Compute the critical value of z for 95% confidence level as follows:

[tex]z_{\alpha/2}=z_{0.05/2}=z_{0.025}=1.96[/tex]

Compute the 95% confidence interval for p as follows:

[tex]CI=\hat p \pm z_{\alpha/2}\times\sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]

     [tex]=0.05 \pm 1.96\times\sqrt{\frac{0.05(1-0.05)}{300}}[/tex]

     [tex]=0.05\pm 0.025\\=(0.025, 0.075)\\[/tex]

Thus, the 95% confidence interval for the proportion defective cups is (2.5%, 7.5%).

(c)

It is provided that Zack has an agreement with his supplier that he is to return lots that are 10% or more defective.

The 95% confidence interval for the proportion defective is (2.5%, 7.5%). This implies that 95% of the lots have 2.5% to 7.5% defective items.

Thus, Zack does not needs to return the lots.

The estimated proportion of defective in the population is 0.05 and the 95 percent confidence interval for the proportion defective is (0.025,0.075).

Given :

a) The formula given below is used in order to determine the estimated proportion of defective in the population.

[tex]\hat{p} = \dfrac{X}{n}[/tex]

[tex]\hat{p} = \dfrac{15}{300}[/tex]

[tex]\hat{p} = 0.05[/tex]

So, the estimated proportion of defective in the population is 0.05.

b) The below formula is used in order to determine the 95 percent confidence interval for the proportion defective.

[tex]CI =\hat{p}\pm z_{\alpha /2}\times \sqrt{ \dfrac{\hat{p}(1-\hat{p})}{n}}[/tex]

Now, substitute the known terms in the above expression.

[tex]CI =0.05 \pm 1.96\times \sqrt{ \dfrac{0.05(1-0.05))}{300}}[/tex]

[tex]CI = 0.05\pm 0.025[/tex]

So, the 95 percent confidence interval for the proportion defective is (0.025,0.075).

c) According to the given data, Zack has an agreement with his supplier that he is to return lots that are 10 percent or more defective.

So, from the above calculation, it can be concluded that he did not have to return the lots.

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Answer:

-8/45

I hope this helped!

Step-by-step explanation:

-2/3 ÷ 3 3/4

Answer:

[tex]n=\frac{0.45(1-0.45)}{(\frac{0.05}{1.96})^2}=380.32[/tex]  

And rounded up we have that n=381

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".  

The margin of error is the range of values below and above the sample statistic in a confidence interval.  

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".  

Solution to the problem

In order to find the critical value we need to take in count that we are finding the interval for a proportion, so on this case we need to use the z distribution. Since our interval is at 95% of confidence, our significance level would be given by [tex]\alpha=1-0.95=0.05[/tex] and [tex]\alpha/2 =0.025[/tex]. And the critical value would be given by:

[tex]z_{\alpha/2}=-1.96, z_{1-\alpha/2}=1.96[/tex]

The margin of error for the proportion interval is given by this formula:  

[tex] ME=z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}[/tex]    (a)  

And on this case we have that [tex]ME =\pm 0.05[/tex] and we are interested in order to find the value of n, if we solve n from equation (a) we got:  

[tex]n=\frac{\hat p (1-\hat p)}{(\frac{ME}{z})^2}[/tex]   (b)  

And replacing into equation (b) the values from part a we got:

[tex]n=\frac{0.45(1-0.45)}{(\frac{0.05}{1.96})^2}=380.32[/tex]  

And rounded up we have that n=381

Final answer:

To estimate the proportion of out-of-state hotel guests with a 95% confidence level and a margin of error of 5%, a sample size of at least 385 guests is needed.

Explanation:

To estimate the proportion of out-of-state guests at a local hotel with a margin of error of no more than 5% and a 95% level of confidence, we can use the formula for determining sample size for a proportion:

n = (Z^2 * p * (1 - p)) / E^2

Where:
- n is the sample size
- Z is the Z-score corresponding to the confidence level (1.96 for 95% confidence)
- p is the preliminary estimate of the proportion (0.45, or 45%, in this case)
- E is the desired margin of error (0.05, or 5%, here)

Substituting the known values, we get:

n = (1.96^2 * 0.45 * (1 - 0.45)) / 0.05^2

n = 384.16

Since we cannot have a fraction of a person, we would round up to the nearest whole number, which gives us a sample size of 385. Therefore, the hotel should sample at least 385 guests to meet their requirements.

Answer:

Probability is the chance of you getting something

for example the chances (or probability) of the dice landing on 2 or 3 is 3 out of 6 (3/6)

hope this helps!

Answer:

The pairs are matched

Step-by-step explanation:

A. A line through the origin that makes an angle of [tex]\pi/6[/tex] with the positive x-axis.

Given a line through the origin that makes an angle of [tex]\pi/6[/tex] with the positive x-axis. The angle which the line makes with the x-axis is [tex]\pi/6[/tex].

B. A vertical line through the point (3, 3).

If a line passes through the point (3,3), x=3 and y=3. The vertical line through the point (3,3) is x=3

C. Given a circle center (h,k) and a center r, the standard form of the equation of the circle is given as:

[tex](x-h)^2+(y-k)^2=r^2[/tex]

Therefore, for a circle with radius 5 and center (2, 3), the standard form equation is:

D. A circle centered at the origin with radius.

For a circle centered at the origin with radius r=4.

The radius of the circle is 4 units.

Answer:

x>-8

Step-by-step explanation:

multiply both sides with -1

Answer:

8

Step-by-step explanation:

8

The correlation coefficient for the data is 0.0091.

This can be obtained by using the formula of correlation coefficient.

The following information is obtained from the table,

∑x  = 10

∑y = 10

∑xy = 42

∑x² = 42

∑y² = 42

The formula for finding the correlation coefficient,

r = n∑xy- ∑x∑y/(n∑x²-(∑x)²)(n∑y²-(∑y)²)

r = (5×42)-(10×10)/((5×42)-10²)((5×42)-10²)

 =[tex]\frac{110}{110.110}[/tex]

=[tex]\frac{1}{110}[/tex]

=0.0091

Thus, the correlation coefficient of the given data is 0.0091.

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Answer:

B) 1

Step-by-step explanation:

Got it right 2024 edg

Answer:

Step-by-step explanation:

The null hypothesis is the hypothesis that is assumed to be true. It is an expression that is the opposite of what the researcher predicts.

The alternative hypothesis what the researcher expects or predicts. It is the statement that is believed to be true if the null hypothesis is rejected.

From the given situation,

The mean number of pounds of fruit on this plot of land with the old fertilizer was 416 pounds. This is the null hypothesis.

H0 : µ = 416

Agriculture scientists believe that the new fertilizer may decrease the yield. This is the alternative hypothesis.

H0 : µ < 416

The capability of the process is determined using the process capability index (Cp). In this case, the process has a capability of 0.833, indicating that it is not meeting the specifications well.

To calculate the capability of the process, we need to use the process capability index (Cp). Cp is calculated by dividing the tolerance range by six times the standard deviation. The tolerance range in this case is 65 - 55 = 10 psi. So, Cp = 10 / (6 * 2) = 10 / 12 = 0.833. Since Cp is a measure of how well the process meets the specifications, a value closer to 1 indicates a better capability.

In this case, the process has a capability of 0.833, which means it is not meeting the specifications very well.

Keywords: capability, process capability index, tolerance range, standard deviation, specifications

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The process capability index (Cpk) for the grease aerosol can pressurization process is 0.67, indicating that the process is not capable of meeting the design specifications as it is less than the acceptable limit of 1.33.

The capability of the process in question relates to its ability to meet design specifications, which can be quantified using statistical measures like the process capability index (Cpk). To compute the Cpk, you need to determine the worst-case process capability scenario by comparing the distance of the process mean to the nearest specification limit in terms of standard deviations. The Cpk can be calculated using the formula:

Cpk = minimum [(USL - x) / (3σ), (x - LSL) / (3σ)]

Where USL is the upper specification limit, LSL is the lower specification limit, x is the process mean, and σ is the standard deviation.

For this process with an average of 61 psi and a standard deviation of 2 psi:

You would calculate two separate indices:

Thus, the Cpk would be the smaller of these two indices which is 0.67. A Cpk of 0.67 indicates that the process is not capable of meeting the design specifications since it is less than the acceptable limit of 1.33 for most industries.

Answer:

[tex]\\\sum_{k=1}^{n+1}\frac{1}{k(k+1)}\\ \\ \\=\sum_{k=1}^n\frac{1}{k(k+1)}+\frac{1}{(n+1)(n+2)} \\ \\ =1-\frac{1}{n+1}+\frac{1}{(n+1)(n+2)}\\ \\ \\ =1+\frac{1-(n+2)}{(n+1)(n+2)} \\ \\ \\ \\\sum_{k=1}^{n+1}\frac{1}{k(k+1)} =1-\frac{1}{n+2}[/tex]

Step-by-step explanation:

The question says; Proof that :

[tex]Let : P\left(n\right)=\sum_{k=1}^n\frac{1}{k(k+1)}=1-\frac{1}{n+1}[/tex]

Base case: P(1) = 1/2.

Inductive step: suppose P(n) has already been proven for some arbitrary n.  The statement P(n+1) is :

[tex]P\left(n+1\right)=\sum_{k=1}^{n+1}\frac{1}{k\left(k+1\right)}=1-\frac{1}{n+2}[/tex]

This concludes the proof by induction.

We Proof that:

The proof abuses the notation P(n) to make reference to the common values of the two sides of the equation to be proved. Moreover, it doesn't makes any sense to define P(n) as the common value of the two sides because it assumes the conclusion that the two sides are equal.

At the very least, the definition of P(n) in the first statement suppose to have be in quote or in parenthesis as shown below.

[tex]Let : P\left(n\right)=(\sum_{k=1}^n\frac{1}{k(k+1)}=1-\frac{1}{n+1})[/tex]

However , P(n) is a statement.

The proof writer confused stating P(n+1) with showing that it must be true; given that P(n) is true.

As such ; the correct proof for P(n+1) is:

[tex]\\\sum_{k=1}^{n+1}\frac{1}{k(k+1)}\\ \\ \\=\sum_{k=1}^n\frac{1}{k(k+1)}+\frac{1}{(n+1)(n+2)} \\ \\ =1-\frac{1}{n+1}+\frac{1}{(n+1)(n+2)}\\ \\ \\ =1+\frac{1-(n+2)}{(n+1)(n+2)} \\ \\ \\ \\\sum_{k=1}^{n+1}\frac{1}{k(k+1)} =1-\frac{1}{n+2}[/tex]

Slope-intercept form:  y = mx + b

(m is the slope, b is the y-intercept or the y value when x = 0 --> (0, y) or the point where the line crosses through the y-axis)

For lines to be parallel, they need to have the same slope.

[tex]y=-\frac{2}{3} x+12[/tex]    The slope is -2/3, so the parallel line's slope is also -2/3.

Now that you know the slope, substitute/plug it into the equation.

y = mx + b

[tex]y=-\frac{2}{3} x+b[/tex]   To find b, plug in the point (6, -11) into the equation, then isolate/get the variable "b" by itself

[tex]-11=-\frac{2}{3}(6)+b[/tex]

-11 = -4 + b     Add 4 on both sides to get "b" by itself

-7 = b

[tex]y=-\frac{2}{3} x-7[/tex]    

The slope-intercept form line that passes through the point (6, -11) and is parallel to the line y = -2/3x + 12 is y = -2/3x - 7.

The subject of this question is Mathematics, specifically algebra and geometry involving slope-intercept form. The question asks for the slope-intercept form of the line that passes through the given point (6, -11) and is parallel to the line y = -2/3x + 12.

In slope-intercept form, y = mx + b, m represents the slope of the line and b represents the y-intercept. We know that parallel lines have the same slope, so the slope of the line in question would be -2/3, same as the provided line.

To find the y-intercept (b), we use the point (6, -11) and the slope -2/3 in the slope-intercept equation: -11 = (-2/3) * 6 + b. Solving for b, we get b = -7. Hence, the equation of the line in slope-intercept form that passes through the given point and is parallel to the given line is y = -2/3x - 7.

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8 (y - 7) = - 16

Divide by 8 on both sides

y - 7 = - 2

Add 7 to both sides

y = 5

Answer:

y = 5

Step-by-step explanation:

8(y-7) = -16

Divide each side by 8

8(y-7)/8 = -16/8

y-7 = -2

Add 7 to each side

y -7+7 = -2+7

y = 5

This question is incomplete, in that the Excel File: data07-11.xlsx a. was not provided, but I was able to get the information on the Excel File: data07-11.xlsx a. from google as below:

57 61 86 74 72 73

20 57 80 79 83 74

The image of the Excel File: data07-11.xlsx a. is also attached below.

Answer:

a) Point estimate of sample mean  = 68

b) Point estimate of standard deviation (4 decimals) = 17.8122

Step-by-step explanation:

a) Point estimate of sample mean, \bar{x}  =  ∑Xi / n = (57 + 61 + 85 + 74 + 73 + 72 + 20 + 58 + 81 + 78 + 84 + 73)/12 = 68

b) Point estimate of standard deviation = sqrt ∑ Xi² - n\bar{x}² / n-1)

= sqrt(((57 - 68)^2 + (61 - 68)^2 + (85 - 68)^2 + (74 - 68)^2 + (73 - 68)^2 + (72 - 68)^2 + (20 - 68)^2 + (58 - 68)^2 + (81 - 68)^2 + (78 - 68)^2 + (84 - 68)^2 + (73 - 68)^2)/11) = 17.8122

Answer:

D. 12.5

Step-by-step explanation:

You can notice that ∠JMK and ∠JML are similar triangles.

That means that JM (from triangle on the left) is similar to KM (from triangle of the right)- the ratio between these must be:

[tex]\frac{6}{8}[/tex], that means that we can multiple KM (from triangle on the right) by this ratio to get ML (from triangle on the right).

[tex]6 * \frac{6}{8}=\frac{36}{8}=4.5[/tex]

Now we add JM to ML to to get JL - so 8 + 4.5 = 12.5

Answer:

12.5

Step-by-step explanation:

Answer:

5(8-x)

Step-by-step explanation:

Fact using the GCF (Greatest Common Factor) of 40 and 5

Factors of 5:

1,5

Factors of 40:

1,2,4,5,8,10,20,40

The greatest factor they both have is 5

Factor by distributing out a 5

40-5x

5(8-x)

The GCF (Greatest Common Factor) of 40 and 5 is 5!

Divide both numbers by 5.

40 / 5 = 8

-5x / 5 = -x

Rewrite the expression using the distributive property.

5(8 - x)

Best of Luck!

Answer:

[tex]\theta_{1} = \frac{\pi}{3} \pm 2\pi\cdot i[/tex], [tex]\forall i \in \mathbb{N}_{O}[/tex]

[tex]\theta_{2} = \frac{5\pi}{3} \pm 2\pi\cdot i[/tex], [tex]\forall i \in \mathbb{N}_{O}[/tex]

Step-by-step explanation:

The critical numbers are found by the First Derivative Test, which consists in differentiating the function, equalizing it to zero and solving it:

[tex]g'(\theta) = 16 - 4\cdot \sec^{2} \theta[/tex]

Following equation needs to be solved:

[tex]16 - 4\cdot \sec^{2}\theta = 0[/tex]

[tex]\sec^{2}\theta = 4[/tex]

[tex]\cos^{2}\theta = \frac{1}{4}[/tex]

[tex]\cos \theta = \frac{1}{2}[/tex]

The solution is:

[tex]\theta = \cos^{-1} \frac{1}{2}[/tex]

Given that cosine is a periodical function, there are two subsets of solution:

[tex]\theta_{1} = \frac{\pi}{3} \pm 2\pi\cdot i[/tex], [tex]\forall i \in \mathbb{N}_{O}[/tex]

[tex]\theta_{2} = \frac{5\pi}{3} \pm 2\pi\cdot i[/tex], [tex]\forall i \in \mathbb{N}_{O}[/tex]

The critical numbers of the function g(θ) = 16θ − 4 tan(θ) can be found by determining where the derivative of the function is zero or undefined. However, specific critical numbers can't be determined from the provided prompt due to complexity of the derivative.

The critical numbers of a function occur when the derivative of the function is equal to zero or undefined. Given the function g(θ) = 16θ − 4 tan(θ), the first step to be done is finding the derivative (g'(θ)) of the function.

This involves applying the rules of differentiation to each term: the coefficient rule (b), and chain rule for the tangent part. After calculating the derivative, we set g'(θ) equal to zero and solve for θ to determine the critical numbers. For the term involving tan(θ), it is undefined at θ = π/2 + πn, where n is an integer. Therefore, these are also considered as critical numbers.

However, due to the complexity of the derivative, finding critical numbers usually requires using algebraic and trigonometric techniques or potentially numerical methods if the equation cannot be solved analytically. As the specific steps for this complex derivative calculation are not provided in the prompt, we can't provide the specific critical numbers.

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Answer:

For this case we want to test if there is a higher incidence of smoking among women than among men in a neighborhood (alternative hypothesis). And we define p1 for men and p2 for women, so for this case the best system of hypothesis are:

Null hypothesis:[tex] p_1- p_2 \geq 0[/tex]

Alternative hypothesis: [tex]p_1 -p_2 <0 [/tex]

And the best option would be:

a) h0: p1-p2 ≥ 0 h1: p1-p2 < 0

Step-by-step explanation:

Previous concepts

A hypothesis is defined as "a speculation or theory based on insufficient evidence that lends itself to further testing and experimentation. With further testing, a hypothesis can usually be proven true or false".  

The null hypothesis is defined as "a hypothesis that says there is no statistical significance between the two variables in the hypothesis. It is the hypothesis that the researcher is trying to disprove".  

The alternative hypothesis is "just the inverse, or opposite, of the null hypothesis. It is the hypothesis that researcher is trying to prove".  

Solution to the problem

For this case we want to test if there is a higher incidence of smoking among women than among men in a neighborhood (alternative hypothesis). And we define p1 for men and p2 for women, so for this case the best system of hypothesis are:

Null hypothesis:[tex] p_1- p_2 \geq 0[/tex]

Alternative hypothesis: [tex]p_1 -p_2 <0 [/tex]

And the best option would be:

a) h0: p1-p2 ≥ 0 h1: p1-p2 < 0

Please provide us with an attachment of image A and B so we can answer your question correctly.

Answer:

Option D) 0.65    

Step-by-step explanation:

We are given the following in the question:

Percentage of executives who read Time magazine = 35%

[tex]P(M) = 0.35[/tex]

Percentage of executives who read U.S. News & World Report = 40%

[tex]P(N) = 0.4[/tex]

Percentage of executives who read both Time magazine and U.S. News & World Report = 10%

[tex]P(M\cap N) = 0.1[/tex]

We have to find the probability that a particular top executive reads either Time or U.S. News & World Report regularly.

Thus, we have to evaluate,

[tex]P(M\cup N) = P(M) + P(N) -P(M\cap N)[/tex]

Putting values, we get,

[tex]P(M\cup N) = 0.35 + 0.4 - 0.1=0.65[/tex]

0.65 is the probability that a particular top executive reads either Time or U.S. News & World Report regularly.

Thus, the correct answer is

Option D) 0.65

Answer:

Pi/3, 2pi/3, 4pi/3, 5pi/3

Step-by-step explanation:

Edge 2021 :)

Answer: after driving 330 miles, the remaining fuel in each tank be the same.

Step-by-step explanation:

Let x represent the number of miles it will take for the the remaining fuel in each tank to be the same.

Jerry has a large car which holds 22 gallons of fuel and gets 20 miles per gallon. It means that the number of gallons needed to drive 1 mile is 1/20. Then the number of gallons needed to drive x miles is

1/20 × x = x/20

If the tank is full, then the number of gallons of fuel left after driving x miles is

22 - x/20

Kate has a smaller car which holds 16.5 gallons of fuel and gets 30 miles per gallon. It means that the number of gallons needed to drive 1 mile is 1/30. Then the number of gallons needed to drive x miles is

1/30 × x = x/30

If the tank is full, then the number of gallons of fuel left after driving x miles is

16.5 - x/30

For the remaining fuel in each tank to be the same, it means that

22 - x/20 = 16.5 - x/30

Multiplying both sides of the equation by 60(LCM), it becomes

1320 - 3x = 990 - 2x

- 2x + 3x = 1320 - 990

x = 330 miles

Answer:

Step-by-step explanation:

Recall that we say that d | a if there exists an integer k for which a = dk. So, let d = gcd(a,b) and let x, y be integers. Let t = ax+by.

We know that [tex]d | a, d | b[/tex] so there exists integers k,m such that a = kd and b = md. Then,

[tex] t = ax+by = (kd)x+(md)y = d(kx+my)[/tex]. Recall that since k,  x, m, y are integers, then (kx+my) is also an integer. This proves that d | t.