What is 6:12 in simplest form?
A.2
B.21
C. 12
D. 12.6

Step-by-step explanation:

The total number of members in the crew  = 20

The number of people able to use meat machine  = 17    ⇒n (M)  = 17

The number of people able to use bread machine  = 18  ⇒n (B)  = 18

The number of people able to use both machine  = 15 ⇒n (M∩B)  = 15

Now, a person is chosen at random:

Let us assume the number of people who cannot use either machine = K

Now,as we know:

[tex]n(A\cup B) = n(A) + n(B) - n(A\cap B)[/tex]

So, the number of people who can use either of the machines  

=n (M) + n(B)  -  n(M∩B)  = 17 + 18 - 15  = 20

So, there are 20 people who can use either of the two machines.

Also, as given the total number of people in the crew = 20.

Hence, from the above statement it is clear that there exists no person who can not use either of the machines.

So, such probability = 0

Without a map or structured dataset, it is not possible to provide the eight-digit grid coordinates for benchmark 86 as requested.

The student asked for the eight-digit grid coordinates for benchmark 86, which is circled in red. To determine the eight-digit grid coordinates, we need a map with grid lines or specific information that provides the exact location of benchmark 86. Unfortunately, the data provided does not contain adequate information to pinpoint the exact eight-digit grid coordinates for this benchmark. Grid coordinates are typically derived from a more structured dataset or map rather than the types of numbers presented in the question. Those appear to be an arbitrary sequence of numbers and are not formatted as grid coordinates, which usually consist of an easting and a northing.

Answer:

The number of ways of selection of 3 items taking 2 at a time in addition to the possible ways in which each girl selects the same style is 9 different ways. Since the list contains 9 different ways in which each of them can select a style, the list is complete.

Step-by-step explanation:

Here we have the number of ways the available styles can be arranged between each girl is given by the number of permutation of 3 items, taking two at a time plus the number of possible selections where both of them select the same style as follows

We note that the formula for permutation is

[tex]_n[/tex]P[tex]_r[/tex] = [tex]\frac{n!}{(n-r)!}[/tex]

Therefore ₃P₂ = [tex]\frac{3!}{(3-2)!}[/tex] = 6 different ways

The number of possible selections where both of them select the same style is given by

(T, T), (s, s), (L, L) = 3 different ways

Total number of ways = 6 + 3 = 9 ways

Number of different ways available in the list = 9 ways

Therefore the list is complete

Answer:

im sorry but that question makes o sense

Step-by-step explanation:

Answer:

The answer to the question is

The kinds of polygons that could be Diedre's quadrilateral are rectangle and square

Step-by-step explanation:

We  note the conditions of the polygon as thus

Number of sides = 4, Which include trapezoid, square, rectangle

Size of angles = 90 ° which eliminates trapezoid

Sizes of opposite sides = Equal opposite sides, includes square and rectangle

With the above we conclude that the possible polygons in Diedre's quadrilateral are rectangle and square

Answer:

0.2696

Step-by-step explanation:

Number of People who responded Yes=175

Number of People who responded No=474

Total Number of Respondents=175+474=649

We want to determine the Probability that a person chosen at random believes that psychic can help solve murder cases.

Now,

[TeX]Probability=\frac{Number of Possible Outcome}{Total Number of Outcomes}[/TeX]

Number who believe psychic can help solve murder cases=175

Total Number=649

Therefore:

P(a person chosen at random person believes some psychics can help solve murder cases) = [TeX]\frac{175}{649}=0.2696[/TeX]

Final answer:

The probability that a randomly chosen person believes some psychics can help solve murder cases is 175 divided by the total number of respondents (649), resulting in approximately 26.96%.

Explanation:

The question asks for the probability that a chosen person believes some psychics can help solve murder cases, given that out of a group, 175 said 'Yes' and 474 said 'No'. To find this probability, divide the number of people who believe psychics can help (175) by the total number of people asked (175 + 474).

Steps to Calculate the Probability:

Calculate the total number of responses: 175 (Yes) + 474 (No) = 649.

Calculate the number of positive responses (people who believe psychics can help): 175.

Divide the number of positive responses by the total number of responses: Probability = 175 / 649.

Perform the division to get the probability.

Therefore, the probability that a person chosen at random believes some psychics can help solve murder cases is 175/649 or approximately 0.2696, which can be expressed as 26.96%.

Answer:

Option C

Step-by-step explanation:

6x² +5x + 1

6x² + 3x + 2x + 1

3x(2x + 1) + 1(2x + 1)

(3x + 1)(2x + 1)

Answer: C

Step-by-step explanation: Answer is C

Question:

Nick and Tasha are buying supplies for a camping trip. They need to buy chocolate bars to make s’mores, their favorite campfire dessert. Each of them has a different recipe for their perfect s’more. Nick likes to use 1/2 of a chocolate bar to make a s’more. Tasha will only eat a s’more that is made with exactly 2/5 of a chocolate bar.

Answer:

The fraction of a chocolate bar will Nick and Tasha will use in total if they each eat one s'more is [tex]\frac{9}{10}[/tex] bar.

Step-by-step explanation:

Here, we are meant to learn how to add fractions and to know about the essence of a common denominator

The portion of the chocolate that Nick makes an s'more with is 1/2

The portion of the chocolate that Tasha makes an s'more with is 2/5

Therefore the fraction of the chocolate bar Nick and Tasha use in total if they each eat one s'more is 1/2 + 2/5

Where [tex]\frac{1}{2}[/tex] + [tex]\frac{2}{5}[/tex]  multiplying the fraction [tex]\frac{1}{2}[/tex] by [tex]\frac{5}{5}[/tex] and  [tex]\frac{2}{5}[/tex]  by  [tex]\frac{2}{2}[/tex] wee have

[tex]\frac{5}{5}[/tex]× [tex]\frac{1}{2}[/tex] + [tex]\frac{2}{2}[/tex]× [tex]\frac{2}{5}[/tex]  =   [tex]\frac{5}{10}[/tex] +  [tex]\frac{4}{10}[/tex] =  [tex]\frac{9}{10}[/tex] bar

Therefore the fraction of the chocolate bar that would be left is

1 bar -  [tex]\frac{9}{10}[/tex] bar = [tex]\frac{1}{10}[/tex] bar.

Here is the complete question

Nick and Tasha are buying supplies for a camping trip. They need to buy chocolate bars to make s’mores, their favorite campfire dessert. Each of them has a different recipe for their perfect s’more. Nick likes to use 1/2 of a chocolate bar to make a s’more. Tasha will only eat a s’more that is made with exactly 2/5 of a chocolate bar.

Answer:

Both Nick and Tacha will use 9/10 of the chocolate bar

Step-by-step explanation:

Nicks share of the chocolate bar is 1/2(fraction of the chocolate bar need to make the s'more

And Tasha requires 2/5 to make her own s'more.

Then if both of them share a single chocolate bar,then it will be:

2/5(Tasha's share) + 1/2(Nick's share)

= 9/10 of a single chocolate bar will be used.

In case they were interested to know the amount remaining, it will be:

1 - 9/10 = 1 1/10

Answer:

Find where the expression  [tex]\frac{3}{(x+2)(x-2)}[/tex]

is undefined.

List all of the vertical asymptotes:

[tex]x=2, -2[/tex]

This is what your graph would look like:

Answer:

1/216

Step-by-step explanation:

n{1111,2222,3333,4444,5555,6666}=6

number of every cases =6×6×6×6

6/6^4=1/6^3=1/216

Answer:

 1/216.

Step-by-step explanation:

The probability that all the 4 dice will show a 1 = (1/6)^4.

There  are also 5 other possible outcomes: 4 2's , 4 3's etc.

So the required probability = 6 * (1/6)^4

= 6 * 1/1296

=  1/216.

Answer:

0.0865

Step-by-step explanation:

Binomial Problem with n = 9 and p(triggered) = 0.99

And the photo below is the full answer,

I could not insert the fomula and hope it will find you well.

The range is 39

The interquartile range is 32

Explanation:

The given data is [tex]57,96,72,63,88[/tex]

Let us arrange the data in ascending order.

Thus, we have,

[tex]57,63,72,88,96[/tex]

We need to determine the range and interquartile range of the data.

Range:

The range of the data is the difference between the highest and the lowest value in the given data.

Highest value = 96

Lowest value = 57

Range = Highest value - Lowest value

           = 96 - 57

           = 39

Thus, the range of the data is 39

Interquartile range:

The interquartile range is the difference between the upper quartile and the lower quartile in the given data.

From, the given data, we have, 3 quartiles. They are [tex]Q_1 , Q_2[/tex] and [tex]Q_3[/tex]

[tex]Q_2=72[/tex]

[tex]Q_1=\frac{57+63}{2} =60[/tex]

[tex]Q_3=\frac{88+96}{2} =92[/tex]

Interquartile range = [tex]Q_3-Q_1[/tex]

                               =  [tex]92-60[/tex]

                               =  [tex]32[/tex]

The interquartile range is 32

List the data set from least to greatest:

57, 63, 72, 88, 96

Range: Highest value - Lowest value = answer

Range: 96 - 57 = 39

(IQR) Interquartile Range: Upper quartile - Lower quartile = answer

(IQR) Interquartile range: 88.5 - 57.5 = 31

Answer:

X=5

Y=-2

Step-by-step explanation:

Answer:

x = 5

y = -2

Step-by-step explanation:

Let's solve the system of equations, this way:

-5y + 6x = 40

3y - 8x = -46

******************

-5y + 6x = 40

6x = 40 + 5y

x = (40 + 5y)/6

******************

Substituting x and solving for y in the 2nd equation:

3y - 8x = - 46

3y - 8 * [(40 + 5y)/6] = - 46

3y - (320 + 40y)/6 = - 46

18y - 320 - 40y = - 276 (Multiplying by 6 at both sides)

-22y = - 276 + 320 (Adding 320 at both sides)

-22y = 44

y = -44/22

y = -2

**************************

Solving for x in the 1st equation:

-5y + 6x = 40

-5 * - 2 + 6x = 40

10 + 6x = 40

6x = 40 - 10 (Subtracting 10 at both sides)

6x = 30

x = 30/6

x = 5

Answer:

  y = -(1/12)(x -6)² -2

Step-by-step explanation:

The vertex of the parabola is halfway between the focus and the directrix, so has y-coordinate (-5+1)/2 = -2. The difference in the y-coordinates between the focus and the vertex is ...

  p = -5 -(-2) = -3

An equation of the parabola with vertex (h, k) and focus-vertex distance p can be written:

  y = 1/(4p)(x -h)² +k

For (h, k) = (6, -2) and p = -3, the equation is ...

  y = (-1/12)(x -6)² -2

Answer:

C) $826.82 to $873.18.

Step-by-step explanation:

Sample mean (M) = $850

Standard deviation (s) = $54

sample size (n) = 36

Z for 99% confidence interval (Z) = 2.576

The confidence interval is determined by the following relationship:

[tex]M \pm Z*\frac{s}{\sqrt{n}}[/tex]

Applying the given values, the lower (L) and upper (U) values are:

[tex]850 \pm 2.576*\frac{54}{\sqrt{36}}\\L=\$826.82\\U=\$873.18[/tex]

The answer is C) $826.82 to $873.18.

Answer:

constitute meeting the competition.

Step-by-step explanation:

Philip Esten action from the above scenario tries to meet up with the new competition he is encountering.

He operates a mail store where he renders services such as delivering of item overnight, and charging of fax service with each page for a fee of $2.00 for clients. He has a competition in the business environment where he now has direct competitors who offers the same service and product type to clients.

In this case, his competitors tend to attract more customers, both new and Esten old customers, because his competitors are rendering the same services at a cheaper fee to build a better brand of product / services for the store and buying the trust of customers now and for the future.

Esten with the knowledge of how business competition an works, reduces his price of charge to $.50 to attract more people or customers back to his store and tries to drive off the  competitors, knowing quite well that this charges will not cover his cost .

Answer:

No, Mutually exclusive events and Independent events are not the same.

Step-by-step explanation:

Mutually exclusive events and Independent events are not the same.

Mutually exclusive events are those events that cannot occur together, i.e. they cannot take place at the same time.

For example, the events of tossing a Head and Tails are mutually exclusive. This is because when a coin is flipped it cannot land on both sides at once.

Independent events are those events that can occur at the same time without affecting each other, i.r. they are not dependent on the occurrence of other events in the sample space.

For example, on rolling two fair die the events of first die rolling 4 and the second die rolling 1 are independent.

Final answer:

Mutually exclusive events cannot occur simultaneously, which means P(A AND B) = 0. Independent events occur independently of each other, and their combined probability is P(A AND B) = P(A)P(B). They are different concepts; being mutually exclusive does not imply events are independent.

Explanation:

No, the statement that mutually exclusive events and independent events are the same is incorrect. Mutually exclusive events are two or more events that cannot happen at the same time. For example, when tossing a coin, the events of landing on heads and tails are mutually exclusive because the coin cannot land on both sides at once. This means that P(A AND B) = 0 for mutually exclusive events A and B.

Independent events, however, are those where the occurrence of one event does not affect the probability of the other event occurring. In other words, events A and B are independent if the probability of A occurring is the same, regardless of whether B has occurred, and vice-versa. This means that P(A AND B) = P(A)P(B) for independent events A and B.

Therefore, while mutually exclusive events must have a combined probability of zero (they cannot both occur), independent events are characterized by one event's occurrence not influencing the probability of the other event occurring. Essentially, mutually exclusive deals with the impossibility of events happening simultaneously, whereas independence is about the absence of any causal connection between them.

Answer:

$207.655

Step-by-step explanation:

Firstly, we calculate the value taken off the cost by the clearance sales tag.

That would be 30/100 * 349 =104.7

We subtract this from the original = 349-104.7 = 244.3

Now, he applied a 15% coupon. That would be 15/100 * 244.3 = 36.645

The cost of the chair before sales tax = 244.3-36.645 = 207.655

Answer:

c=f(x)−ax^2−bx

Step-by-step explanation:

Answer:

C=1

Step-by-step explanation:

Put the first point in the equation y=ax^2+bx+c where x=0 and y=1 you get

1=a*0+b*0+c

1=c

Answer:

Population next year = 1.075p

Step-by-step explanation:

The current population can be represented by 1 p the increase in the population number by next year is 7.5% p or 7.5/100 p that equals to 0.075p. The expression that represents the expected population by next year is:

Population next year = This year population + expected growth

Population next year = 1p + 0.075p = 1.075p

Given Information:

Population rate = 7.5 %

Current population = p

Required Information:

Population next year = ?

Answer:

Population next year = 1.075p

Step-by-step explanation:

The rate of increase in the population is given as 7.5 %

We know that the current population is equal to p

The population in the next year would be the sum of current population and 7.5 % of current population.

Population next year = current population + 7.5% of current population

Population next year = p + 7.5% of p

Population next year = p + 0.075*p

Population next year = p(1 + 0.075)

Population next year = p(1.075)

Population next year = 1.075p

Answer:

Her employer should withhold from her check for social security is d) 26.01

Step-by-step explanation:

Given:

Kate Alexander worked 40 hours last week. Her pay rate is $8.50 per hour. Assuming 7.65% social security withholding.

Now, to find the amount her employer should withhold from her check for social security.

Total number of hours = 40.

Pay rate per hour = $8.50.

Now, the total amount of total number of hours she worked in accordance of her pay rate:

[tex]40\times 8.50\\\\=\$340.[/tex]

Social security withholding = 7.65%.

So, to get the amount of social security withholding:

[tex]7.65\%\ of\ \$340[/tex]

[tex]=\frac{7.65}{100} \times 340[/tex]

[tex]=\frac{2601}{100}[/tex]

[tex]=\$26.01.[/tex]

The amount of social security withholding is $26.01.

Therefore, her employer should withhold from her check for social security is d) 26.01

Answer:

3:10

Step-by-step explanation:

Solution to the question is in the attached picture.

Answer:

3:10 trust

Step-by-step explanation:

Question:

A prism with a base area of 8 cm² and a height of 6 cm is dilated by a factor of 5/4 .

What is the volume of the dilated prism?

Enter your answer, as a decimal, in the box.

Answer:

The volume of the dilated prism is [tex]93.75 \ {cm}^{3}[/tex]

Explanation:

A prism with a base area of 8 cm² and a height of 6 cm

The volume of the prism can be determined by the formula, [tex]V=Bh[/tex]

Volume of the prism is given by

[tex]V=Bh[/tex]

[tex]V=(8)(6)[/tex]

[tex]V=46\ cm^3[/tex]

Thus, the volume of the prism is [tex]46 \ {cm}^{3}[/tex]

It is also given that the volume of the dilated prism is dilated by a factor of [tex]\frac{5}{4}[/tex]

Hence, the new volume is given by

[tex]Volume = 48(\frac{5}{4} )^3[/tex]

             [tex]=48(\frac{125}{64} )[/tex]

             [tex]=48(1.953125)[/tex]

             [tex]=93.75 \ {cm}^{3}[/tex]

Thus, the volume of the dilated prism is [tex]93.75 \ {cm}^{3}[/tex]

Final answer:

The volume of the dilated prism is 93.75 cm³ after applying the dilation factor of 5/4 to the original volume.

Explanation:

To find the volume of the dilated prism, we must apply the dilation factor to the original dimensions of the prism. Since a volume is a three-dimensional measure, the dilation affects all three dimensions (the base area and the height). The dilation factor is given as 5/4.

We must raise this factor to the third power when dealing with volumes, since we are dealing with three dimensions. Therefore, the original volume, V, of the prism is the base area multiplied by the height, which is:

V = 8 cm² × 6 cm = 48 cm³.

Applying the dilation factor, the volume V' of the dilated prism is:

V' = V × (5/4)³.

First, calculate:

(5/4)³3 = 125/64.

Then multiply the original volume by this factor:
V' = 48 cm³ × (125/64) = (48 × 125) / 64 = 6000 / 64 = 93.75 cm³.

The system of equations is inconsistent, meaning there is no solution for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfies both equations simultaneously.

The given information can be translated into equations:

1. "The difference of the product of 10 and (x) and the product of 6 and y is -8":

[tex]\[10x - 6y = -8\][/tex]

2. "The sum of the product of -5 and x and the product of 3 and y is -1":

[tex]\[-5x + 3y = -1\][/tex]

Now, we have a system of two equations with two variables:

[tex]\[ \begin{cases} 10x - 6y = -8 \\ -5x + 3y = -1 \end{cases} \][/tex]

Let's solve this system step by step:

Step 1: Eliminate one variable from one of the equations.

Multiply the second equation by 2 to make the coefficients of [tex]\(y\)[/tex] in both equations opposites:

[tex]\[ \begin{cases} 10x - 6y = -8 \\ -10x + 6y = -2 \end{cases} \][/tex]

Now, add the two equations to eliminate [tex]\(y\)[/tex]:

[tex]\[ 0 = -10 \][/tex]

The result is a contradiction, indicating that the system is inconsistent. This means there is no solution that satisfies both equations simultaneously.

Conclusion:

The system of equations is inconsistent, meaning there is no solution for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfies both equations simultaneously. The given information might contain an error or contradiction. Please double-check the equations and conditions provided.

The statement that is: a. ∠1 and ∠5 are alternate interior angles.

What is the alternate interior angles ?

A pair of angles that are inside the two parallel lines and on opposing sides of the transversal line are known as alternate internal angles. 1 and 5 are situated on opposite sides of the transversal line and inside the parallel lines in the diagram. They are thus different internal angles.

The fact that alternate inner angles have the same measure and are congruent is significant.

Therefore the correct option is A.

The volume of the cylinder is [tex]4000.36 \ cm^3[/tex]

Explanation:

Given that the radius of the cylinder is 7 cm

The height of the cylinder is 26 cm

We need to find the volume of the cylinder.

The volume of the cylinder can be determined using the formula,

[tex]Volume = \pi r^2 h[/tex]

Let us substitute the values [tex]\pi= 3.14[/tex] , [tex]r=7[/tex] and [tex]h=26[/tex] in the above formula.

Thus, we have,

[tex]Volume = (3.14)(7)^2(26)[/tex]

Simplifying the terms, we get,

[tex]Volume = (3.14)(49)(26)[/tex]

Multiplying the terms, we have,

[tex]Volume = 4000.36 \ cm^3[/tex]

Thus, the approximate volume of the cylinder is [tex]4000.36 \ cm^3[/tex]

Answer:

30,000 phone numbers.

Step-by-step explanation:

If the first three numbers are already defined then it means that only the other 4 digits remain to be defined.

The solution would be the number of numbers between 0000 and 9999, because the order here matters, since each number is different.

Therefore between these two numbers there are a total of 10,000 numbers, that is to say that for each of the three initial numbers there are 10,000 numbers, therefore in the town there are a total of 30,000 numbers.

Final answer:

The small town has 30,000 available telephone numbers.

Explanation:

To calculate the number of available telephone numbers in the small town, we need to consider that local phone numbers have seven digits, with the first three being the telephone exchange numbers. The town uses three different telephone exchanges: 313, 359, and 387. Since the last four digits of a phone number can range from 0000 to 9999, that means there are 10,000 possible combinations for the last four digits.

Since there are three possible telephone exchanges, each with 10,000 combinations for the remaining four digits, we calculate the total number of available phone numbers by multiplying the number of exchanges by the number of combinations per exchange:

Number of exchanges x 10,000 (combinations per exchange) = Total number of phone numbers

3 x 10,000 = 30,000

Therefore, the small town has 30,000 available telephone numbers.

Answer: x = - 2

y = 0

Step-by-step explanation:

The given system of linear equations is expressed as

x + 4y = - 2 - - - - - - - - - - - - - 1

- 5x + 5y = 10- - - - - - - - - - - - - 2

We would eliminate x by multiplying equation 1 by 5 and equation 2 by 1. It becomes

5x + 20y = - 10

- 5x + 5y = 10

Adding both equations, it becomes

25y = 0

Dividing both sides by 25, it becomes

y = 0/25 = 0

Substituting y = 0 into equation 1, it becomes

x + 4 × 0 = - 2

x = - 2

The system of equations have only one solution.

Answer:

The mistake stems from the assumption that angle dab and abc are both 90 and ad = bc and that the perpendicular bisector of dc is different from the perpendicular bisector to ab because they are the same and abcd is a rectangle.

Step-by-step explanation:

If ∡dab = ∡abc and side ab is equal to side bc which are opposite sides, ten then ab is parallel to bc which means the quadrilateral is  parallelogram. Also since two angles of the four angles of the parallelogram are 90 degrees then the parallelogram is a rectangle.

The bisector of one side of a rectangle will also bisect the opposite side of the rectangle. Therefore the bisector of dc is the same as the bisector of ab and it meets ab at the midpoint of ab. Therefore p is now at the midpoint of ab and there are no triangles pad and pbc.

Answer:

x  = 12  hours

Step-by-step explanation:

If one pipe fills store pool in 12 hours   in 1 hour will fill 1/12 of the pool

If one pipe fills a store pool in 6 hours in 1 hour will fill 1/6 of the pool

Let call " x " number of hours needed by the third pipe to fills the pool

then in 1 hour  will fill 1/x

The three pipes working together in 1 hour will fill

1/12  +  1/6  + 1/x

Now we know that the three pipes take 3 hours to fill the pool then in 1 hour three pipes will fill 1/3 of the pool, therefore

1/12  +  1/6  + 1/x  = 1/3

( x + 2*x + 12) /12*x  = 1/3   ⇒ ( 3*x +12 )/ 12*x = 1/3

9*x  + 36  = 12*x

- 3*x  = - 36

x  = 36/3

x  = 12  hours

Final answer:

The third pipe, when operating alone, takes 12 hours to fill the pool. This is found by calculating the combined rate of the first two pipes, then the rate of all three, and deducting to find the third pipe's alone rate.

Explanation:

The question asks how long it takes for a third pipe to fill a pool when utilized alongside two other pipes, where one fills the pool in 12 hours, another in 6 hours, and all three together in 3 hours. This is a rate of work problem, commonly encountered in algebra and practical mathematics. To solve, we first find the rates at which each individual pipe fills the pool and then calculate the third pipe's rate.

Determine the rate at which each pipe fills the pool. The first pipe fills the pool at a rate of 1/12 of the pool per hour, and the second at 1/6 of the pool per hour.

Add these rates to determine the combined rate when the two pipes are working together: (1/12) + (1/6) = 1/12 + 2/12 = 3/12 = 1/4. So, the two pipes fill the pool at a rate of 1/4 of the pool per hour together.

When the third pipe is added, all three fill the pool in 3 hours, which means their combined rate is 1/3 of the pool per hour. The rate of the third pipe alone can be found by subtracting the rate of the first two pipes from this combined rate: (1/3) - (1/4) = 4/12 - 3/12 = 1/12.

Therefore, the third pipe fills the pool at a rate of 1/12 of the pool per hour on its own. To find the time it takes for the third pipe to fill the pool by itself, we take the reciprocal of its rate, resulting in 12 hours.