MATH

Four coins are tossed in a line. What is the probability that three coins come up heads and one coin comes up tails?

ince there are 4 coins, there are 4 possible ways for one of the coins to be tails: THHH, HTHH, HHTH, and HHHT. Consider the probability of THHH. The probability for either H or T is 1/2 in either case, so: The same is true for HTHH, HHTH, and HHHT. Thus, the probability of getting three heads and one tail is:

Bill can mow a lawn in one hour and twenty minutes. If Bill and Dave can mow the lawn together in a half hour, how long does it take for Dave to mow the lawn alone?

1/80 + 1/x= 1/30 1/80-1/30=1/x solve and get 48 min

How many distinct ways can you select 2 people from a group of 10, if the order in which they are picked matters?

10!/(10-2)! 90

How many groups of 3 puppies can be picked from a group of 10 puppies?

10!/3!(10-3)! 120

How many ways can you choose 7 people from a group of 10?

10!/7!(10-7)!

How many lines can be drawn between a set of 7 distinct points?

6+5+4+3+2+1+0=21

Isotope A has a 1/4 chance of decaying in an hour, while Isotope B has a 1/9 chance of decaying in an hour. In one hour, what is the chance that Isotope A decays and Isotope B does not decay?

1- 1/9= 8/9 1/4x8/9= 2/9

A farmer redistributes his cows among three pastures. The first has 10 cows less than half of the total, the second has 8 cows less than one third the total, and the third has 6 less than one fourth the total. How many cows does he have?

(x/2-10)+(x/3-8)+(x/4-6)=x x/12=24 x=288

A normal basketball has a radius of 5 inches. It must have 122 dots per square inch on its surface. Knowing this, about how many dots does a normal basketball have?

38,300 dots Recall that the surface area A of a sphere with radius r is given by A = 4πr2. Thus, the surface area of the basketball is A = 4π(52) = 100π in2. Because there are 122 dots per square inch, there is a total of 122 × 100π ≈ 38,327 dots.

Joanna must choose 3 friends from a group of 5 to play a game with. How many different groups of 3 friends can she make?

5!/3!(5-3)! 10

How many ways can you select four playing cards from ten different cards, where the order in which the cards are selected matters?

5,040 ways N!/N-R!

How many different combinations of 3 apples can be picked from a set of 7 apples?

7!/3!(7-3)!

If both x and y are odd, which of the following is always even?

A simple solution here is to assign test values for x and y, and plug them into each choice. If we let x = 1 and y = 3, we will see that the only even answer is x + y = 4

A population of rabbits grows by 10% every month. If the population begins with 1,000 rabbits, how many rabbits will there be in 6 months?

A= 1000(1+0.1/1)SQ1(6)

There are 8 marbles in a bag: 4 are blue, 3 are green, and 1 is red. What is the probability that of two randomly chosen marbles without replacement, 1 is blue and 1 is red?

First, let's find the probability of drawing a blue marble first and then red. The probability of getting a blue marble on the first draw is 4/8, or 1/2. Since there is no replacement, there are 7 marbles left after a blue marble is removed, so the probability of a red marble next is 1/7. The probability of both events occurring is 1/2 x 1/7 = 1/14. Next, let's find the probability of drawing a red marble first and then blue. The probability of getting a red marble on the draw is 1/8. The probability of a blue marble next is 4/7. The probability of both events occurring is 1/8 x 4/7 = 4/56 = 1/14. Using the addition rule, the probability that either of the two possibilities occurs is: 1/14 + 1/14=1/7

Given 8 ≤ |4x - 12| ≤ 48, find a solution set for x.

C. 5 ≤ x ≤ 15

If f(x) = x2 and g(x) = x + 2, what is f(g(f(x)))?

For nested equation problems like this, always work from the inside out. First, determine g(f(x)) by substituting f(x) for x in g(x): g(f(x)) = g(x2) = x2 + 2 We will now take the expression we have just reached and substitute it for x in the f(x) equation to solve f(g(f(x))): f(g(f(x))) = f(x2 + 2) = (x2 + 2)2

How many different outfits can be made from 10 shirts and 6 pairs of pants?

Each shirt can be paired with 6 different pants, and since there are 10 different shirts, there are a total of 10 × 6 = 60 pairings.

Ed is walking from his house to his friend's house. There is a tree between the two houses, located 50 feet from Ed's house to his friend's. After he has traveled 80 feet towards his friend's house, he is located one third of the way between the tree and his friend's house. What is the distance between Ed's house and his friend's house?

Ed is 80 - 50 = 30 ft past the tree. Since this is one third of the distance between the tree and his friend's house, his friend's house is 3 × 30 = 90 ft away from the tree. Since Ed's house is 50 feet away from the tree, his house is a total of 50 + 90 = 140 ft from his friend's house.

Corey pays $100 for an order of 1,000 stickers. If 20% of the stickers get lost in shipping, how much should Corey charge per sticker to make a 100% profit?

First, let's find the number of stickers that Corey receives from his order: 80% of 1,000 is 800 stickers. Now, let's calculate the amount that Corey needs to make from selling them. 100% profit on his initial investment of $100 would be $100 in profits, so Corey must sell the stickers for $200 total. Therefore, he should sell each sticker for $200/800 = $0.25 per sticker.

Two regular ten-sided dice are labeled with the numbers 1 to 10. If one ten-sided die has its even numbers painted blue, and another has its perfect squares painted blue, what is the probability that both dice will roll a blue number?

For the die with even numbers painted blue, there are 5 blue numbers (2, 4, 6, 8, 10) out of 10 numbers. Thus: For the die with perfect squares painted blue, there are 3 blue numbers (1, 4, 9) out of 10 numbers. Thus: By the multiplication rule, the chance of rolling an even number and then a perfect square is: 1/2 x 3/10= 3/20

Which of the following are possible functions forming the composite function "G(F(x)) = (x + 1/x)2"?

From observation, a straightforward solution would be if F(x) were the function inside of the square F(x) = x + 1/x and G(x) = x2.

In a group of cats, 37% have long tails, 11% have differently colored eyes, and 3% have both. What is the chance that a randomly selected cat has either a long tail or differently colored eyes, but not both?

If 37% of cats have long tails and 3% have both long tails and differently colored eyes, then 37% - 3% = 34% have only long tails. Similarly, if 11% of cats have differently colored eyes and 3% have both long tails and differently colored eyes, then 11% - 3% = 8% have only differently colored eyes. The chance a cat has a long tail or differently colored eyes, but NOT both, is therefore 34% + 8% = 42%.

If 40% of the population owns a bike, 10% of the population owns a skateboard, and 5% of the population owns both, what percent of the population owns either a skateboard or a bike or both?

If 40% of people own a bike and 5% of people own both a bike and skateboard, then 40 - 5 = 35% of people only own a bike. Similarly, if 10% of people own a skateboard and 5% of people own both, then 10 - 5 = 5% of people only own a skateboard. The total number of people that own only a bike, only a skateboard, OR both would be 35% + 5% + 5% = 45.

Connie can file 100 papers an hour, while Eric can file 80 papers an hour. Working together, how long will it take them to file 900 papers?

If Connie files 100 papers per hour and Eric files 80 papers per hour, together they file 100 + 80 = 180 papers per hour. To file 900 papers, they would take 900 / 180 = 5 hours.

Casey is selling 1 orange for x dollars. How many oranges can you buy for 80 cents?

If each orange costs x dollars, then the cost of oranges per dollar is 1/x oranges per dollar. 80 cents is 8/10 = 4/5 dollar, so the number of oranges we can buy is: 4/5x oranges

A bathtub can hold a maximum of 50 gallons of water. Water can be drained out of the tub at a rate of 2 gallons per minute. If the tub is initially completely filled by a faucet at a rate of 1 gallon per minute, how long will it take to drain the full tub if the drain is opened but the faucet is kept on?

If the tub is being drained at 2 gallons per minute but being filled at 1 gallon per minute at the same time, it loses water at a rate of 2 - 1 = 1 gallon per minute. To drain 50 gallons, it will take 50 minutes.

If the probability that an athlete likes swimming is 0.38, the probability of those who like running is 0.28, and the probability of those who like either is 0.43, what is the probability of those who like both?

If we were to simply add together the probabilities of athletes who like swimming and athletes who like running to find the probability of those who like either, we have 0.38 + 0.28 = 0.66. However, this would count the athletes who like both twice. We know that the actual probability of athletes who enjoy either swimming or running is 0.43. If we subtract this from the sum above, we would be left with the probability of athletes who like both. Therefore, the portion of those who like both is 0.66 - 0.43 = 0.23.

If the probability that a chef likes carrots is 0.13, the probability of those who like broccoli is 0.72, and the probability of those who like neither is 0.22, what is the probability of those who like both?

If we were to simply add together the probabilities of chefs who like carrots and chefs who like broccoli to find the probability of those who like either, we have 0.13 + 0.72 = 0.85. However, this counts the chefs who like both twice. Since we know the probability of chefs who like neither is 0.22, the probability of a chef who likes either broccoli or carrots is 1 - 0.22 = 0.78. If we subtract this from the sum above, we would be left with the probability of chefs who like both. Therefore, the portion of those who like both is 0.85 - 0.78 = 0.07.

13 soccer players all have to shake each others' hands. How many handshakes will take place in total? Assume that each soccer player only shakes hands with everyone once.

Imagine all the soccer players in a line, and start with the first one. The first soccer player performs 12 handshakes with all the other soccer players down the line (excluding herself). The second soccer player must also go down the line, but since she already shook hands with the first, she only performs a total of 11 handshakes. Following this pattern, 1 less handshake happens each time until the 13th soccer player has 0 handshakes left to perform. Therefore, the total number of handshakes is 12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 + 0 = 78 handshakes.

If x is a random variable whose probability distribution function is given by a normal distribution with a mean of 7 and a standard deviation of 2, which of the following values of x is least likely?

In a normal distribution, values farther from the mean have a lower probability of occurring. Out of all the choices, 2.5 is the farthest away from the mean of 7

Half of the population of Cool Town owns a bicycle, and 25% of the population owns a car. If 10% of the population owns both a car and a bicycle, what is the probability that a person chosen at random from Cool Town owns either a car or a bicycle or both?

In order to make this problem easier, let's assume Cool Town has a population of 100 people. Therefore, 50 people own a bike, 25 people own a car, and 10 people own both. We can also say that 50 - 10 = 40 people only own a bike and 25 - 10 = 15 people only own a car. The total amount of people that would own only a car, only a bicycle, OR both would be 40 + 15 + 10 = 65. Since the population of Cool Town is 100, the total probability would be 65/100, or 65%.

If the z-score of Jamie's test score is -0.23, what is the best representation of this z-score?

Jamie's score is below the mean.

If Corey gave 70 stickers to Jared, and Jared gave 10 stickers to Megan, all three kids would have the same amount of stickers. How many more stickers does Corey have than Megan?

Let C represent the number of stickers Corey has and M represent the number of stickers Megan has. We want the value of C - M. We know that Corey and Megan would have the same number of stickers after Corey gives 70 to Jared, and Megan receives 10 from Jared. Therefore: C - 70 = M + 10 C - M = 80

Daniel is four years older than Greg. In 8 years, Daniel will be two years older than three times Greg's current age. How old is Greg now?

Let D represent Daniel's age and G represent Greg's age. Since Daniel is 4 years older than Greg: D = G + 4 In 8 years, Daniel's age (D + 8) will be 2 years older than 3 times Greg's current age (3G), so: D + 8 = 3G + 2 Now that we have a system of linear equations, we can solve for D and G. Substitute G + 4 for D in the second equation to solve for G: G + 4 + 8 = 3G + 2 10 = 2G G = 5 Therefore, Greg is 5 years old,

Richard is three times as old as Bryan. In six years, Richard will be twice as old as Bryan will be in six years. How old is Richard right now?

Let R represent Richard's age and B represent Bryan's age. From the question stem we derive this linear system of equations: R = 3B R + 6 = 2(B + 6) Substitute 3B for R in the second equation to solve for B: 3B + 6 = 2B + 12 B = 6 Because R = 3B, we have R = 3(6) = 18 years old.

Jacob is astonished to find the value of his home is $150,000. He calculates that the value of his car, increased by half, is 30% the value of his home. What is the value of his car?

Let c be the value of his car. The value of his car increased by half is the same as 1.5c, so we have the following equation: 1.5c = 0.3(150,000) c = 0.3(100,000) = $30,000

The price of corn is expected to rise by 5% each year. What will be the percent change in the price of corn in 2 years compared to now?

Let the current price of corn be x. In 1 year, the price will be 1.05x. In 2 years, it'll be 1.05(1.05x) = 1.1025x. This is an increase of 0.1025 x 100% = 10.25%.

If the complement of an angle and the supplement of the same angle summed together equals 100 degrees, find the angle.

Let the x be the measure of the angle in degrees. Its complement is 90 - x and its supplement is 180 - x. The complement and supplement sum to 100, so we have: 90 - x + 180 - x = 100 270 - 2x = 100 170 = 2x x = 85º

Todd has a rectangular garden with an area of 30 square feet. If the garden is 13 feet longer than it is wide, what is the perimeter of the garden (in feet)?

Let w represent the width of the garden. Since the length of the garden is 13 feet longer than it is wide, the length is w + 13. Using the area, we can solve for w: w(w + 13) = 30 w2 + 13w = 30 w2 + 13w - 30 = 0 (w + 15) (w - 2) = 0 w = -15 OR 2 Since width must be positive, we know w = 2. Therefore, the length is 13 + 2 = 15, and the perimeter is: 2(15) + 2(2) = 30 + 4 = 34 feet

The sum of two numbers is 168. If the larger number is divided by the smaller number, the quotient is 6 with a remainder of 7. What is the smaller number?

Let x be the smaller number. The larger number must be 6x + 7. Since they sum to 168: x + 6x + 7 = 168 7x = 161 x = 23

A commuter drives 30 miles each way between home and work every day. When traveling home from work, traffic causes his commute to take triple the amount of time it takes compared to the trip driving to work. If traveling to and from work takes a total of 2 hours, what is the commuter's average speed traveling to work?

Let x represent the amount of time it takes the commuter to travel to work: x + 3x = 2 4x = 2 x = 0.5 Since the commuter travels 30 miles to get to work, his speed is: 30mi/0.5hr= 60 mph

There are 30 employees at Boogle. Of the 30, some employees have dogs, some have cats, some have both, and some have none. If there are 2 more employees with only cats than employees with only dogs, 3 more employees with only cats than employees with no pets, and 5 employees with both, how many employees have only cats?

Let x represent the number of employees that only have cats, y the number of employees with only dogs, and z the number of employees with no pets. Since 5 people have both cats and dogs, we can subtract that number from the total of 30 to find the number of employees with only cats, only dogs, or no pets. x + y + z = 30 - 5 x + y + z = 25 From the question stem, we also have: x = y + 2 x = z + 3 Now that we have a system of three linear equations in three variables, we can solve for x, the number of employees with only cats. Rewriting y and z in terms of x, we have: x - 2 = y x - 3 = z Plugging those expressions for y and z into the first equation, we have: x + (x - 2) + (x - 3) = 25 3x - 5 = 25 3x = 30 x = 10

A passcode for a safe must be two distinct letters followed by three distinct digits. How many different combinations for a passcode exist?

Let's consider the total possibilities at each position in the password. There are 26 letters in the alphabet, so there are 26 possibilities for the first letter. Since letters must be distinct, there are only 25 possibilities for the second letter. We can use the same logic for the three distinct digits. There are 10 possibilities for the first digit (0-9), 9 for the second digit, and 8 for the third digit. Therefore, the total number of possibilities for all passcodes is 26 * 25 * 10 * 9 * 8.

The odd numbers on a regular die are painted red. On a second regular die, the numbers that are one less than a perfect square are painted red. If both dice are thrown, what is the chance that both come up red?

On the first die, only odd numbers are painted red. Therefore 1, 3, and 5 are red. Since 3 of the 6 numbers on the die are red, the probability of getting a red number using the first die is 1/2 (or 3/6). On the second die, only the numbers that are one less than a perfect square are painted red. A perfect square is a number that is the square of another whole number. Some examples are 1, 4, 9, 16, 25 (or 12, 22, 32, 42, 52). The only perfect squares that appear on a die are 1 and 4, so the only number one less than a perfect square on a die is 4 - 1 = 3. Since only 1 of the 6 numbers on the second die is red, the probability of getting a red number using the second die is 1/6. Using the multiplication rule for the probability of independent events, the probability that both dice come up red is: 1/2 x 1/6 = 1/12

f you reach in a jar containing the letters INTERNET and grab two letters with replacement, what is the probability that you picked out 2 consonants?

Out of the 8 letters in INTERNET, 5 are consonants (NTRNT). The probability of getting a consonant the first time is 5/8. Since the first letter is replaced, the probability of getting a consonant on the second time is still 5/8. Thus, the probability of getting a consonant two times is 5/8 × 5/8 = 25/64

If F(x)= 10x2 and G(x)= 2 + x, find a possible value of x given that F(G(x)) = 10.

Plug in F(x) and G(x) and into F(G(x)) = 10, and work backwards to solve for x: 10(2 + x)2 = 10 10(4 + 4x + x2) = 10 4 + 4x + x2 = 1 Rearrange the quadratic to standard form: x2 + 4x + 3 = 0 Finally, factor to solve for x: (x + 1)(x + 3) = 0 x = -3 OR x = -1

Find f(g(x)) if f(x) = x2 and g(x) = x + 1.

Plug the expression for g(x) into f(x) for the variable x: f(x) = x2 f(g(x)) = (x + 1)2 Simplifying this expression (with FOIL) gives: f(g(x)) = x2 + x + x + 1 f(g(x)) = x2 + 2x + 1

If an angle is two more than three times its complement, what is the measure of this angle?

Recall that an angle and its complement sum to 90º. If our angle is x, its complement is 90 - x. From the information in the problem, we can set up an equation to solve for x: x = 3(90 - x) + 2 x = 270 - 3x + 2 4x = 272 x = 68º

The standard deviation of {10, 15, 20, 25, 30} can be best approximated by which of the following?

Recall that the standard deviation is a measure of how far data is from the mean, so we can approximate it as the average distance of each data point from the mean. Since this data set is symmetrical, the middle number 20 is the mean. If we find each number's distance from the mean, we get the following list: {10, 5, 0, 5, 10}. The average of these values is (10 + 5 + 0 + 5 + 10) / 5 = 30 / 5 = 6.

If a polygon has six sides, the sum of all of its six interior angles must equal ___.

Recall that the sum of all the internal angles of a polygon with n sides is 180(n - 2)º. For a 6 sided polygon, this is: 180(6 - 2) = 180(4) = 720º

If you have two spheres, one with diameter 3, and the other with diameter 9, what is the ratio of the volumes of the largest sphere to the smallest sphere?

Recall that the volume of a sphere is given by V = 4/3πr3. However, instead of calculating both of their volumes, we can realize that the volume is directly proportional to the radius cubed. Thus, to find the ratio of their volumes, we can find the ratio of their radii cubed. Since the ratio of their diameters is the same as the ratio of their radii, we can find the ratio of their diameters cubed. The ratio of the large sphere's diameter to the small one's diameter is 9/3 = 3, so the ratio of their volumes is 33 = 27.

According to a survey, the average number of keys kept on a given person's keychain is normally distributed with a mean of 4 and a standard deviation of 1. If a person is chosen at random, what is the probability that this person has less than 3 keys on their key chain?

Recall that with a normal distribution, 68% of the data lies within one standard deviation from the mean. Since our mean is 4 and our standard deviation is 1, 68% of people have between 3 and 5 keys on their keychain. Therefore, 100% - 68% = 32% of people have less than 3 OR more than 5 keys. Due to the symmetry of a normal curve, we know that 32% / 2 = 16% have less than 3 keys

According to a survey, the number of patients in a given dental office in a given month is normally distributed with a mean of 1,100 patients and a standard deviation of 100 patients. If a dental office is chosen at random, what is the probability that more than 1,400 patients visit this dental office?

Recall that with a normal distribution, 99.7% of the data lies within three standard deviations from the mean. Since the mean is 1,100 and the standard deviation is 100, 99.7% of dental offices have between 1,100 - 3(100) = 800 and 1,100 + 3(100) = 1,400 patients in the given month. Thus, 100% - 99.7% = 0.3% of dental offices have less than 800 OR more than 1,400 patients in the month. Due to symmetry of the normal curve, 0.3% / 2 = 0.15% of offices have more than 1,400 patients.

If a bird saw a hawk flying towards it and flew to its nest 28 meters away at 7 meters per second, and the hawk reached the nest 3 seconds after the bird, how fast was the hawk flying towards the bird? Assume that the hawk travels the same distance as the bird.

Recall the relationship between distance d, rate (or speed) r, and time t: Since the bird flies 28 meters at a speed of 7 meters per second, it takes 28/7 = 4 seconds to reach the nest. Therefore, the hawk takes 4 + 3 = 7 seconds to arrive at the nest. Since the hawk also travels 28 meters, its speed is 28/7 = 4 meters per second.

f a bird saw an eagle flying towards it and it flew to its nest in a tree 72 feet away at 9 feet per second, and the eagle arrived at the tree 4 seconds after the bird, how fast was the eagle flying towards the bird? Assume that the eagle travels the same distance as the bird.

Recall the relationship between distance d, rate (or speed) r, and time t: Since the bird flies 72 feet at a speed of 9 feet per second, it takes 72/9 = 8 seconds to reach the nest. Therefore, the eagle takes 8 + 4 = 12 seconds to arrive at the tree. Since the eagle also travels 72 feet, its speed is 72/12 = 6 feet per second.

Ray's house in South Florida is 300 miles away from Gainesville. Ray's friend Ron lives in Ocala. If Ray's house is 10 times further away from Gainesville than Ray's house is from Ron's house, how long will it take for Ray to travel to Ron's house if he travels at 60 miles per hour?

Recall the relationship between distance d, rate (or speed) r, and time t: d = rt Since the distance from Ray's house to Gainesville (300 miles) is 10 times the distance between his house to Ron's, the distance between Ray and Ron's houses is 30 miles. If he travels at a speed of 60 miles per hour, it'll take him: 30mi/60mi= .5 hr

Solve the inequality: |3 - x| < 14

Rewrite the absolute value inequality as a combined inequality that accounts for both positive and negative cases: |3 - x | < 14 -14 < 3 - x < 14 Then, solve for x: -17 < -x < 11 17 < x < -11 -11 < x < 17

In a group of people, 30 of them wear glasses and 17 of them wear earrings. If 12 people wear both glasses and earrings, how many people wear only earrings or only glasses?

Since 12 people wear both glasses and earrings, 30 - 12 = 18 people wear only glasses, and 17 - 12 = 5 people wear only earrings. The total number of people who wear only glasses OR only earrings is 18 + 5 = 23 people.

The number of hours a person works per day is normally distributed with an average of 7 hours and a standard deviation of 1 hour. What is the probability that a randomly chosen person works between 7 and 9 hours?

Since 95% of the data is contained within 2 standard deviations from the mean, 95% of people work from 5 to 9 hours. Due to the symmetry of a normal distribution, 95% ÷ 2 = 47.5% of people work from 7 to 9 hours

The number of jackets a college student owns is normally distributed with a mean of 4 and a standard deviation of 1. What is the probability that a student has 6 or fewer jackets?

Since 95% of the data is contained within two standard deviations from the mean, 95% of college students have between 2 to 6 jackets. Therefore, 100% - 95% = 5% of college students fall outside of this range and have either less than 2 more than 6 jackets. Due to the symmetry of a normal distribution, 5% ÷ 2 = 2.5% have less than 2 jackets. Since we want to include this in the percentage of students who have 6 jackets or less, we find that 95% + 2.5% = 97.5% of college students have 6 jackets or less.

Gear X and Gear Y have equally sized and spaced teeth that mesh so that when one gear turns the other is also forced to turn without slipping. Gear X has a total of 20 teeth and Gear Y has a total of 30 teeth. If Gear X turns 15 times, how many times does Gear Y turn?

Since Gear X and Y are connected, the number of gear teeth that turn in 15 rotations for Gear X must also equal the number of teeth that turn for Gear Y: 20 (15) = 30x x = 300/30 = 10

A regular hexagon has a perimeter of 12. What is the area of the hexagon?

Since the length of the hypotenuse is 2 and the length of the base is 1, the height must be √3. Now we can find the area of the triangle: A= 1/2(2)(sq3)= Sq3 Therefore, the total area of the hexagon is 6√3.

The average of a large data set is 60. Which of the following must be greater than or equal to 60?

Since the mean is the same thing as the average, the mean is equal to the average.

A military aircraft is traveling from Norfolk to McKinley, a 50-mile, straight line journey, with a refueling station on the route between the two. 30 miles into the flight, the pilot is recalled back to Norfolk, but must refuel. Given that the refueling station is in the direction of Norfolk and that the distance from the pilot's current position to the refueling station is 1/4 the remaining distance to McKinley, how far is the refueling station from Norfolk?

Since the plane has traveled 30 miles out of the 50-mile trip, there are 20 miles until McKinley. Therefore, the plane is 20 × 1/4 = 5 miles from the refueling station. Now, we can see that the distance from Norfolk to the Refuel station is 30 - 5 = 25 miles.

Two normal dice are rolled separately. What is the probability of getting a sum of 7 or 11?

Since there are 6 sides to a die, the total number of possible combinations from rolling two dice is 6 × 6 = 36 combinations. Now, we need to count all possible ways we can get a sum of 7 or 11 with two dice: Sum of 7: (1, 6), (6, 1), (2, 5), (5, 2), (3, 4), (4, 3) Sum of 11: (5, 6), (6, 5) There are 6 ways to get a sum of 7 and 2 ways to get a sum of 11, so there are 6 + 2 = 8 ways to get a sum of 7 or 11. Therefore the probability is 8/36 = 2/9.

Which of the following is a solution to "x + 8 = |3x - 2|"?

Split the absolute value into both it's positive and negative cases: x + 8 = 3x - 2 AND x + 8 = -(3x - 2) 10 = 2x AND -6 = 4x x = 5 AND x = -3/2

If ab + 2 = 9 - 4a, what is the value of a?

Start by isolating the terms including a on one side of the equation: ab + 2 = 9 - 4a ab + 4a = 7 Factor out a: a(b + 4) = 7 Solve for a: a = 7 / (b + 4)

If (4x + 3)2 = 289, what is a possible value of x - 3?

Start by taking the square root of both sides: (4x+3)^2=289 4x+3= +- sq289 4x+3=17 or 4x+3=-17 Next, solve both equations for x: Finally, subtract 3 from both values of x to find the values of x - 3: One of the values for x - 3 is 1/2

If an engineer combines 1 L of 30% by volume solution and 4 L of 60% by volume solution, what's the percent by volume of the resulting solution?

The amount of solute in the 1 L of 30% solution is 1(0.3) = 0.3 L, and the amount of solute in the 4 L of 60% solution is 4(0.6) = 2.4 L. Thus, there is a total of 0.3 + 2.4 = 2.7 L of solute in the final solution. The total volume of the final solution itself is 1 + 4 = 5 L. Therefore, the percent by volume of the final solution is 2.7/5 = 0.54, or 54%

If a farmer combines 4 L of 25% by volume solution and 4 L of 75% by volume solution, what's the percent by volume of the resulting solution?

The amount of solute in the 4 L of 25% solution is 4(0.25) = 1 L, and the amount of solute in the 4 L of 75% solution is 4(0.75) = 3 L. Thus, there is a total of 1 + 3 = 4 L of solute in the final solution. The total volume of the final solution itself is 4 + 4 = 8 L. Therefore, the percent by volume of the final solution is 4/8 = 0.5, or 50%.

A standard deck of playing cards has 52 cards, half of which are black and the other half are red. If Angela picks a card at random, then picks another card at random in another deck of cards, what is the probability that both cards are of the same color?

The probability of drawing a card of -either color on the first card 52/52. Once a card is pulled and a color is selected, the probability of drawing the same color from a separate deck of cards is 26/52. Therefore, the probability of pulling two cards of the same color in a row is (52/52)(26/52).

A standard deck of playing cards has 52 cards, half of which are black and half of which are red. If a person picks a card at random, replaces it, then picks another card at random, what is the chance that both cards are red?

The probability of getting a red card on the first is 1/2. Since the card is replaced, the probability of getting a red card on the second draw is still 1/2 × 1/2 = 1/4.

Three dice are rolled. What is the probability of getting all even numbers

The probability of getting an even number when rolling a die is 3/6 or 1/2. Thus, the probability of getting all even numbers on 3 dice is: 1/2 × 1/2 × 1/2 = 1/8.

A survey was conducted to determine the number of televisions in a household. The data was found to be normally distributed with a mean of 3 televisions and a standard deviation of 1 television. Approximately what percent of the population owns more than 5 televisions in their households?

The households follow a normal distribution with a mean of 3 televisions and a standard deviation of 1 television: Since 95% of the data is contained within two standard deviations from the mean, 95% of households have between 1 to 5 televisions. Therefore, 100% - 95% = 5% of households fall outside of this range, and have either less than 1 more than 5 televisions. Due to the symmetry of a normal distribution, 5% ÷ 2 = 2.5% have more than 5 televisions.

The mean score on a physics test was 49.5 points. What would be the effect on the mean and standard deviation if the teacher decides to add 10 points to every student's test score?

The mean is the average of a dataset, and since adding 10 points to every score will increase the average score, the mean increases. The standard deviation is a measure of spread or variation in a dataset, and since adding the same value to each score will not change the variation among the scores, standard deviation remains the same.

If 25% of a group of people are taller than six feet, what is the probability that if 4 people are chosen at random from that group with replacement, at least one of them is taller than six feet?

The probability of choosing at least one person taller than six feet equals one minus the probability of choosing zero people taller than six feet. The probability of choosing a person taller than six feet is 25%, or 1/4. Therefore, the probability of choosing a person shorter than six feet is 1 - 1/4 = 3/4. The probability of choosing 4 people shorter than six feet is: (3/4) (3/4) (3/4) (3/4) = 81/256 Thus, the probability of choosing at least one taller than 6 feet is: 1 - 81/256 = 175/256

2 cards are drawn without replacement from a deck of playing cards; the deck contains no jokers. What is the probability that the first card is a face card (a king, queen, or jack) and the second card is the ace of spades?

The probability of the first card being a face card is 12/52. Since the cards are drawn without replacement, the probability of the second card being the ace of spades is 1/51. 12/52 x 1/51= 3/13 x 1/51= 1/13 x 1/17= 1/221

Assume 15% of the population has blue eyes. If examining two people, what is the probability that exactly one of them has blue eyes?

The probability that a person has blue eyes is 0.15, so the probability of a person not having blue eyes is 1 - 0.15 = 0.85. Say we have two people, Person A and Person B. The probability that Person A has blue eyes and Person B does not 0.15 × 0.85 = 0.1275, and the probability that Person B has blue eyes but Person A does not is 0.85 × 0.15 = 0.1275 Therefore, the probability exactly one person has blue eyes is 0.1275 + 0.1275 = 0.255. Turning this into a fraction, we have 0.255 = 255/1000 = 51/200

The probability of Plane A leaving on time is 3/8, while the probability of Plane B leaving on time is 5/6. What is the probability of Plane A leaving on time and Plane B leaving late?

The probability that plane A will leave on time is 3/8. If the probability that plane B will leave on time is 5/6, then the probability that plane B will leave late is 1 - 5/6 = 1/6. By the multiplication rule, the probability of both events is: 3/8 x 1/6= 1/16

A nickel, a dime, and a quarter are tossed in the air. What is the probability that the dime and quarter come up heads and the nickel comes up tails?

The probability the dime comes up heads is 1/2, the probability that the quarter comes up heads is 1/2, and the probability that the nickel comes up tails is 1/2.. Thus, the final probability is 1/2 × 1/2 × 1/2 = 1/8.

If x > 10 and must be an integer number, which of the following expressions is the largest?

The simplest way to solve this equation is to plug in a possible value for x into the answer choices, and see which choice is largest. The only condition for x is that it is an integer and larger than 10. If we let x = 20, we will find that [A] has the largest value at 24.

Approximately what is the measure of each interior angle in a regular 13-gon?

The sum of all the internal angles of a polygon with n sides is 180(n - 2)º. For a regular 13-gon, this is: 180(13 - 2) = 180(11) = 1980º Since the 13-gon is a regular polygon, all sides and angles are equal. Therefore, the measure of a single internal angle is: 1980º / 13 ≈ 152º

A geography final exam's scores are normally distributed with a mean of 52 and a standard deviation of x. If x = 8, what is the z-score of a test grade of 64?

The z-score of a measurement X is found by subtracting the mean µ and dividing by the standard deviation σ: Z = (X - µ) / σ The z-score for a test score of 64 is therefore: Z = (64 - 52) / 8 = 1.5

Five people are applying for the positions of president and treasurer. How many different ways can the two positions be filled if each position can only be filled by one person?

There are 5 possibilities for choosing the first person to be president. After that, there are 4 possibilities for choosing a second person to be treasurer. Thus, the total number of ways to fill the two positions from 5 people is 5 × 4 = 20.

How many distinct ways can a person choose 2 different cards from a standard deck of 52 playing cards without replacement?

There are 52 possibilities for the first card choice. After one card has been selected, there are only 51 possibilities for the second card choice. By the multiplication rule, the total number of possible ways of drawing the two cards is: 52 × 51 = 2,652 possibilities.

If two dice are rolled separately, what is the probability that the difference between the two dice is 3?

There are 6 possible outcomes that would result in a difference of 3: (4, 1), (1, 4), (5, 2), (2, 5), (3, 6), and (6, 3). Since the total number of outcomes when rolling two dice is 6 × 6 = 36, the probability is 6/36 = 1/6.

About how many doubles would you expect to see if throwing 2 dice simultaneously 100 times?

There are 6 possible ways to get doubles: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6). Therefore, the probability of getting a double is 6/36, or 1/6. Out of 100 roles, the approximate number of doubles would be 1/6 × 100 ≈ 16.67, or about 17 doubles.

Kevin's grades from last semester went as follows: Physics (3 credits), A; Chemistry (4 credits), B; Theater Appreciation (2 credits), A; Scuba diving (1 credit), C. If an A is assigned 4 grade points, a B is 3 grade points, and a C is 2 grade points, what is Kevin's grade point average?

To find the GPA, we take a weighted average. Each grade is weighted by the number of credits it was worth, and then divided by the total number of credits to find the grade point average. For example, since Physics is worth 3 credits and Kevin earned 4 grade points (an A) in Physics, he earns 3 × 4 = 12 points towards his GPA. There are a total of 3 + 4 + 2 + 1 = 10 credits. Thus, his GPA is: 3.4

In a forest, 30% of the butterflies have orange wings, and 41% of the butterflies are longer than 3 inches, with 7% belonging to both classifications. What is the probability that a randomly selected butterfly either has orange wings or is longer than 3 inches, but not both?

We are told that 7% of butterflies have BOTH orange wings and a length greater than 3 inches. Since 30% of butterflies have orange wings, 30% - 7% = 23% of butterflies have only orange wings. Similarly, since 41% of butterflies are longer than 3 inches, 41% - 7% = 34% of butterflies are only longer than 3 inches. The portion of butterflies that have orange wings OR are longer than 3 inches (but NOT both) is 23% + 34% = 57%.

An investment in Pear Computers has an initial value of $5,000. A second investment in Macrosoft Computers has an initial value of $7,500. The Pear stock falls by the same percentage as the Macrosoft stock rises. If the new combined investment value is $13,000, by what percentage did the Pear stock fall and the Macrosoft stock rise?

We can set up the following equation to describe the value of the combined investment and use x to represent the percentage that Pear stock falls (and Macrosoft rises). 13000 = 5000(1.0 - x) + 7500(1.0 + x) Simplify and solve for x: 13000 = 5000 - 5000x + 7500 + 7500x 13000 = 12500 + 2500x 500 = 2500x x = 0.2 Therefore, Pear Stock fell by 20%, and Macrosoft stock rose by 20%.

How many different ways (orders) can you arrange 6 different playing cards?

We have 6 different cards so there are 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720 orderings.

If the probability that a hygienist likes dental floss is 0.55, the probability of those who like both dental floss and water picks is 0.33, and the probability of those who like neither is 0.05, what is the probability of those who like only water picks?

We know that the probability of hygienists who like neither floss nor picks is 0.05, which means the probability of hygienists who like either floss or picks is 1 - 0.05 = 0.95. Hygienists who like either floss or picks include those who like only floss, only picks, and both. The probability of hygienists who like floss (0.55) includes those who like only floss and those who like both floss and picks, so if we subtract that from the probability of hygienists who like either, we are left with hygienists who like only picks. Therefore, the probability of hygienists who like only picks is 0.95 - 0.55 = 0.4.

What is the probability of 5 consecutive heads or 5 consecutive tails in five tosses of a fair coin?

We want the possibility of either 5 heads OR 5 tails in a row. First, consider the probability of getting 5 heads in a row. Since the probability of getting one head is 1/2, we have: P(5 heads) = 1/2 × 1/2 × 1/2 × 1/2 × 1/2 = 1/32 Since the chance of getting trails is also the same as getting heads, the probability of getting 5 tails in a row is: P(5 tails) = 1/32 By the addition rule, the probability of either 5 heads or 5 tails is: 1/32 + 1/32 = 1/16

In a village, 22% of the population has a cat and 40% of the population has a dog. If 12% of the population has both cats and dogs, what is the probability that a person chosen randomly from the village has a cat only or a dog only, or both?

We want to find the total percentage of people who have only a cat, only a dog, or both. If 22% of people have a cat and 12% of people have both a cat and dog, then 22% - 12% = 10% of people have only a cat. Likewise, if 40% of people have a dog and 12% of people have both, then 40% - 12% = 28% of people have only a dog. We already know that 12% of people have both. Thus, the total percentage is 10% + 28% + 12% = 50%.

Train A leaves a city traveling at 100 miles per hour. Train B leaves from the same city in the same direction of Train A traveling at 150 miles per hour. If Train A leaves 3 hours before Train B, how long will it take for Train B to catch up to Train A?

When Train B catches up to Train A, the trains are at the same distance from the start. Therefore, we can solve for t by setting the distances of the two trains equal to each other: 100(3) + 100t = 150t 300 + 100t = 150t 300 = 50t t = 6 hours

If 3 coins are tossed in the air, what is the chance that exactly 1 of the coins lands on heads?

When we toss 3 coins, there are 23 = 8 possible combinations of heads and tails. Since order is not important, all the possible ways we can have 1 head are: HTT THT TTH There are 3 possible ways, so the probability is 3/8. Answer

Julia took 4 courses at her college this past semester. She earned an A in Chemistry (3 credits), a B in Chemistry Lab (1 credit), an A in Spanish (4 credits), and a C in Fruit Science (1 credit). If grades are weighted such that an A is four grade points, a B is three grade points, and a C is two grade points, what is Julia's grade point average from this past semester?

total grade points = 4(3) + 3(1) + 4(4) + 2(1) = 33 Summing up the credits from each class gives us the total number of credits: total credits = 3 + 1 + 4 + 1 = 9 33/9= 3.67

Since the region is below the straight line and above the exponential, the answer is in the form

y ≤ "straight line" and y ≥ "exponential function".