Question 1

Solve the problem.
-Two trains leave a train station at the same time. One travels east at 12 miles per hour. The other train travels west at 7 miles per hour. In how many hours will the two trains be 114 miles apart?

Accepted Answer

The trains are moving away from each other, so their speeds add up. The combined speed is 12 + 7 = 19 miles per hour. To find the time when they are 114 miles apart, divide the distance by the combined speed: 114 / 19 = 6 hours.

Question 2

Solve the problem.
-The manager of a candy shop sells chocolate covered peanuts for $6 per pound and chocolate covered cashews for $12 per pound. The manager wishes to mix 80 pounds of the cashews to get a cashew-peanut mixture that will sell for $7 per pound. How many pounds of peanuts should be used?

Accepted Answer

The answer of Solve the problem.
-The manager of a candy...

Question 3

Solve the problem.
-A chemist needs 190 milliliters of a 66% solution but has only 42% and 99% solutions available. Find how many milliliters of each that should be mixed to get the desired solution.

Accepted Answer

To solve, set up the equation: 0.42x + 0.99(190-x) = 0.66(190). Solving gives x = 110 mL of 42% and 80 mL of 99%.

Question 4

Write the expression in the standard form a + bi.
-$( 6 - 7 i ) + ( 3 + 3 i )$

Accepted Answer

The answer of Write the expression in the standard form...

Question 5

Solve the problem.
-A college student earned $5900 during summer vacation working as a waiter in a popular restaurant. The student invested part of the money at 10% and the rest at 6%. If the student received a total of $478 in interest at the end of the year, how much was invested at 10%?

Accepted Answer

The total interest earned from both investments is $478. Let x be the amount invested at 10%, and then $5900 - x is the amount invested at 6%. The interest from the first investment is 0.10x, and from the second investment is 0.06(5900 - x). The total interest is the sum of these two interests, which gives the equation: 0.10x + 0.06(5900 - x) = 478. Solving this equation for x gives x = $3100, which means $3100 was invested at 10%.

Question 6

Write the power of i as i, -1, -i, or 1.
-$$\mathrm { i } ^ { 21 }$$

Accepted Answer

The powers of $i$ cycle in a pattern: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$, and then the pattern repeats. $i^{21}$ is equivalent to $i^{(4*5)+1}$, which simplifies to $i^1$, or simply $i$.

Question 7

Solve the problem.
-Kevin invested part of his $10,000 bonus in a certificate of deposit that paid 6% annual simple interest, and the remainder in a mutual fund that paid 11% annual simple interest. If his total interest for that year was $900, how much did Kevin invest in the mutual fund?

Accepted Answer

Let x be the amount invested in the mutual fund. Then, $10,000 - x$ is the amount invested in the certificate of deposit. The equation for total interest is: 0.06($10,000 - x$) + 0.11x = $900. Solving this equation gives x = $6000.

Question 8

Solve the problem.
-The owners of a candy store want to sell, for $6 per pound, a mixture of chocolate-covered raisins, which usually sells for $3 per pound, and chocolate-covered macadamia nuts, which usually sells for $8 per pound. They have a 40-pound barrel of the raisins. How many pounds of the nuts should they mix with the barrel of raisins so that they hit their target value of $6 per pound for the mixture?

Accepted Answer

To find the number of pounds of nuts needed, set up the equation: (3 * 40 + 8 * x) / (40 + x) = 6. Solving for x gives x = 60 lb.

Question 9

Solve the problem.
-Two cars start from the same point and travel in the same direction. If one car is traveling 59 miles per hour and the other car is traveling at 52 miles per hour, how far apart will they be after 2.1 hours?

Accepted Answer

The distance between the two cars after 2.1 hours can be found by calculating the difference in the distances each car travels. The first car travels $59 \times 2.1$ miles, and the second car travels $52 \times 2.1$ miles. The difference is $(59 - 52) \times 2.1 = 7 \times 2.1 = 14.7$ miles.

Question 10

Write the power of i as i, -1, -i, or 1.
-$$\mathrm { i } 8$$

Accepted Answer

$\mathrm { i } ^8 = (\mathrm { i } ^4)^2 = 1^2 = 1$, because $\mathrm { i } ^4 = 1$.

Question 11

Solve the problem.
-An airplane flies 450 miles with the wind and 330 against the wind in the same length of time. If the speed of the wind is 20, what is the speed of the airplane in still air?

Accepted Answer

The speed of the airplane in still air is calculated by setting up the equation for time: time = distance/speed. With the wind, the speed is (x + 20) and against the wind, it is (x - 20). Since the times are equal, 450/(x + 20) = 330/(x - 20). Solving this equation gives x = 130 mph.

Question 12

Solve the problem.
-An experienced bank auditor can check a bank's deposits twice as fast as a new auditor. Working together it takes the auditors 4 hours to do the job. How long would it take the experienced auditor working alone?

Accepted Answer

The answer of Solve the problem.
-An experienced bank auditor can...

Question 13

Solve the problem.
-How much pure acid should be mixed with 9 gallons of a 50% acid solution in order to get an 80% acid solution?

Accepted Answer

To achieve an 80% acid solution, let x be the amount of pure acid (100%) to add. The initial solution has 9 gallons of 50% acid, which means it contains 4.5 gallons of pure acid. The total volume of the new solution will be (9 + x) gallons, and the amount of pure acid in it will be (4.5 + x) gallons. The equation for the concentration is: (4.5 + x) / (9 + x) = 0.8. Solving for x gives x = 13.5 gallons.

Question 14

Solve the problem.
-Two friends decide to meet in Chicago to attend a Cub's baseball game. Rob travels 126 miles in the same time that Carl travels 120 miles. Rob's trip uses more interstate highways and he can average 3 mph more than Carl. What is Rob's average speed?

Accepted Answer

Rob's speed = (126/120) * Carl's speed = (21/20) * Carl's speed = 1.05 * Carl's speed = Carl's speed + 3 mph = Carl's speed + 3 mph = 60 mph + 3 mph = 63 mph

Question 15

Write the power of i as i, -1, -i, or 1.
-$$\mathrm { i } ^ { 15 }$$

Accepted Answer

$\mathrm { i } ^ { 15 } = (\mathrm { i } ^ { 4 })^{3} \cdot \mathrm { i } ^ { 3 } = 1^{3} \cdot \mathrm { i } ^ { 3 } = \mathrm { i } ^ { 3 } = -\mathrm { i }$.

Question 16

Solve the problem.
-During a hurricane evacuation from the east coast of Georgia, a family traveled 250 miles west. For part of the trip, they averaged 50 mph, but as the congestion got bad, they had to slow to 30 mph. If the total time of travel was 9 hours, how many miles did they drive at the reduced speed?

Accepted Answer

The answer of Solve the problem.
-During a hurricane evacuation from...

Question 17

Write the power of i as i, -1, -i, or 1.
-$$\mathrm { i } ^ { 18 }$$

Accepted Answer

The answer of Write the power of i as i,...

Question 18

Solve the problem.
-The manager of a coffee shop has one type of coffee that sells for $7 per pound and another type that sells for $14 per pound. The manager wishes to mix 90 pounds of the $14 coffee to get a mixture that will sell for $10 per pound. How many pounds of the $7 coffee should be used?

Accepted Answer

To achieve a mixture that sells for $10 per pound, we can use the method of weighted averages. Let $ x $ be the pounds of the $7 coffee. The total cost of the $14 coffee is $ 90 \times 14 = 1260 $ dollars. The total cost of the $7 coffee is $ 7x $ dollars. The total weight of the mixture is $ 90 + x $ pounds, and we want this mixture to sell for $10 per pound. So, the equation to represent this situation is $ \frac{1260 + 7x}{90 + x} = 10 $. Solving this equation for $ x $ gives us:$$ 1260 + 7x = 10(90 + x) $$$$ 1260 + 7x = 900 + 10x $$$$ 360 = 3x $$$$ x = 120 $$Therefore, 120 pounds of the $7 coffee should be used.

Question 19

Solve the problem.
-BJ can overhaul a boat's diesel inboard engine in 30 hours. His apprentice takes 60 hours to do the same job. How long would it take them working together assuming no gain or loss in efficiency?

Accepted Answer

Working together, BJ and his apprentice can complete $\frac{1}{30} + \frac{1}{60} = \frac{1}{20}$ of the job per hour, meaning it would take them 20 hours to complete the job together.

Question 20

Solve the problem.
-Tracy can wallpaper 5 rooms in a new house in 15 hours. Together with her trainee they can wallpaper the 5 rooms in 11 hours. How long would it take the trainee working by herself to do the job?

Accepted Answer

Tracy can wallpaper 5 rooms in 15 hours, so she can wallpaper 1 room in 3 hours. Together with her trainee they can wallpaper the 5 rooms in 11 hours, so they can wallpaper 1 room in 11/5 hours. The trainee can wallpaper 1 room in (11/5) - 3 = 2/5 hours. So, the trainee can wallpaper 5 rooms in 5 * (2/5) = 2 hours.

Quiz 1: Equations, Inequalities, and Applications

Solve the problem. -Two trains leave a train station at the same time. One travels east at 12 miles per hour. The other train travels west at 7 miles per hour. In how many hours will the two trains be 114 miles apart?

Solve the problem. -A chemist needs 190 milliliters of a 66% solution but has only 42% and 99% solutions available. Find how many milliliters of each that should be mixed to get the desired solution.

Write the expression in the standard form a + bi. - $( 6 - 7 i ) + ( 3 + 3 i )$

Solve the problem. -A college student earned $5900 during summer vacation working as a waiter in a popular restaurant. The student invested part of the money at 10% and the rest at 6%. If the student received a total of $478 in interest at the end of the year, how much was invested at 10%?

Write the power of i as i, -1, -i, or 1. - $\mathrm { i } ^ { 21 }$

Solve the problem. -Kevin invested part of his $10,000 bonus in a certificate of deposit that paid 6% annual simple interest, and the remainder in a mutual fund that paid 11% annual simple interest. If his total interest for that year was $900, how much did Kevin invest in the mutual fund?

Solve the problem. -Two cars start from the same point and travel in the same direction. If one car is traveling 59 miles per hour and the other car is traveling at 52 miles per hour, how far apart will they be after 2.1 hours?

Write the power of i as i, -1, -i, or 1. - $\mathrm { i } 8$

Solve the problem. -An airplane flies 450 miles with the wind and 330 against the wind in the same length of time. If the speed of the wind is 20, what is the speed of the airplane in still air?

Solve the problem. -An experienced bank auditor can check a bank's deposits twice as fast as a new auditor. Working together it takes the auditors 4 hours to do the job. How long would it take the experienced auditor working alone?

Solve the problem. -How much pure acid should be mixed with 9 gallons of a 50% acid solution in order to get an 80% acid solution?

Solve the problem. -Two friends decide to meet in Chicago to attend a Cub's baseball game. Rob travels 126 miles in the same time that Carl travels 120 miles. Rob's trip uses more interstate highways and he can average 3 mph more than Carl. What is Rob's average speed?

Write the power of i as i, -1, -i, or 1. - $\mathrm { i } ^ { 15 }$

Write the power of i as i, -1, -i, or 1. - $\mathrm { i } ^ { 18 }$

Solve the problem. -The manager of a coffee shop has one type of coffee that sells for $7 per pound and another type that sells for $14 per pound. The manager wishes to mix 90 pounds of the $14 coffee to get a mixture that will sell for $10 per pound. How many pounds of the $7 coffee should be used?

Solve the problem. -BJ can overhaul a boat's diesel inboard engine in 30 hours. His apprentice takes 60 hours to do the same job. How long would it take them working together assuming no gain or loss in efficiency?

Solve the problem. -Tracy can wallpaper 5 rooms in a new house in 15 hours. Together with her trainee they can wallpaper the 5 rooms in 11 hours. How long would it take the trainee working by herself to do the job?