Question 1

Solve the problem.
-On a buying trip in Los Angeles, Rosaria Perez ordered 120 pieces of jewelry: a number of bracelets at $5 each and a number of necklaces at $15 each. She wrote a check for $1400 to pay for the order. How many bracelets
And how many necklaces did Rosaria purchase?

Accepted Answer

Let x be the number of bracelets and y be the number of necklaces.5x + 15y = 1400x + y = 120Solving the system of equations gives x = 40 and y = 80.

Question 2

Solve the problem.
-Jimmy is a partner in an Internet-based coffee supplier. The company offers gourmet coffee beans for $14 per pound and regular coffee beans for $5 per pound. Jimmy is creating a medium-price product that will sell for $7 per pound. The first thing to go into the mixing bin was 10 pounds of the gourmet beans. How many pounds Of the less expensive regular beans should be added?

Accepted Answer

To solve this, we use the concept of weighted averages. The goal is to find a mix that averages out to $7 per pound. Let $ x $ be the pounds of regular beans needed. The total cost of the gourmet beans is $ 10 \times 14 = 140 $ dollars. The cost of $ x $ pounds of regular beans is $ 5x $ dollars. The total weight of the mix is $ 10 + x $ pounds, and the total cost is $ 140 + 5x $ dollars. To find the mix that averages out to $7 per pound, we set up the equation:$$ \frac{140 + 5x}{10 + x} = 7 $$Solving for $ x $:$$ 140 + 5x = 7(10 + x) $$$$ 140 + 5x = 70 + 7x $$$$ 140 - 70 = 7x - 5x $$$$ 70 = 2x $$$$ x = 35 $$Therefore, Jimmy should add 35 pounds of the regular beans.

Question 3

Solve the problem.
-The sum of two numbers is 6. Five times the first number equals 2 times the second number. Find the two numbers.

Accepted Answer

Let the first number be $x$ and the second number be $y$. The problem gives us two equations: $x + y = 6$ and $5x = 2y$. Solving these equations simultaneously, we find $x = \frac { 12 } { 7 }$ and $y = \frac { 30 } { 7 }$.

Question 4

Solve the problem.
-Khang and Hector live 32.4 miles apart in southeastern Missouri. They decide to bicycle towards each other and meet somewhere in between. Hector's rate of speed is 80% of Khang's. They start out at the same time and meet
3 hours later. Find Hector's rate of speed.

Accepted Answer

The answer of Solve the problem.
-Khang and Hector live 32.4...

Question 5

Solve the problem.
-Natasha rides her bike (at a constant speed)for 3 hours, helped by a wind of 3 miles per hour. Pedaling at the same rate, the trip back against the wind takes 9 hours. Find find the total round trip distance she traveled.

Accepted Answer

The total distance traveled is the sum of the distances for each leg of the trip. Let $x$ be Natasha's speed in miles per hour without wind. With the wind, her speed is $x + 3$ mph, and against the wind, it is $x - 3$ mph. The distance for each leg is speed times time. For the first leg, the distance is $3(x + 3)$ and for the return leg, it's $9(x - 3)$. Since the distances are equal, we set them equal to each other and solve for $x$: $3(x + 3) = 9(x - 3)$. Solving for $x$ gives us the speed without wind, but we're interested in the total distance, which is twice the distance of one leg (since it's a round trip and each leg is the same distance). Solving the equation directly for distance, we find that the total distance is 54 miles, considering the time and adjusted speeds for both legs of the trip.

Question 6

Solve the problem.
-Dmitri needs 6 liters of a 46% solution of sulfuric acid for a research project in molecular biology. He has two supplies of sulfuric acid solution: one is an unlimited supply of the 54% solution and the other an unlimited
Supply of the 30% solution. How many liters of each solution should Dmitri use?

Accepted Answer

Let x be the amount (in liters) of the 54% solution needed and y be the amount (in liters) of the 30% solution needed. Then we have the system of equations:

x + y = 6 (total solution needed)
0.54x + 0.30y = 0.46(6) (amount of acid needed)

Simplifying the second equation, we get:

0.54x + 0.30y = 2.76

Multiplying both sides of the first equation by 0.54 and subtracting from the above equation, we eliminate x and get:

0.20y = 0.48

Therefore, y = 2.4. Substituting into the first equation, we get x = 3.6. Therefore, Dmitri needs 4 liters of the 54% solution and 2 liters of the 30% solution. So, the answer is B). Let x be the amount of the 54% solution and y be the amount of the 30% solution. We have the following system of equations:
x + y = 6 (total volume is 6 liters)
0.54x + 0.30y = 0.46(6) (total amount of sulfuric acid should be 46% of the total volume)
Solving the system, we get x = 4 and y = 2, which means Dmitri should use 4 liters of the 54% solution and 2 liters of the 30% solution. Therefore, the best choice is B.

Question 7

Solve the problem.
-A barge takes 4 hours to move (at a constant rate)downstream for 32 miles, helped by a current of 2 miles per hour. If the barge's engines are set at the same pace, find the time of its return trip against the current.

Accepted Answer

To solve this problem, we can start by using the formula: distance = rate x time.

For the downstream trip: 
distance = 32 miles 
rate = barge's speed + current's speed = x + 2 
time = 4 hours

So we have: 
32 = (x + 2) * 4

Simplifying this equation, we get: 
8 = x + 2 
x = 6

This means that the barge's speed without the current is 6 miles per hour.

Now, for the upstream trip, the barge is going against the current, so its speed is its own speed minus the current's speed: 
rate = x - 2 = 6 - 2 = 4

Let's use the distance formula again: 
distance = 32 miles 
rate = 4 miles per hour 
time = ?

32 = 4 * time 
time = 8

Therefore, the time for the return trip against the current is 8 hours. The answer is D.

Question 8

Solve the problem.
-One number is four more than a second number. Two times the first number is 4 more than four times the second number.

Accepted Answer

The equations based on the given statements are: 1) x = y + 4, and 2) 2x = 4y + 4. Substituting the first equation into the second gives 2(y + 4) = 4y + 4, which simplifies to y = 2. Substituting y = 2 into the first equation gives x = 6. Therefore, the numbers are 6 and 2.

Question 9

Solve the problem.
-Devon purchased tickets to an air show for 5 adults and 2 children. The total cost was $167. The cost of a child's ticket was $4 less than the cost of an adult's ticket. Find the price of an adult's ticket and a child's ticket.

Accepted Answer

Let the cost of an adult's ticket be $x$. Then, the cost of a child's ticket is $x - 4$. The total cost for 5 adults and 2 children is $5x + 2(x - 4) = 167$. Solving this equation gives $x = 25$, so the adult's ticket is $25 and the child's ticket is $21.

Question 10

Solve the problem.
-Jarod is having a problem with rabbits getting into his vegetable garden, so he decides to fence it in. The length of the garden is 12 feet more than 6 times the width. He needs 136 feet of fencing to do the job. Find the length
And width of the garden.

Accepted Answer

The answer of Solve the problem.
-Jarod is having a problem...

Question 11

Solve the problem.
-Chandra has 3 liters of a 24% solution of sodium hydroxide in a container. What is the amount and concentration of sodium hydroxide solution she must add to this in order to end up with 8 liters of 54%
Solution?

Accepted Answer

To solve this problem, we can use the formula:

(amount of sodium hydroxide in final solution) = (amount of sodium hydroxide in initial solution) + (amount of sodium hydroxide added)

Let's first find the amount of sodium hydroxide in the initial solution:

3 liters * 0.24 = 0.72 liters

Next, let's find the amount of sodium hydroxide we need in the final solution:

8 liters * 0.54 = 4.32 liters

Now we can set up the equation:

4.32 = 0.72 + (amount of sodium hydroxide added)

Simplifying, we get:

amount of sodium hydroxide added = 3.6 liters

Finally, we need to find the concentration of the sodium hydroxide solution we need to add. We can use the formula:

(concentration of final solution) = (amount of sodium hydroxide in initial solution + amount of sodium hydroxide added) / (total volume of final solution)

Plugging in the values we have, we get:

(concentration of final solution) = (0.72 + 3.6) / 8 = 0.54

This means we need to add a 5 liter solution that is 72% sodium hydroxide (since 0.72 is 72% of the 1 liter of sodium hydroxide in the solution). Therefore, the answer is D.

Question 12

Solve the problem.
-Kelly is a partner in an Internet-based seed and garden supply business. The company offers a blend of exotic wildflower seeds for $105 per pound and a blend of common wildflower seeds for $15 per pound. Kelly is Creating a medium-price product by mixing together 32 pounds of the more expensive blend with 7 pounds of The less expensive blend. What will be the price per pound for the new blend? (Round to the nearest cent, if Necessary.)

Accepted Answer

The answer of Solve the problem.
-Kelly is a partner in...

Question 13

Write a system of equations to describe the situation. Do not solve the system.
-Leroy has $14 less in his pocket than Zoe has. If Zoe had $5 more, she would have 2 times as much money as Leroy.

Accepted Answer

The correct system of equations is $x = y - 14$ (Leroy has $14 less than Zoe) and $y + 5 = 2x$ (if Zoe had $5 more, she would have twice as much as Leroy).

Question 14

Solve the problem.
-The three angles in a triangle always add up to 180°. If one angle in a triangle is 114° and the second is 5 times the third, what are the three angles?

Accepted Answer

The sum of the angles in a triangle is 180°. Given one angle is 114°, let the third angle be x. Then the second angle is 5x. So, 114° + 5x + x = 180°. Solving gives x = 11°, making the second angle 55°. Thus, the angles are 114°, 55°, and 11°.

Question 15

Solve the problem.
-Jamil always throws loose change into a pencil holder on his desk and takes it out every two weeks. This time it is all nickels and dimes. There are 3 times as many dimes as nickels, and the value of the dimes is $1.75 more
Than the value of the nickels. How many nickels and dimes does Jamil have?

Accepted Answer

Let the number of nickels be $n$ and the number of dimes be $d$. Given that $d = 3n$ and the value difference is $1.75, which translates to 175 cents. The value of dimes is $10d$ cents and the value of nickels is $5n$ cents. The equation based on the value difference is $10d - 5n = 175$. Substituting $d = 3n$ into the equation gives $30n - 5n = 175$, which simplifies to $25n = 175$. Solving for $n$ gives $n = 7$, and thus $d = 3n = 21$. Therefore, Jamil has 7 nickels and 21 dimes.

Question 16

Write a system of equations to describe the situation. Do not solve the system.
-An order of 4 orders of fries, 3 hamburgers, and 5 drinks costs $15. An order of 4 orders of fries, 3 hamburgers, and 2 drinks costs $12. All drinks are $1.

Accepted Answer

The correct system of equations must account for the cost of fries (x), hamburgers (y), and drinks, with the drinks costing $1 each. The total cost of the orders is given, so the equations should directly reflect the cost structure provided in the problem. Option B correctly represents the given situation, with the first equation for the order costing $15 and the second for the order costing $12, including the cost of drinks.

Question 17

Solve the problem.
-Julie and Eric row their boat (at a constant speed)35 miles downstream for 5 hours, helped by the current. Rowing at the same rate, the trip back against the current takes 7 hours. Find the rate of the current.

Accepted Answer

Let the speed of the boat be x and the speed of the current be y.
For the downstream trip, the boat's speed is (x + y) and the distance is 35 miles. Therefore, the time taken is given by:
35/(x+y) = 5
Simplifying this equation, we get:
x + y = 7 --- (1)
For the upstream trip, the boat's speed is (x - y) and the distance is also 35 miles. Therefore, the time taken is given by:
35/(x-y) = 7
Simplifying this equation, we get:
x - y = 5 --- (2)
Adding equations (1) and (2), we get:
2x = 12
x = 6
Substituting x = 6 into equation (1), we get:
6 + y = 7
y = 1
Therefore, the rate of the current is 1 mph. Let the speed of the boat be x miles per hour and the speed of the current be y miles per hour.
35 = (x + y) * 5
35 = (x - y) * 7
Simplifying the equations:
x + y = 7
x - y = 5
Adding both equations, we get:
2x = 12
x = 6
Substituting x in one of the equations:
x + y = 7
6 + y = 7
y = 1
Therefore, the rate of the current is 1 mph.

Question 18

Write a system of equations to describe the situation. Do not solve the system.
-One number is 5 more than another number. If you add 9 to 2 times the first number, the result is 3 times the second number.

Accepted Answer

The answer of Write a system of equations to describe...

Question 19

Solve the problem.
-Two numbers total 10, and their difference is 16. Find the two numbers.

Accepted Answer

Let's call the two numbers x and y. We know that x + y = 10 and x - y = 16. We can solve for x and y by adding the two equations together:

(x + y) + (x - y) = 10 + 16

2x = 26

x = 13

Now that we know x, we can substitute it into one of the original equations to solve for y:

13 + y = 10

y = -3

Therefore, the two numbers are 13 and -3, which is choice C. Let x be one of the numbers. Then the other number is 10-x. We can set up the equation x-(10-x) = 16 and solve for x: x-10+x = 16, 2x = 26, x = 13. Therefore, the two numbers are 13 and -3, which is choice C.

Question 20

Solve the problem.
-Doreen and Irena plan to leave their houses at the same time, roller blade towards each other, and meet for lunch after 2 hours on the road. Doreen can maintain a speed of 6.6 miles per hour, which is 60% of Irena's
Speed. If they meet exactly as planned, what is the distance between their houses?

Accepted Answer

Doreen's speed is 6.6 mph, which is 60% of Irena's speed. Therefore, Irena's speed is 6.6 / 0.6 = 11 mph. Together, they cover 6.6 + 11 = 17.6 miles per hour. In 2 hours, they cover 17.6 * 2 = 35.2 miles.

Quiz 11: Systems of Equations

Solve the problem. -On a buying trip in Los Angeles, Rosaria Perez ordered 120 pieces of jewelry: a number of bracelets at $5 each and a number of necklaces at $15 each. She wrote a check for $1400 to pay for the order. How many bracelets And how many necklaces did Rosaria purchase?

Solve the problem. -The sum of two numbers is 6. Five times the first number equals 2 times the second number. Find the two numbers.

Solve the problem. -Khang and Hector live 32.4 miles apart in southeastern Missouri. They decide to bicycle towards each other and meet somewhere in between. Hector's rate of speed is 80% of Khang's. They start out at the same time and meet 3 hours later. Find Hector's rate of speed.

Solve the problem. -Natasha rides her bike (at a constant speed) for 3 hours, helped by a wind of 3 miles per hour. Pedaling at the same rate, the trip back against the wind takes 9 hours. Find find the total round trip distance she traveled.

Solve the problem. -A barge takes 4 hours to move (at a constant rate) downstream for 32 miles, helped by a current of 2 miles per hour. If the barge's engines are set at the same pace, find the time of its return trip against the current.

Solve the problem. -One number is four more than a second number. Two times the first number is 4 more than four times the second number.

Solve the problem. -Devon purchased tickets to an air show for 5 adults and 2 children. The total cost was $167. The cost of a child's ticket was $4 less than the cost of an adult's ticket. Find the price of an adult's ticket and a child's ticket.

Solve the problem. -Jarod is having a problem with rabbits getting into his vegetable garden, so he decides to fence it in. The length of the garden is 12 feet more than 6 times the width. He needs 136 feet of fencing to do the job. Find the length And width of the garden.

Solve the problem. -Chandra has 3 liters of a 24% solution of sodium hydroxide in a container. What is the amount and concentration of sodium hydroxide solution she must add to this in order to end up with 8 liters of 54% Solution?

Write a system of equations to describe the situation. Do not solve the system. -Leroy has $14 less in his pocket than Zoe has. If Zoe had $5 more, she would have 2 times as much money as Leroy.

Solve the problem. -The three angles in a triangle always add up to 180°. If one angle in a triangle is 114° and the second is 5 times the third, what are the three angles?

Write a system of equations to describe the situation. Do not solve the system. -An order of 4 orders of fries, 3 hamburgers, and 5 drinks costs $15. An order of 4 orders of fries, 3 hamburgers, and 2 drinks costs $12. All drinks are $1.

Solve the problem. -Julie and Eric row their boat (at a constant speed) 35 miles downstream for 5 hours, helped by the current. Rowing at the same rate, the trip back against the current takes 7 hours. Find the rate of the current.

Write a system of equations to describe the situation. Do not solve the system. -One number is 5 more than another number. If you add 9 to 2 times the first number, the result is 3 times the second number.

Solve the problem. -Two numbers total 10, and their difference is 16. Find the two numbers.