Question 1

Solve the problem.
-In a chemistry class, 3 liters of a 4% silver iodide solution must be mixed with a 10% solution to get a 6% solution. How many liters of the 10% solution are needed?

Accepted Answer

Let x be the amount (in liters) of the 10% solution needed.
We can set up the following equation based on the desired concentration:
0.04(3) + 0.1x = 0.06(3+x)
0.12 + 0.1x = 0.18 + 0.06x
0.04x = 0.06
x = 1.5
Therefore, 1.5 liters of the 10% solution are needed. Let x be the number of liters of the 10% solution.
The amount of silver iodide in the 4% solution is 0.04*3 = 0.12 L.
The amount of silver iodide in the 10% solution is 0.1x.
The amount of silver iodide in the final 6% solution is 0.06(3+x).
So we can write the following equation:
0.12 + 0.1x = 0.06(3+x)
Simplifying and solving for x, we get:
0.12 + 0.1x = 0.18 + 0.06x
0.04x = 0.06
x = 1.5

Therefore, 1.5 liters of the 10% solution are needed.

Question 2

Solve the problem.
-Ellen wishes to mix candy worth $1.08 per pound with candy worth $3.05 per pound to form 24 pounds of a mixture worth $2.64 per pound. How many pounds of the more expensive candy should she use?

Accepted Answer

Let x be the number of pounds of candy worth $1.08 per pound, then 24 - x is the number of pounds of candy worth $3.05 per pound. We can set up the equation:
1.08x + 3.05(24 - x) = 2.64(24)
Simplifying and solving for x, we get:
x = 19
Therefore, Ellen should use 19 pounds of candy worth $1.08 per pound and 5 pounds of candy worth $3.05 per pound to form the mixture. Let x be the number of pounds of candy worth $3.05 per pound that Ellen mixes. Then, the number of pounds of candy worth $1.08 per pound that she mixes is 24 - x.

We know that the total cost of the mixture is equal to the cost of the $3.05 per pound candy plus the cost of the $1.08 per pound candy.

So we can write:

3.05x + 1.08(24-x) = 2.64(24)

Simplifying this equation:

3.05x + 25.92 - 1.08x = 63.36

1.97x = 37.44

x ≈ 19

Therefore, Ellen should mix 19 pounds of candy worth $3.05 per pound and 5 pounds of candy worth $1.08 per pound.

Question 3

Solve the problem.
-Mrs. Boyd has a desk full of quarters and nickels. If she has a total of 13 coins with a total face value of $2.25, how many of the coins are nickels?

Accepted Answer

Let the number of nickels be $n$ and the number of quarters be $q$. We have two equations based on the given information: $n + q = 13$ (total coins) and $0.05n + 0.25q = 2.25$ (total value in dollars). Solving these equations simultaneously, we find that $n = 5$ and $q = 8$, meaning there are 5 nickels.

Question 4

Solve the problem.
-A sum of money amounting to $5.50 consists of dimes and quarters. If there are 31 coins in all, how many are quarters?

Accepted Answer

Let the number of quarters be q and the number of dimes be d. We have two equations: q + d = 31 and 0.25q + 0.10d = 5.50. Solving these equations, we find q = 16.

Question 5

Solve the problem.
-There were 630 people at a play. The admission price was $2 for adults and $1 for children. The admission receipts were $960. How many adults and how many children attended?

Accepted Answer

Let's assume x to be the number of adults and y to be the number of children. We can form two equations from the given information:
x + y = 630  ---(1)
2x + y = 960  ---(2)

We can solve this system of equations through elimination or substitution method.

Using the elimination method, we can subtract equation (1) from equation (2) to get:
x = 330

Substituting x=330 in equation (1), we can get:
330 + y = 630
y = 300

Therefore, the number of adults and children attended the play are 330 and 300 respectively, which is option B.

Question 6

Solve the problem.
-Walt made an extra $5000 last year from a part-time job. He invested part of the money at 6% and the rest at 9%. He made a total of $420 in interest. How much was invested at 9%?

Accepted Answer

Let x be the amount invested at 9%. Then, the amount invested at 6% is $5000 - x. The interest earned from the 6% investment is 0.06(5000 - x) = 300 - 0.06x. The interest earned from the 9% investment is 0.09x. The total interest earned is 300 - 0.06x + 0.09x = 420. Solving for x, we get x = $4000.

Question 7

Solve the problem.
-Andy has 23 coins made up of quarters and half dollars, and their total value is $7.75. How many quarters does he have?

Accepted Answer

Andy has 15 quarters. To solve this, let's use a system of equations where x is the number of quarters and y is the number of half dollars. We have two equations: x + y = 23 (since there are 23 coins in total) and 0.25x + 0.5y = 7.75 (since the total value of the coins is $7.75). Solving these equations simultaneously, we find that x = 15, meaning Andy has 15 quarters.

Question 8

Solve the problem.
-How many liters of a 20% alcohol solution must be mixed with 90 liters of a 90% solution to get a 30% solution?

Accepted Answer

The answer of Solve the problem.
-How many liters of a...

Question 9

Solve the problem.
-A merchant has coffee worth $20 a pound that she wishes to mix with 80 pounds of coffee worth $80 a pound to get a mixture that can be sold for $60 a pound. How many pounds of the $20 coffee should be used?

Accepted Answer

To find the amount of $20 coffee needed, set up the equation: 20x + 80(80) = (x + 80)60. Solving for x gives x = 40 pounds.

Question 10

Solve the problem.
-Anne and Nancy use a metal alloy that is 17% copper to make jewelry. How many ounces of an alloy that is 11% copper must be mixed with an alloy that is 22% copper to form 55 ounces of the desired alloy?

Accepted Answer

Let x be the amount of 11% copper alloy. Then, 55 - x is the amount of 22% copper alloy. The equation is 0.11x + 0.22(55 - x) = 0.17(55). Solving for x gives x = 25 ounces.

Question 11

Solve the problem.
-Ron and Kathy are ticket-sellers at their class play, Ron handling student tickets that sell for $2.00 each and Kathy selling adult tickets for $6.50 each. If their total income for 22 tickets was $102.50, how many did Ron sell?

Accepted Answer

Let x be the number of student tickets Ron sold, and y be the number of adult tickets Kathy sold. We have two equations: x + y = 22 (total tickets) and 2x + 6.5y = 102.50 (total income). Solving these equations, we find x = 9.

Question 12

Solve the problem.
-A woman made a deposit of $215. If her deposit consisted of 79 bills, some of them one-dollar bills and the rest being five-dollar bills, how many one-dollar bills did she deposit?

Accepted Answer

The answer of Solve the problem.
-A woman made a deposit...

Question 13

Solve the problem.
-Tim and Judy mix two kinds of feed for pedigreed dogs. They wish to make 26 pounds of feed worth $0.54 per pound by mixing one kind worth $0.34 per pound with another worth $0.99 per pound. How many pounds of
The cheaper kind should they use in the mix?

Accepted Answer

Let x be the number of pounds of the cheaper kind to use in the mix.
Then, the number of pounds of the more expensive kind to use in the mix is (26-x).
The cost of the mix is:

0.34x + 0.99(26-x) = 0.54(26)

Simplifying and solving for x, we get:

0.34x + 25.74 - 0.99x = 14.04
-0.65x = -11.7
x = 18

Therefore, they should use 18 pounds of the cheaper kind in the mix.

Question 14

Solve the problem using the elimination method.
-The sum of two numbers is 104. The second number is three times as large as the first number. What are the numbers?

Accepted Answer

The system of equations based on the problem is: x + y = 104 (where x is the first number and y is the second number) and y = 3x. Substituting the second equation into the first gives x + 3x = 104, which simplifies to 4x = 104. Solving for x gives x = 26. Substituting x = 26 into y = 3x gives y = 78. Therefore, the numbers are 26 and 78.

Question 15

Solve the problem using the elimination method.
-The sum of two numbers is 28. The larger number minus the smaller number is 8. What are the numbers?

Accepted Answer

Let x be the smaller number and y be the larger number. We have:
x + y = 28 (equation 1)
y - x = 8 (equation 2)
We can add equation 1 and equation 2 to eliminate x:
(x + y) + (y - x) = 28 + 8
2y = 36
y = 18
Substituting y = 18 into equation 1, we get:
x + 18 = 28
x = 10
Therefore, the numbers are 10 and 18, and the best choice is C.

Question 16

Solve the problem.
-A contractor mixes concrete from bags of pre-mix for small jobs. How many bags with 6% cement should he mix with 4 bags of 19.5% cement to produce a mix containing 9% cement?

Accepted Answer

Let x be the number of bags with 6% cement that the contractor needs to mix.

The problem can be solved using the following equation:

0.06x + 0.195(4) = 0.09(x + 4)

Simplifying this equation yields:

0.06x + 0.78 = 0.09x + 0.36

0.03x = 0.42

x = 14

Therefore, the contractor needs to mix 14 bags with 6% cement to produce a mix containing 9% cement. Let the number of bags with 6% cement be x. 
We can set up the equation: 
0.06x + 4(0.195) = (x + 4)(0.09) 
Simplifying this equation gives us: 
0.06x + 0.78 = 0.09x + 0.36 
0.03x = 0.42 
x = 14

Therefore, the contractor should mix 14 bags with 6% cement to produce a mix containing 9% cement.

Question 17

Solve the problem.
-Mardi received an inheritance of $70,000. She invested part at 11% and deposited the remainder in tax-free bonds at 7%. Her total annual income from the investments was $6900. Find the amount invested at 11%.

Accepted Answer

Let x be the amount invested at 11%. Then, the amount invested at 7% is 70000 - x. The annual income from the 11% investment is 0.11x, and the annual income from the 7% investment is 0.07(70000 - x). The total annual income is 6900, so we have the equation:0.11x + 0.07(70000 - x) = 6900Solving for x, we get:x = 50000Therefore, the amount invested at 11% is $50,000.

Question 18

Solve the problem using the elimination method.
-In a basketball game, Will scored 36 points, consisting only of three-point shots and two-point shots. He made a total of 15 shots. How many shots of each type did he make?

Accepted Answer

Let x be the number of two-point shots made by Will, and y be the number of three-point shots made by Will.

From the problem, we know:

x + y = 15 (total number of shots)
2x + 3y = 36 (total number of points)

To solve for x and y, we can use the elimination method. We want to eliminate one variable (either x or y) from the equations above, so we'll multiply the first equation by -2 to get:

-2x - 2y = -30

Then we'll add this equation to the second one to get:

2x + 3y = 36
-2x - 2y = -30
--------------
y = 6

Now that we know y = 6, we can substitute this value into the first equation to solve for x:

x + y = 15
x + 6 = 15
x = 9

Therefore, Will made 9 two-point shots and 6 three-point shots, which matches option D.

Question 19

Solve the problem using the elimination method.
-The sum of two numbers is 8. Four times the larger number plus two times the smaller number is 66. Find the numbers.

Accepted Answer

Let's call the larger number x and the smaller number y. 
From the first sentence, we know that: 
x + y = 8 
From the second sentence, we know that: 
4x + 2y = 66 
Now, we can use the elimination method by multiplying the first equation by 2 and subtracting it from the second equation:

(4x + 2y = 66) - 2(x + y = 8) 
Simplifying: 
4x + 2y = 66 - 2x - 2y = 62 
Combining like terms: 
2x = 4 
x = 2

Now, we can substitute x=2 into either of the two original equations to solve for y:

2 + y = 8 
y = 6

So the two numbers are x=2 and y=6. However, this is not one of the answer choices. We need to check if one of the answer choices can be obtained by switching the values of x and y (since the problem did not specify which number is larger).

If we switch x and y, we get: 
x + y = 8 
2x + 4y = 66

Multiplying the first equation by 2 gives: 
2x + 2y = 16

Subtracting this from the second equation gives: 
2y = 50 
y = 25

Now, we can substitute y=25 into either of the two original equations to solve for x:

x + 25 = 8 
x = -17

So the two numbers are x=-17 and y=25, which matches answer choice C. Let x be the larger number and y be the smaller number. From the first equation, we have:
x + y = 8
From the second equation, we have:
4x + 2y = 66
We can eliminate y by multiplying the first equation by -2 and adding it to the second equation:
-2(x + y) + 4x + 2y = -16 + 66
2x = 50
x = 25
Substituting x = 25 into the first equation, we have:
25 + y = 8
y = -17
Therefore, the numbers are 25 and -17, which corresponds to choice C.

Question 20

Solve the problem using the elimination method.
-A vendor sells hot dogs and bags of potato chips. A customer buys 3 hot dogs and 5 bags of potato chips for $9.50. Another customer buys 2 hot dogs and 5 bags of potato chips for $8.00. Find the cost of each item.

Accepted Answer

The answer of Solve the problem using the elimination method.
-A...

Quiz 3: Systems of Equations

Solve the problem. -In a chemistry class, 3 liters of a 4% silver iodide solution must be mixed with a 10% solution to get a 6% solution. How many liters of the 10% solution are needed?

Solve the problem. -Ellen wishes to mix candy worth $1.08 per pound with candy worth $3.05 per pound to form 24 pounds of a mixture worth $2.64 per pound. How many pounds of the more expensive candy should she use?

Solve the problem. -Mrs. Boyd has a desk full of quarters and nickels. If she has a total of 13 coins with a total face value of $2.25, how many of the coins are nickels?

Solve the problem. -A sum of money amounting to $5.50 consists of dimes and quarters. If there are 31 coins in all, how many are quarters?

Solve the problem. -There were 630 people at a play. The admission price was $2 for adults and $1 for children. The admission receipts were $960. How many adults and how many children attended?

Solve the problem. -Walt made an extra $5000 last year from a part-time job. He invested part of the money at 6% and the rest at 9%. He made a total of $420 in interest. How much was invested at 9%?

Solve the problem. -Andy has 23 coins made up of quarters and half dollars, and their total value is $7.75. How many quarters does he have?

Solve the problem. -How many liters of a 20% alcohol solution must be mixed with 90 liters of a 90% solution to get a 30% solution?

Solve the problem. -A merchant has coffee worth $20 a pound that she wishes to mix with 80 pounds of coffee worth $80 a pound to get a mixture that can be sold for $60 a pound. How many pounds of the $20 coffee should be used?

Solve the problem. -Anne and Nancy use a metal alloy that is 17% copper to make jewelry. How many ounces of an alloy that is 11% copper must be mixed with an alloy that is 22% copper to form 55 ounces of the desired alloy?

Solve the problem. -Ron and Kathy are ticket-sellers at their class play, Ron handling student tickets that sell for $2.00 each and Kathy selling adult tickets for $6.50 each. If their total income for 22 tickets was $102.50, how many did Ron sell?

Solve the problem. -A woman made a deposit of $215. If her deposit consisted of 79 bills, some of them one-dollar bills and the rest being five-dollar bills, how many one-dollar bills did she deposit?

Solve the problem. -Tim and Judy mix two kinds of feed for pedigreed dogs. They wish to make 26 pounds of feed worth $0.54 per pound by mixing one kind worth $0.34 per pound with another worth $0.99 per pound. How many pounds of The cheaper kind should they use in the mix?

Solve the problem using the elimination method. -The sum of two numbers is 104. The second number is three times as large as the first number. What are the numbers?

Solve the problem using the elimination method. -The sum of two numbers is 28. The larger number minus the smaller number is 8. What are the numbers?

Solve the problem. -A contractor mixes concrete from bags of pre-mix for small jobs. How many bags with 6% cement should he mix with 4 bags of 19.5% cement to produce a mix containing 9% cement?

Solve the problem. -Mardi received an inheritance of $70,000. She invested part at 11% and deposited the remainder in tax-free bonds at 7%. Her total annual income from the investments was $6900. Find the amount invested at 11%.

Solve the problem using the elimination method. -In a basketball game, Will scored 36 points, consisting only of three-point shots and two-point shots. He made a total of 15 shots. How many shots of each type did he make?

Solve the problem using the elimination method. -The sum of two numbers is 8. Four times the larger number plus two times the smaller number is 66. Find the numbers.

Solve the problem using the elimination method. -A vendor sells hot dogs and bags of potato chips. A customer buys 3 hot dogs and 5 bags of potato chips for $9.50. Another customer buys 2 hot dogs and 5 bags of potato chips for $8.00. Find the cost of each item.