Question
A movie theater sold 125 tickets for a a movie. A children’s ticket costs $6 and an adult ticket costs $12. If the theater sold $1,170 worth of tickets, how many adult tickets were sold?
Answers
GPT-4o mini
To solve for the number of adult tickets sold, we can set up a system of equations based on the information provided.
Let:
- \( x \) be the number of adult tickets sold
- \( y \) be the number of children's tickets sold
We have two equations based on the problem's details:
1. The total number of tickets sold:
\[
x + y = 125
\]
2. The total revenue from ticket sales:
\[
12x + 6y = 1170
\]
Now we can solve these equations step by step.
From the first equation, we can express \( y \) in terms of \( x \):
\[
y = 125 - x
\]
Next, we will substitute this expression for \( y \) into the second equation:
\[
12x + 6(125 - x) = 1170
\]
Now, simplify the equation:
\[
12x + 750 - 6x = 1170
\]
Combine like terms:
\[
6x + 750 = 1170
\]
Now, subtract 750 from both sides:
\[
6x = 420
\]
Now, divide by 6:
\[
x = 70
\]
So, the number of adult tickets sold is \( \boxed{70} \).
To find the number of children's tickets sold, we substitute \( x \) back into the equation for \( y \):
\[
y = 125 - 70 = 55
\]
Now we can double-check our solution by verifying the revenue:
- Adult tickets sold = 70; Revenue from adult tickets = \( 70 \times 12 = 840 \)
- Children's tickets sold = 55; Revenue from children’s tickets = \( 55 \times 6 = 330 \)
- Total revenue = \( 840 + 330 = 1170 \)
The values check out, confirming that the answer is indeed correct: 70 adult tickets were sold. Thus, the final answer is \( \boxed{70} \).
Let:
- \( x \) be the number of adult tickets sold
- \( y \) be the number of children's tickets sold
We have two equations based on the problem's details:
1. The total number of tickets sold:
\[
x + y = 125
\]
2. The total revenue from ticket sales:
\[
12x + 6y = 1170
\]
Now we can solve these equations step by step.
From the first equation, we can express \( y \) in terms of \( x \):
\[
y = 125 - x
\]
Next, we will substitute this expression for \( y \) into the second equation:
\[
12x + 6(125 - x) = 1170
\]
Now, simplify the equation:
\[
12x + 750 - 6x = 1170
\]
Combine like terms:
\[
6x + 750 = 1170
\]
Now, subtract 750 from both sides:
\[
6x = 420
\]
Now, divide by 6:
\[
x = 70
\]
So, the number of adult tickets sold is \( \boxed{70} \).
To find the number of children's tickets sold, we substitute \( x \) back into the equation for \( y \):
\[
y = 125 - 70 = 55
\]
Now we can double-check our solution by verifying the revenue:
- Adult tickets sold = 70; Revenue from adult tickets = \( 70 \times 12 = 840 \)
- Children's tickets sold = 55; Revenue from children’s tickets = \( 55 \times 6 = 330 \)
- Total revenue = \( 840 + 330 = 1170 \)
The values check out, confirming that the answer is indeed correct: 70 adult tickets were sold. Thus, the final answer is \( \boxed{70} \).
nut
Alisha sells fruit at a farmer’s market. In one day, she sold a total of 200 pounds of apples and bananas. She sold apples for $2 per pound and bananas for $3 per pound and made $480 total. How many pounds of apples did Alisha sell?
GPT-4o mini
To find out how many pounds of apples Alisha sold, we can set up a system of equations based on the information given.
Let:
- \( x \) be the number of pounds of apples sold.
- \( y \) be the number of pounds of bananas sold.
From the problem, we have the following equations:
1. The total weight of apples and bananas:
\[
x + y = 200
\]
2. The total revenue from selling apples and bananas:
\[
2x + 3y = 480
\]
Now we can solve this system of equations.
From the first equation, solve for \( y \):
\[
y = 200 - x
\]
Now substitute this expression for \( y \) into the second equation:
\[
2x + 3(200 - x) = 480
\]
Simplifying the equation results in:
\[
2x + 600 - 3x = 480
\]
Combine like terms:
\[
-1x + 600 = 480
\]
Now subtract 600 from both sides:
\[
-x = 480 - 600
\]
\[
-x = -120
\]
Now, multiply by -1:
\[
x = 120
\]
So, Alisha sold \( \boxed{120} \) pounds of apples.
To find out how many pounds of bananas she sold, substitute \( x \) back into the equation for \( y \):
\[
y = 200 - 120 = 80
\]
Now, let's double-check our solution by verifying the total revenue:
- Revenue from apples = \( 120 \times 2 = 240 \)
- Revenue from bananas = \( 80 \times 3 = 240 \)
- Total revenue = \( 240 + 240 = 480 \)
The values check out, confirming that the number of pounds of apples sold is indeed \( \boxed{120} \).
Let:
- \( x \) be the number of pounds of apples sold.
- \( y \) be the number of pounds of bananas sold.
From the problem, we have the following equations:
1. The total weight of apples and bananas:
\[
x + y = 200
\]
2. The total revenue from selling apples and bananas:
\[
2x + 3y = 480
\]
Now we can solve this system of equations.
From the first equation, solve for \( y \):
\[
y = 200 - x
\]
Now substitute this expression for \( y \) into the second equation:
\[
2x + 3(200 - x) = 480
\]
Simplifying the equation results in:
\[
2x + 600 - 3x = 480
\]
Combine like terms:
\[
-1x + 600 = 480
\]
Now subtract 600 from both sides:
\[
-x = 480 - 600
\]
\[
-x = -120
\]
Now, multiply by -1:
\[
x = 120
\]
So, Alisha sold \( \boxed{120} \) pounds of apples.
To find out how many pounds of bananas she sold, substitute \( x \) back into the equation for \( y \):
\[
y = 200 - 120 = 80
\]
Now, let's double-check our solution by verifying the total revenue:
- Revenue from apples = \( 120 \times 2 = 240 \)
- Revenue from bananas = \( 80 \times 3 = 240 \)
- Total revenue = \( 240 + 240 = 480 \)
The values check out, confirming that the number of pounds of apples sold is indeed \( \boxed{120} \).
nut
Kevin is four years older than his sister, June. The sum of their ages is 14. How old is Kevin?
GPT-4o mini
Let's denote Kevin's age as \( K \) and June's age as \( J \).
According to the problem, we can set up the following equations:
1. Kevin is four years older than June:
\[
K = J + 4
\]
2. The sum of their ages is 14:
\[
K + J = 14
\]
Now, we can substitute the expression for \( K \) from the first equation into the second equation:
\[
(J + 4) + J = 14
\]
Now, combine like terms:
\[
2J + 4 = 14
\]
Next, subtract 4 from both sides:
\[
2J = 10
\]
Now, divide by 2:
\[
J = 5
\]
Now that we have June's age, we can find Kevin's age using the first equation:
\[
K = J + 4 = 5 + 4 = 9
\]
Thus, Kevin is \( \boxed{9} \) years old.
According to the problem, we can set up the following equations:
1. Kevin is four years older than June:
\[
K = J + 4
\]
2. The sum of their ages is 14:
\[
K + J = 14
\]
Now, we can substitute the expression for \( K \) from the first equation into the second equation:
\[
(J + 4) + J = 14
\]
Now, combine like terms:
\[
2J + 4 = 14
\]
Next, subtract 4 from both sides:
\[
2J = 10
\]
Now, divide by 2:
\[
J = 5
\]
Now that we have June's age, we can find Kevin's age using the first equation:
\[
K = J + 4 = 5 + 4 = 9
\]
Thus, Kevin is \( \boxed{9} \) years old.
nut
Is (0,0)
(
0
,
0
)
a viable solution for the following system? Write either 1 for viable or 2 for nonviable as your response.
2y≥−3x−2
2
𝑦
≥
−
3
𝑥
−
2
2y<−x+2
(
0
,
0
)
a viable solution for the following system? Write either 1 for viable or 2 for nonviable as your response.
2y≥−3x−2
2
𝑦
≥
−
3
𝑥
−
2
2y<−x+2
GPT-4o mini
To determine if the point \((0, 0)\) is a viable solution for the given system of inequalities:
1. **First inequality:** \(2y \geq -3x - 2\)
Substituting \((0,0)\):
\[
2(0) \geq -3(0) - 2 \implies 0 \geq -2
\]
This is **true**.
2. **Second inequality:** \(2y < -x + 2\)
Substituting \((0,0)\):
\[
2(0) < -(0) + 2 \implies 0 < 2
\]
This is **true**.
Since both inequalities are satisfied by the point \((0, 0)\), it is a viable solution.
Thus, the response is \( \boxed{1} \) for viable.
1. **First inequality:** \(2y \geq -3x - 2\)
Substituting \((0,0)\):
\[
2(0) \geq -3(0) - 2 \implies 0 \geq -2
\]
This is **true**.
2. **Second inequality:** \(2y < -x + 2\)
Substituting \((0,0)\):
\[
2(0) < -(0) + 2 \implies 0 < 2
\]
This is **true**.
Since both inequalities are satisfied by the point \((0, 0)\), it is a viable solution.
Thus, the response is \( \boxed{1} \) for viable.
nut
A vendor sells fruit at a farmer’s market. In one day, they sold a total of 175 oranges and pears. They sold oranges for $1.50 each and pears for $2 each, making a total of $305 dollars. If x represents the number of oranges sold and y represents the number of pears sold, is (90,85)
(
90
,
85
)
a viable solution to the system? Write either 1 for viable or 2 for nonviable as your response.
(
90
,
85
)
a viable solution to the system? Write either 1 for viable or 2 for nonviable as your response.
GPT-4o mini
To determine if \((90, 85)\) is a viable solution to the system of equations based on the given problem, we will use the following equations derived from the information provided:
1. The total number of oranges and pears sold:
\[
x + y = 175
\]
2. The total revenue from selling the oranges and pears:
\[
1.5x + 2y = 305
\]
Here, \(x\) is the number of oranges sold, and \(y\) is the number of pears sold.
Let's substitute \((x, y) = (90, 85)\) into these equations.
1. **First equation:**
\[
90 + 85 = 175
\]
This is **true**.
2. **Second equation:**
\[
1.5(90) + 2(85) = 305
\]
Calculate each term:
\[
1.5 \times 90 = 135
\]
\[
2 \times 85 = 170
\]
Now sum these values:
\[
135 + 170 = 305
\]
This is **true**.
Since both equations are satisfied by the point \((90, 85)\), it is a viable solution.
Therefore, the response is \( \boxed{1} \) for viable.
1. The total number of oranges and pears sold:
\[
x + y = 175
\]
2. The total revenue from selling the oranges and pears:
\[
1.5x + 2y = 305
\]
Here, \(x\) is the number of oranges sold, and \(y\) is the number of pears sold.
Let's substitute \((x, y) = (90, 85)\) into these equations.
1. **First equation:**
\[
90 + 85 = 175
\]
This is **true**.
2. **Second equation:**
\[
1.5(90) + 2(85) = 305
\]
Calculate each term:
\[
1.5 \times 90 = 135
\]
\[
2 \times 85 = 170
\]
Now sum these values:
\[
135 + 170 = 305
\]
This is **true**.
Since both equations are satisfied by the point \((90, 85)\), it is a viable solution.
Therefore, the response is \( \boxed{1} \) for viable.
x
Alisha sells fruit at a farmer’s market. In one day, she sold a total of 200 pounds of apples and bananas. She sold apples for $2 per pound and bananas for $3 per pound and made $480 total. How many pounds of apples did Alisha sell?
GPT-4o mini
To determine how many pounds of apples Alisha sold, we can set up a system of equations based on the information provided.
Let:
- \( x \) be the number of pounds of apples sold.
- \( y \) be the number of pounds of bananas sold.
From the problem, we have the following equations:
1. The total weight of apples and bananas:
\[
x + y = 200
\]
2. The total revenue from selling apples and bananas:
\[
2x + 3y = 480
\]
Now, we can solve this system of equations.
### Step 1: Solve the first equation for \( y \)
From the first equation:
\[
y = 200 - x
\]
### Step 2: Substitute \( y \) in the second equation
Now substitute \( y \) in the second equation:
\[
2x + 3(200 - x) = 480
\]
### Step 3: Distribute and simplify
Distributing the 3:
\[
2x + 600 - 3x = 480
\]
Combine like terms:
\[
-1x + 600 = 480
\]
### Step 4: Solve for \( x \)
Subtract 600 from both sides:
\[
-x = 480 - 600
\]
\[
-x = -120
\]
Multiply by -1:
\[
x = 120
\]
Thus, Alisha sold **120 pounds** of apples.
### Step 5: Find the number of pounds of bananas sold
Using the first equation:
\[
y = 200 - x = 200 - 120 = 80
\]
So, Alisha also sold **80 pounds** of bananas.
### Summary
Alisha sold \( \boxed{120} \) pounds of apples.
### Verification
To verify, we check the total revenue:
- Revenue from apples = \( 120 \times 2 = 240 \)
- Revenue from bananas = \( 80 \times 3 = 240 \)
- Total revenue = \( 240 + 240 = 480 \)
Everything checks out correctly, confirming that Alisha sold \( \boxed{120} \) pounds of apples.
Let:
- \( x \) be the number of pounds of apples sold.
- \( y \) be the number of pounds of bananas sold.
From the problem, we have the following equations:
1. The total weight of apples and bananas:
\[
x + y = 200
\]
2. The total revenue from selling apples and bananas:
\[
2x + 3y = 480
\]
Now, we can solve this system of equations.
### Step 1: Solve the first equation for \( y \)
From the first equation:
\[
y = 200 - x
\]
### Step 2: Substitute \( y \) in the second equation
Now substitute \( y \) in the second equation:
\[
2x + 3(200 - x) = 480
\]
### Step 3: Distribute and simplify
Distributing the 3:
\[
2x + 600 - 3x = 480
\]
Combine like terms:
\[
-1x + 600 = 480
\]
### Step 4: Solve for \( x \)
Subtract 600 from both sides:
\[
-x = 480 - 600
\]
\[
-x = -120
\]
Multiply by -1:
\[
x = 120
\]
Thus, Alisha sold **120 pounds** of apples.
### Step 5: Find the number of pounds of bananas sold
Using the first equation:
\[
y = 200 - x = 200 - 120 = 80
\]
So, Alisha also sold **80 pounds** of bananas.
### Summary
Alisha sold \( \boxed{120} \) pounds of apples.
### Verification
To verify, we check the total revenue:
- Revenue from apples = \( 120 \times 2 = 240 \)
- Revenue from bananas = \( 80 \times 3 = 240 \)
- Total revenue = \( 240 + 240 = 480 \)
Everything checks out correctly, confirming that Alisha sold \( \boxed{120} \) pounds of apples.