Asked by nut
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
Answered by
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?
Answered by
nut
Kevin is four years older than his sister, June. The sum of their ages is 14. How old is Kevin?
Answered by
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
Answered by
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.
Answered by
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?
Answered by
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 \( \boxed{70} \).
Answered by
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 \( \boxed{120} \).
Answered by
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.
Answered by
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.
Answered by
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.
Answered by
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.