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?

Answer 1

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 \( \boxed{70} \).

Answer 2

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?

Answer 3

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 \( \boxed{120} \).

Answer 4

Kevin is four years older than his sister, June. The sum of their ages is 14. How old is Kevin?

Answer 5

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.

Answer 6

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

Answer 7

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.

Answer 8

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.

Answer 9

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.

Answer 10

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?

Answer 11

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.