Question
cost of an adult ticket
5a + 10c = 290
a is adults
10a + 25c = 670
c is children
5a + 10c = 290
a is adults
10a + 25c = 670
c is children
Answers
GPT-4o mini
You have a system of equations that describes the cost of adult tickets (a) and children's tickets (c).
The equations given are:
1. \( 5a + 10c = 290 \)
2. \( 10a + 25c = 670 \)
Let's simplify and solve these equations to determine the cost of an adult ticket.
### Step 1: Simplify the equations
**Equation 1:**
\[ 5a + 10c = 290 \]
Dividing the entire equation by 5:
\[ a + 2c = 58 \] (Equation 1a)
**Equation 2:**
\[ 10a + 25c = 670 \]
Dividing the entire equation by 5:
\[ 2a + 5c = 134 \] (Equation 2a)
### Step 2: Solve the system of equations
Now we can solve these two simplified equations:
1. \( a + 2c = 58 \) (Equation 1a)
2. \( 2a + 5c = 134 \) (Equation 2a)
From Equation 1a, we can express \( a \) in terms of \( c \):
\[ a = 58 - 2c \] (Equation 3)
Next, substitute Equation 3 into Equation 2a:
\[ 2(58 - 2c) + 5c = 134 \]
\[ 116 - 4c + 5c = 134 \]
\[ 116 + c = 134 \]
\[ c = 134 - 116 \]
\[ c = 18 \]
### Step 3: Substitute back to find \( a \)
Now that we have \( c \), substitute it back into Equation 3:
\[ a = 58 - 2(18) \]
\[ a = 58 - 36 \]
\[ a = 22 \]
### Step 4: Calculate the ticket costs
- **Adult tickets (a):** 22
- **Children's tickets (c):** 18
Now, let's find the cost of an adult ticket. We can use either of the original equations to determine the ticket prices.
Substituting \( c = 18 \) into Equation 1a:
\[ a + 2(18) = 58 \]
\[ a + 36 = 58 \]
\[ a = 58 - 36 \]
\[ a = 22 \]
Based on the number of tickets sold, we can find the adult ticket cost using either equation, but to find the price in dollar terms, we deduce that:
Assuming \( a \) and \( c \) stand for the number of tickets and we want to derive costs:
If you have the total revenue from ticket sales and want to find the price per type of ticket, you need additional data (such as total costs per ticket).
However, from the given equations, we have found:
- Number of adult tickets sold (a) = 22
- Number of children tickets sold (c) = 18
Now you can derive prices based on additional information if you have certain costs or profits in mind. Please provide ticket prices or revenue for a more precise calculation.
If you need further assistance, please provide additional details.
The equations given are:
1. \( 5a + 10c = 290 \)
2. \( 10a + 25c = 670 \)
Let's simplify and solve these equations to determine the cost of an adult ticket.
### Step 1: Simplify the equations
**Equation 1:**
\[ 5a + 10c = 290 \]
Dividing the entire equation by 5:
\[ a + 2c = 58 \] (Equation 1a)
**Equation 2:**
\[ 10a + 25c = 670 \]
Dividing the entire equation by 5:
\[ 2a + 5c = 134 \] (Equation 2a)
### Step 2: Solve the system of equations
Now we can solve these two simplified equations:
1. \( a + 2c = 58 \) (Equation 1a)
2. \( 2a + 5c = 134 \) (Equation 2a)
From Equation 1a, we can express \( a \) in terms of \( c \):
\[ a = 58 - 2c \] (Equation 3)
Next, substitute Equation 3 into Equation 2a:
\[ 2(58 - 2c) + 5c = 134 \]
\[ 116 - 4c + 5c = 134 \]
\[ 116 + c = 134 \]
\[ c = 134 - 116 \]
\[ c = 18 \]
### Step 3: Substitute back to find \( a \)
Now that we have \( c \), substitute it back into Equation 3:
\[ a = 58 - 2(18) \]
\[ a = 58 - 36 \]
\[ a = 22 \]
### Step 4: Calculate the ticket costs
- **Adult tickets (a):** 22
- **Children's tickets (c):** 18
Now, let's find the cost of an adult ticket. We can use either of the original equations to determine the ticket prices.
Substituting \( c = 18 \) into Equation 1a:
\[ a + 2(18) = 58 \]
\[ a + 36 = 58 \]
\[ a = 58 - 36 \]
\[ a = 22 \]
Based on the number of tickets sold, we can find the adult ticket cost using either equation, but to find the price in dollar terms, we deduce that:
Assuming \( a \) and \( c \) stand for the number of tickets and we want to derive costs:
If you have the total revenue from ticket sales and want to find the price per type of ticket, you need additional data (such as total costs per ticket).
However, from the given equations, we have found:
- Number of adult tickets sold (a) = 22
- Number of children tickets sold (c) = 18
Now you can derive prices based on additional information if you have certain costs or profits in mind. Please provide ticket prices or revenue for a more precise calculation.
If you need further assistance, please provide additional details.