Meha is helping her aunt to plan a birthday party for her cousin at the science center. Her aunt

Question

tells her that the party will cost $150, plus $10.50 per child and $5.50 per adult. There needs to be
at least one adult and at least one child at the party. She has budgeted $300 for the party and that
is the maximum amount she wants to spend.
A. Write a two-variable inequality to represent this situation. Be sure to define the variables
you use in your inequality.

Answer 1

To represent the situation, let's define the variables:

• Let \( c \) represent the number of children at the party.
• Let \( a \) represent the number of adults at the party.

The total cost for the party can be expressed as:

\[ \text{Total Cost} = 150 + 10.50c + 5.50a \]

According to the information provided, the total cost must be less than or equal to the budget of $300. Therefore, we can write the inequality as:

\[ 150 + 10.50c + 5.50a \leq 300 \]

To simplify this, we can subtract 150 from both sides:

\[ 10.50c + 5.50a \leq 150 \]

Additionally, the constraints for the number of children and adults can be represented as:

\[ c \geq 1 \quad \text{(at least one child)} \] \[ a \geq 1 \quad \text{(at least one adult)} \]

So, the complete set of inequalities representing the situation is:

1. \( 10.50c + 5.50a \leq 150 \)
2. \( c \geq 1 \)
3. \( a \geq 1 \)

Answer 2

make it simpler

Answer 3

Sure! Let's simplify the explanation and present the key components clearly.

Variables:

• Let \( c \) = number of children
• Let \( a \) = number of adults

Cost Formula:

The total party cost is given by: \[ \text{Total Cost} = 150 + 10.50c + 5.50a \]

Budget Constraint:

Since the total cost must not exceed $300, we write the inequality: \[ 150 + 10.50c + 5.50a \leq 300 \]

Simplified Cost Inequality:

Subtracting 150 from both sides gives: \[ 10.50c + 5.50a \leq 150 \]

Additional Constraints:

There must be at least one child and one adult:

1. \( c \geq 1 \) (at least one child)
2. \( a \geq 1 \) (at least one adult)

Final Representation:

The final set of inequalities is:

1. \( 10.50c + 5.50a \leq 150 \)
2. \( c \geq 1 \)
3. \( a \geq 1 \)

This succinctly represents the party planning budget and requirements!

Answer 4

will this do 150+10.50x+5.50x=300

Answer 5

The equation you provided, \(150 + 10.50x + 5.50x = 300\), combines the costs for children and adults into one term using only one variable \(x\). However, in the context of this problem, children and adults should be represented by separate variables since they can have different quantities.

To clarify:

1. In your equation, you are assuming that \(x\) is both the number of children and the number of adults, which is not accurate because the party requires at least one of each.

2. Instead, you should keep the variables separate for clarity:

   • Let \(c\) = number of children
   • Let \(a\) = number of adults

The correct approach involves using: \[ 150 + 10.50c + 5.50a \leq 300 \] and the inequalities: \[ c \geq 1 \quad \text{(at least one child)} \] \[ a \geq 1 \quad \text{(at least one adult)} \]

If you want to express the equation without inequalities, you could set up an equation to find the costs for specific values of \(c\) and \(a\), but you wouldn’t be able to simply use one variable for both unless you were assuming there is only one type of guest (which there is not in this scenario).

Summary:

The equation \(150 + 10.50x + 5.50x = 300\) is not appropriate for this situation because it does not reflect that children and adults are different variables. You need to keep them as distinct variables \(c\) and \(a\).

Answer 6

what do i do then

Answer 7

To represent the birthday party costs accurately while adhering to the budget constraints given, you should:

1. Define the Variables:

   • Use \( c \) for the number of children.
   • Use \( a \) for the number of adults.

2. Set Up the Cost Equation: Since the total cost is made up of a fixed cost plus variable costs for children and adults, write it as: \[ \text{Total Cost} = 150 + 10.50c + 5.50a \]

3. Establish the Budget Inequality: Since you want the total cost to be at most $300, you can create the inequality: \[ 150 + 10.50c + 5.50a \leq 300 \]

4. Simplify the Inequality: Subtract 150 from both sides: \[ 10.50c + 5.50a \leq 150 \]

5. Add Minimum Requirements: According to the party plan, there should be at least one child and one adult, which gives you the following inequalities:

   • For children: \( c \geq 1 \)
   • For adults: \( a \geq 1 \)

Final Summary:

The inequalities representing the birthday party costs and requirements are:

1. Cost Inequality: \[ 10.50c + 5.50a \leq 150 \]
2. Minimum Children Requirement: \[ c \geq 1 \]
3. Minimum Adult Requirement: \[ a \geq 1 \]

Example of How to Use the Inequalities:

You can choose different values for \( c \) and \( a \) that satisfy all these inequalities. For example:

• If you set \( c = 3 \) (3 children) and \( a = 2 \) (2 adults):
  • Calculate the total cost: \[ 150 + 10.50(3) + 5.50(2) = 150 + 31.50 + 11 = 192.50 \]
  • Since \( 192.50 \leq 300 \), this choice works.

Continue adjusting the values of \( c \) and \( a \) to find different combinations that will satisfy the budget while ensuring at least one adult and one child.

Answer 8

so is the answer this 192.50<300

Answer 9

Not exactly. The value \(192.50 < 300\) is a specific example of a total cost that meets the budget requirement, but it doesn’t fully encapsulate the problem you are trying to represent.

Correct Formulation:

To summarize clearly, the full answer to your original question regarding how to express the situation remains the inequalities, which are necessary to define the scenario:

1. Cost Inequality: \[ 10.50c + 5.50a \leq 150 \]

2. Minimum Requirements for Children and Adults: \[ c \geq 1 \quad \text{(at least one child)} \] \[ a \geq 1 \quad \text{(at least one adult)} \]

Example Interpretation:

You used a specific example with \( c = 3 \) and \( a = 2 \) leading to: \[ \text{Total Cost} = 150 + 10.50(3) + 5.50(2) = 192.50 \] This shows that a potential total cost of $192.50 is indeed less than the budget of $300, which indicates you are under budget, but this is just one scenario among many possible values for \( c \) and \( a \) that maintain the conditions you've set.

Conclusion:

The inequalities are what you need to present the general situation and constraints. The statement \(192.50 < 300\) is true but it’s just one example of a solution—what you really want is to keep the inequalities so you can explore various combinations of \( c \) (children) and \( a \) (adults) that will fit within your budget constraints.

Answer 10

can you tell me specifically what the answer is please.

Answer 11

Sure! The specific answer to your question about how to represent the situation, including the costs for a party within the budget constraint, can be summarized as follows:

Final Answer:

1. Define the Variables:

   • Let \( c \) = number of children
   • Let \( a \) = number of adults

2. Cost Inequality: The total cost of the party must not exceed the budget of $300. This can be expressed as: \[ 150 + 10.50c + 5.50a \leq 300 \]

   Simplifying this gives: \[ 10.50c + 5.50a \leq 150 \]

3. Minimum Requirements: There must be at least one child and one adult at the party, which can be represented as: \[ c \geq 1 \] \[ a \geq 1 \]

Complete Set of Inequalities:

The complete answer is articulated through these inequalities:

1. Cost Constraint: \[ 10.50c + 5.50a \leq 150 \]
2. Children Constraint: \[ c \geq 1 \]
3. Adults Constraint: \[ a \geq 1 \]

This is the structured way to represent the problem you are trying to solve. From this point, you can use specific values for \( c \) and \( a \) to explore potential party arrangements that adhere to the budget.

Example Calculation:

If, for example:

• You choose \( c = 3 \) (3 children) and \( a = 2 \) (2 adults), the total cost would be: \[ 150 + 10.50(3) + 5.50(2) = 150 + 31.50 + 11 = 192.50 \] This example cost of \( 192.50 \) satisfies \( 192.50 < 300 \) and adheres to the requirement of at least one child and one adult.

Feel free to ask if you need any further specifics or examples!