Question

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

Give a recommendation to Meha as to how many adults and children you think she should invite to the party. Justify your recommendation.

Answer 1

To determine how many adults and children Meha can invite to the party within her budget of $300, we need to set up the total cost equation based on the information provided.

1. **Fixed costs**: $150 (the base cost of the party).
2. **Variable costs**:
- $10.50 per child (let's denote the number of children as \( c \)).
- $5.50 per adult (let's denote the number of adults as \( a \)).

The total cost \( T \) can be represented as:
\[
T = 150 + 10.50c + 5.50a
\]

Meha's aunt has set a budget of $300, so we want:
\[
150 + 10.50c + 5.50a \leq 300
\]

To simplify, we can subtract 150 from both sides:
\[
10.50c + 5.50a \leq 150
\]

Next, we can further simplify by dividing all terms by 5.50 to make the numbers easier to work with:
\[
\frac{10.50}{5.50}c + a \leq \frac{150}{5.50}
\]
Calculating the division gives us:
\[
1.909c + a \leq 27.27
\]

Since the number of children and adults must be whole numbers, we can multiply through by 100 to eliminate the decimals:
\[
190.9c + 100a \leq 2727
\]

Now, we need to find integer values for \( a \) and \( c \) such that \( a \geq 1 \), \( c \geq 1 \), and the budget constraint holds.

### Recommendations:

Let’s try a few integer combinations:

1. **Case 1**: Assume \( c = 1 \) (1 child)
\[
190.9(1) + 100a \leq 2727
\]
\[
190.9 + 100a \leq 2727
\]
\[
100a \leq 2536.1
\]
\[
a \leq 25.36
\]
This means Meha can invite a maximum of 25 adults if there is 1 child.

2. **Calculating total costs** with \( c = 1 \) and \( a = 25 \):
\[
T = 150 + 10.50(1) + 5.50(25) = 150 + 10.50 + 137.50 = 298
\]
This combination fits the budget.

3. **Case 2**: Let’s try with \( c = 2 \) (2 children)
\[
190.9(2) + 100a \leq 2727
\]
\[
381.8 + 100a \leq 2727 \implies 100a \leq 2345.2 \implies a \leq 23.45
\]
The maximum adults here can be 23.
\[
T = 150 + 10.50(2) + 5.50(23) = 150 + 21 + 126.50 = 297.50
\]
This also fits the budget.

### Conclusion:

To maximize the number of people at the party while staying within the budget, Meha could consider the following options:

- **Option 1**: 25 adults and 1 child ($298 total).
- **Option 2**: 23 adults and 2 children ($297.50 total).
- **Option 3**: 20 adults and 5 children, for an estimate ($295 total) if she prefers a balanced group.

Based on this analysis, Meha should invite **at least 2 children and 20 adults** as it's a balanced option within the budget and allows for a fun party atmosphere without sacrificing adult supervision or child participation. This way she could ensure everyone has a delightful time!