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

Answer 1

To determine how many children and adults can attend the party while staying within the budget of $300, let's set up an equation based on the given costs. Let \( x \) be the number of children, and \( y \) be the number of adults attending the party. The total cost of the party can be represented by the equation:

\[ 150 + 10.50x + 5.50y = 300 \]

First, let's simplify the equation by subtracting the flat fee of $150 from both sides to find out how much is left for paying per person:

\[ 10.50x + 5.50y = 300 - 150 \]
\[ 10.50x + 5.50y = 150 \]

Let's simplify this equation by dividing all terms by 5.50 to make it easier to solve:

\[ \frac{10.50x}{5.50} + \frac{5.50y}{5.50} = \frac{150}{5.50} \]
\[ 1.909x + y = 27.273 \]

Now we need to manipulate the equation to solve for either \( x \) or \( y \). Let's solve for \( y \):

\[ y = 27.273 - 1.909x \]

Since \( y \) has to be a whole number because you can't have a fraction of an adult attending, and there must be at least one adult, we will need to choose values of \( x \) (the number of children) that result in \( y \) being a whole number equal to or greater than 1.

Let's try different values for \( x \) to see which ones would work:

With 1 child (\( x = 1 \)):
\[ y = 27.273 - 1.909 \cdot 1 \]
\[ y = 27.273 - 1.909 \]
\[ y \approx 25.364 \]

With 2 children (\( x = 2 \)):
\[ y = 27.273 - 1.909 \cdot 2 \]
\[ y = 27.273 - 3.818 \]
\[ y \approx 23.455 \]

We can continue this pattern until we reach a combination that goes over budget. However, we need to remember that the $300 budget is the maximum amount to spend, so the actual combination of children and adults must result in a total cost that does not exceed $300.

Each whole number of adults reduces the total sum available for children because each adult is $5.50 cheaper than a child. To maximize the number of children, the number of adults should be kept to a minimum, but we have to ensure we meet the condition of at least one adult.

Let's try \( x = 10 \) (just as an example):

\[ y = 27.273 - 1.909 \cdot 10 \]
\[ y = 27.273 - 19.090 \]
\[ y \approx 8.183 \]

Since we're looking for whole numbers, let's round \( y \) down to keep the total cost within the budget:

\[ y = 8 \]

Now let's calculate the cost with this combination:

\[ Total cost = 150 + 10.50 \cdot 10 + 5.50 \cdot 8 \]
\[ Total cost = 150 + 105 + 44 \]
\[ Total cost = 299 \]

With 10 children and 8 adults, the cost would be $299, which is just under the budget. Now, remember that we rounded \( y \) down. The calculation won't be exact for other values of \( x \) without rounding. The key is to find how many children can be included, with at least one adult, while staying under the $300 budget by using trial and error or an automated method to systematically check for valid pairs of \( x \) and \( y \).