Cracking the Puzzle: Solving for Adult and Kids Meal Prices

Imagine a family dining scenario where the precise pricing of meals is the puzzle you need to solve. Today, we’ll dive into a real-world problem, exploring how to determine the pricing of adult and kids meals based on the total cost of meals on two different days. This vacation booking problem isn’t just a fun exercise; it’s a practical application of algebraic equations. Let’s get started!

The Scenario

Two adults and three kids go to a restaurant twice. On the first day, they spend $47.50, and on the second day, they spend $72.50. Adults are charged differently than kids, with kids being less expensive. How much is an adult meal and how much is a kids meal at that restaurant?

The Ambiguities and Considerations

This problem, at its core, involves a system of linear equations. However, it comes with a few ambiguities that make a definitive solution challenging. Let x be the price of an adult meal, and y be the price of a kids meal. The given information translates to two equations:

2x 3y 47.50 2x 3y 72.50

The equations are identical, which means the problem can’t be solved with the information provided. This is because the total spent is the same on both days, but different numbers of adults and kids could eat, or different meals might have been ordered, which would change the total expenditure.

Exploring Other Possibilities

Let’s explore three other scenarios that could explain the discrepancies in the total cost spent:

No Change in the Number of Adults and Kids, but Different Meals

One possibility is that the family ordered different combinations of meals on the two days. For example:

Day 1: 1 adult meal and 3 kids meals, totaling $47.50

Day 2: 2 adult meals and 2 kids meals, totaling $72.50

Using the same equations, we can set up a new system of equations with different numbers of meals:

1x 3y 47.50 2x 2y 72.50

Solving this system of equations will give a different solution than the original one.

Change in the Number of Adults and Kids

Another possibility is that the number of adults and kids changed between the two days:

For example:

Day 1: 2 adults and 3 kids

Day 2: 3 adults and 3 kids

Using the original equations:

2x 3y 47.50 3x 3y 72.50

Solving this system will give different meal prices for adults and kids.

Solving any of these systems of equations can give a different result, but without more specific information, we cannot definitively determine the meal prices. Let’s explore one potential solution based on the original equations:

A Proposed Solution

Let’s assume the family ordered the same number of meals each day, but the pricing changed. One person proposed a solution where they got:

$11.75 for adults on the first day $7.92 for kids on the first day $18.13 for adults on the second day $12.08 for kids on the second day

Let’s verify if these prices work with the given totals:

Verification

For the first day:

2x 3y 47.50

Substituting the proposed prices:

2(11.75) 3(7.92) 23.50 23.76 47.26 (close, but not exact)

For the second day:

2x 3y 72.50

Substituting the proposed prices:

2(18.13) 3(12.08) 36.26 36.24 72.50 (exact)

The second day’s prices work, but the first day’s prices do not. This indicates that the price of an adult meal likely increased, and the price of a kids meal remained relatively the same, but not exactly as proposed.

Conclusion and Feedback

The proposed solution shows that meal prices are not constant and can change. Without more specific information, it is impossible to provide a definitive answer. The most likely scenario is that the prices adjusted between the two days or the number of meals ordered changed.

If you’re working on this for an Algebra 2 performance task, your solution could be partially correct or missing some details. Your professor might be looking for the exploration of different possibilities such as changes in meal pricing or changes in the number of adults and kids. Providing these possibilities could demonstrate a deeper understanding of the problem.

Feel free to share your solution with your teacher and discuss the different scenarios and how they might affect the meal prices. This approach will not only help you solve the problem but also showcase your analytical skills.