Solvenick And Isaac Are At A School Fair They Want To Collec

Solvenick And Isaac Are At A School Fairthey Want To Collect Point

Solvenick and Isaac are at a school fair. They want to collect points to exchange for prizes. The prizes and their point values are: Pencil (30 points), Notebook (50 points), Crayons (100 points), Pencil holder (200 points), Stationery set (500 points), Backpack (1000 points). At the fair, Nick has 215 points and Isaac has 78 points. They combine their points for a total of 293 points. The goal is to determine two sets of three prizes that add up to or below their combined points of 293.

Create all possible combinations of three prizes that they can exchange for, such that the total points of each combination do not exceed 293 points. Identify two distinct sets of three prizes that meet this criterion.

Paper For Above instruction

Solvenick and Isaac are attempting to maximize their prize exchange based on their combined points at a school fair. They possess a total of 293 points, obtained by summing Nick’s 215 points and Isaac’s 78 points. Their goal is to select two different sets of three prizes each, where the total points of each set do not surpass their combined points of 293. This problem involves analyzing the point values of available prizes and systematically finding combinations that fit within their total points, ensuring both sets are distinct and feasible.

Introduction

The process of prize selection at a school fair often involves combining points collected through various games or activities. In this scenario, students Nick and Isaac have accumulated points, which they plan to exchange for prizes. The challenge is to determine two different groups of three prizes each, for which the total point value does not exceed their combined points of 293. Solving this problem requires understanding the point values assigned to each prize and employing combinatorial methods to identify suitable prize sets.

Understanding the Prize Values and Constraints

The available prizes and their respective points are as follows: Pencil (30 points), Notebook (50 points), Crayons (100 points), Pencil holder (200 points), Stationery set (500 points), Backpack (1000 points). Given the combined total of 293 points, the feasible selections must consider the point constraints, ensuring that the sum of the three prizes in each set is less than or equal to 293 points. Since some prizes exceed this limit individually, only items priced at or below 200 points can be part of the combinations, as higher-priced items alone surpass the total points.

Methodology

To identify the two sets, the process involves enumerating all possible three-prize combinations where the total points are within the 293-point limit. Starting with the lower-value prizes, possible combinations should be systematically checked, ensuring no duplicate sets. Considerations include combining the lowest-value prizes and gradually including higher-value items, always verifying the total does not exceed the available points.

Possible Prize Combinations

Possible combinations under 293 points include:

• Pencil (30) + Notebook (50) + Crayons (100) = 180 points
• Pencil (30) + Notebook (50) + Pencil holder (200) = 280 points
• Pencil (30) + Crayons (100) + Pencil holder (200) = 330 points – exceeds limit, discard
• Notebook (50) + Crayons (100) + Pencil holder (200) = 350 points – exceeds limit, discard
• Pencil (30) + Stationery set (500) = exceeds 293, discard

Given the above, feasible combinations include:

1. Pencil (30) + Notebook (50) + Crayons (100) = 180 points

2. Pencil (30) + Notebook (50) + Pencil holder (200) = 280 points

3. Crayons (100) + Pencil holder (200) + Pencil (30) = 330 points – exceeds 293, discard

Now, find a second combination different from the first but also within the point limit:

- Pencil (30) + Crayons (100) + Pencil holder (200) = 330 – discard

- Notebook (50) + Crayons (100) + Pencil holder (200) = 350 – discard

- Pencil (30) + Notebook (50) + Stationery set (500) – discard

Alternative options focus on combining the lower-point items:

- Pencil (30) + Notebook (50) + Crayons (100) = 180 points

- Pencil (30) + Pencil holder (200) + Crayons (100) = 330 – discard

- Notebook (50) + Pencil holder (200) + Crayons (100) = 350 – discard

Thus, the plausible combinations are:

a. Pencil, Notebook, Crayons (180 points)

b. Pencil, Notebook, Pencil holder (280 points)

Both are within the total point limit, and offer distinct prize sets.

Conclusion

Based on the analysis, two feasible prize sets consist of:

1. Set a: Pencil, Notebook, Crayons
2. Set b: Pencil, Notebook, Pencil holder

These selections satisfy the point constraints and provide variety in their choices. The approach demonstrates effective combinatorial reasoning and problem-solving within set constraints, highlighting how students can optimize their prize exchanges based on their accumulated points at events such as school fairs.

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