Solving Word Problems Penelope's Candy Bar Fundraiser

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Introduction

In this article, we'll delve into a word problem involving Penelope's fundraising efforts to support her church mission trip. Penelope sold three types of candy bars: Choconut Bars, Almond-Bliss Bars, and Fudge-Brownie Bars. Each type of candy bar had a different price, and she sold three times as many Almond-Bliss Bars as Choconut Bars. We need to determine how many of each type of candy bar she sold. This problem requires careful reading, setting up equations, and solving them systematically. By breaking down the problem into smaller parts, we can effectively find the solution. Understanding the relationship between the quantities and prices is crucial in solving this real-world mathematical problem.

Problem Statement

Penelope sold three different types of candy bars for a fundraiser to go on a church mission trip. Choconut Bars cost $2, Almond-Bliss Bars cost $1.50, and Fudge-Brownie Bars cost $1. She ended up selling three times as many Almond-Bliss Bars as Choconut Bars.

To solve this, we need additional information. The problem statement appears to be incomplete. We need more information, such as:

  • The total number of candy bars sold.
  • The total amount of money she raised.
  • A relationship between the number of Fudge-Brownie Bars and the other types.

Without additional information, we cannot determine a unique solution for the number of each type of candy bar sold. Let’s assume we have a total revenue amount to work with. For example, let's assume Penelope raised $200 in total. With this additional piece of information, we can set up equations and solve for the number of each type of candy bar sold. This will give us a complete and solvable problem that we can use to demonstrate the problem-solving process.

Setting Up the Equations

To begin solving this word problem, we must first define our variables. Let's denote the number of Choconut Bars sold as x. Since Penelope sold three times as many Almond-Bliss Bars as Choconut Bars, the number of Almond-Bliss Bars sold can be represented as 3x. We'll denote the number of Fudge-Brownie Bars sold as y. Understanding these variables is the first step in translating the word problem into mathematical equations. We are now ready to translate the given information into algebraic expressions that will help us find the values of x and y. The careful assignment of variables and the correct interpretation of the relationships between them is essential for accurately representing the problem. This foundational step allows us to proceed with confidence in setting up and solving the equations.

The prices of the candy bars are as follows:

  • Choconut Bars: $2 each
  • Almond-Bliss Bars: $1.50 each
  • Fudge-Brownie Bars: $1 each

If we assume that Penelope raised a total of $200, we can set up an equation based on the total revenue:

2x + 1.50(3x) + 1y = 200

This equation represents the total revenue from the sales of each type of candy bar. The term 2x represents the revenue from Choconut Bars, 1.50(3x) represents the revenue from Almond-Bliss Bars, and 1y represents the revenue from Fudge-Brownie Bars. The sum of these revenues equals the total amount raised, which we've assumed to be $200. Now, we can simplify the equation:

2x + 4.5x + y = 200

Combining the terms with x, we get:

6.5x + y = 200

This equation is a crucial step in solving the problem. It relates the number of each type of candy bar sold to the total revenue. However, we still have two variables (x and y) and only one equation. To solve for x and y, we need another independent equation. Let’s assume we know the total number of candy bars sold. For instance, if Penelope sold a total of 100 candy bars, we can create a second equation:

x + 3x + y = 100

This equation represents the total number of candy bars sold. It is the sum of the number of Choconut Bars (x), Almond-Bliss Bars (3x), and Fudge-Brownie Bars (y). Simplifying this equation gives us:

4x + y = 100

Now we have a system of two equations with two variables:

    1. 5x + y = 200
  2. 4x + y = 100

With this system of equations, we can proceed to solve for x and y. This setup allows us to use algebraic methods such as substitution or elimination to find the values of x and y, which will give us the number of each type of candy bar sold.

Solving the System of Equations

To solve the system of equations, we can use the method of elimination. We have two equations:

    1. 5x + y = 200
  2. 4x + y = 100

To eliminate y, we can subtract the second equation from the first equation:

(6.5x + y) - (4x + y) = 200 - 100

This simplifies to:

  1. 5x = 100

Now, we can solve for x by dividing both sides by 2.5:

x = 100 / 2.5 x = 40

So, Penelope sold 40 Choconut Bars. Next, we can substitute the value of x into one of the equations to solve for y. Let's use the second equation:

4x + y = 100

Substitute x = 40:

4(40) + y = 100

160 + y = 100

Subtract 160 from both sides:

y = 100 - 160 y = -60

However, we have a problem. The number of Fudge-Brownie Bars cannot be negative. This indicates that our assumption of the total candy bars sold (100) or the total revenue ($200) might be incorrect, or there may be an error in the problem statement or our calculations. This highlights the importance of checking the validity of the solutions in the context of the problem. A negative number of candy bars sold doesn't make sense in this scenario. It is crucial to review the given information and the equations to identify any possible mistakes.

Let's re-evaluate the problem and consider a different approach or a different set of assumptions. Suppose, instead of assuming a total number of candy bars, we had another piece of information, such as the number of Fudge-Brownie Bars being a certain fraction of the Almond-Bliss Bars. Without additional and consistent information, the problem remains unsolvable. The accuracy of the input and the completeness of the problem statement are vital for arriving at a meaningful solution.

Revisiting the Problem with a Modified Scenario

Since the initial problem setup led to an impossible solution, let’s modify the scenario slightly to make it solvable. We'll keep the original information: Choconut Bars cost $2, Almond-Bliss Bars cost $1.50, and Fudge-Brownie Bars cost $1. Penelope sold three times as many Almond-Bliss Bars as Choconut Bars. Let's now assume that Penelope sold a total of 80 candy bars and raised $150. This modification gives us a different set of constraints that we can use to find a valid solution.

As before, let x represent the number of Choconut Bars sold, so the number of Almond-Bliss Bars sold is 3x. Let y represent the number of Fudge-Brownie Bars sold. We now have two equations:

  1. Revenue Equation: 2x + 1.50(3x) + y = 150
  2. Total Candy Bars Equation: x + 3x + y = 80

First, let’s simplify the equations:

Revenue Equation:

2x + 4.5x + y = 150

  1. 5x + y = 150

Total Candy Bars Equation:

4x + y = 80

Now we have a system of two equations:

    1. 5x + y = 150
  2. 4x + y = 80

We can use the elimination method again. Subtract the second equation from the first:

(6.5x + y) - (4x + y) = 150 - 80

This simplifies to:

  1. 5x = 70

Now, solve for x:

x = 70 / 2.5 x = 28

So, Penelope sold 28 Choconut Bars. Now, find the number of Almond-Bliss Bars:

  1. x = 3 * 28 = 84

Next, substitute x into the second equation to find y:

4x + y = 80

4(28) + y = 80

112 + y = 80

y = 80 - 112 y = -32

Again, we encounter a negative value for y, which means our modified scenario still leads to an impossible situation. This highlights the critical nature of the data provided in word problems. If the information is inconsistent, no valid solution exists. It’s essential to ensure that the given conditions align and that the problem is well-posed. The inconsistency suggests that either the total amount raised or the total number of candy bars sold (or both) are not correctly aligned with the other conditions of the problem.

Conclusion: The Importance of Complete Information

Through our attempts to solve this candy bar fundraising problem, we've encountered the crucial importance of having complete and consistent information. In both the initial problem statement and the modified scenario, we ran into situations where the given data led to impossible solutions (a negative number of candy bars). This underscores the fundamental principle that a well-defined problem must provide sufficient and coherent information to arrive at a meaningful answer. Without adequate data, mathematical models, and equations cannot produce realistic results.

In real-world applications, this translates to the need for thorough data collection and validation before attempting to solve problems using mathematical techniques. Whether it’s in business, science, or everyday life, having accurate and complete information is essential for effective problem-solving and decision-making. Our journey through this problem serves as a reminder that the power of mathematics is maximized when applied to well-defined and data-supported scenarios. Word problems, like the one we've explored, are valuable tools for honing our problem-solving skills and emphasizing the importance of critical thinking and data analysis.