Problem 8 - Introductory Algebra for College Students

Question

It's probably not a good idea if you want to look like Mr. Universe or Kate Winslet. The graph shows the four candy bars with the highest fat content, representing grams of fat and calories in each bar. (GRAPH CAN NOT COPY) One Snickers bar and two Reese's Peanut Butter Cups contain 737 calories. Two Snickers bars and one Reese's Peanut Butter Cup contain 778 calories. Find the caloric content of each candy bar.

Step-by-Step Solution

Answer

A Snickers bar contains 273 calories and a Reese's Peanut Butter Cup contains 232 calories.

Step 1: Identify and define the unknowns

Let \(S\) represent the calories in a Snickers bar and \(R\) represent the calories in a Reese's Peanut Butter Cup.

Step 2: Setting up the system of equations

The problem gives us two equations involving Snickers and Reese's calories. They are: \(1*S + 2*R = 737\) (one Snickers bar and two Reese's Peanut Butter Cups contain 737 calories), and \(2*S + 1*R = 778\) (two Snickers bars and one Reese's Peanut Butter Cup contain 778 calories).

Step 3: Solve the system of equations

Start by multiplying the first equation by 2 and the second by 1, giving: \(2*S + 4*R = 1474\) and \(2*S + 1*R = 778\). Next, subtract the second equation from the first to get \(3*R = 696\). Solving for \(R\), we find that \(R = 232\). Substituting \(R = 232\) into the second equation gives \(2*S + 232 = 778\), which simplifies to \(2*S = 546\), so \(S = 273\).

Step 4: Interpret the result

The result means that a Snickers bar contains 273 calories and a Reese's Peanut Butter Cup contains 232 calories.

Key Concepts

Understanding System of Equations

A system of equations is a set of two or more equations that have shared variables. In this case, the variables are the calorie content of Snickers and Reese's Peanut Butter Cups. To solve such systems, we need to find values for these variables that satisfy all equations simultaneously. An important tool for solving systems of equations is elimination or substitution, which allows us to isolate one variable and solve for it. For this exercise:

• The system is given as \(1S + 2R = 737\) and \(2S + 1R = 778\).
• These equations represent the combined calorie counts of different combinations of the candy bars.

We can solve by either substitution or elimination (as shown in the example), helping us find the exact calorie content of each candy bar.

Calorie Calculation Using Equations

Calorie calculation in this context involves using equations to determine the calorie content of each candy bar based on the total calories provided by their combinations. Here's how it can be broken down:

• Each candy bar combination has a known calorie total. For example, one Snickers and two Reese's cups have 737 calories.
• By understanding this, we can set up equations where unknowns \(S\) for Snickers and \(R\) for Reese's represent their respective calories.
• The task is then to solve these equations to uncover the calorie content per candy bar, helping us understand more about nutrition based on these mathematical principles.

Problem-Solving Steps Explained

In tackling a problem like this, following a clear set of steps ensures we can reliably find a solution. First, we identify and define unknowns. Here, defining \(S\) and \(R\) as the calorie content for different candy bars was crucial. Next, setting up the system involves translating the given scenario into mathematical equations. This is directly linked to the information given, so it's important to read carefully and set the equations correctly.Afterward, we solve the system. This can be through elimination, as performed here: transforming one equation to allow subtraction to isolate a variable. This step is crucial for finding values for \(S\) and \(R\).Lastly, interpreting the results allows us to understand the meaning of our solution in the context of the problem. This reinforces the importance of mathematical steps, providing clarity on calorie counts in a real-world scenario.