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Chapter 3: Problem 1

A party: You have \(\$ 36\) to spend on refreshments for a party. Large bags of chips cost \(\$ 2.00\) and sodas cost \(\$ 0.50\). You need to buy five times as many sodas as bags of chips. How many bags of chips and how many sodas can you buy?

Short Answer

Expert verified
You can buy 8 bags of chips and 40 sodas.

Step by step solution

01

Define Variables

Let \( x \) represent the number of bags of chips you buy. According to the problem, you need to buy five times as many sodas as bags of chips, so let \( y = 5x \) represent the number of sodas.
02

Write the Cost Equation

The total cost for the chips is \( 2x \) dollars since each bag costs \( \\(2.00 \). The cost for the sodas is \( 0.5y \) dollars since each soda costs \( \\)0.50 \). The total amount you can spend is \( \$36 \), so the equation is: \[ 2x + 0.5y = 36 \]
03

Substitute the Sodas Equation

From Step 1, we have \( y = 5x \). Substitute \( y \) in the cost equation:\[ 2x + 0.5(5x) = 36 \] simplifying gives:\[ 2x + 2.5x = 36 \]
04

Solve for Bags of Chips (x)

Combine like terms:\[ 4.5x = 36 \]Divide both sides by 4.5 to solve for \( x \):\[ x = \frac{36}{4.5} = 8 \]
05

Solve for Number of Sodas (y)

Since \( y = 5x \), substitute \( x = 8 \):\[ y = 5 \times 8 = 40 \]
06

Verify the Solution

The cost for 8 bags of chips is \( 8 \times 2 = \\(16 \), and the cost for 40 sodas is \( 40 \times 0.5 = \\)20 \). Total cost is \( 16 + 20 = \$36 \), which matches your budget.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Systems of Equations
When solving problems like the one mentioned, we deal with something called "systems of equations." This is a set of two or more equations that share two or more unknowns that we need to solve for. In our party planning example, the unknowns are the number of bags of chips and the number of sodas. By creating equations from the problem's conditions, we can find the values for these unknowns.

In this exercise:
  • The first equation revolves around the cost: the total amount of money spent on chips and sodas must equal $36.
  • The second equation comes from the constraint about the number of sodas being five times the number of bags of chips.
These equations work together as a system. Solving them gives us a complete picture of how the budget should be divided.
Substitution Method
One common way to solve systems of equations is the substitution method. This approach involves expressing one variable in terms of another from one of the equations and then replacing it in the other equation. This reduces the number of variables and makes solving the system easier.

In our example:
  • We determined that the number of sodas is five times the number of bags of chips, so we wrote this as \( y = 5x \).
  • We substituted this expression for \( y \) into the cost equation: \( 2x + 0.5y = 36 \) becomes \( 2x + 0.5(5x) = 36 \).
The substitution simplifies solving, as it reduces equations to one variable, making it straightforward to find the solution step by step.
Linear Equations
Linear equations are a key component of algebra that form the backbone of our problem. They are straightforward equations where each term is either a constant or the product of a constant and a single variable. In our example, we have linear relationships between money spent on chips and sodas.

Consider our cost equation:
  • It's linear because the relationships between total cost, the number of chips, and the number of sodas are all direct.
  • The variables just appear to the first power, meaning no squares, cubes, or other higher powers.
Learning to handle linear equations is crucial, especially in real-life situations like budgeting or resource allocation.
Budgeting and Constraints
Budgeting involves managing financial constraints and deciding how best to allocate resources within those constraints. Our original exercise is an excellent example of this process. We had a budget, in this case, $36, that must not be exceeded while fulfilling certain conditions.

Here is how the budgeting and constraints were applied:
  • The money must cover the cost of both chips and sodas.
  • We had the additional constraint that the number of sodas must be five times the number of chips.
By prioritizing and aligning spending with these constraints, we achieve the desired outcome 鈥� buying the maximum possible within the given limitations.

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Most popular questions from this chapter

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