Problem 38 - Algebra Form and Function

Question

Apples are 99 cents a pound, and pears are \(\$ 1.25\) a pound. If I spend \(\$ 4\) and the weight of the apples I buy is twice the weight of the pears, how many pounds of pears do I buy?

Step-by-Step Solution

Answer

Answer: Approximately 1.24 pounds of pears were bought.

Step 1: Define Variables and Equations

Let "A" be the number of pounds of apples and "P" be the number of pounds of pears. We know that the weight of apples is twice the weight of pears, and the total cost is $4. So, we can set up the following equations: 1. A = 2 * P (Weight relationship) 2. 0.99A + 1.25P = 4 (Total cost)

Step 2: Solve for the first variable

We can substitute the first equation into the second equation to eliminate one variable. In this case, we will substitute "A = 2P" into the total cost equation: 0.99(2 * P) + 1.25P = 4

Step 3: Simplify the Equation

Now, let's simplify the equation: 1.98P + 1.25P = 4

Step 4: Find the Value of "P"

Combine like terms and solve for "P": 3.23P = 4 Divide both sides by 3.23: P = 4/3.23 P ≈ 1.24

Step 5: Interpret the Result

The number of pounds of pears bought is approximately 1.24 pounds.

Key Concepts

Linear Equations

Linear equations are a fundamental concept in algebra that involve expressions set in a form where the highest exponent of the variable is one. They are essential because they help us model real-world scenarios, like the cost and weight of items in this exercise.
In our problem, we defined two linear equations based on the given conditions:

• Equation 1: This shows the relationship between the weights of apples and pears as \(A = 2P\), where \(A\) is the pounds of apples, and \(P\) is the pounds of pears. This represents a dependency between the two variables.
• Equation 2: It represents the total cost of the fruits as \(0.99A + 1.25P = 4\). This equation incorporates both the weight and price of apples and pears.

Linear equations like these help us simplify and solve problems by stacking different conditions mathematically and finding the intersection of these constraints.

Substitution Method

The substitution method is a technique used to solve systems of equations, whereby you solve one of the equations for one variable and substitute this value into the other equation. This effectively reduces the system to a single equation that is easier to solve.
In our specific problem, we use the substitution method to replace \(A\) in the cost equation with \(2P\) (from \(A = 2P\)), giving us a formula only in terms of one variable, \(P\):

• Substitute \(A = 2P\) into \(0.99A + 1.25P = 4\), resulting in \(0.99(2P) + 1.25P = 4\).

This substitution turns the original two-variable equation into a single-variable equation, making it much easier to solve for \(P\). By handling one variable at a time, this strategy provides a clear path to finding a solution.

Cost Analysis

Cost analysis is a crucial step in determining the feasibility and affordability of purchase scenarios like the one given. It involves breaking down the total cost into contributing factors and understanding how changes affect the outcome.
For our particular word problem, we consider two factors:

• The cost per pound of apples, 99 cents (or \(0.99\) dollars).
• The cost per pound of pears, \(1.25\) dollars.

The total expenditure must equal \(4\) dollars, as derived from the total cost equation: \(0.99A + 1.25P = 4\). Because of this constraint, we needed to ensure that our combinations of apples and pears maintained this total.
By solving for \(P\) using a cost constraint and substitution, we ultimately determine that the maximum amount of pears, given the cost limits and the weight relationship with apples, is approximately 1.24 pounds. This solution reflects a nuanced understanding of pricing and quantity within prescribed budgetary conditions.