Problem 190 - Elementary Algebra

Question

Hector spent \(\$ 33.20\) at the post office. He bought four times as many \(\$ 0.41\) stamps as \(\$ 0.02\) stamps. How many of each did he buy?

Step-by-Step Solution

Answer

Hector bought 20 \(0.02\) stamps and 80 \(0.41\) stamps.

Step 1: Define variables

Let the number of \(\$0.02\) stamps be \(x\). Since Hector bought four times as many \(\$0.41\) stamps, let the number of \(\$0.41\) stamps be \(4x\).

Step 2: Set up the total cost equation

The total cost can be represented by the equation \[0.02x + 0.41(4x) = 33.20\].

Step 3: Simplify the equation

Simplify the equation to combine like terms: \[0.02x + 1.64x = 33.20\].

Step 4: Combine like terms

Add the terms with \(x\): \[1.66x = 33.20\].

Step 5: Solve for x

Divide both sides of the equation by \(1.66\) to find \(x\): \[x = \frac{33.20}{1.66} ≈ 20\].

Step 6: Find the number of \( \$0.41\) stamps

Since the number of \(0.41\) stamps is \(4x\), calculate \(4x = 4 \times 20 = 80\). Hector bought \(80\) \(0.41\) stamps.

Key Concepts

Simple Equations

In algebra, simple equations are the building blocks for solving problems. A simple equation is a mathematical statement that shows the equality of two expressions. For example, in the problem where Hector spends money on stamps, we use a simple equation to find out how many stamps he bought.
The general form of these equations looks like this: \[a + b = c\].
By defining the variables and translating the word problem into this form, we can easily find a solution. Simple equations are crucial because understanding them is the first step to tackling more complex algebraic problems.

Variable Definition

Defining variables is an essential step in solving algebra word problems. A variable is a symbol, usually a letter like \( x \) or \( y \), that represents an unknown value in an equation.
In our example about Hector and his stamps, we let \( x \) represent the number of \( \$0.02 \) stamps. By clearly defining \( x \), we can then use it to express other quantities in the problem.
Variables help us turn word problems into mathematical equations, making it easier to solve for unknown values. Properly defining variables ensures that our work is organized and understandable.

Combining Like Terms

Combining like terms is a method used to simplify equations. Like terms are terms that have the same variable raised to the same power. By combining them, we can make our equations simpler and easier to solve.
In the given problem, after setting up the equation\(0.02x + 0.41(4x) = 33.20\), we simplify it by combining like terms: \(0.02x + 1.64x = 33.20\).
This helps us reduce the left side of the equation to \(1.66x\). Combining like terms minimizes the number of terms in an equation, making it more straightforward to find solutions.

Solving Linear Equations

After simplifying the equation, the next step is to solve the linear equation. A linear equation is an equation in which the highest power of the variable is one. In our example, the simplified equation was\(1.66x = 33.20\).
To solve for \( x \), we divide both sides by \( 1.66 \):\(x = \frac{33.20}{1.66} \).
Solving linear equations often involves isolating the variable on one side of the equation. This process may include dividing, multiplying, adding, or subtracting. Understanding how to solve linear equations is fundamental in algebra, as it allows us to find the value of unknown variables efficiently.