Problem 189 - Elementary Algebra

Question

Maria spent \(\$ 12.50\) at the post office. She bought three times as many \(\$ 0.41\) stamps as \(\$ 0.02\) stamps. How many of each did she buy?

Step-by-Step Solution

Answer

Maria bought 10 \(0.02 stamps and 30 \)0.41 stamps.

Step 1: Identify Variables

Let the number of \(0.02 stamps be denoted as x. Then the number of \)0.41 stamps will be 3x.

Step 2: Set Up the Equation

Since Maria spent \(\textbackslash\) 12.50 in total, create the equation based on the total cost:\[3x \times 0.41 + x \times 0.02 = 12.50\]

Step 3: Simplify the Equation

Combine like terms: \[1.23x + 0.02x = 12.50\] which simplifies to \[1.25x = 12.50\]

Step 4: Solve for x

Divide both sides by 1.25 to find x: \[x = \frac{12.50}{1.25} = 10\]

Step 5: Find the Number of Each Stamp

Since x = 10, the number of \(0.02 stamps is 10, and the number of \)0.41 stamps is 3 times 10, which is 30.

Key Concepts

variable identification

In any algebra word problem, the first step is to clearly identify the variables. This means finding the unknowns we need to solve for. In Maria's problem:

• We'll call the number of \(0.02 stamps she bought **x**.
• Since she bought three times as many \)0.41 stamps, we denote this quantity as **3x**.

Identifying variables is crucial for creating equations that reflect the problem scenario accurately and are easier to solve.

equation setup

The next step is to set up an equation based on the given information. This converts the word problem into a math problem:

Maria spent a total of \(12.50 on stamps. We need to represent this spending mathematically:
• The cost of \)0.02 stamps = **0.02 \times x**.
• The cost of \(0.41 stamps = **0.41 \times 3x**.
Since she bought both types, we combine these costs to match the total amount spent:
\[3x \times 0.41 + x \times 0.02 = 12.50\]
This equation is set up to find the total cost of both types of stamps and equate it to \)12.50.

solving linear equations

Solving the equation involves simplifying and then finding the value of the variable. Starting from the setup equation:
\[3x \times 0.41 + x \times 0.02 = 12.50\]

• First, multiply and combine like terms:
\[1.23x + 0.02x = 12.50\]
which simplifies to\[1.25x = 12.50\]

• Next, solve for **x** by dividing both sides by 1.25:
\[x = \frac{12.50}{1.25} = 10\]

This tells us that **x** equals 10. Solving linear equations often involves simple arithmetic and basic algebraic manipulation.

arithmetic operations

In steps to solve this problem, several arithmetic operations are used:

• **Multiplication**: Used to calculate the total cost of each type of stamp.
• **Addition**: Combining the multiple costs to form one total cost.
• **Division**: Used to isolate the variable and find its value.

Here’s a quick breakdown:
1. Multiplying costs: \[3x \times 0.41 + x \times 0.02\] = \[1.23x + 0.02x\].
2. Adding costs: \[1.23x + 0.02x = 1.25x\].
3. Dividing to isolate the variable: \[x = \frac{12.50}{1.25} = 10\].
Mastering these arithmetic operations will make solving similar algebra problems simpler.

unit price calculations

Finally, understanding unit price calculations is essential. In this problem:

• Each \(0.02 stamp costs **0.02**.
• Each \)0.41 stamp costs **0.41**.

When solving the problem:
• Total cost for \(0.02 stamps: **0.02 \times x**.
• Total cost for \)0.41 stamps: **0.41 \times 3x**.

These concepts help to correctly set up the equation and understand the total cost calculation:\[0.02 \times 10 + 0.41 \times 30 = 12.50\].
By using unit prices directly in calculations, you can easily form and solve equations related to real-world problems.