Problem 188 - Elementary Algebra

Question

Jason went to the post office and bought both \(\$ 0.41\) stamps and \(\$ 0.26\) postcards and spent \(\$ 10.28\). The number of stamps was four more than twice the number of postcards. How many of each did he buy?

Step-by-Step Solution

Answer

Jason bought 20 stamps and 8 postcards.

Step 1: Define Variables

Let the number of stamps be denoted by \( s \) and the number of postcards be denoted by \( p \).

Step 2: Set Up Equations

We know two things:1. The total cost is \(10.28\).2. The number of stamps is four more than twice the number of postcards. From the cost, we get: \[ 0.41s + 0.26p = 10.28 \] From the number relation, we get: \[ s = 2p + 4 \]

Step 3: Substitute the Second Equation into the First

Substitute \( s = 2p + 4 \) into the first equation: \[ 0.41(2p + 4) + 0.26p = 10.28 \]

Step 4: Simplify and Solve for \( p \)

Expand and combine like terms: \[ 0.82p + 1.64 + 0.26p = 10.28 \] Combine like terms: \[ 1.08p + 1.64 = 10.28 \] Solve for \( p \): \[ 1.08p = 10.28 - 1.64 \] \[ 1.08p = 8.64 \] \[ p = 8 \]

Step 5: Find \( s \)

Substitute \( p = 8 \) back into the equation \( s = 2p + 4 \): \[ s = 2(8) + 4 \] \[ s = 16 + 4 \] \[ s = 20 \]

Step 6: Verify the Solution

Verify the solution by checking if the total cost matches: \[ 0.41s + 0.26p = 0.41(20) + 0.26(8) \] \[ 8.20 + 2.08 = 10.28 \] The solution is correct.

Key Concepts

Systems of Equations

A system of equations is a collection of two or more equations with the same set of variables. The goal is to find the values of these variables that satisfy all equations. When solving word problems, systems of equations can be very useful.
This exercise involves a real-world scenario where we are given two relationships about stamps and postcards.
First, their total cost and second, the relationship between the number of stamps and postcards.
Let's see how we can use this information to set up our system of equations:

• Define variables: Let the number of stamps be denoted by \( s \) and the number of postcards by \( p \).
• Set up equations: Based on the problem we get:
  1. Total cost: \[ 0.41s + 0.26p = 10.28 \]
  2. Number relation: \[ s = 2p + 4 \]

This forms our system of equations:
\[ \begin{cases} 0.41s + 0.26p = 10.28 \ s = 2p + 4 \end{cases} \]We will use this system to find the number of stamps and postcards.

Substitution Method

The substitution method is a common technique used to solve systems of equations. You solve one equation for one variable and then substitute this expression into the other equation.
Let's apply the substitution method to our current system:

• First, we solve the second equation for \( s \):
  \[ s = 2p + 4 \]
• Next, we substitute \( s = 2p + 4 \) into the first equation:
  \[ 0.41(2p + 4) + 0.26p = 10.28 \]This equation now only has one variable (\( p \)).

We then simplify and solve for \( p \):
\[ 0.82p + 1.64 + 0.26p = 10.28 \]

Combine like terms:
\[ 1.08p + 1.64 = 10.28 \]

Solve for \( p \):
\[ 1.08p = 10.28 - 1.64 \]
\[ 1.08p = 8.64 \]
\[ p = 8 \]

With \( p \) found, we substitute this value back into the second equation to find \( s \):
\[ s = 2(8) + 4 \]
\[ s = 20 \]

Problem Solving in Algebra

Problem solving in algebra involves translating real-world scenarios into mathematical equations.
The process usually involves understanding the problem, defining variables, setting up equations, solving the equations, and verifying the results.

• Understand the problem: Read the problem carefully and identify what you need to find and what information is given.
• Define variables: Assign letters to the quantities you need to find. For example, let \( s \) be the number of stamps and \( p \) be the number of postcards.
• Set up equations: Use the information given to write equations that relate the variables. In our case, the equations were:
  \[ 0.41s + 0.26p = 10.28 \]
  \[ s = 2p + 4 \]
• Solve the equations: Use appropriate methods such as substitution or elimination (we used substitution here) to find the values of the variables.
• Verify the results: Substitute the values found back into the original equations to ensure they satisfy all equations. This step confirms the solution is correct.

By following these steps, you can systematically tackle and solve algebraic word problems with confidence.