Problem 50 - Algebra for College Students

Question

Solve each problem involving consecutive integers. Find three consecutive odd integers such that the sum of the least integer and the greatest integer is 13 more than the middle integer.

Step-by-Step Solution

Answer

The three consecutive odd integers are 11, 13, and 15.

Step 1 - Define Variables

Let the consecutive odd integers be represented as: First integer: \( x \) Second integer: \( x + 2 \) Third integer: \( x + 4 \)

Step 2 - Set Up the Equation

According to the problem, the sum of the least integer and the greatest integer is 13 more than the middle integer. This gives us the equation: \( x + (x + 4) = (x + 2) + 13 \)

Step 3 - Simplify the Equation

Combine like terms and simplify the equation: \( x + x + 4 = x + 2 + 13 \) \( 2x + 4 = x + 15 \)

Step 4 - Solve for x

Isolate \( x \) by subtracting \( x \) from both sides: \( 2x + 4 - x = 15 \) \( x + 4 = 15 \) Subtract 4 from both sides: \( x = 11 \)

Step 5 - Find the Consecutive Integers

Now that we have \( x \), calculate the three consecutive odd integers: First integer: \( 11 \) Second integer: \( 11 + 2 = 13 \) Third integer: \( 11 + 4 = 15 \)

Step 6 - Verification

Verify that the solution fits the condition stated in the problem: Sum of the least and greatest integers: \( 11 + 15 = 26 \) The middle integer plus 13: \( 13 + 13 = 26 \) Since both are equal, the solution is correct.

Key Concepts

Solving Equations

Solving equations is a fundamental part of algebra. This problem uses algebraic equations to find unknown values. Here's a step-by-step approach:
First, we define variables to represent our unknowns. In this case, the consecutive odd integers are represented by: \ First integer: \( x \)
Second integer: \( x + 2 \)
Third integer: \( x + 4 \)
Next, we set up an equation based on the problem's description. We are told that the sum of the least and greatest integers is 13 more than the middle integer. This translates to the equation: \ \( x + (x + 4) = (x + 2) + 13 \) Combine like terms to simplify:
\( 2x + 4 = x + 15 \)
Solve for \( x \) by isolating it on one side of the equation: \ \( x + 4 = 15 \)
Subtract 4 from both sides:
\( x = 11 \)
This step-by-step method makes it easy to solve similar problems.

Consecutive Integers

Consecutive integers follow a sequence without any gaps. For consecutive odd integers, the difference between each number is 2. In the original problem: \ Let the first integer be \( x \).
Then the next consecutive odd integer is \( x + 2 \) and the one after that is \( x + 4 \).
It's essential to set up your variables correctly to solve the problem accurately.
Remember:

• The first integer is \( x \).
• The second integer is \( x + 2 \).
• The third integer is \( x + 4 \).

This sequence pattern will help you when solving for the variables.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators. In our problem, expressions represent the consecutive odd integers: \ First integer: \( x \).
Second integer: \( x + 2 \).
Third integer: \( x + 4 \).
The problem's condition gives us a relational expression:
\( x + (x + 4) = (x + 2) + 13 \). This translates the problem statement into an equation that we can manipulate mathematically. It's crucial to properly set up these expressions as they form the foundation of your equations.

Problem-Solving Steps

Effective problem-solving in algebra involves several clear steps:

• Define the variables.
• Set up an equation based on the given conditions.
• Simplify the equation by combining like terms.
• Solve the simplified equation for the unknown variable.
• Use the value of the variable to find the other quantities.
• Verify your solution to ensure it meets the problem's conditions.

By following these steps systematically, you can solve even complex algebraic problems. Always verify your solution as the final step to confirm your answer is correct.