Problem 14 - Algebra and Trigonometry

Question

Number Problem Find four consecutive odd integers whose sum is 416 .

Step-by-Step Solution

Answer

The integers are 101, 103, 105, and 107.

Step 1: Define Variables for Odd Integers

Let's represent the first odd integer as \( x \). Since we are dealing with consecutive odd integers, the next three integers would be \( x+2 \), \( x+4 \), and \( x+6 \). Odd numbers are spaced by 2 units.

Step 2: Set up the Equation

According to the problem, the sum of these four consecutive odd integers is 416. We can write this as an equation:\[ x + (x + 2) + (x + 4) + (x + 6) = 416 \]

Step 3: Simplify the Equation

Combine all the \( x \) terms and the constant terms:\[ 4x + 12 = 416 \]Now, we need to solve for \( x \).

Step 4: Solve for x

Subtract 12 from both sides of the equation to isolate the term with \( x \):\[ 4x = 416 - 12 \]\[ 4x = 404 \]Next, divide both sides by 4 to solve for \( x \):\[ x = \frac{404}{4} = 101 \]

Step 5: Find All Four Integers

Now that we have the first integer as \( x = 101 \), the four consecutive odd integers are:1. \( x = 101 \)2. \( x + 2 = 103 \)3. \( x + 4 = 105 \)4. \( x + 6 = 107 \)

Key Concepts

Summing Integers

When we talk about summing integers, particularly consecutive odd integers, it's important to visualize how these numbers are structured. Consecutive odd integers are numbers in sequence that follow immediately after each other, like 1, 3, 5, and 7. The key feature is that they all have a difference of 2 between each number. This creates a clear pattern that helps in adding them together.

To find the sum of consecutive odd integers, identify the pattern and add them. For example: To find four consecutive odd integers whose sum is 416, think about what each number looks like. If the first is \( x \), the next ones would be \( x+2 \), \( x+4 \), and \( x+6 \).

In this specific problem, the sum is expressed as:

• \( x + (x+2) + (x+4) + (x+6) = 416 \)

By adding these, you get a straightforward way of handling and manipulating such sums.

Algebraic Equations

Algebraic equations are a fundamental tool in solving problems involving unknown numbers. They allow us to set relationships between known and unknown values, creating a pathway to find the solution. When dealing with a problem that involves finding integers, an equation is used to connect the sum of these numbers to a given value.

Consider the equation for the problem of four consecutive odd integers:

• \( x + (x+2) + (x+4) + (x+6) = 416 \)
• This equation uses simple addition to represent the integers.

To solve it, you simplify. Begin by combining like terms, which in this instance means adding all the \( x \) terms together and then all the constant numbers together. This moves us from:

• \( x + (x+2) + (x+4) + (x+6) = 416 \)
• To \( 4x + 12 = 416 \)

Algebraic manipulations such as these enable us to restate complex relationships in a solvable form.

Variable Representation

Representing a number with a variable is one of the key aspects of solving algebra problems. A variable is a symbol, often \( x \), used to represent unknowns within an equation. By setting the first integer as \( x \), we can express others relative to it. For consecutive odd integers, imagine \( x \) is the smallest integer.

Then you define the others based on \( x \):

• The second integer becomes \( x+2 \)
• The third is \( x+4 \)
• The fourth follows as \( x+6 \)

This uniformity allows for the equation to neatly summarize relationships and give a straightforward way to explore all numbers. Such representation converts a theoretical problem into practical calculation, like solving for \( x \) to discover all those integers.