Problem 58 - Algebra A Combined Function

Question

To make an international telephone call, you need the code for the country you are calling. The codes for Mali Republic, Côte d'Ivoire, and Niger are three consecutive odd integers whose sum is \(675 .\) Find the code for each country.

Step-by-Step Solution

Answer

The country codes are 223, 225, and 227.

Step 1: Define the Variables

Let's denote the three consecutive odd integers as \( x, x+2, \text{ and } x+4 \). These three represent the country codes for Mali Republic, Côte d'Ivoire, and Niger.

Step 2: Set Up the Equation

According to the problem, the sum of these three odd integers is 675. Therefore, we can write the equation:\[x + (x + 2) + (x + 4) = 675\]

Step 3: Simplify the Equation

Combine like terms in the equation to simplify it:\[3x + 6 = 675\]

Step 4: Solve for x

Subtract 6 from both sides of the equation to isolate terms involving \( x \):\[3x = 669\]Now, divide both sides by 3 to find \( x \):\[x = 223\]

Step 5: Find the Three Codes

Now that we have \( x = 223 \), calculate the consecutive odd integers:- First code: \( x = 223 \)- Second code: \( x + 2 = 225 \)- Third code: \( x + 4 = 227 \)

Step 6: Assign Codes to Countries

Given no specific association in the problem text, let's assign the codes arbitrarily: - Mali Republic: 223 - Côte d'Ivoire: 225 - Niger: 227

Key Concepts

Consecutive odd integers

Consecutive odd integers are numbers that follow each other in order with a difference of 2 between each pair of numbers. They maintain their 'odd' nature, meaning they cannot be divided evenly by 2. For example, 3, 5, and 7 are consecutive odd integers because there is a consistent pattern of adding 2 to the previous integer to get the next. In problems like the one we are addressing, recognizing that the sequence follows a consistent pattern helps to simplify the problem-solving process. In our example, the consecutive odd integers represent country codes, so we begin by setting the first integer as \(x\), the second as \(x+2\), and the third as \(x+4\). Understanding this pattern allows you to effectively manage and solve problems involving sequences or lists of numbers.

Equation setup

Equation setup is a crucial step in solving algebra problems as it translates a word problem into a mathematical form that can be solved. Here, our goal is to find the sum of the given consecutive odd integers. Given the problem statement, we know that:

• The integers sum to 675
• The sequence of integers is \(x, x+2, x+4\)

The setup process involves creating an equation from these integers and their total sum. By representing them algebraically, we get: \[ x + (x + 2) + (x + 4) = 675 \] This equation serves as the starting point for finding the solution.

Integer sum

After setting up the equation, we need to address the integer sum. This crucial step involves organizing and manipulating the numbers in the equation to simplify it. Here, we focus on combining like terms. Like terms are algebraic expressions that have identical variable parts. Thus, in our example, we combine:

• \(x\)
• \(x+2\)
• \(x+4\)

All together, this equates to \(3x + 6\). This simplification helps pave the way for solving the variable \(x\), ultimately allowing us to establish the real-world numbers, or codes, in this scenario.

Solve for x

Once the equation has been simplified to \(3x + 6 = 675\), the next task is solving for \(x\). Solving for \(x\) involves isolating the variable on one side of the equation to find its value. This is done by two main steps:

• First, subtract 6 from both sides of the equation, resulting in \(3x = 669\)
• Then, divide both sides by 3, yielding \(x = 223\)

This isolation of \(x\) reveals it as the first of our odd integers. With this value, you can determine the subsequent consecutive odd integers by plugging back \(x\) into each respective expression \(x + 2\) and \(x + 4\), giving you 225 and 227. Thus, the solution to the problem is complete, and we find the country codes to be 223, 225, and 227.