Problem 33 - Algebra A Combined Function

Question

To make an international telephone call, you need the code for the country you are calling. The codes for Belgium, France, and Spain are three consecutive integers whose sum is \(99 .\) Find the code for each country. (Source: The World Almanac and Book of Facts)

Step-by-Step Solution

Answer

Belgium: 32, France: 33, Spain: 34.

Step 1: Define the Variables

Let's call the code for Belgium \( x \). Since the codes are consecutive integers, the code for France will be \( x + 1 \) and the code for Spain will be \( x + 2 \).

Step 2: Setup the Equation

We know that the sum of the codes for Belgium, France, and Spain is 99. Therefore, we set up the equation: \[ x + (x + 1) + (x + 2) = 99 \]

Step 3: Simplify the Equation

Combine like terms in the equation: \[ 3x + 3 = 99 \]

Step 4: Solve for x

First, subtract 3 from both sides of the equation to isolate the term with \( x \): \[ 3x = 96 \] Next, divide both sides by 3 to solve for \( x \): \[ x = 32 \]

Step 5: Find the Codes

Now that we have \( x = 32 \), the code for Belgium is 32. The code for France is \( 32 + 1 = 33 \). The code for Spain is \( 32 + 2 = 34 \).

Step 6: Verify the Solution

Add the three codes to confirm their sum is 99: \[ 32 + 33 + 34 = 99 \] The calculation confirms our codes are correct.

Key Concepts

Consecutive Integers

When dealing with problems like this, it's important to understand the concept of consecutive integers. These are numbers that follow each other in sequence, one after the other. For instance, 4, 5, and 6 are consecutive integers. In our problem, Belgium, France, and Spain have telephone codes that are consecutive. This means that if Belgium's code is represented by \( x \), then France's has to be \( x + 1 \), and Spain's \( x + 2 \). Recognizing patterns such as these consecutive numbers is crucial in simplifying and solving algebra word problems efficiently. When you spot consecutive numbers, always remember to define them using variables and expressions that highlight their position in sequence.

Equation Setup

Setting up the equation is a vital next step after identifying your variables. In algebra word problems, equations are often constructed based on the relationships between the quantities involved.

For this exercise, once we've identified that the telephone codes for Belgium, France, and Spain are consecutive integers represented by \( x, x + 1, \) and \( x + 2 \), respectively, we move onto setting up an equation. The problem states that the sum of these codes is 99. Therefore, the equation becomes: - \[ x + (x + 1) + (x + 2) = 99 \] This equation captures the cumulative total of the three consecutive integers. Setting up this equation correctly allows us to solve for one variable and ultimately find the required consecutive numbers. Remember that forming the right equation involves translating the given word problem clues into mathematical expressions.

Solving Equations

Once the equation is set up, solving it involves a series of logical steps aimed at isolating the unknown variable, which in this case is \( x \).

Let's look at the equation from our exercise: - \[ x + (x + 1) + (x + 2) = 99 \] Combine like terms to simplify: - \[ 3x + 3 = 99 \] Now, to solve for \( x \), you can follow these steps:

• Subtract 3 from both sides to help isolate the term with \( x \): \( 3x = 96 \)
• Divide each side by 3 to solve for \( x \): \( x = 32 \)

Once we find that \( x = 32 \), we can easily calculate that the codes for Belgium, France, and Spain are 32, 33, and 34, respectively. Verifying your solution by adding the numbers back together ensures that the problem's conditions are met. The sum should equal 99, confirming the correctness of the solution.