Problem 37 - Algebra A Combined Function

Question

The sum of three numbers is 40 . The first number is five more than the second number. It is also twice the third. Find the numbers.

Step-by-Step Solution

Answer

The numbers are 18, 13, and 9.

Step 1: Define Variables

Let's represent the numbers with variables. Let the second number be \(x\). Then the first number can be represented as \(x + 5\) since it is five more than the second number. The third number, being half of the first, becomes \(\frac{x+5}{2}\).

Step 2: Formulate the Equation

According to the problem, the sum of the three numbers equals 40. Therefore, we write an equation based on the expressions for the numbers: \((x + 5) + x + \frac{x+5}{2} = 40\).

Step 3: Simplify the Equation

First, we find a common denominator for the terms, which will be 2. The equation becomes: \[2(x + 5) + 2x + (x + 5) = 80\]

Step 4: Combine Like Terms

Distribute the terms and combine like terms: \[2x + 10 + 2x + x + 5 = 80\] which simplifies to \[5x + 15 = 80\].

Step 5: Solve for \(x\)

Subtract 15 from both sides to isolate the terms with \(x\): \[5x = 65\]. Divide both sides by 5 to solve for \(x\): \[x = 13\].

Step 6: Find the Other Numbers

Use the value of \(x\) to find the other numbers. The first number is \(x + 5 = 18\), and the third number is \(\frac{x+5}{2} = \frac{18}{2} = 9\).

Step 7: Verify the Solution

Check if the numbers sum to 40: \(18 + 13 + 9 = 40\). This confirms the solution is correct.

Key Concepts

Understanding Variables Definition

In algebra word problems, defining variables is an essential step. It provides a clear framework for translating a problem's words into mathematical expressions. In the given exercise, we're dealing with three distinct numbers. A smart way to simplify problems with multiple items is to define one item as a variable and express the others in terms of it.

Here, we selected the second number to be our variable, represented by \(x\). This selection is practical because it's directly involved in the dependency relationships with the first and third numbers.

• The first number is defined as \(x + 5\), since it is five more than the second number.
• The third number is defined as \(\frac{x+5}{2}\), which is half of the first number.

By defining these relationships, we create a system that can systematically solve for the unknowns.

Equation Formulation Explained

Formulating an equation is like piecing together a puzzle. In algebra, this means converting verbal descriptions into algebraic sentences. Once we've defined our variables, as done in the previous section, the next logical step is to capture their relationships as equations.

In this particular problem, we know that the sum of the three numbers equals 40. Therefore, we craft our equation as follows:

• The full expression is \((x + 5) + x + \frac{x+5}{2}\).
• The equation representing the sum is: \[(x + 5) + x + \frac{x+5}{2} = 40\]

This step is crucial as it lays the groundwork for subsequent solving. It encapsulates all the given information in a single mathematical expression that can be manipulated to find the needed values.

Ensuring Accuracy with Solution Verification

Arguably, one of the most critical steps in solving algebra word problems is verifying your solution. It's the part where you check whether your answers satisfy the original conditions of the problem.

In this problem, after computing our variables and finding the specific numbers, the final algebraic statements should hold true according to the problem description. After solving, we calculate:

• First number: \(x + 5 = 18\)
• Second number: \(x = 13\)
• Third number: \(\frac{x+5}{2} = 9\)

To verify, substitute these back:

• Check if \(18 + 13 + 9 = 40\)

Correctly obtaining a sum of 40 confirms our solution. This step ensures that our computations and logic align perfectly with the given problem constraints. Always verify to eliminate errors and confirm the reliability of your answers.