Problem 51 - College Algebra

Question

Three even numbers sum up to 108 . The smaller is half the larger and the middle number is \(\frac{3}{4}\) the larger. What are the three numbers?

Step-by-Step Solution

Answer

The three numbers are 24, 36, and 48.

Step 1: Define the Variables

To solve the problem, we start by defining variables for the three numbers. Let the largest even number be \( x \). According to the problem, the smallest number is half the largest, so it is \( \frac{x}{2} \), and the middle number is \( \frac{3}{4}x \).

Step 2: Write the Equation

Now, we write an equation based on the problem. The sum of these three numbers should be 108. This gives us: \[ \frac{x}{2} + \frac{3}{4}x + x = 108 \].

Step 3: Solve the Equation

First, simplify and combine the terms in the equation. Find a common denominator to combine the fractions: \[ \frac{x}{2} = \frac{2x}{4}, \frac{3}{4}x = \frac{3x}{4}, \text{then}\ \frac{2x}{4} + \frac{3x}{4} + \frac{4x}{4} = 108.\] Combine these fractions:\[ \frac{2x + 3x + 4x}{4} = 108 \] which simplifies to:\[ \frac{9x}{4} = 108. \]

Step 4: Solve for x

Multiply both sides of the equation by 4 to eliminate the fraction: \[ 9x = 108 \times 4 \].Calculate the right side: \[ 9x = 432 \].Now, divide by 9 to solve for \( x \):\[ x = \frac{432}{9} = 48. \]

Step 5: Find All Three Numbers

With \( x = 48 \), which is the largest even number, calculate the other two:- Smallest number: \( \frac{x}{2} = \frac{48}{2} = 24 \).- Middle number: \( \frac{3}{4}x = \frac{3}{4} \times 48 = 36 \).Thus, the three numbers are 24, 36, and 48.

Key Concepts

Even Numbers

Even numbers are integers that are divisible by 2 without a remainder. This means that if you divide an even number by 2, you will get another whole number.
These numbers follow a pattern, increasing by 2 as you go up or down the number line. Examples of even numbers include 2, 4, 6, 8, and so on.

• Key property: Even numbers always end in 0, 2, 4, 6, or 8.
• If you add or subtract any two even numbers, the result is always an even number.
• The product of even numbers is also even.

Understanding the nature of even numbers is essential for solving problems that specify even values, like the exercise here where we are asked to find three even numbers that add up to a given sum.

Equation Solving

Equation solving involves finding the value of unknown variables that make an equation true. This process requires following some logical steps to manipulate the equation.
In the context of our exercise, creating and solving an equation allowed us to sum the even numbers correctly and find their values.
Start with transforming an equation to isolate the variable of interest. Typical steps include combining like terms, using inverse operations (adding, subtracting, multiplying, dividing), and simplifying complex fractions or expressions.

• Combine terms: Gather all terms related to the variable on one side.
• Apply inverse operations: Undo operations to isolate the variable.
• Check: Verify the solution by plugging it back into the original equation.

Solving the equation in our problem required understanding these steps to find that the variable equals 48.

Variable Definition

Defining variables is a critical first step in problem-solving, especially in algebra. It involves designating symbols to represent unknown or changing quantities.
For this exercise, we defined "\(x\)" as our largest even number. This choice laid the groundwork for setting up the relationships stated in the problem.

• The smallest number as \(\frac{x}{2}\): Indicating the smallest is half the largest.
• The middle number as \(\frac{3}{4}x\): Showing it's three-quarters of the largest.

Having a clear definition for each variable makes it easier to substitute and form equations, ultimately simplifying the solution process.

Fraction Operations

Working with fractions is a fundamental skill in algebra. It is essential for solving problems where quantities are given in parts of a whole.
Fractions consist of a numerator (top part) and a denominator (bottom part). Operations with fractions include addition, subtraction, multiplication, and division.

• Addition/Subtraction: Find a common denominator before combining fractions.
• Multiplication: Multiply numerators and denominators together.
• Division: Invert the second fraction and multiply.

In the exercise, fractions like \(\frac{x}{2}\) and \(\frac{3}{4}x\) needed to be combined into a single equation to solve for the unknown. This required finding a common denominator, which simplified the equation and made it workable. After converting and merging, solving the equation allowed us to find the exact value of each term.