Problem 51
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
Verified Answer
The three numbers are 24, 36, and 48.
1Step 1: Identify Variables
Let's represent the largest of the three numbers as \(x\). According to the problem, the smallest number is half of the largest, so we can denote it as \(\frac{x}{2}\). The middle number is \(\frac{3}{4}\) of the largest, so it is represented as \(\frac{3x}{4}\).
2Step 2: Formulate the Equation
The sum of the three numbers is 108. Therefore, we have:\[\frac{x}{2} + \frac{3x}{4} + x = 108\] This equation represents the sum of the smallest, middle, and largest numbers.
3Step 3: Clear Fractions
To eliminate the fractions, multiply the entire equation by 4 to clear the denominators:\[4 \times \left(\frac{x}{2} + \frac{3x}{4} + x\right) = 4 \times 108\]This simplifies to:\[ 2x + 3x + 4x = 432 \]
4Step 4: Combine and Simplify
Combine the like terms in the equation:\[ 9x = 432 \]
5Step 5: Solve for the Largest Number
Divide both sides of the equation by 9 to solve for \(x\):\[ x = \frac{432}{9} = 48 \] Thus, the largest number is 48.
6Step 6: Find the Other Numbers
Using \(x = 48\):- The smallest number is \(\frac{48}{2} = 24\).- The middle number is \(\frac{3 \times 48}{4} = 36\).Therefore, the three numbers are 24, 36, and 48.
Key Concepts
Understanding Even NumbersBasics of Equation SolvingHandling Fractions in EquationsThe Role of Variable Representation
Understanding Even Numbers
Even numbers are integers that are divisible by 2 without leaving a remainder. In simple terms, if you can divide a number by 2 and get a whole number, it's an even number.
Some examples of even numbers include 2, 4, 6, 10, and so on. In our problem, all three numbers are even because they add up to 108, an even number itself. When working with algebra problems involving even numbers, it's often helpful to understand that evenly spaced even numbers will each have factors of 2.
In this problem, identifying the even nature of these numbers helps clarify the relationships between them, such as one being half of another.
Some examples of even numbers include 2, 4, 6, 10, and so on. In our problem, all three numbers are even because they add up to 108, an even number itself. When working with algebra problems involving even numbers, it's often helpful to understand that evenly spaced even numbers will each have factors of 2.
In this problem, identifying the even nature of these numbers helps clarify the relationships between them, such as one being half of another.
Basics of Equation Solving
In this exercise, equation solving is about finding the values of unknown variables that make a given mathematical statement true. Here, we formulate an equation based on the relationships given in the problem.
Our step-by-step approach begins with identifying variables: assigning one of the numbers as the reference point (in this case, the largest number is x). We then express other variables in terms of this primary variable (e.g., smallest as \(\frac{x}{2}\), and middle as \(\frac{3x}{4}\)).
Next, we set up an equation that represents the sum of these numbers to match a specific total, which is given as 108:
Our step-by-step approach begins with identifying variables: assigning one of the numbers as the reference point (in this case, the largest number is x). We then express other variables in terms of this primary variable (e.g., smallest as \(\frac{x}{2}\), and middle as \(\frac{3x}{4}\)).
Next, we set up an equation that represents the sum of these numbers to match a specific total, which is given as 108:
- The smallest: \( \frac{x}{2} \)
- The middle: \( \frac{3x}{4} \)
- The largest: \( x \)
Handling Fractions in Equations
Fractions can sometimes make equations look complex, but their role in representing parts of a whole or ratios is crucial. In problems like this, incorporating fractions provides a precise mathematical representation of relationships between numbers.
To ease calculations, we can clear fractions by multiplying every term in the equation by a common denominator. In our solution, multiplying through by 4 eliminated denominators, allowing us to focus on integer arithmetic:
To ease calculations, we can clear fractions by multiplying every term in the equation by a common denominator. In our solution, multiplying through by 4 eliminated denominators, allowing us to focus on integer arithmetic:
- Original form: \( \frac{x}{2} + \frac{3x}{4} + x = 108 \)
- After multiplying by 4: \( 2x + 3x + 4x = 432 \)
The Role of Variable Representation
Variable representation is a core component in solving algebra problems. It involves using symbols to stand in for unknown values, which allows you to set up equations based on relationships described in word problems.
In the given problem, we begin by letting \(x\) represent the largest even number. Using this variable allows you to relate all three numbers to each other through multiplication and division (e.g., the smallest number becomes \(\frac{x}{2}\), and the middle number \(\frac{3x}{4}\)).
Using variables effectively:
In the given problem, we begin by letting \(x\) represent the largest even number. Using this variable allows you to relate all three numbers to each other through multiplication and division (e.g., the smallest number becomes \(\frac{x}{2}\), and the middle number \(\frac{3x}{4}\)).
Using variables effectively:
- Provides a clear and concise way to express relationships between components of the problem.
- Makes solving equations systematic and allows for manipulation to find exact values.
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