Problem 42 - Algebra for College Students

Question

Use an algebraic approach to solve each problem. One-half of a number plus three-fourths of the number is 2 more than four- thirds of the number. Find the number.

Step-by-Step Solution

Answer

The number is -24.

Step 1: Define the Variable

Let's define the unknown number as \( x \). This number is what we need to solve for.

Step 2: Setup the Equation

Using the problem statement, we can write the equation. One-half of the number is represented as \( \frac{1}{2}x \) and three-fourths of the number is \( \frac{3}{4}x \). The sum of these is equal to "2 more than four-thirds of the number," which can be expressed as \( \frac{4}{3}x + 2 \). Therefore, the equation is:\[ \frac{1}{2}x + \frac{3}{4}x = \frac{4}{3}x + 2 \]

Step 3: Combine Like Terms on the Left

Combine the terms on the left side of the equation:\[ \frac{1}{2}x + \frac{3}{4}x = \frac{2}{4}x + \frac{3}{4}x = \frac{5}{4}x \]So our equation becomes:\[ \frac{5}{4}x = \frac{4}{3}x + 2 \]

Step 4: Eliminate the Fractions

To eliminate the fractions, find the least common multiple (LCM) of the denominators 4 and 3, which is 12. Multiply every term by 12:\[ 12 \times \frac{5}{4}x = 12 \times \frac{4}{3}x + 12 \times 2 \]This simplifies to:\[ 15x = 16x + 24 \]

Step 5: Solve for x

Rearrange the equation to isolate \( x \) on one side by subtracting \( 16x \) from both sides:\[ 15x - 16x = 24 \]This simplifies to:\[ -x = 24 \]Multiply both sides by -1:\[ x = -24 \]

Key Concepts

Variables in Algebra

In algebra, a variable is a symbol (often a letter) that represents an unknown value. In the given problem, the variable is used to represent the unknown number that we need to find. We labeled this unknown with the letter \( x \). This helps transform the word problem into a mathematical equation that can be solved. Using variables, we translate statements into expressions easier. For example:- "one-half of a number" becomes \( \frac{1}{2}x \).- "three-fourths of a number" is written as \( \frac{3}{4}x \).By using \( x \), we create an equation that mirrors the problem's conditions, making it easier to solve mathematically.

Fractions in Equations

Fractions often appear in equations and can make them look complex. Breaking them down makes them easier to work with. In our problem, fractions such as \( \frac{1}{2}x \), \( \frac{3}{4}x \), and \( \frac{4}{3}x \) are involved. To work with these fractions:

• Recognize that they represent parts of the variable \( x \).
• Write each fraction as a coefficient followed by \( x \), making it part of an equation expression.

This helps in forming the equation correctly and sets the groundwork for solving it.

Combining Like Terms

Combining like terms is an important step in simplifying equations. It means adding or subtracting terms that have the same variable, making the equation easier to handle. In our problem, both terms on the left side of the equation share the same variable, \( x \), and can be combined:\[ \frac{1}{2}x + \frac{3}{4}x = \frac{5}{4}x \]

Why is this useful?

When you combine like terms:

• The equation becomes shorter and more manageable.
• It aligns terms with common denominators, which is useful in further steps, such as eliminating fractions.

Eliminating Fractions

Eliminating fractions is a pivotal step in solving equations smoothly. Fractions can be cleared by finding and using the least common multiple (LCM) of their denominators as a multiplier. In this problem, the denominators are 4 and 3. Their LCM is 12. By multiplying every term of the equation by 12, we eliminate the fractions:\[ 12 \times \frac{5}{4}x = 12 \times \frac{4}{3}x + 12 \times 2 \]Results in:\[ 15x = 16x + 24 \]

What does this achieve?

• It transforms a complex equation with fractions into a simpler, whole number equation.
• Makes the equation easier to solve through basic operations like addition or subtraction.