Problem 59 - Elementary and Intermediate Algebra

Question

Solve each proportion. $$ \frac{x+1}{4}=\frac{3 x}{8} $$

Step-by-Step Solution

Answer

The solution to the proportion is \( x = 2 \).

Step 1: Understand the Proportion

The equation \( \frac{x+1}{4}=\frac{3x}{8} \) is a proportion, meaning that two ratios are equal. To solve for \( x \), we need to find the value of \( x \) that satisfies this equality.

Step 2: Cross Multiply

To eliminate the fractions, use cross multiplication. Multiply the numerator of the first fraction by the denominator of the second fraction and set it equal to the numerator of the second fraction multiplied by the denominator of the first fraction: \((x+1) \times 8 = 4 \times 3x\)

Step 3: Simplify the Equation

Perform the multiplication for both sides of the equation:\[ 8(x+1) = 12x \] Distribute the 8 on the left side: \[ 8x + 8 = 12x \]

Step 4: Isolate the Variable

To solve for \( x \), get all terms involving \( x \) on one side. Subtract \( 8x \) from both sides:\[ 8 = 12x - 8x \]This simplifies to:\[ 8 = 4x \]

Step 5: Solve for x

Divide both sides by 4 to solve for \( x \):\[ x = \frac{8}{4} \]This gives:\[ x = 2 \]

Key Concepts

Proportions

A proportion is an equation stating that two ratios or fractions are equal. These are sometimes used to compare quantities, and they help us understand how values relate to each other. In mathematics, proportions are essential because they provide a way to solve for unknown values when given comparative information. For example, if you have a problem like \( \frac{a}{b} = \frac{c}{d} \), it means the ratio of \( a \) to \( b \) is the same as the ratio of \( c \) to \( d \).

Understanding proportions begins with recognizing that the cross products of these ratios are equal. Therefore, knowing how to work with fractions and ratios is vital. It is particularly useful in everyday situations like cooking recipes, scale modeling, or converting units. Always remember to ensure that both ratios you are dealing with are in the same units for accuracy.

Cross Multiplication

Cross multiplication is a powerful tool for solving proportions. It helps eliminate fractions and simplifies the process of finding unknown values in an equation. With cross multiplication, you multiply diagonally across the equal sign:

• Multiply the numerator of the first fraction by the denominator of the second fraction.
• Multiply the numerator of the second fraction by the denominator of the first fraction.

For a proportion like \( \frac{x+1}{4}=\frac{3x}{8} \), cross multiplication means you perform \((x+1) \times 8 \) and \(4 \times 3x\), setting them equal:

\(8(x+1) = 12x\). This step removes the fractions and transforms the equation into a more straightforward algebraic expression. Cross multiplication is particularly valuable because it sets up simpler equations that are easier to solve using basic arithmetic operations.

Solving Equations

To solve an equation, a step-by-step approach is essential. The goal is to isolate the variable, often x, on one side of the equation to find its value. Let's consider the equation after cross multiplication: \(8(x+1) = 12x\). Here is the process to solve it:

• First, distribute the 8 on the left side: \(8x + 8 = 12x\).
• Next, gather like terms. Move all terms involving x to one side and constant terms to the other. This means subtracting \(8x\) from both sides: \(8 = 12x - 8x\).
• Further simplifying, we find \(8 = 4x\).
• To solve for \(x\), divide both sides by 4: \(x = \frac{8}{4}\), which simplifies to \(x = 2\).

Solving equations requires patience and careful arithmetic. Each step should simplify the equation incrementally, leading to a straightforward solution.