Problem 66 - Intermediate Algebra

Question

Solve each proportion. $$ \frac{2}{5}=\frac{1.5 x-2}{0.25} $$

Step-by-Step Solution

Answer

x = 1.4

Step 1: Set up the Proportion Equation

We are given the proportion equation \( \frac{2}{5} = \frac{1.5x - 2}{0.25} \). Our goal is to find the value of \( x \) that makes this proportion true.

Step 2: Cross-Multiply

In a proportion, the product of the means equals the product of the extremes. Multiply diagonally across the equals sign: \( 2 \times 0.25 = 5 \times (1.5x - 2) \). This simplifies to \( 0.5 = 7.5x - 10 \).

Step 3: Move Constants to One Side

Add 10 to both sides of the equation to isolate the term with \( x \): \( 0.5 + 10 = 7.5x \). This simplifies to \( 10.5 = 7.5x \).

Step 4: Solve for x

Divide both sides by 7.5 to solve for \( x \): \( x = \frac{10.5}{7.5} \). Simplify the expression to find \( x = 1.4 \).

Key Concepts

Cross-Multiplication

Cross-multiplication is a fundamental concept in solving certain types of algebraic equations, especially proportions. The technique involves multiplying the numerator of one fraction by the denominator of the other, across the equals sign. This method is particularly useful when dealing with an equation that sets two fractions equal to each other, known as a proportion.

Cross-multiplication allows us to eliminate the fractions and transform the equation into a simpler one that is easier to handle. For instance, in the given proportion \( \frac{2}{5} = \frac{1.5x - 2}{0.25} \), we apply cross-multiplication by multiplying 2 by 0.25 on one side and 5 by \(1.5x - 2\) on the other.

• Calculate \( 2 \times 0.25 = 0.5 \)
• Calculate \( 5 \times (1.5x - 2) = 7.5x - 10 \)

Thus, the original complex proportion simplifies to a more straightforward algebraic equation: \( 0.5 = 7.5x - 10 \).
The simplicity and efficiency of cross-multiplication make it a go-to method in proportion problems.

Proportions

Proportions are equations that declare two ratios to be equal. They appear frequently in algebra and are essential for comparing quantities and solving many real-life problems.

A proportion takes the form \( \frac{a}{b} = \frac{c}{d} \), where \( a, b, c, \) and \( d \) are numbers, and \( b \) and \( d \) are non-zero. The concept is based on the idea that the product of the means (the inner terms) equals the product of the extremes (the outer terms).

In the given exercise, the proportion \( \frac{2}{5} = \frac{1.5x - 2}{0.25} \) helps us establish an equation that can be solved to find the value of \( x \). By emphasizing the balance of ratios, proportions ensure that the changes applied to one side must equivalently reflect on the other.

• The left-hand side ratio is \( \frac{2}{5} \).
• The right-hand side ratio involving the variable \( x \) is \( \frac{1.5x-2}{0.25} \).

By solving these proportions, we maintain equality and discover unknown values such as \( x \), exemplifying how proportions provide a valuable tool in algebra.

Solving for x

Solving for \( x \) is a crucial skill in algebra, which involves isolating the variable to find its value that satisfies the equation. After applying cross-multiplication and simplifying the equation, the task becomes straightforward.

Once the equation from the current problem is simplified to \( 0.5 = 7.5x - 10 \), solving for \( x \) involves a few systematic steps. First, we isolate the term containing \( x \). Add 10 to both sides to remove the constant from the equation's right side:

• \( 0.5 + 10 = 7.5x \)
• Simplifies to \( 10.5 = 7.5x \)

The next step is to divide both sides by 7.5 to get \( x \) alone:

• \( x = \frac{10.5}{7.5} \)
• Which simplifies further to \( x = 1.4 \)

By following this process, students can solve for \( x \) in a variety of equations, not just proportions. This methodical approach ensures clarity and accuracy, enhancing problem-solving skills in algebra.