Problem 45 - Elementary Algebra

Question

Use proportions to change each common fraction to a percent. $$\frac{3}{2}$$

Step-by-Step Solution

Answer

\(\frac{3}{2}\) is 150%.

Step 1: Understand the Concept of Percent

A percent is a way of expressing a number as a fraction of 100. To convert a common fraction like \(\frac{3}{2}\) to a percent, we need to set up a proportion where the fraction equals \(\frac{x}{100}\), with \(x\) being the value in percent.

Step 2: Set Up the Proportion

Write the proportion equation: \(\frac{3}{2} = \frac{x}{100}\). The goal is to find the value of \(x\), which represents the percentage equivalent of the fraction \(\frac{3}{2}\).

Step 3: Cross-Multiply to Solve for x

To solve for \(x\), cross-multiply in the proportion: \[3 \times 100 = 2 \times x\]This gives:\[300 = 2x\]

Step 4: Isolate x

To find \(x\), divide both sides of the equation by 2:\[x = \frac{300}{2}\]This simplifies to:\[x = 150\]

Step 5: Conclusion

The value of \(x\) is 150, which means \(\frac{3}{2}\) is equivalent to 150%.

Key Concepts

Proportion Method

The proportion method is a fundamental mathematical tool used to convert fractions to percentages. Understanding proportions begins with recognizing a percent as a ratio or fraction out of 100. Let's consider any fraction, like \(\frac{3}{2}\), where we aim to express this fraction as a percent, \(\frac{x}{100}\). Here, \(x\) is the unknown value representing our fraction as a percentage.

To set up a proportion, write it as an equation: \(\frac{3}{2} = \frac{x}{100}\). This setup establishes a relationship between our given fraction and the percentage.

• Remember, the numerator of the given fraction corresponds to \(x\), and the denominator (100) is a constant in percentage conversion.
• Proportion method is versatile and can handle a vast range of fractional conversion tasks.

Once the proportion is set, you're ready for the next step, cross-multiplication, to further simplify and solve the problem.

Cross-Multiplication

Cross-multiplication is a critical step in solving equations that involve proportions. It allows you to eliminate the fractions and simplify the equation to find the unknown variable. In our case, we have set up the equation from the proportion method as \(\frac{3}{2} = \frac{x}{100}\).

Now, apply cross-multiplication: Multiply the numerator of one fraction by the denominator of the other fraction, and do the same for the remaining terms. Thus, the equation transforms into two products: \(3 \times 100 = 2 \times x\).

• This calculates to \(300 = 2x\).
• Cross-multiplication is a straightforward technique that works well for solving problems involving proportions and fractional conversions.

After cross-multiplying, the goal is to isolate \(x\), giving us a direct pathway to finding the percent value.

Percent Calculation

Percent calculation is the final and satisfying step of converting a fraction to a percent. After cross-multiplying, you have an equation like \(300 = 2x\), which can be solved to find \(x\).

To isolate \(x\), divide both sides of the equation by 2, giving you \(x = \frac{300}{2}\). This simplifies to \(x = 150\).

Thus, the fraction \(\frac{3}{2}\) is equivalent to 150%. Converting to a percent is about expressing the total relationship of any given fraction with 100 parts.

• This process reinforces your understanding of fractions, ratios, and percentages.
• Percent calculations are omnipresent in daily arithmetic, financial tasks, and data interpretation.

Understanding how to convert fractions into percentages is both practical and empowering, as it broadens your mathematical capabilities and problem-solving skills.