Problem 46 - Elementary Algebra

Question

Use proportions to change each common fraction to a percent. $$\frac{5}{4}$$

Step-by-Step Solution

Answer

125%

Step 1: Understand the Conversion

To convert a fraction to a percent, you need to find an equivalent fraction with a denominator of 100. This fraction can then be read as a percent where the numerator is the percentage value.

Step 2: Set Up the Proportion

We are given the fraction \(\frac{5}{4}\). Set up a proportion to find the equivalent fraction with the denominator as 100: \(\frac{5}{4} = \frac{x}{100}\), where \(x\) represents the percentage.

Step 3: Cross-Multiply and Solve for x

Cross-multiply to solve the proportion: \(5 \times 100 = 4 \times x\). This simplifies to \(500 = 4x\).

Step 4: Divide to Find the Percentage

Divide both sides of the equation by 4 to solve for \(x\): \(x = \frac{500}{4}\). Calculating this gives \(x = 125\).

Step 5: State the Result

The fraction \(\frac{5}{4}\) is equivalent to 125%. So, 125% is the percentage representation of \(\frac{5}{4}\).

Key Concepts

Proportions

Proportions are a way to express the equivalence of two ratios or fractions. They state that two fractions are equal in value, even if their numerators and denominators are different. Applying this concept helps in understanding relationships between numbers, especially when converting forms like fractions to percents. When you set up a proportion in the context of percent conversion, you want to relate your given fraction to a new fraction whose denominator is 100. This is because percentages are based on a 100 part whole. It's essential to remember:

• The numerator represents the part you are dealing with,
• The denominator represents the whole,
• In a percent, the denominator is always 100.

This helps in translating a fraction directly into a percentage by just adjusting the values to maintain equivalency in proportion.

Percent Conversion

Percent conversion involves transforming a given fraction into a percentage format. Percent means "per hundred," so when you convert a fraction to a percent, you are essentially figuring out how many parts out of 100 the fraction represents. For instance, to convert the fraction \(\frac{5}{4}\), you want an equivalent fraction like \(\frac{x}{100}\). Here, \(x\) designates the percentage value you aim to find. Remember that:

• Multiplying or dividing both the numerator and denominator of a fraction by the same number won't change the fraction's value,
• Your goal is to adjust the values so that the denominator becomes 100.

This method allows you to express the original fraction in an understandable percent format, helping clear up the relative size of numbers in a familiar way.

Cross-Multiplication

Cross-multiplication is a technique used to solve proportions and find unknown variables swiftly. When a proportion is established, like \(\frac{5}{4} = \frac{x}{100}\), cross-multiplication helps reveal the unknown by multiplying across the equal sign. You perform cross-multiplication by following these steps:

• Multiply the numerator of one fraction by the denominator of the other: \(5 \times 100\),
• Multiply the numerator of the opposite fraction by the denominator of the first: \(4 \times x\),
• Set these two products equal to each other: \(500 = 4x\).

After obtaining this equation, you can solve for the unknown \(x\) by isolating it. Divide both sides by the remaining coefficient (in this case, 4) to find \(x = 125\). This method is straightforward and provides exact results, confirming the equivalence of the given fraction in percentage form.