Problem 42 - Elementary Algebra

Question

Use proportions to change each common fraction to a percent. $$\frac{5}{7}$$

Step-by-Step Solution

Answer

\( \frac{5}{7} \) is approximately 71.43%.

Step 1: Understand What a Proportion Is

A proportion is an equation that states two ratios are equivalent. In this problem, we will set up a proportion to find out what percent \( \frac{5}{7} \) is equivalent to.

Step 2: Set Up the Proportion

To convert \( \frac{5}{7} \) to a percent, set up the proportion \( \frac{5}{7} = \frac{x}{100} \), where \( x \) is the percent value we want to find. In this setup, 100 is the whole, representing 100%.

Step 3: Solve for x

Cross-multiply to solve for \( x \): \[ 5 \times 100 = 7 \times x \] Simplifying gives us: \[ 500 = 7x \] Now, divide both sides by 7 to isolate \( x \): \[ x = \frac{500}{7} \]

Step 4: Calculate the Value of x

Perform the division \( \frac{500}{7} \) to get the value of \( x \). The result is approximately \( 71.43 \).

Step 5: Write the Final Answer as a Percent

Since \( x = 71.43 \), \( \frac{5}{7} \) is equivalent to approximately 71.43%. Remember to round to two decimal places if needed for exactness.

Key Concepts

Percent Conversion

Percent conversion helps us express a part out of 100, which can be much easier to understand. This is achieved using a proportion, as seen in converting \( \frac{5}{7} \) to a percentage. The aim is to find what number is equivalent when the denominator is 100.

• To begin with, fractions like \( \frac{5}{7} \) are parts of a whole.
• To convert them to percentages, equate the fraction to \( \frac{x}{100} \) where \( x \) represents the percentage.
• "100" is used because percentages are a way to express something as a part of 100.

This concept simplifies comparisons, making fractions more relatable as percentages neatly fit into our base-10 number system. By following these steps, we can easily handle percent conversions in various mathematical problems and real-world scenarios.

Cross-Multiplication

Cross-multiplication is a powerful tool in solving proportions. When you have a proportion like \( \frac{a}{b} = \frac{c}{d} \), you can solve it using cross-multiplication:

• Multiply \( a \) by \( d \).
• Multiply \( b \) by \( c \).
• Set the two products equal to each other: \( a \times d = b \times c \).

In our fraction to percent problem of converting \( \frac{5}{7} \) to percent, we initially set up the proportion as \( \frac{5}{7} = \frac{x}{100} \).
Cross-multiplying gives us:
\[ 5 \times 100 = 7 \times x \]
By performing these steps, we clear the fractions, making it easier to solve for \( x \) and find the required percentage value inexpensively.

Fractions to Percent

Converting fractions to percent is quite simple once you understand the mechanics as demonstrated. The conversion uses the basic idea of scaling the fraction's denominator to 100. Here's how you go about it:

• Express the fraction, \( \frac{5}{7} \), as equivalent to \( \frac{x}{100} \).
• Use cross-multiplication to solve for \( x \).
• The solution of \( x = \frac{500}{7} \) gives us the equivalent percent.

After computing \( \frac{500}{7} \), you get approximately \( 71.43\% \).
This gives the percent form, offering clarity on what portion 5 is out of 7 when considered in terms of 100.

Equivalent Ratios

Understanding equivalent ratios is crucial for setting up and solving proportions. Equivalent ratios represent the same relational quantity, even if the numbers look different. When we work with these, we state that two quantities are consistent in size or scale.

• For instance, \( \frac{5}{7} = \frac{x}{100} \) shows two ratios being equivalent.
• This implies that the base relationship or proportion between 5 and 7 is the same as that between \( x \) and 100.
• Finding equivalent ratios allows us to seamlessly transition from one format to another, such as from fractions to percents.

This concept is an essential building block in mathematics, helping you perform meaningful conversions and solve problems relating quantities in a coherent manner.