Problem 74 - Algebra 1: Concepts and Skills

Question

Complete the statement using \(<,>,\) or \(=\). \(\frac{6}{8} ? 0.75\)

Step-by-Step Solution

Answer

\(\frac{6}{8} = 0.75\)

Step 1: Convert the fraction into a decimal

To compare a fraction with a decimal, one or the other must be converted. Converting the fraction \(6/8\) into a decimal is easier. It follows as: \[ \frac{6}{8} = 0.75 \]

Step 2: Compare the decimal numbers

Now, both numbers to be compared are in a decimal format. Therefore, 0.75 can be compared with 0.75.

Step 3: Determine the correct mathematical symbol

Because both numbers are equal, the correct mathematical symbol to use in the equation is \(=\).

Key Concepts

Comparing Fractions and Decimals

When comparing fractions and decimals, it's essential to present both numbers in the same format. This makes it easier to evaluate them. You have two choices:

• Convert the decimal to a fraction.
• Convert the fraction to a decimal.

For many, converting the fraction to a decimal is straightforward, especially if the fraction has a denominator like 8, which is easily divisible. For instance, converting \(\frac{6}{8}\) into a decimal involves dividing 6 by 8, resulting in 0.75.
Once both numbers are in decimal form, comparison is straightforward. You can directly see if one number is greater, smaller, or equal to the other. This method avoids potential errors in comparing mixed formats and simplifies the mathematical process.

Mathematical Symbols in Equations

In mathematical equations, symbols like \(<\), \(>\), and \(=\) are fundamental. They help describe the relationship between two values:

• \(<\): Indicates that the left side is less than the right side.
• \(>\): Shows that the left side is greater than the right side.
• \(=\): Denotes that both sides have equal value.

Every comparison task requires choosing the correct symbol. For our problem, after converting both the fraction and the decimal to the same format, we must determine if \(0.75\) is less than, greater than, or equal to \(0.75\). Since the values are equivalent, the symbol \(=\) is the perfect choice.
Understanding these symbols is crucial for developing mathematical literacy and solving equations accurately. They form the basic language of mathematics and help convey precise relationships.

Decimal Numbers

Deciding whether to work with decimal numbers can simplify many mathematical operations. Decimal numbers express values using powers of 10, allowing for easy comparison and arithmetic operations.

• Decimal places represent tenths, hundredths, thousandths, etc.
• In our problem, \(0.75\) has two decimal places, equivalent to \(\frac{75}{100}\) in fractional form.

Decimals make operations clear, especially when precision is key. When you convert fractions like \(\frac{6}{8}\) into decimals, the results can be easily compared to other decimals without worrying about simplifying fractions or adjusting denominators.
Moreover, decimals are often used in everyday contexts such as money, measurements, and percentages. This makes them a handy format for performing quick and clear calculations in both academic exercises and real-life situations.