Problem 44 - Prealgebra

Question

Replace each \(\circ\) with \(<,>,\) or \(=\) to make a true statement. $$0.714 \circ \frac{5}{7}$$

Step-by-Step Solution

Answer

0.714 < \frac{5}{7}.

Step 1: Convert Fraction to Decimal

First, we need to convert the fraction \( \frac{5}{7} \) into a decimal. To do this, divide the numerator (5) by the denominator (7): \( \frac{5}{7} \approx 0.714285714\ldots \). This decimal is a repeating decimal, represented as \( 0.714285\ldots \).

Step 2: Compare Decimals

Now that we have both numbers as decimals, we need to compare them. We have \( 0.714 \) and the repeating decimal \( 0.714285\ldots \). To compare, consider the first few decimal places: \( 0.714000 \) versus \( 0.714285\ldots \).

Step 3: Identify the Correct Symbol

Since \( 0.714000 \) is less than \( 0.714285\ldots \), we need to replace the \( \circ \) with the \( < \) symbol to make the statement true.

Key Concepts

Fraction Conversion

Converting fractions into decimals is a vital math skill. It helps us understand and work with numbers in different forms. Here's a simple way to convert a fraction into a decimal:

• Divide the numerator (the top number) by the denominator (the bottom number).

For example, to convert the fraction \( \frac{5}{7} \), you divide 5 by 7. The result is a repeating decimal, approximately \( 0.714285714\ldots \).
Repeating decimals don't end, and they have a pattern that recurs. Understanding this process allows you to handle fractions and decimals interchangeably, which is often necessary in solving math problems.

Repeating Decimal

Repeating decimals are intriguing because they are never-ending but have a repeating pattern. For instance, when you converted \( \frac{5}{7} \) into a decimal, you saw it is approxiamtely \( 0.714285714\ldots \).

To identify them, look for a series of digits after the decimal point that repeats indefinitely. Mathematicians like to denote this with a bar above the repeating digits, so \( 0.714285\ldots \) is written as \( 0.\overline{714285} \).
Knowing how to work with repeating decimals is beneficial because it allows comparisons with other numbers, helping us decide which is larger or smaller in problems.

Inequality Symbols

Inequality symbols like \(<\), \(>\), and \(=\) are essential for comparing numbers and showing relationships between them:

• \(<\) means "less than"
• \(>\) means "greater than"
• \(=\) means "equal to"

When comparing decimals, such as \( 0.714 \) and \( 0.714285\ldots \), it is important to compare place by place as needed until you find a difference.
In this example, \( 0.714 \) is less than \( 0.714285\ldots \), so we use the \(<\) symbol to show that \( 0.714 \) is indeed smaller. Having a strong grasp of inequality symbols makes you competent in evaluating the size and order of various quantities.