Problem 32 - Prealgebra

Question

Replace each \(\circ\) with \(<,>,\) or \(=\) to make a true sentence. $$0.3 \circ \frac{1}{2}$$

Step-by-Step Solution

Answer

0.3 < 0.5, thus use '<'. 0.3 < 1/2.

Step 1: Convert the Fraction to Decimal

First, convert the fraction \(\frac{1}{2}\) into a decimal to easily compare it with 0.3. The fraction \(\frac{1}{2}\) is equivalent to 0.5 in decimal form.

Step 2: Compare the Numbers

Now, compare the two decimals: 0.3 and 0.5. Since 0.3 is less than 0.5, you will use the \(<\) symbol to complete the inequality.

Step 3: Write the Inequality

Replace the \(\circ\) symbol with the correct comparison symbol. The true sentence is: $$0.3 < 0.5$$Hence, in the original problem, replace \(\circ\) with \(<\).

Key Concepts

Decimal Conversion

Understanding decimal conversion is essential in comparing fractions with decimals. Fractions represent a part of a whole number, while decimals are another way to express these parts.
To convert a fraction like \( \frac{1}{2} \) into a decimal, you can divide the numerator by the denominator. In this case, 1 divided by 2 equals 0.5.

• This conversion allows for easier direct comparison of values.
• Once converted to decimals, you can easily line up the numbers and see which is larger or smaller.

When converting fractions to decimals, it's crucial to perform accurate division for precise comparison results.

Fraction Comparison

Comparing fractions directly can sometimes be challenging. This is where converting them to decimals can simplify the process. When you compare, you assess which value is larger, equal, or smaller.
This approach makes it consistently straightforward because:

• Decimals are often easier to visualize and compare at a glance.
• It removes the need for finding common denominators which can be tedious.

Once fractions are converted to decimals, compare their digits from left to right, just like comparing whole numbers. For example, 0.3 versus 0.5 shows that 0.3 has fewer tenths than 0.5, confirming that it is smaller.

Inequalities

Inequalities involve expressions that do not necessarily equate to one another, using symbols such as \( <, >, \) and \( = \) to define their relationship.
These are crucial in determining how values relate to one another. The symbols used to represent these relations are:

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

When comparing 0.3 and 0.5, we use \( < \) because 0.3 is less than 0.5. Understanding and using these symbols allows for meaningful mathematical statements and can aid in solving complex problems that involve comparisons or expressions needing solutions.