Problem 29 - Prealgebra

Question

Replace each \(\circ\) with \(<,>,\) or \(=\) to make a true sentence. $$0.3 \circ \frac{1}{4}$$

Step-by-Step Solution

Answer

0.3 > \(\frac{1}{4}\)

Step 1: Convert Fractions to Decimals

Start by converting the fraction to a decimal. The fraction \(\frac{1}{4}\) is equal to 0.25. Now the comparison is between 0.3 and 0.25.

Step 2: Compare Decimals

To compare 0.3 and 0.25, note that 0.3 is larger than 0.25 because 3 is greater than 2 in the tenths place.

Step 3: Choose the Correct Symbol

Since 0.3 is greater than 0.25, replace \(\circ\) with \(>\). Thus, the complete expression is \(0.3 > \frac{1}{4}\).

Key Concepts

Decimal Conversion

Decimal conversion is an important part of understanding how different numbers compare to one another, especially when dealing with fractions. To convert a fraction into a decimal, you need to perform a division operation. For example, if you take the fraction \(\frac{1}{4}\), you need to divide 1 by 4. When you do this division, you get 0.25 as a result.

• Begin with the numerator (the top number of the fraction) and divide it by the denominator (the bottom number).
• Use long division or a calculator to find the decimal value.
• This decimal can then easily be used to compare it with other decimals or numbers.

Once you have the decimal form of a fraction, it becomes much easier to compare it with any other decimal number.

Fraction to Decimal Conversion

To master the skill of converting fractions to decimals, it's helpful to understand the process behind it. Converting a fraction into a decimal requires dividing the numerator by the denominator.

• Identify the numerator and denominator in your fraction. In \(\frac{1}{4}\), 1 is the numerator and 4 is the denominator.
• Perform the division: divide 1 by 4 to get 0.25.
• Certain simple fractions like \(\frac{1}{2}\) and \(\frac{3}{4}\) should be memorized for their common decimal equivalents: 0.5 and 0.75.

This conversion is useful because decimal numbers are often easier to compare and compute with than fractions.

Number Comparison

Once you have converted both numbers to decimals, comparing them becomes straightforward. When comparing decimals, look at the numbers from left to right, starting with the tenths place. In the case of 0.3 and 0.25, you start by comparing the tenths place.

• Observe the digit in the tenths place for each number. Here, 0.3 has a 3 in the tenths place, while 0.25 has a 2.
• The higher number in the tenths place indicates the larger decimal value. Since 3 is greater than 2, 0.3 is greater than 0.25.
• Use appropriate symbols: '>' for greater than, '<' for less than, and '=' for equal, to complete the comparison.

Remember, this method can be applied to compare any decimal numbers. The key is to go digit by digit, left to right, until a difference is found.