Problem 69 - Algebra 1: Concepts and Skills

Question

Complete the statement using \(<,>,\) or \(=\). \(0.30 ? \frac{2}{3}\)

Step-by-Step Solution

Answer

The correct symbol to complete the statement is '<', making the complete statement \(0.30 < \frac{2}{3}\).

Step 1: Convert the fraction into decimal

To compare the decimal with the fraction, we can convert the fraction into a decimal. The fraction is \(\frac{2}{3}\). To convert this to decimal, divide the numerator which is 2 by the denominator which is 3. Performing this division 2 ÷ 3, we get around 0.6667.

Step 2: Compare the decimal numbers

Now that we have converted the fraction to a decimal, we need to compare it with the given decimal number, \(0.30\), to find out if it is less than, equal to or greater than \(\frac{2}{3}\). By observing the numbers, we can clearly see that \(0.30\) is smaller than \(0.6667\).

Step 3: Fill in the appropriate symbol

Seeing that \(0.30\) is less than \(0.6667\), we can now directly substitute the less than symbol in the original problem, giving us \(0.30 < \frac{2}{3}\).

Key Concepts

Converting Fractions to Decimals

Converting fractions to decimals is a valuable skill that helps in comparing numbers more easily. To do this, you divide the numerator by the denominator. Think of the fraction as a division problem. For instance, the fraction \(\frac{2}{3}\) means you divide 2 by 3. Once you do this division, you get a repeating decimal, approximately 0.6667.

• The numerator is the top number in the fraction, which shows how many parts you have.
• The denominator is the bottom number, which tells how many equal parts make up a whole.

Repeating decimals occur because some simple fractions don’t convert into exact decimals. It's common to round them in practical uses. However, it's ideal to use enough decimal places to keep the comparison as accurate as needed for the problem at hand.

Comparing Numbers

Comparing numbers means determining which number is smaller, larger, or if they are equal. You can compare numbers by looking at their value, magnitude, or even the number of decimal places they have. In the case of comparing decimals, like 0.30 and 0.6667, you align them from the decimal point and compare digit by digit.
For exact comparisons:

• Look at each digit post-decimal if the whole number part is the same.
• Start comparing from leftmost non-zero digit.
• If one decimal number finishes before the other or has fewer digits, assume it is smaller for practical reasons if the continuing digits are zeros.

In the example with 0.30 and \(\frac{2}{3}\), a comparison shows 0.30 is less because, when expanded, 0.6667 clearly has digits that show it's greater than 0.3.

Inequalities in Mathematics

Inequalities are expressions used to show the relationship between two values that are not equal. They use symbols like \(<, >,\) and \(=\) to denote these relationships. Knowing how to select and use these symbols is crucial for expressing mathematical ideas effectively.
When facing a problem that requires filling in such a symbol:

• Use \(<\) when the first number is smaller.
• Use \(>\) when the first number is larger.
• Use \(=\) when both numbers are exactly the same.

In the statement \(0.30 ? \frac{2}{3}\), the correct inequality symbol is \(<\), because 0.30 is less than 0.6667. Inequalities allow us to communicate and solve mathematical problems by pinpointing precisely how numbers relate.