Problem 92 - Algebra 1: Concepts and Skills

Question

Complete the statement using \(<,>,\) or \(=.\) $$ 20 \% ? 0.25 $$

Step-by-Step Solution

Answer

20% < 0.25.

Step 1: Convert the Percentage to a Decimal

To convert a percentage to a decimal, divide the percentage by 100. So, \(20\% = \frac{20}{100} = 0.2\).

Step 2: Compare the Decimals

Now that we have two decimals, we can compare them directly. We see that \(0.2 < 0.25\).

Key Concepts

Converting Percentages to Decimals

Understanding how to convert percentages into decimals is a key mathematical skill that makes comparing figures much simpler. To do this, you simply divide the percentage by 100, as a percentage is actually a fraction with a denominator of 100. This means that 20% is equivalent to \( \frac{20}{100} \), which simplifies to 0.2 when divided.

Always ensure you move the decimal point two places to the left when converting a percentage to a decimal. For example, 55% becomes 0.55 or 7.5% becomes 0.075. This technique is particularly useful because it streamlines the comparison process, as decimals are often easier to work with than percentages.

Decimal Comparison

Comparing decimals is straightforward once you align them by their decimal points. Start by comparing digits in the highest place value and work your way to the right. If the digits differ, you have your answer - the number with the larger digit is greater. If they're the same, move to the next place value.

For instance, with 0.2 and 0.25, start at the tenths place. Both numbers have '2', so you move to the next place - the hundredths. Here 0.2 has '0' while 0.25 has '5', so 0.25 is greater than 0.2. Remember, a missing digit after the decimal point is equal to zero, which makes 0.2 equivalent to 0.20.

Mathematical Inequalities

Mathematical inequalities are expressions that define the relative size or order of two values. The symbols \( < \) (less than), \( > \) (greater than), and \( = \) (equals) are integral in showcasing how two values compare to each other. If a number on the left side of the inequality is smaller than the number on the right, we use the \( < \) symbol. For example, since 0.2 is smaller than 0.25, we write \( 0.2 < 0.25 \).

These symbols are fundamental in the realm of mathematics for establishing value relationships, and it's crucial to comprehend their meanings to properly translate and solve inequality expressions.