Problem 95 - Algebra 1: Concepts and Skills

Question

Complete the statement using \(<,>,\) or \(=.\) $$ 80 \% ? 1.8 $$

Step-by-Step Solution

Answer

The solution is \(80\% < 1.8\)

Step 1: Percentage to Decimal Conversion

Firstly, we need to convert \(80\%\) into decimal format. A percentage is simply a fraction of 100, so to convert it to a decimal, we can divide by 100, which gives us \(0.8\).

Step 2: Comparison

Now, we compare the two numbers \(0.8\) and \(1.8\). Since \(0.8 < 1.8\), hence, it is clear that \(80\% < 1.8\) when both numbers are compared in the same format.

Key Concepts

Percentage to Decimal Conversion

Converting percentages to decimals is a fundamental skill in mathematics that helps in making comparisons more straightforward. Let's take the example of converting 80% into a decimal. When dealing with percentages, it’s important to remember that percent means "per hundred." So, 80% can also be expressed as the fraction \( \frac{80}{100} \). To convert a percentage to a decimal, you simply divide the number by 100. For 80%, you perform the calculation: \[ 80\% = \frac{80}{100} = 0.8 \]. This conversion allows you to work with decimals, which is often easier and more applicable in various mathematical operations. Keep these conversion steps in mind:

• Identify the percentage you want to convert.
• Divide that number by 100.
• Your result is the decimal equivalent.

Understanding this conversion is not only useful in solving mathematical problems but also in real-life scenarios, like calculating discounts or interpreting statistical data.

Inequality Symbols

Inequality symbols are mathematical symbols used to compare two numbers or expressions. The key symbols you'll encounter are \( < \), \( > \), and \( = \). These symbols help indicate the relationship between numbers:- \( < \) means "less than."- \( > \) means "greater than."- \( = \) means "equal to."In the exercise, the symbol used was \( < \), denoting that the first number is smaller than the second. It's important to always read the inequality from left to right. For example, in the statement \( 0.8 < 1.8 \), we understand that 0.8 is less than 1.8. With inequalities, the positioning of numbers is crucial, and flipping an inequality changes its meaning. To correctly apply these symbols:

• Identify both numbers or expressions you are comparing.
• Determine the size or magnitude of each.
• Use the correct inequality symbol to show their relationship.

Mastering these symbols is essential for solving problems that require ordering numbers or expressions.

Mathematical Comparison

Mathematical comparison is a method of evaluating two numbers to determine their relative size. This process involves the use of inequality symbols which clarify the relationship between the numbers. The step-by-step solution uses comparison to determine which quantity is larger or smaller.When you compare numbers, especially those with different formats like percentages and decimals, follow these steps:

• Convert all numbers to the same format, such as all decimals.
• Place the numbers on a number line if it helps visualize scale easily.
• Assess which number is greater, less, or if they are equal, then use the appropriate symbol \((<, >, = )\).

In our example, converting 80% to 0.8 allowed an accurate comparison with 1.8. The result was that 0.8 is less than 1.8, hence, \(80\% < 1.8\). Doing these comparisons helps greatly in various applications, ensuring accurate interpretations of quantities whether in academics, finance, or daily calculations. Understanding how to effectively compare numbers builds a solid foundation for advanced mathematics.