Problem 79 - Algebra 1: Concepts and Skills

Question

Complete the statement using \(<,>,\) or \(=.\) $$ 40 \% ? 0.35 $$

Step-by-Step Solution

Answer

The complete statement using the correct relation symbol is: 40% > 0.35.

Step 1: Convert the Percentage into Decimal

First, convert 40% into decimal form. To do this, divide 40 by 100, as there are 100 percentage points in a whole. So, the decimal form of 40% is \(\frac{40}{100} = 0.40.\)

Step 2: Compare the Decimal Forms

Now that we have 0.40 and 0.35 in their decimal forms, we can easily compare them. 0.40 is more than 0.35. So, 40% is greater than 0.35.

Step 3: Write the Complete Statement

As 40% (or 0.40) is greater than 0.35, we can complete the statement as: 40% > 0.35.

Key Concepts

Decimal Conversion

Extending your understanding beyond basic arithmetic is key when dealing with decimals. While the move from whole numbers to decimals can seem daunting, the process is straightforward. Decimal conversion involves changing numbers from one format to another, such as from percentages to decimals.

Most commonly, you’ll see this with money, measurements, and percentages, where precision up to fractions of a unit is required. To convert a percentage to a decimal, you must understand that percentages represent a part out of one hundred. Therefore, to move from a percentage to a decimal, you simply divide the percentage by 100. This action effectively removes the percent symbol and places the decimal point correctly.

For instance, converting 75% to a decimal involves dividing 75 by 100 to get 0.75. It’s a practical skill that feeds into more complex math, such as calculating discounts, interest rates, and statistical data.

Percentage to Decimal

Turning a percentage into a decimal is an essential step in many mathematical calculations, particularly when you need to compare values that are expressed in different forms. The process is simple but bears repeating until it becomes second nature. As mentioned, you convert percentages to decimals by dividing by 100—essentially moving the decimal point two places to the left.

Why do this? Because percentages, at their root, express hundredths. So, when you see 50%, think of \(\frac{50}{100}\) or 0.5. For the visual learners, imagine the percent sign as a gate that, once opened by division, lets the numbers flow into a decimal form. Remembering this visualization can help you swiftly convert percentages like 20%, which becomes 0.20, or 125%, which becomes 1.25.

Inequalities

Inequalities are at the heart of comparing quantities. They're statements that relate numbers or expressions using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Understanding inequalities lets you determine the relative size of two numbers.

In practice, you often see inequalities used to compare numbers, where determining the greater or lesser value is essential—similar to a seesaw where one side is heavier (greater) than the other. When comparing decimals, line them up by their decimal points and compare digit by digit, starting from the leftmost non-zero digit. The first difference you encounter will usually determine the inequality.

It’s crucial to keep in mind that symbols pointing to the left (<) indicate 'less than', while those pointing to the right (>) signify 'greater than'. Think of the symbol as an arrow pointing towards the smaller number, or a hungry alligator that always eats the larger number. Visual cues like these can be helpful learning aids.