Problem 89 - Algebra 1: Concepts and Skills

Question

Complete the statement using \(<,>,\) or \(=.\) $$ 1.5 ? 150 \% $$

Step-by-Step Solution

Answer

The correct symbol to insert into the given statement is \(=\), making the final statement \(1.5 = 150\%\).

Step 1: Convert Percent to Decimal

To convert the percentage to a decimal, simply divide by 100. Therefore, 150% becomes \( \frac{150}{100} = 1.50 \).

Step 2: Comparing the two numbers

Now, since both numbers are in decimal form, it's clear that \(1.5 = 1.50\). So, the given statement with the comparison symbol inserted is \(1.5 = 150 \%\)

Key Concepts

Percent to Decimal Conversion

Understanding how to convert percentages to decimals is crucial in various fields of mathematics and real-life applications. The process is quite straightforward. As you saw in the exercise, to convert a percentage to a decimal form, you divide by 100. Why is that? Because the term 'percent' implies 'per hundred'. Therefore, 150% as a decimal becomes \[\begin{equation}\frac{150}{100} = 1.50.\end{equation}\]This simplifies the process of comparing it with other numbers that may already be in decimal form. It's like converting different measuring units to the same one before you compare or add them. Always remember to align decimal places when you write down your converted number, as this will help to avoid any confusion in the next steps of your comparisons.

Inequalities

Inequalities are a way of expressing the relationship between two values that are not equal. When dealing with inequalities, it's essential to determine which of the two numbers is larger or whether they are indeed equal. There are three primary symbols we use to show this:

• The 'greater than' (\(>\)) symbol indicates that the number on the left is larger than the one on the right.
• The 'less than' (\(<\)) symbol means the opposite — that the number on the left is smaller.
• The 'equals' (\(=\)) sign is used when two values are precisely the same.

In the context of the original exercise, deciding which symbol to use involves converting the percent to a decimal and then directly comparing the numbers. Recognizing and properly using inequalities is vital, as they often represent real-world situations, like comparing prices, distances, or measurements, and form the foundation for solving algebraic equations.

Mathematical Comparisons

When you compare numbers mathematically, it's like holding a race to see which number wins in terms of size or value. After converting percentages to decimals, comparing them is just like comparing any other numbers. You look at each number's place value, starting from the left, to determine their relationship.In our question with 1.5 and 1.50, once converted to the same format, you'll see that they're equal. This might seem surprising because 150 seems much larger than 1.5, but the decimal point changes everything. Remember that a zero at the end of a decimal number doesn't alter its value but can be crucial when aligning place values for complex calculations. When inserting mathematical comparisons into statements or equations, ensure you're using the correct symbol that represents the relationship you have uncovered through careful evaluation. This precision is a fundamental skill, as math is often used to interpret data, predict outcomes, and make decisions.