Problem 80 - Algebra 1: Concepts and Skills

Question

Complete the statement using \(<,>,\) or \(=.\) $$ 110 \% ? 110 $$

Step-by-Step Solution

Answer

The correct relationship is 110% < 110.

Step 1: Translate the Percentage to a Decimal Value

A percentage represents a number out of 100. Therefore, 110% can be written as \( \frac{110}{100} \) in fractions or 1.10 in decimal form.

Step 2: Compare Decimal Value to 110

Now that we have converted 110% to a decimal (1.10), we can compare it to 110. Clearly, 1.10 is less than 110.

Key Concepts

Converting Percentages to Decimals

Understanding how to convert percentages to decimals is a basic but essential skill in mathematics. A percentage is essentially a fraction with a denominator of 100. For example, to convert 110% to a decimal, you divide 110 by 100, resulting in 1.10.

To make this process even more straightforward, simply move the decimal point two places to the left. For instance:

• 150% becomes 1.50 (150/100 = 1.50)
• 75% becomes 0.75 (75/100 = 0.75)

Keep in mind that any percentage greater than 100% will translate to a decimal greater than 1, signaling that it's more than the whole. By mastering this conversion, comparing percentages and numbers becomes much simpler.

Inequalities with Percentages

When comparing inequalities with percentages, it helps to make a uniform basis for comparison. By converting percentages to decimals or to their equivalent fraction form, you can compare them with other numbers effectively. This is particularly useful in situations where one number is represented as a percentage and the other is a different form, such as an integer or a fraction.

For instance, consider the inequality 110% ? 110. After converting 110% to a decimal (1.10), it becomes clear that 1.10 is significantly smaller than 110. Therefore, in the context of this exercise, the correct symbol to complete the statement is '<', representing that 1.10 (or 110%) is less than 110.

Basic Algebra Concepts

Algebra involves using symbols and letters to represent numbers and quantities in mathematical expressions and equations. One of the foundational concepts in algebra is understanding the equality and inequality of expressions. Inequalities show the relationship between expressions that are not necessarily equal and can be represented using symbols such as < (less than), > (greater than), and = (equal to).

Basic algebra teaches how to manipulate these expressions to solve for unknown values or to compare values, as in the exercise where we compare a percentage with a natural number. By applying the concept of converting percentages to decimals and understanding inequalities, we can solve algebraic expressions and equations that involve percentages efficiently and effectively.