Problem 94 - Algebra 1: Concepts and Skills

Question

Complete the statement using \(<,>,\) or \(=.\) $$ 2.25 ? 250 \% $$

Step-by-Step Solution

Answer

The correct symbol to complete the expression is \(<\). So, the complete statement is \(2.25 < 250\%\).

Step 1: Convert the percentage to a decimal

To convert a percentage into a decimal, you have to divide the percentage by 100. So, 250% is converted to a decimal by doing \(250 \div 100 = 2.5\)

Step 2: Compare the decimal numbers

Now that both numbers are in decimal form, they can be compared. It's clear that 2.25 is less than 2.5.

Key Concepts

Converting Percentages to Decimals

Understanding how to convert percentages to decimals is crucial in mathematics, as it allows for the comparison between different forms of expressing quantities. The conversion process is straightforward: to convert a percentage into a decimal, simply divide the percentage by 100.

This method removes the percent symbol, which essentially means 'per hundred', and converts the value into a decimal that can be easily worked with in various calculations. For example, if we take 250%, we divide it by 100 (250 ÷ 100) and get 2.5 as a decimal.

• 50% becomes 0.50 or just 0.5
• 125% becomes 1.25
• 250% becomes 2.5

Remember, moving the decimal point two places to the left is the same as dividing by 100. This simple yet essential technique facilitates greater understanding and manipulation of figures in mathematical problems.

Decimal Comparison

Comparing decimals is another key concept in mathematics. When comparing two decimals, you look at the highest place value first and move to the right until a difference is found. For numbers with different whole number parts, like 2.25 and 2.5, it's clear that 2.5 is greater because the whole number part of both is 2, but the tenth place in 2.5 is greater than the tenth place in 2.25.

Here's a simple method for comparing decimals:

• Align the numbers by their decimal point.
• Compare the highest place value columns first.
• If those are equal, move to the next place value to the right, and so on.

Applying this method, it's evident that 2.25 is less than 2.5 since the former has a 2 in the tenths place, while the latter has a 5.

Percentages in Math

Percentages are a fundamental part of mathematics and are used to express how one quantity relates to another on a scale of 100. The term 'percent' means 'out of 100', and is symbolized by the '%'. This representation is extremely useful for comparing relative quantities or expressing ratios.

Different ways to encounter percentages include:

• Finding the percentage of a number.
• Calculating discounts or interest rates.
• Converting between fractions, decimals, and percentages.

Understanding percentages thus becomes indispensable not just for academic pursuits in math, but for practical, everyday uses like determining sales tax, calculating tips, or understanding batting averages in baseball.