Problem 86 - Algebra 1: Concepts and Skills

Question

Complete the statement using \(<,>,\) or \(=.\) $$ 0.75 ? 85 \% $$

Step-by-Step Solution

Answer

0.75 < 85 \%

Step 1: Convert decimal into a percentage

First convert the decimal 0.75 into a percentage. This can be done by multiplying by 100 and adding a percentage sign (%) to the value. So, \(0.75 = 0.75 * 100 \% = 75 \% \).

Step 2: Compare the two percentages

Next, compare the two percentages, \(75 \% \) and \(85 \% \). The percentage \(75 \% \) is less than \(85 \% \).

Key Concepts

Decimal to Percentage Conversion

Understanding how to convert a decimal to a percentage is fundamental in math and everyday life. Here's how it works: to convert a decimal into a percentage, you simply multiply the decimal by 100. That transformation turns the decimal into a value representative of a part out of a whole, which is what percent means — per hundred.

For example, the decimal 0.75 can be converted into a percentage by calculating \(0.75 \times 100\). This gives us 75, and we add the percentage sign to get 75%. This means that 0.75 is the same as 75% when we talk about proportions of a whole. Remember, moving the decimal point two places to the right can help in quick conversion without a calculator.

Exercise Improvement Advice

• Emphasize that the multiplication by 100 is essentially shifting the decimal point two places to the right.
• Highlight that adding the percentage sign is a crucial step to denote that the number is now in percentage form.

Comparing Percentages

Once decimals are converted to percentages, it becomes much easier to compare them because you are dealing with like terms. Comparing percentages is all about understanding which of the two figures represents a larger part of a whole.

A common mistake is to overlook that percentages are inherently comparative. For instance, when comparing 75% and 85%, it's clear that 85% is larger because each percentage is viewed in terms of 100 parts; 85 out of 100 is more than 75 out of 100. Thus, in this context, \(75\% < 85\%\).

Exercise Improvement Advice

• Use real-life examples to explain that a greater percentage signifies a greater amount or portion relative to the whole.
• Demonstrate the comparison visually, perhaps by showing pie charts to give a graphical representation of the percentages.

Percentage Arithmetic

Working with percentages often involves more than just conversion and comparison. Percentage arithmetic includes addition, subtraction, multiplication and division of percentages, which can be applied to various real-life scenarios like calculating discounts, interest rates, and statistical data.

To effectively use percentage arithmetic, always remember to first convert all the terms you are working with into percentages. This ensures consistency and accuracy in your calculations. For instance, when asked to find out how much a 10% increase from 50% is, we first change the 10% increase to a decimal by dividing by 100, making it 0.10, and then add it to 0.50 (50% as a decimal) which sums up to 0.60 or 60%.

Exercise Improvement Advice

• Show step-by-step how each arithmetic operation applies to percentages and how it parallels operations with whole numbers.
• Provide examples of how these operations would be used in practical scenarios, for example, calculating the final price after a discount and tax are applied.