Problem 87 - Prealgebra

Question

Complete Exercises 86 and 87 without actually finding the percents. Does 0.055 represent a percent greater than \(1 \%\) or less than \(1 \% ?\)

Step-by-Step Solution

Answer

0.055 represents a percent greater than 1%.

Step 1: Understanding Percent Representations

A percent is a way to express a number as a part of a hundred. Therefore, a value of \(1\%\) is equivalent to the decimal \(0.01\). To determine if \(0.055\) represents a percent greater than or less than \(1\%\), we need to recognize where \(0.055\) stands in relation to \(0.01\).

Step 2: Compare 0.055 with 0.01

Now that we know \(1\%\) equals \(0.01\), we compare \(0.055\) to \(0.01\). Since \(0.055\) is greater than \(0.01\), it represents a number greater than \(1\%\).

Step 3: Verify Comparison

For further clarity, think of both decimals in terms of percentages: \(0.055\) is the same as \(5.5\%\) and \(0.01\) is equal to \(1\%\). Clearly, \(5.5\%\) is greater than \(1\%\).

Key Concepts

Decimal Comparison

When dealing with decimals, understanding how to compare them is vital. Each decimal has a specific value, generally interpreted based on its fractional parts divided by powers of ten. A simple way to compare decimals like 0.055 and 0.01 is to look at their digits from left to right, focusing on the most significant place value.

For instance, start comparing numbers from the decimal point. Here, 0.055 can be seen as a number with 0 in the unit hundredth and 5 in the thousandth place. While 0.01 has 0 in the unit hundredth place and 1 in the thousandth position. You can compare as:

• If one is further to the right than the other in value, it's larger (0.055 as 55/1000 is greater than 0.010 as 10/1000).
• Visualize these comparisons to understand which decimal represents a bigger or smaller quantity.

This approach helps in identifying which value is larger or smaller, preparing you better for comparing percentages in different contexts.

Percentage Representation

Percentages offer a helpful way to express decimal numbers as parts of a whole, specifically out of 100. This means that each percentage directly correlates to a decimal when divided by 100. Thus, understanding how decimals translate into percent can help you in different calculations.

For instance, let's convert 0.055 into a percentage:

• Multiply the decimal by 100 to transform it into a percent: 0.055 × 100 = 5.5%.

Similarly, for 0.01, the percentage is:

• 0.01 × 100 = 1%.

These percentages make clear representations: 5.5% appears larger than 1%, helping decide that 0.055 is greater. This thinking is particularly useful when needing to understand or solve questions about what percentages mean in real-world scenarios.

Prealgebra Concepts

Prealgebra is vital for building a foundation in mathematics, covering basic number theory, operations, and relationships. One core concept is manipulating decimals and percentages, which bridges arithmetic and algebra.

Through prealgebra, one learns:

• Identifying relationships between decimals and percentages, enabling easier conversion.
• The importance of place value is to understand comparative sizes in decimals.
• Simplifying expressions using basic operations, which paves the way to grasp more advanced mathematical concepts, such as algebra.

By mastering these concepts, one can solve real-world problems more effectively, such as comparing data to interpret financial information or changes in quantities. Approaching these problems with an understanding gained through prealgebra not only improves mathematical fluency but also boosts overall confidence in handling numbers.