Problem 49 - Algebra 1: Concepts and Skills

Question

Write the numbers in increasing order. \(4.8,-2.6,0,-\frac{7}{2}, \frac{1}{2},-\frac{1}{2}\)

Step-by-Step Solution

Answer

-\(\frac{7}{2}\), -2.6, -\(\frac{1}{2}\), 0, \(\frac{1}{2}\), 4.8

Step 1: Convert fractions into decimals

In order to compare fractions and decimals, it is easier if all numbers are in the same format. By converting \(-\frac{7}{2}\), \(\frac{1}{2}\), and \(-\frac{1}{2}\) into decimals, we obtain -3.5, 0.5, and -0.5, respectively.

Step 2: List all numbers

Now all numbers are decimals and can be easily compared: 4.8, -2.6, 0, -3.5, 0.5, -0.5.

Step 3: Arrange in increasing order

In increasing order (from smallest to largest), the numbers are: -3.5, -2.6, -0.5, 0, 0.5, 4.8. Since these were derived from the original list by simple conversions, this is also the order of the original list when arranged in increasing order.

Key Concepts

Converting Fractions to Decimals

Fractions and decimals are simply two ways to represent parts of a whole. To easily compare fractions and decimals, it’s useful to convert all numbers to one format—decimals, in most cases.

Converting fractions to decimals involves division. The fraction tells you to divide the numerator (top number) by the denominator (bottom number). For example:

• For \(-\frac{7}{2}\), divide -7 by 2, resulting in -3.5.
• For \(\frac{1}{2}\), divide 1 by 2, which gives 0.5.
• For -\(\frac{1}{2}\), divide -1 by 2, yielding -0.5.

Once you've completed these divisions, you'll have the equivalent decimal form of these fractions. These conversions make comparisons easier since decimals align with our base-10 system.

Comparing Decimals

Once all your numbers are in decimals, comparing them becomes straightforward. Decimals are compared using two main considerations:

The significant digit to the left, and subsequent digits to the right as you move from left to right. In a list of decimals, smaller numbers appear to the left, and larger numbers appear to the right. Consider the decimal places sequentially:

• Compare the first digit of each decimal; smaller values take precedence.
• If the initial digits are the same, compare the next digit to the right.

For example, among -3.5, -2.6, -0.5, 0, 0.5, and 4.8:

• -3.5 is smaller than -2.6, as -3 is smaller than -2.
• Similarly, -2.6 is smaller than any positive number.

By comparing these decimals step-by-step, you can easily determine their relative sizes.

Increasing Order of Numbers

Arranging numbers in increasing order means placing them from the smallest to the largest. This is an important skill, as it assists in understanding number values relative to each other. Let's see how this applies after converting fractions to decimals and comparing them.

The steps for ordering a series of numbers are simple:

• First, convert any fractions to decimals if needed.
• Next, compare each number, start from the lowest value based on the decimals.
• Lastly, list the numbers in order, taking special care with negative numbers, as they are smaller than positive numbers.

Using our decimals: -3.5, -2.6, -0.5, 0, 0.5, and 4.8, these would be arranged in increasing order. Simply start with -3.5 (the smallest) and work your way to 4.8 (the largest).