Problem 53 - Prealgebra

Question

Order each group of numbers from least to greatest. $$-1 . \overline{1},-1 \frac{1}{8},-1 \frac{1}{10}$$

Step-by-Step Solution

Answer

Order: \(-1 \frac{1}{8}, -1.\overline{1}, -1 \frac{1}{10}\).

Step 1: Convert Mixed Numbers to Improper Fractions

The numbers given are: - \(-1\overline{1}\), - \(-1 \frac{1}{8}\), - \(-1 \frac{1}{10}\).First, we convert the mixed numbers to improper fractions. - For \(-1 \frac{1}{8}\): Convert to improper fraction: \(- \left(\frac{8+1}{8}\right) = -\frac{9}{8}\).- For \(-1 \frac{1}{10}\): Convert to improper fraction: \(- \left(\frac{10+1}{10}\right) = -\frac{11}{10}\).

Step 2: Convert Repeating Decimal to Fraction

Now, we convert the repeating decimal \(-1.\overline{1}\) into a fraction.- Let \(x = -1.\overline{1}\).- Multiply by 10 to shift the decimal: \(10x = -11.\overline{1}\).- Subtract these two equations: \(10x - x = -11.\overline{1} - (-1.\overline{1})\) \(9x = -10\)- Solve for \(x\): \(x = -\frac{10}{9}\).

Step 3: Find a Common Denominator

To compare \(-\frac{9}{8}\), \(-\frac{11}{10}\), and \(-\frac{10}{9}\), we find a common denominator.1. Compute the least common multiple (LCM) of the denominators 8, 9, and 10. - LCM is 360.2. Convert each fraction:- \(-\frac{9}{8} = -\frac{9 \times 45}{360} = -\frac{405}{360}\)- \(-\frac{11}{10} = -\frac{11 \times 36}{360} = -\frac{396}{360}\)- \(-\frac{10}{9} = -\frac{10 \times 40}{360} = -\frac{400}{360}\).

Step 4: Order the Fractions

Now that all fractions have a common denominator, we can order them:- \(-\frac{405}{360}\)- \(-\frac{400}{360}\)- \(-\frac{396}{360}\)From least to greatest:1. \(-\frac{405}{360}\) which is \(-\frac{9}{8}\) or \(-1 \frac{1}{8}\)2. \(-\frac{400}{360}\) which is \(-\frac{10}{9}\) or \(-1.\overline{1}\)3. \(-\frac{396}{360}\) which is \(-\frac{11}{10}\) or \(-1 \frac{1}{10}\).

Key Concepts

Converting Mixed Numbers to Improper Fractions

Mixed numbers, such as \(-1 \frac{1}{8}\) or \(-1 \frac{1}{10}\), combine a whole number with a fraction. To make them easier to work with, especially when comparing, converting them into improper fractions is a good starting plan. Improper fractions have numerators larger than their denominators, offering a consistent way to handle numbers, whether positive or negative.
To convert a mixed number:

• Calculate the total numerator by multiplying the whole number by the fraction's denominator and then adding the fraction's numerator.
• For instance, to convert \(-1 \frac{1}{8}\) to an improper fraction: \(-1\) means a total part of 8 divided pieces. Add the \(1\) piece on top for \(-\frac{9}{8}\).
• Apply the same steps to other mixed numbers you encounter, like \(-1 \frac{1}{10}\), ending with \(-\frac{11}{10}\).

Converting to improper fractions simplifies the comparison process since you can more easily find a common basis for comparison.

Repeating Decimals to Fractions

Repeating decimals, like \(-1.\overline{1}\), can look overwhelming at first. However, they can be converted into fractions to simplify the comparison process with other numbers.
A step-by-step process to convert a repeating decimal to a fraction works as follows:

• Let's consider \(x = -1.\overline{1}\), indicating the decimal 1 repeats indefinitely.
• Multiply this by 10 to shift the repeat, giving you \(10x = -11.\overline{1}\).
• Subtract the original equation from this new one, effectively removing the repeating part: \(10x - x = -11.\overline{1} - (-1.\overline{1})\).
• This equates to \(9x = -10\). Solving it gives you \(x = -\frac{10}{9}\).

Getting from a repeating decimal to a simple fraction helps align all our terms into compatible formats for direct comparison methods.

Least Common Multiple

To effectively compare fractions, particularly those with different denominators, finding the least common multiple (LCM) is key. The LCM is the smallest number that all denominators can divide into without remainder. Calculating the LCM keeps our fraction operations aligned.
For denominators 8, 9, and 10 as in our exercise:

• First, list the multiples of each denominator.
• Identify the smallest shared multiple across all lists: 360 is the result in this scenario.

Converting each fraction to have this common denominator allows direct number comparison. This is done by calculating the necessary multiplier for each original fraction denominator to create equivalent fractions, establishing a consistent footing for ordering.

Comparing Fractions

Once fractions share a common denominator, comparing them becomes straightforward.

• Align all fractions to have this shared denominator.
• Directly compare numerators, determining which is greater or smaller.

In our case:

• \(-\frac{9}{8}\), which becomes \(-\frac{405}{360}\), has the smallest numerator value among our adjusted fractions making it the smallest original fraction.
• \(-\frac{10}{9} = -\frac{400}{360}\), is larger.
• \(-\frac{11}{10} = -\frac{396}{360}\), is the largest.

Ensure your fractions are in the same format before proceeding to order them easily, thus avoiding any misinterpretation of value and magnitude.