Problem 282 - Intermediate Algebra

Question

In the following exercises, convert each fraction to a decimal. $$ -\frac{18}{11} $$

Step-by-Step Solution

Answer

-1.\overline{63}

Step 1 - Understand the fraction

The given fraction to convert to a decimal is \(-\frac{18}{11}\). The negative sign indicates that the decimal number will also be negative.

Step 2 - Perform the division

Divide the numerator by the denominator: \(-18 \,÷\ 11\).

Step 3 - Calculate the decimal

Carrying out the division we get: \(-1.6363636\ldots\). The decimal is repeating as the digits '63' repeat indefinitely.

Step 4 - Write the repeating decimal properly

Represent the repeating part with a bar over the repeating digits: \(-1.\overline{63}\).

Key Concepts

Negative Fractions

When dealing with fractions, the negative sign plays an important role. A fraction is composed of two parts: the numerator (top number) and the denominator (bottom number). In the given fraction \(-\frac{18}{11}\), the negative sign means the overall value is less than zero. This means your final decimal will also be negative.

To solve any negative fraction:

• Ignore the negative sign initially and focus on converting the fraction.
  
• Once the fraction is in decimal form, add the negative sign back.
  

For example, without worrying about the negative sign, \(\frac{18}{11}\) would convert to \(1.6363636\ldots\). By reintroducing the negative sign, the final answer is \(-1.6363636\ldots\).

So, whenever you see a negative fraction, remember your final decimal must also carry that negative sign.

Repeating Decimals

When dividing numbers, sometimes you'll find that the decimal part keeps repeating forever. These are called repeating decimals.

In our example, the fraction \(-\frac{18}{11}\) when converted, gives \(-1.6363636\ldots\). Notice how the '63' keeps repeating? This is a classic repeating decimal situation.

To write repeating decimals properly:

• Identify the repeating sequence of digits. In our example, it is '63'.
  
• Use a bar (\overline) over the repeating digits. This is how you show that the sequence repeats indefinitely.
  

Therefore, \(-\frac{18}{11}\) becomes \(-1.\overline{63}\). Such notation makes it clear and concise that '63' keeps repeating.

Division of Fractions

Converting fractions to decimals involves the process of division. The numerator is divided by the denominator. For \(-\frac{18}{11}\), you're performing \(-18 ÷ 11\).

Here's how to handle the division:

• Write the fraction as a division problem: \(-18 ÷ 11\).
  
• Perform the division just like any other integer division. You may need to use long division if the numbers don't divide perfectly.
  
• If your result is a decimal that keeps going, check for a repeating pattern and apply the repeating decimal rules.
  

Performing the division \(-18 ÷ 11\) gives \(-1.6363636\ldots\), and marking the repeating '63' alters it to \(-1.\overline{63}\). Regular practice with such exercises will make these steps second nature.