Problem 25 - Prealgebra

Question

Write each fraction or mixed number as a decimal. Use a bar to show a repeating decimal. $$-\frac{5}{11}$$

Step-by-Step Solution

Answer

The decimal is -0.\overline{45}.

Step 1: Set Up Division

To convert \(-\frac{5}{11}\) into a decimal, we need to perform division. This means dividing the numerator (-5) by the denominator (11). Begin by setting up the division: 5 divided by 11. Since the fraction is negative, the decimal will be negative as well.

Step 2: Perform Long Division

Start dividing 5 by 11: 1. 11 goes into 5 zero times, so write 0. 2. Add a decimal point to the 0 and extend a 0 to make it 50. 3. 11 goes into 50 four times (11 * 4 = 44), write 4 after the decimal and subtract to get 6. 4. Bring down another 0 to make 60. 5. 11 goes into 60 five times (11 * 5 = 55), write 5 next to the 4 and subtract to get 5. 6. Bring down another 0 to make 50 and repeat. Notice the repeating pattern.

Step 3: Identify the Repeating Pattern

The division results in 0.454545..., as the remainders cycle between 6 and 5 leading the digits after the decimal to repeat in a pattern of '45'. This indicates that the decimal is repeating.

Step 4: Write the Repeating Decimal

Express the repeating decimal using a bar notation. Since '45' repeats, the decimal representation of \(-\frac{5}{11}\) is written as -0.\overline{45}.

Key Concepts

Understanding Long Division

Long division is a method used to divide numbers and obtain a decimal result, especially when the division doesn't result in a whole number. In essence, long division helps break down complex division problems into smaller, manageable steps. To convert a fraction like \(-\frac{5}{11}\) into a decimal, follow the steps of long division:

• Divide the numerator (5) by the denominator (11). Start by noting that since 5 is smaller than 11, you initially deal with 0.
• Add a decimal point and extend with zeroes; think of 5 as 5.0000, which allows you to proceed with the division.
• Now, divide 50 by 11, getting parts of the quotient one digit at a time, working from left to right.

It’s essential to align the decimals correctly and understand that the operation continues until you either repeat a cycle or reach an endpoint.

Identifying Repeating Decimals

Repeating decimals are decimals that exhibit a repeating pattern. They're especially common when converting fractions with denominators that do not evenly divide into the numerator. For example, when you convert \(-\frac{5}{11}\) into a decimal, you notice that after performing long division steps, a recurring pattern emerges: the digits '45' continually repeat.

• Observe the process of long division: once you start getting a repeated sequence of remainders, the pattern of digits starts to repeat as well.
• This repeating pattern is often signified by placing a bar over the digits that repeat. For \(-\frac{5}{11}\), the decimal form is written as \(-0.\overline{45}\).

Recognizing this pattern helps in accurately expressing non-terminating decimals and is crucial for solving mathematical problems involving fractions converted to decimals.

Dealing with Negative Numbers in Division

Negative numbers can change the outcome of a calculation, especially when dealing with fractions. When converting fractions like \(-\frac{5}{11}\) into decimal form, it is important to remember the rules about signs in division. Here are some essential points to remember:

• If one number in the division is negative, the decimal result will also be negative. Given \(-5 \div 11\), the outcome \(-0.\overline{45}\) is negative because the numerator was negative.
• If both numbers were negative, the result would have been positive. This correlates with the rule that a negative divided by a negative results in a positive.

Keeping these rules in mind ensures you get the correct sign for your decimal, reinforcing your understanding of basic arithmetic operations involving negative numbers.