Problem 104 - Algebra 1: Concepts and Skills

Question

Write the fraction as a terminating or repeating decimal. $$ \frac{6}{11} $$

Step-by-Step Solution

Answer

The fraction \(\frac{6}{11}\) as a repeating decimal is 0.54 (repeating).

Step 1: Division

The fraction \(\frac{6}{11}\) can be interpreted as the division 6 ÷ 11. When performed, this operation returns roughly 0.5454545454545454.

Step 2: Identify if the decimal is terminating or repeating

Looking at the result, it can be seen that the digits 54 are repeating indefinitely. This makes the fraction a repeating decimal.

Key Concepts

Converting Fractions to Decimals

Converting fractions to decimals is a fundamental math skill that is incredibly useful in a variety of real-world and academic situations. When we have a fraction, such as \(\frac{6}{11}\), it represents a division problem where the numerator (6) is divided by the denominator (11). We want to express this division as a decimal number.

• Start by setting up a long division, where 6 is divided by 11.
• The goal is to find the decimal equivalent.
• This process often reveals whether the decimal is terminating or repeating.

Converting fractions to decimals can sometimes seem daunting, but it's really just straightforward division. Once you know the steps, it becomes easy to determine the type of decimal you are dealing with.

Understanding Division for Fractions

To convert a fraction into a decimal using division, you'll need to perform long division. Let's consider the example of converting \(\frac{6}{11}\) into a decimal. Here's what happens in a step-by-step breakdown:

• Set up 6 as the dividend and 11 as the divisor.
• Since 11 cannot go into 6, you place a decimal point and add a zero, making it 60.
• 11 goes into 60 five times, and this gives 55 when multiplied (since 5 × 11 = 55).
• Subtract 55 from 60, which leaves you with 5, then bring down another zero to make it 50.
• Repeat the process: 11 goes into 50 four times (4 × 11 = 44).
• Subtract, bring down zeros, and continue this pattern. You will notice a repeating cycle of the digits '54'.

With division, each step brings you closer to the decimal representation. When the digits start repeating, you've found a repeating decimal!

What are Terminating and Repeating Decimals?

Decimals derived from fractions can be categorized into terminating and repeating decimals. This is crucial for understanding the nature of the fraction when expressed in a decimal form.

• Terminating Decimals: These decimals have a finite number of digits after the decimal point. For instance, \(\frac{1}{4} = 0.25\) is a terminating decimal since it ends after two digits.
• Repeating Decimals: These have one or more repeating digits after the decimal point. For example, in \(\frac{6}{11}\), the sequence '54' repeats continuously, making it a repeating decimal.

Determining whether a decimal is terminating or repeating helps in understanding the precision and representation of numbers in mathematical calculations and real-life applications.