Problem 58 - Prealgebra

Question

Find a terminating and a repeating decimal between \(\frac{1}{6}\) and \(\frac{8}{9} .\) Explain how you found them.

Step-by-Step Solution

Answer

A terminating decimal is 0.25, and a repeating decimal is 0.3333... (\(\frac{1}{3}\)).

Step 1: Understand Terminating and Repeating Decimals

A decimal is terminating if it has a finite number of digits after the decimal point. A repeating decimal has one or more digits that repeat infinitely. For example, 0.5 is a terminating decimal, while 0.666... (which represents two-thirds) is a repeating decimal.

Step 2: Convert Fractions to Decimals

Convert the fractions \( \frac{1}{6} \) and \( \frac{8}{9} \) to decimal form.\[ \frac{1}{6} \approx 0.1666... \quad \text{(repeating decimal)} \ \frac{8}{9} = 0.8888... \quad \text{(repeating decimal)}\]These forms confirm the boundaries as repeating decimals, with \( \frac{1}{6} \) as 0.1666... and \( \frac{8}{9} \) as 0.8888... .

Step 3: Find a Terminating Decimal Between the Bounds

We can choose any number between 0.1666... and 0.8888..., ensuring it terminates. For example, 0.25 is between these values. 0.25 is a terminating decimal as it has a finite number of digits.

Step 4: Find a Repeating Decimal Between the Bounds

To find a repeating decimal between 0.1666... and 0.8888..., we look for a simple fraction whose repeating nature is evident. For example, \( \frac{1}{3} = 0.3333... \) is a repeating decimal that resides in the desired range.

Key Concepts

Terminating Decimals

Terminating decimals are numbers that come to a clear end after a few places. They do not go on forever. For example, 0.5, 2.75, and 7.125 are all terminating decimals because they stop after a certain number of digits.

One powerful trick to recognize a terminating decimal is to examine its fraction form. If the fraction, in its simplest form, has a denominator with only the prime factors of 2 and/or 5, it will convert into a terminating decimal. This happens because our number system is based on ten, which is the product of 2 and 5.

• Example: The fraction \( \frac{1}{4} \), when converted, results in the decimal 0.25, which is terminating.
• Example: Similarly, \( \frac{5}{8} \) translates to the terminating decimal 0.625.

Understanding this can help you quickly determine if a fraction will result in a decimal that ends.

Repeating Decimals

Repeating decimals differ from terminating decimals in that they go on forever, with a set of digits continually repeating. These repeating sections are often denoted with a bar above the repeating digits. For instance, \( 0.333... \) can be written as \( 0.\overline{3} \), indicating that the 3 is repeating.

Many fractions, when converted to decimal form, become repeating decimals if their denominator has prime factors other than 2 and 5. This extended repetition happens because of the inability to completely divide them cleanly in base ten.

• Example: \( \frac{1}{3} \) turns into \( 0.333... \).
• Example: \( \frac{2}{11} \) results in \( 0.1818... \) or \( 0.\overline{18} \).

Recognizing and writing repeating decimals is crucial when converting fractions or working with non-terminating decimals.

Fraction to Decimal Conversion

Converting a fraction to a decimal can transform a simple math challenge into an understandable numeric form. This process involves division: the numerator (top number) is divided by the denominator (bottom number). The resulting quotient is the decimal form. Here's how you do it:

1. **Division**: Perform long division of the numerator by the denominator.

• Example: To convert \( \frac{3}{4} \) to a decimal, divide 3 by 4, resulting in 0.75, a terminating decimal.

2. **Noticing the Pattern**: Some fractions, upon conversion, will show a repeating pattern if they divide into non-ending decimals.

• Example: \( \frac{7}{9} \) becomes \( 0.777... \), indicating a repeating decimal \( 0.\overline{7} \).

3. **Simplifying the Fraction**: Always simplify the fraction first to predict whether it results in a terminating or repeating decimal.

• Example: Simplifying \( \frac{10}{20} \) to \( \frac{1}{2} \), you can determine its decimal \( 0.5 \), which is terminating.

Mastering this conversion process helps in tackling various mathematical problems, revealing whether each fraction turns into a clear-cut or an endless decimal.