Problem 103 - Algebra 1: Concepts and Skills

Question

Write the fraction as a terminating or repeating decimal. $$ \frac{8}{15} $$

Step-by-Step Solution

Answer

The decimal representation of the fraction \( \frac{8}{15} \) is a repeating decimal, specifically 0.53.

Step 1: Division

The basic method to convert a fraction into a decimal is by dividing the numerator by the denominator. Here, divide the numerator 8 by the denominator 15. The result is 0.5333333333.

Step 2: Identifying the Decimal Type

Notice that the decimal form contains the digit '3' repeating, and the pattern appears to continue indefinitely. This indicates that this decimal is a repeating decimal.

Key Concepts

Repeating Decimals

Repeating decimals, often referred to as recurring decimals, are decimals in which a sequence of one or more digits repeat endlessly. This means that after a certain point, the same set of digits appears over and over again. In some cases, the entire decimal part may repeat. In others, a mix of different numbers and repeating sections may be observed. A classic example is when you divide 1 by 3, which equals 0.3333..., or simply 0.\overline{3}. Another example is the fraction \( \frac{8}{15} \), which converts to a decimal that appears as 0.5333..., but can be expressed more compactly as 0.5\overline{3}. This notation indicates the digit '3' repeats indefinitely.

Distinguishing between repeating and terminating decimals:

• If the same digits continuously repeat, it’s a repeating decimal.
• The repeating sequence is indicated by a line (bar) over the repeating digit(s).
• Repeating decimals represent rational numbers, which can be expressed as a fraction.

Terminating Decimals

Terminating decimals come to an end after a finite number of digits. They are precise and easy to work with since they don't continue indefinitely. When converting fractions to decimals, if the division of the numerator by the denominator ends without a remainder, the result is a terminating decimal. For instance, the fraction \( \frac{1}{4} \) equals 0.25, a perfect example of a terminating decimal because it stops after two decimal places.

Recognizing Terminating Decimals:

• They simply conclude after a certain number of decimal digits without any repetition.
• All terminating decimals are rational numbers since they can be written as fractions.
• They frequently arise when a fraction's denominator (in its simplest form) only contains prime factors of 2 or 5.

Long Division Method

The long division method is a traditional technique used to divide two numbers to find their quotient, and it's essential for converting fractions to decimals. To use this method, divide the numerator of the fraction by the denominator. For instance, when converting \( \frac{8}{15} \) into a decimal, divide 8 by 15. Set up the division as 8.000 divided by 15.

Steps of Long Division:

• Setup: Place 8 inside the division bracket and 15 outside.
• Divide: Since 15 can't go into 8, place a 0 in the quotient and bring down a 0 to make it 80.
• Continue: 15 goes into 80 five times (as close as possible without exceeding), which equals 75. Subtract 75 from 80 to get 5, then bring down the next zero.
• Repeat: Continue this process, bringing down zeros and repeating divisions. You’ll notice after a few steps the digits repeat, leading to a repeating decimal.

Understanding long division provides a solid foundation for working with both terminating and repeating decimals, helping clarify how fractions convert into these forms.