Problem 107 - Algebra 1: Concepts and Skills

Question

Write the fraction as a terminating or repeating decimal. $$ \frac{5}{16} $$

Step-by-Step Solution

Answer

The fraction \( \frac{5}{16} \) as a decimal is 0.3125

Step 1: Understand the Fraction

The given fraction is \(\frac{5}{16}\). The numerator is 5 and the denominator is 16. The goal is to express this fraction as a decimal.

Step 2: Perform the Division

The main idea behind the transformation of fractions into decimals is to perform a division. The numerator should be divided by the denominator, i.e., \(5 ÷ 16\).

Step 3: Write Down the Decimal Result

Performing the division \(5 ÷ 16\) results in the decimal 0.3125

Key Concepts

Terminating Decimals

When a fraction is expressed as a decimal, it can either come to an end or continue indefinitely. Terminating decimals are those that end at some point. In simple terms, after performing the division, the decimal does not continue infinitely. For example, in the fraction \( \frac{5}{16} \), performing the division \(5 \div 16\) results in the decimal 0.3125, a number that stops with no further digits. This means it's a terminating decimal.
A key feature of terminating decimals is that the denominator of the fraction, once in its simplest form, only has 2 and/or 5 as prime factors. This is what allows for the decimal to "terminate" or stop after a certain number of digits.

Repeating Decimals

Sometimes, when you convert fractions to decimals, you end up with a decimal that never ends but instead, develops a repeating pattern. These are known as repeating decimals. For instance, consider the fraction \(\frac{1}{3}\). If you divide 1 by 3, the result is 0.333..., with the digit 3 continuing forever. This is a classic example of a repeating decimal.

• A repeating decimal is represented by placing a line (called a vinculum) over the repeating digits. For example, 0.333... can be represented as \(0.\overline{3}\).
• Repeating decimals occur when the denominator (in its simplest form) of the fraction contains any prime factors other than 2 or 5.

Understanding repeating decimals is important when classifying fractions presented as decimals because it aids in distinguishing between numbers that end (terminate) and those that have an endless, cyclical nature.

Division of Fractions

Converting a fraction to a decimal generally involves division. Here’s how it works: the top number (numerator) is divided by the bottom number (denominator). Take the fraction \(\frac{5}{16}\). In this case, you would divide 5 by 16.

• This process involves performing long division as you might have learned in elementary math. You may find it helpful to practice this skill, as it is fundamental for converting fractions to decimals.
• Sometimes, the division might end quickly, leading to a terminating decimal, as we saw with \(\frac{5}{16}\).
• Other times, the division might not end neatly, resulting in a repeating decimal.

By mastering the division of fractions, you will gain greater insights into the relationship between fractions and their decimal counterparts, facilitating a deeper understanding of mathematical concepts.