Problem 26 - Introductory and Intermediate Algebra for College Students 4th

Question

Express each rational number as a decimal. $$\frac{5}{16}$$

Step-by-Step Solution

Answer

The decimal form of \( \frac{5}{16} \) is 0.3125

Step 1: Identify the rational number

The given rational number is \( \frac{5}{16} \). A rational number is a number that can be expressed as the quotient of two integers where the denominator is not zero.

Step 2: Divide the numerator by the denominator

To convert a fraction to a decimal, you divide the numerator by the denominator. Therefore, divide 5 by 16.

Step 3: Perform the division

By performing the division, 5 divided by 16 gives the decimal 0.3125.

Key Concepts

Rational Numbers

A rational number is any number that can be expressed as the quotient or division of two integers, where the denominator is not zero. This means that you can write a rational number in the form \( \frac{a}{b} \), where \(a\) and \(b\) are integers, and \(b eq 0\). These numbers include fractions, integers, and finite or repeating decimals.

Examples of rational numbers include:

• \( \frac{3}{4} \) - This is a typical fraction.
• 7 - This is a whole number, which can be expressed as \( \frac{7}{1} \).
• 0.5 - This decimal can be converted into the fraction \( \frac{1}{2} \).
• -0.75 - This can be expressed as \( \frac{-3}{4} \).

Understanding rational numbers is crucial because it sets the foundation for learning how to manipulate fractions and decimals. Recognizing that every rational number can be expressed in multiple forms helps deepen mathematical understanding.

Convert Fractions to Decimals

Converting fractions to decimals is a common task in math. This involves dividing the numerator by the denominator. For example, to convert \( \frac{5}{16} \) to a decimal, you will perform the division of 5 by 16.

This conversion is useful because decimals often make it easier to compare numbers or perform calculations. For instance, using decimals simplifies arithmetic operations like addition, subtraction, multiplication, and division.

To perform the conversion:

• Write down the numerator and the denominator of your fraction.
• Divide the numerator (top number) by the denominator (bottom number).
• The result will be a decimal representation of the fraction.

Sometimes, the division might not end and the decimal may repeat. In such cases, knowing the repeated sequence helps you express it accurately, such as writing 0.333... as 0.3̅.

Numerator and Denominator

In any fraction, there are two main parts: the numerator and the denominator. It's key to understand what these terms mean because they determine the value of the fraction.

- **Numerator**: This is the top part of a fraction. It indicates how many parts are considered out of the whole. For instance, in \( \frac{5}{16} \), the numerator is 5, meaning we are focusing on 5 parts.- **Denominator**: This is the bottom part of a fraction. It shows the total number of equal parts that make up the whole. In our example of \( \frac{5}{16} \), the denominator is 16, indicating the entire is divided into 16 equal parts.
Understanding these components is crucial:

• The denominator gives the context for the fraction and shows how it compares to a whole.
• The numerator tells you what fraction of the whole you're dealing with.

This clear understanding aids in operations involving fractions and converting them into decimals.

Long Division Process

The long division process is a systematic approach for dividing numbers that is especially useful for converting fractions into decimals.

To convert \( \frac{5}{16} \) to a decimal using long division, follow these steps:

• Set up the division by writing 5 as the dividend inside the division symbol and 16 as the divisor outside.
• Since 5 is smaller than 16, you need to add a decimal point to the quotient and a zero to 5, making it 50.
• Divide 50 by 16, which goes 3 times (since 16 x 3 = 48), and write 3 above the division symbol.
• Subtract 48 from 50 to get 2, then bring down another 0, making it 20.
• Continue the process by dividing 20 by 16, which goes 1 time (since 16 x 1 = 16). Write 1 in the quotient.
• Repeat the process until you reach desired decimal places or recognize a repeating pattern.

Following these steps results in the decimal 0.3125 for \( \frac{5}{16} \). Being precise with each step ensures accurate conversion.