Problem 24 - Introductory Algebra for College Students

Question

Express each rational number as a decimal. $$\frac{3}{20}$$

Step-by-Step Solution

Answer

\(\frac{3}{20} = 0.15\)

Step 1 - Identify the Numerator and the Denominator

The number \(3\) is the numerator and \(20\) is the denominator. We will need to divide the numerator by the denominator.

Step 2 - Perform the Division

Now perform the division. When you perform \(3 \div 20\), you get \(0.15\).

Step 3 - Write the Result as a Decimal

The result \(0.15\) is already in decimal form. Thus, \( \frac{3}{20} = 0.15 \).

Key Concepts

Numerator and Denominator

When working with rational numbers, a basic concept to understand is that of the numerator and the denominator. A rational number is a number that can be expressed as a fraction where both the numerator and the denominator are integers. In the fraction \( \frac{3}{20} \), the number 3 is known as the numerator. This is the number that appears on top of the fraction. It indicates how many parts of the whole are being considered. The number 20, on the other hand, is the denominator. It is placed below the line in the fraction and it indicates the total number of equal parts the whole is divided into.

This concept is crucial because the manner of dividing these numbers directly affects how we interpret or express the rational number in different forms. The numerator and denominator need to be correctly identified to move forward with calculations such as division or conversion to other forms, such as decimal numbers.

Decimal Conversion

Converting a rational number into a decimal format can make it easier to understand and use in calculations. The decimal form is typically more intuitive for many applications in daily life and practical scenarios. This process involves dividing the numerator by the denominator. For the fraction \( \frac{3}{20} \):

• The decimal form of \( \frac{3}{20} \) is found by calculating how many times 20 fits into the number 3, which in practice means performing a division.


• The conversion gives us a clearer picture: for example, \( \frac{3}{20} \) translates into 0.15, which can be more easily interpreted and compared with other decimal numbers.

Understanding decimal conversion is useful in many contexts. It helps in making fractions comparable through a unified scale - the decimal system.

Division Process

The division process is essential for converting a fraction into a decimal. It involves performing simple division where the numerator is divided by the denominator. When you divide 3 by 20, you determine how much of one unit (of the denominator) the numerator represents. Let's look at the mechanics:

• Begin with the numerator 3 and divide by 20.
• In this case, since 3 is less than 20, you know you are going to get a result less than 1. So, you place a decimal point and start adding zeros to the numerator.
• Proceed by converting this to 30, 300, etc., checking each time how many times 20 fits in. As you divide the "extended" number, 20 fits into 60 one time with a remainder of 0, giving you 0.15.

This division process results in a decimal, helping us work with smaller, more intuitive formats in everyday reasoning and calculation.