Problem 15 - Prealgebra

Question

Write each fraction or mixed number as a decimal. Use a bar to show a repeating decimal. $$\frac{1}{5}$$

Step-by-Step Solution

Answer

\( \frac{1}{5} = 0.2 \) with no repeating decimal.

Step 1: Understand the Fraction

The fraction given is \( \frac{1}{5} \). This means "1 divided by 5." To convert this fraction into a decimal, we need to perform the division.

Step 2: Set Up the Division

Set up the long division where 1 is the dividend (inside the division bracket) and 5 is the divisor (outside the division bracket). We seek to determine how many times 5 can be subtracted from 1 including decimal places.

Step 3: Perform the Division

Since 5 does not go into 1, we extend 1 to 10 by adding a decimal point and zeros. We express 1 as 1.0 to begin the division.- Divide 10 by 5, which equals 2.- Place the 2 in the tenths place of the decimal, giving us 0.2.- Since there's no remainder, the division ends here, and \( \frac{1}{5} \) converts to 0.2.

Key Concepts

Long Division

Long division is a method used to divide larger numbers, especially when converting fractions to decimals. In the context of fractions, long division helps us understand how a numerator is divided by a denominator. To begin, we typically write the numerator (top number of the fraction) inside the division symbol and the denominator (bottom number) outside the symbol. The numerator, in this case, is the dividend, whereas the denominator is the divisor.

Here's how it works:

• Verify if the divisor can go into the dividend. If not, add a decimal point to the dividend and increase it by appending one or more zeros.
• Divide the newly formed number by the divisor, and write the result as part of your answer.
• If you have a remainder, bring down another zero if necessary and repeat the division.

This step-by-step breakdown allows fractions to be expressed as decimals, making calculations easier for various applications.

Repeating Decimals

When performing long division, sometimes the division doesn’t come to an end. This happens when the same sequence of digits continues indefinitely, which is known as a repeating decimal. For example, when dividing 1 by 3, we get 0.333..., where the digit 3 repeats forever.

To represent repeating decimals succinctly, we use a bar, called a vinculum, over the repeating digit or digits, such as \(0.\overline{3}\). This indicates that the number repeats infinitely.

Examples of repeating decimals include:

• \(\frac{1}{3} = 0.\overline{3}\)
• \(\frac{2}{9} = 0.\overline{2}\)
• \(\frac{5}{11} = 0.\overline{45}\)

Recognizing repeating decimals is crucial because it helps in understanding the approximate nature of these numbers and how they behave in arithmetic operations.

Mixed Numbers

Mixed numbers combine whole numbers with fractions. They are used to express quantities greater than one in a more intuitive way. For instance, 2\(\frac{1}{4}\) translates to 2 whole units plus a quarter. To convert a mixed number into a decimal, begin by separately dealing with its whole and fractional components.

Here's the conversion process:

• Keep the whole number as is.
• Convert the fractional part to a decimal using long division.
• Add both parts – the whole number and the decimal equivalent of the fraction.

For instance, in 3\(\frac{3}{5}\), the whole number is 3, and \(\frac{3}{5}\) as a decimal is 0.6 based on division (3 divided by 5), leading to a complete decimal of 3.6.

Understanding mixed numbers and their conversion to decimals can be beneficial in real-life situations such as measuring, cooking, or when engaging in tasks where precision is essential.