Problem 65 - Prealgebra

Question

Write each decimal as an equivalent proper fraction or mixed number. 0.8

Step-by-Step Solution

Answer

0.8 as a fraction is \( \frac{4}{5} \).

Step 1: Understanding the Decimal

First, we need to recognize that a decimal like 0.8 represents the tenths place. This means 0.8 can be interpreted as 8 tenths.

Step 2: Write as a Fraction

Next, express 0.8 as a fraction. Since it is 8 tenths, we can write it as \( \frac{8}{10} \).

Step 3: Simplify the Fraction

Now, simplify the fraction \( \frac{8}{10} \) by finding the greatest common divisor of 8 and 10, which is 2. Divide both the numerator and the denominator by 2 to get \( \frac{4}{5} \).

Step 4: Result as a Proper Fraction

The simplified fraction \( \frac{4}{5} \) is already a proper fraction since the numerator is less than the denominator. Thus, 0.8 is equivalent to \( \frac{4}{5} \).

Key Concepts

Equivalent Fractions

Creating equivalent fractions is like dressing numbers in different clothes: they look different but represent the same value. When converting decimals to fractions, finding equivalency is crucial. For example, with the decimal 0.8, it can be written as the fraction \(\frac{8}{10}\) by placing 8 over a denominator of 10, reflecting its place value. However, you can create fractions that represent the same quantity by multiplying or dividing both the numerator and denominator by the same number. Consider \(\frac{8}{10}\). If we multiply both the numerator and the denominator by 2, we get \(\frac{16}{20}\). Despite looking different, \(\frac{8}{10}\) and \(\frac{16}{20}\) are equivalent. How we choose to express these numbers doesn't change their value, but simplifying them makes it cleaner and often easier to work with.

Simplifying Fractions

Simplifying fractions is like tidying up. It makes numbers neater and easier to handle. To simplify, you need to find the greatest common divisor (GCD), which is the largest number that can evenly divide both the numerator and the denominator. For example, take \(\frac{8}{10}\). Find the GCD of 8 and 10, which is 2. Divide both the numerator and the denominator by 2:

• The numerator: \(8 \div 2 = 4\)
• The denominator: \(10 \div 2 = 5\)

This gives us \(\frac{4}{5}\), the simplified version of \(\frac{8}{10}\). Simplifying fractions makes them easier to manage, especially in calculations or when comparing with other fractions. Remember, the goal is to have no common factors other than 1 in the numerator and the denominator.

Proper Fractions

Proper fractions might sound fancy, but they're straightforward. They're fractions where the numerator (top number) is smaller than the denominator (bottom number). This ensures that the fraction represents a value less than 1. When we simplified 0.8 and found \(\frac{4}{5}\), we noted that 4 is smaller than 5, making it a proper fraction. Proper fractions are helpful in comparing and understanding portions or pieces of a whole. They are always less than one full circle—imagine slicing a pizza. If you have \(\frac{4}{5}\) of a pizza, you have less than a full pie, illustrating why the fraction is "proper." This concept contrasts with improper fractions or mixed numbers, which represent values greater than or equal to 1. Embracing proper fractions ensures we understand how fractions fit within a whole.