Problem 5 - Prealgebra

Question

Write each decimal as a fraction or mixed number in simplest form. $$0.8$$

Step-by-Step Solution

Answer

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

Step 1: Understand decimal to fraction conversion

The decimal 0.8 can be converted into a fraction by recognizing it as 8 tenths. This is because the digit 8 is in the tenths place after the decimal point.

Step 2: Write the decimal as a fraction

Convert 0.8 into fraction form by placing 8 over 10, since 8 is in the tenths place. So, we have \(\frac{8}{10}\).

Step 3: Simplify the fraction

To simplify \(\frac{8}{10}\), find the greatest common divisor (GCD) of 8 and 10, which is 2. Divide both the numerator and the denominator by their GCD: \(\frac{8 \div 2}{10 \div 2} = \frac{4}{5}\).

Step 4: Final Answer in simplest form

The decimal 0.8 converted to a fraction in simplest form is \(\frac{4}{5}\).

Key Concepts

Simplifying Fractions

Simplifying fractions is a crucial step in understanding how to work with fractions effectively. When we simplify a fraction, we express it in its most basic form, where the numerator and the denominator have no common factors other than 1. This makes the fraction easier to work with in further calculations or comparisons.
To simplify a fraction, follow these steps:

• Identify the greatest common divisor (GCD) of both the numerator and denominator.
• Divide both the numerator and the denominator by this GCD.

For example, if we start with the fraction \( \frac{8}{10} \), we first find that the GCD is 2. We then divide both 8 and 10 by 2, which results in the simplified fraction \( \frac{4}{5} \). Simplifying ensures that fractions are easier to interpret and compare.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor (GCF), is the largest number that divides two numbers without leaving a remainder. Finding the GCD helps in simplifying fractions.
Here's how to find the GCD:

• List the factors of each number.
• Identify the highest factor common to both lists.

Let's take the numbers 8 and 10.
The factors of 8 are: 1, 2, 4, 8.
The factors of 10 are: 1, 2, 5, 10.
The common factors are 1 and 2, with 2 being the greatest. Therefore, the GCD of 8 and 10 is 2. Using the GCD, we can simplify the fraction \( \frac{8}{10} \) to \( \frac{4}{5} \) by dividing both the numerator and the denominator by 2.

Decimal Place Value

Understanding decimal place value is essential when converting decimals to fractions or vice versa. Place value helps us recognize the fractional equivalent of a decimal. Each position in a decimal number has a place value, depending on its position relative to the decimal point.
For example:

• The first position to the right of the decimal point is the tenths place.
• The second position is the hundredths place.
• The third position is the thousandths place, and so forth.

In the decimal 0.8, the digit 8 is in the tenths place. This means 0.8 is equivalent to \( \frac{8}{10} \). Recognizing this place value allows us to write decimals as fractions more effectively. Once in fraction form, we can simplify as needed to ensure it is in its simplest form.