Problem 18 - Prealgebra

Question

Write each decimal as a fraction or mixed number in simplest form. $$0.4$$

Step-by-Step Solution

Answer

0.4 as a fraction is \( \frac{2}{5} \).

Step 1: Identify the Decimal Place Value

The decimal 0.4 has the digit 4 in the first decimal place, which represents tenths.

Step 2: Write the Decimal as a Fraction

Since 0.4 represents 4 tenths, we can write it as a fraction: \( \frac{4}{10} \).

Step 3: Simplify the Fraction

To simplify \( \frac{4}{10} \), we identify that both the numerator and the denominator can be divided by their greatest common divisor, which is 2.\( \frac{4 \div 2}{10 \div 2} = \frac{2}{5} \).

Step 4: Conclusion

The simplest form of 0.4 as a fraction is \( \frac{2}{5} \).

Key Concepts

Decimal Place Value

Decimal place value helps us understand where each digit in a decimal number is located and what value they represent. Each position to the right of the decimal point holds a place value that is a power of ten, but it becomes a fraction of ten.
Here's how you can break it down:

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

Since 0.4 has the digit 4 in the tenths place, it means it represents four-tenths or \( \frac{4}{10} \). This understanding is foundational for converting decimals to fractions, as each digit corresponds to a fractional value depending on its place.

Simplifying Fractions

Once a decimal is expressed as a fraction, it is essential to simplify the fraction to its simplest form. Simplifying a fraction means finding an equivalent fraction where the numerator and the denominator are as small as possible but still represent the same value.

Here’s how to simplify fractions:

• First, determine the greatest common divisor (GCD) of both the numerator and the denominator.
• Divide both the numerator and the denominator by their GCD.
• This results in a simpler, equivalent fraction.

For example, with the fraction \( \frac{4}{10} \), the GCD of 4 and 10 is 2. Dividing both by 2 gives us \( \frac{2}{5} \), which is the simplest form. Simplified fractions make it easier to work with numbers, especially in calculations.

Numerator and Denominator

In any fraction, the numerator and the denominator are the two key components that tell us what part of a whole we are considering.

• The **numerator** is the top number of a fraction and indicates how many parts of the whole we have.
• The **denominator** is the bottom number and shows into how many parts the whole is divided.

For \( \frac{4}{10} \), 4 is the numerator, telling us we have four parts out of ten. The denominator, 10, informs us that these parts make up a whole that is divided into ten equal pieces.
After simplifying, the fraction becomes \( \frac{2}{5} \), with 2 as the numerator and 5 as the denominator, which reflects the same proportion of the whole. Understanding both the numerator and the denominator is crucial not only for simplifying fractions but also for performing operations such as addition or subtraction of fractions.