Problem 58 - Prealgebra

Question

Write each fraction or mixed number as a decimal. Use a bar to show a repeating decimal. $$\frac{2}{5}$$

Step-by-Step Solution

Answer

The decimal form of \( \frac{2}{5} \) is 0.4.

Step 1: Understand the Problem

Recognize that the goal is to convert the fraction \( \frac{2}{5} \) into its decimal form.

Step 2: Divide the Numerator by the Denominator

Perform the division \( 2 \div 5 \). Since 2 is smaller than 5, consider it as 20 and perform the division.

Step 3: Perform Long Division

When dividing 2 by 5, 5 goes into 20 (since we're considering 2 as 20) exactly 4 times because \( 5 \times 4 = 20 \). Subtract 20 from 20 to get 0, indicating that there is no remainder.

Step 4: Write the Result as a Decimal

The quotient of the division \( 2 \div 5 \) is 0.4. There is no remainder, so the decimal is not repeating.

Key Concepts

Fractions to Decimals

Converting fractions to decimals is a straightforward process. Essentially, you're breaking down how many times the denominator fits into the numerator to find the equivalent decimal value. In simple terms, a fraction like \( \frac{2}{5} \) asks how many times does 5 fit into 2.

To convert a fraction like \( \frac{2}{5} \) to a decimal, you initiate a division problem: 2 divided by 5. In many cases, fractions result in clean, terminating decimals. But the essence of the conversion remains the same—divide the numerator by the denominator.

Here, converting \( \frac{2}{5} \) yields 0.4 because 5 fits into 2, zero times initially, but when extended (think of it as 20), it fits perfectly and evenly four times.

Long Division

Long division is a useful method for converting fractions to decimals, especially when dealing with more complex numerators and denominators. It systematically breaks down the division process step by step, ensuring no stone is left unturned.

Here's how it works:

• Setup the division where the numerator is inside the division bracket, and the denominator is outside.
• Begin by determining how many times the denominator fits into the adjusted numerator. If your numerator is less than the denominator, consider adding zeros and turning it into a higher number for division.
• Subtract the product of the divisor and the determined quotient from the current adjusted number.
• Continue the process until reaching zero, or decide that it repeats.

In our example, \( 2 \div 5 = 0.4 \), with the 2 adjusted as 20 (considering 0.20), demonstrating the beauty and precision of long division.

Non-repeating Decimals

Not all fractions result in repeating decimals. When you encounter a fraction like \( \frac{2}{5} \), you achieve a non-repeating decimal as the outcome of your division.

A non-repeating decimal is essentially a terminating decimal, which means the division results in a decimal with a finite number of digits. Here, after realizing \( 5 \times 4 = 20 \), you subtract and reach an exact 0, signaling no further numbers are needed.

These decimals are simpler to handle because they do not stretch into infinity. Understanding non-repeating decimals is essential in mathematics to distinguish between numbers that end neatly and those that don't.

When dividing and you find no remainder at the correct decimal place, rejoice—this means your number is clean, clear, and completely non-repeating!