5.33333 as a fraction

2 min read 25-02-2025

Meta Description: Learn how to convert the repeating decimal 5.33333... into a fraction. This comprehensive guide provides a step-by-step explanation and explores the underlying mathematical principles. Discover the simple method to solve similar repeating decimal problems and master fraction conversion.

The number 5.33333... is a decimal number with a repeating decimal part. Converting this repeating decimal into a fraction might seem tricky, but it's a straightforward process using a simple algebraic technique. This article will guide you through the steps.

Understanding Repeating Decimals

Before we begin, let's clarify what we mean by a repeating decimal. A repeating decimal is a decimal number where one or more digits repeat infinitely. In our case, the digit 3 repeats endlessly after the decimal point. We often represent this with a bar over the repeating digits: 5. $\overline{3}$ .

Converting 5.33333... to a Fraction: A Step-by-Step Approach

Here's how to convert the repeating decimal 5.33333... (or 5. $\overline{3}$ ) into a fraction:

Step 1: Set up an equation.

Let x = 5.33333...

Step 2: Multiply to shift the repeating part.

Multiply both sides of the equation by 10 to shift the repeating part to the left of the decimal point:

10x = 53.33333...

Step 3: Subtract the original equation.

Subtract the original equation (x = 5.33333...) from the equation in Step 2:

10x - x = 53.33333... - 5.33333...

This simplifies to:

9x = 48

Step 4: Solve for x.

Divide both sides of the equation by 9 to isolate x:

x = 48/9

Step 5: Simplify the fraction.

Simplify the fraction by finding the greatest common divisor (GCD) of the numerator (48) and the denominator (9). The GCD of 48 and 9 is 3. Divide both the numerator and denominator by 3:

x = 16/3

Therefore, the fraction equivalent of 5.33333... is 16/3.

Why This Method Works

This method works because multiplying by powers of 10 shifts the decimal point. Subtracting the original equation eliminates the repeating decimal part, leaving a simple equation that can be solved for x. The resulting fraction represents the exact value of the repeating decimal.

Other Examples of Converting Repeating Decimals to Fractions

Let's look at another example to solidify your understanding. Consider the repeating decimal 0.6666... (or 0. $\overline{6}$ ):

Let x = 0.6666...
Multiply by 10: 10x = 6.6666...
Subtract: 10x - x = 6.6666... - 0.6666... => 9x = 6
Solve for x: x = 6/9 = 2/3

Therefore, 0.6666... is equivalent to 2/3.

Practice Makes Perfect

Try converting other repeating decimals to fractions using this method. The more you practice, the more confident you'll become in handling these types of conversions. Remember, the key steps are setting up an equation, multiplying to shift the repeating digits, subtracting to eliminate the repeating part, solving for x, and simplifying the fraction.

Conclusion

Converting a repeating decimal like 5.33333... to a fraction is a valuable skill in mathematics. By following the steps outlined above, you can accurately and efficiently convert any repeating decimal to its fractional equivalent. Remember the 16/3 represents the exact value, unlike the approximation provided by the decimal. This method is a useful tool for understanding the relationship between decimals and fractions.