what fraction is 5.3333

2 min read 26-12-2024

The number 5.333... (where the 3s repeat infinitely) presents a common question: how do we express this as a fraction? Understanding how to convert repeating decimals to fractions is a fundamental skill in mathematics. Let's break down the process step-by-step.

Understanding Repeating Decimals

A repeating decimal, also known as a recurring decimal, is a decimal number that has a digit or group of digits that repeat infinitely. In our case, the repeating digit is 3. We often represent this with a bar over the repeating part: 5. $\overline{3}$ .

Converting 5.333... to a Fraction

Here's how to convert the repeating decimal 5.333... into a fraction:

1. Set up an equation:

Let 'x' represent the decimal:

x = 5.333...

2. Multiply to shift the decimal:

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

10x = 53.333...

3. Subtract the original equation:

Now, subtract the original equation (x = 5.333...) from the equation we just created (10x = 53.333...):

10x - x = 53.333... - 5.333...

This simplifies to:

9x = 48

4. Solve for x:

Divide both sides by 9:

x = 48/9

5. Simplify the fraction:

Both 48 and 9 are divisible by 3, so we can simplify:

x = 16/3

Therefore, the fraction equivalent of 5.333... is 16/3.

Checking Your Work

To verify our answer, you can perform long division: 16 divided by 3 equals 5.333... This confirms our conversion is correct.

Other Repeating Decimals

This method works for other repeating decimals. The key is to multiply by a power of 10 that shifts the repeating portion to align for subtraction. For example, a decimal like 0.121212... would require multiplication by 100 before subtraction.

Fractions with Terminating Decimals

It's worth noting that not all decimals can be expressed as simple fractions. Decimals that terminate (end) can always be converted into fractions. For example, 0.75 is equivalent to 3/4. However, irrational numbers like Pi (π) cannot be expressed as a fraction.

Conclusion: Mastering Decimal to Fraction Conversions

Converting repeating decimals like 5.333... to fractions requires a systematic approach. By following these steps, you can confidently convert any repeating decimal into its fractional equivalent, expanding your understanding of number representation. Remember to always simplify your final fraction to its lowest terms.