4.135 repeating as a fraction

2 min read 23-02-2025

Meta Description: Learn how to convert the repeating decimal 4.135135... into a fraction. This step-by-step guide simplifies the process, explaining the method and providing a clear solution. Master the technique for similar repeating decimal conversions.

Understanding Repeating Decimals

A repeating decimal, also known as a recurring decimal, is a decimal that has a digit or group of digits that repeats infinitely. In this case, we're dealing with 4.135135135..., where the digits "135" repeat endlessly. Knowing how to convert this into a fraction is a valuable skill in mathematics.

Step-by-Step Conversion: 4.135135... to a Fraction

Here's how to transform 4.135135... into its fractional equivalent:

1. Set up an Equation

First, we represent the repeating decimal as 'x':

x = 4.135135135...

2. Multiply to Shift the Repeating Part

Next, we multiply both sides of the equation by a power of 10 that shifts the repeating block to the left of the decimal point. Since the repeating block is three digits ("135"), we multiply by 1000:

1000x = 4135.135135135...

3. Subtract the Original Equation

Now, we subtract the original equation (x) from the multiplied equation (1000x):

1000x - x = 4135.135135... - 4.135135...

This cleverly eliminates the repeating decimal part:

999x = 4131

4. Solve for x

Finally, we solve for 'x' by dividing both sides by 999:

x = 4131/999

5. Simplify the Fraction (If Possible)

In this case, we can simplify the fraction by finding the greatest common divisor (GCD) of 4131 and 999. The GCD of 4131 and 999 is 27. Dividing both the numerator and denominator by 27 gives us the simplified fraction:

x = 153/37

Therefore, the fraction equivalent of the repeating decimal 4.135135... is 153/37.

Checking Your Work

You can always verify your answer by performing long division with the fraction. Dividing 153 by 37 will indeed result in the repeating decimal 4.135135...

Converting Other Repeating Decimals

This method works for any repeating decimal. The key is to identify the repeating block of digits and multiply by the appropriate power of 10 to shift it. Remember to always simplify the resulting fraction to its lowest terms.

Frequently Asked Questions (FAQs)

Q: What if the repeating decimal starts after a non-repeating part?

A: If the repeating decimal has a non-repeating part, you need to adjust the equations accordingly. For example, if you had 2.135135..., you would follow the same steps but handle the "2" separately.

Q: What is the easiest way to find the greatest common divisor (GCD)?

A: You can use the Euclidean algorithm or online GCD calculators to easily find the GCD of two numbers.

Q: Are there any other methods to convert repeating decimals to fractions?

A: While this method is generally the most efficient, there are alternative approaches involving geometric series, but they are more advanced.

By following these steps, you can confidently convert repeating decimals like 4.135135... into their fractional representations. Remember to practice and you'll master this useful mathematical skill.