1.66666 repeating as a fraction

2 min read 26-02-2025

Meta Description: Discover how to easily convert the repeating decimal 1.66666... into a fraction. This simple guide breaks down the process step-by-step, explaining the underlying math and offering helpful tips for similar conversions. Learn the technique and never struggle with repeating decimals again!

The number 1.66666… (where the 6s repeat infinitely) is a repeating decimal. Many people find converting repeating decimals to fractions tricky. But it's actually a straightforward process using a bit of algebra. This guide will show you how to convert 1.66666... into a fraction.

Understanding Repeating Decimals

Repeating decimals, also known as recurring decimals, are numbers with digits that repeat infinitely after the decimal point. In our case, the digit 6 repeats endlessly after the decimal. Understanding this repetition is key to our conversion.

Converting 1.66666... to a Fraction: Step-by-Step

Here's how to transform the repeating decimal 1.66666… into its fractional equivalent:

Step 1: Set up an Equation

Let's represent the repeating decimal as 'x':

x = 1.66666…

Step 2: Multiply to Shift the Decimal

Multiply both sides of the equation by 10, shifting the repeating digits to the left of the decimal point:

10x = 16.66666…

Step 3: Subtract the Original Equation

Subtracting the original equation (x = 1.66666…) from the equation in Step 2 eliminates the repeating part:

10x - x = 16.66666… - 1.66666…

This simplifies to:

9x = 15

Step 4: Solve for x

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

x = 15/9

Step 5: Simplify the Fraction

Finally, simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 3:

x = 5/3

Therefore, 1.66666… is equivalent to the fraction 5/3.

Why This Method Works

This method works because multiplying by powers of 10 shifts the repeating decimal portion, allowing us to subtract it out and leave a whole number. This whole number can then be easily expressed as a fraction.

Other Examples of Converting Repeating Decimals

This method isn't limited to 1.66666…. You can use the same process for other repeating decimals. For instance:

0.33333…: Let x = 0.33333…; 10x = 3.33333…; 10x - x = 3; x = 3/9 = 1/3
0.142857142857…: This requires multiplying by a higher power of 10 depending on the length of the repeating sequence.

Practice Makes Perfect

Converting repeating decimals to fractions may seem daunting initially. However, by following the steps outlined above and practicing with various examples, you'll master this useful skill in no time. The more you practice, the easier it will become!

Conclusion

Converting the repeating decimal 1.66666… to a fraction is a valuable exercise in understanding decimal-fraction relationships. The steps involved, though simple, illustrate a powerful algebraic technique applicable to a wide range of repeating decimals. Remember the steps, practice with other repeating decimals, and you'll confidently navigate these conversions in the future. Now you know that 1.66666... is simply another way of representing the fraction 5/3!