Repeating Decimals
Grade 8 math lesson on repeating decimals and converting them to fractions. Students learn to identify the repeating block in a decimal, use algebraic techniques to convert repeating decimals to exact fractions, and understand why some fractions produce repeating decimals.
Key Concepts
Fractions can convert to decimals with endlessly repeating digits. We use a bar over this repeating part, called the repetend, to write them efficiently.
The repeating digits are called the repetend . We can indicate repeating digits with a bar over the repetend. We write $0.272727...$ as $0.\overline{27}$. What’s next.
Common Questions
What is a repeating decimal?
A repeating decimal is a decimal in which one or more digits repeat infinitely. For example, 1/3 = 0.333... (the 3 repeats forever) and 1/7 = 0.142857142857... (the block 142857 repeats).
How do you convert a repeating decimal to a fraction?
Let x equal the repeating decimal. Multiply both sides by a power of 10 to shift the repeating part. Subtract the original equation to eliminate the repeating part. Solve for x. For example, for 0.333...: x = 0.333, 10x = 3.333, so 9x = 3 and x = 1/3.
Which fractions produce repeating decimals?
A fraction produces a repeating decimal when its denominator (in lowest terms) has prime factors other than 2 and 5. Fractions like 1/3 and 1/7 repeat, while 1/4 and 1/8 terminate because their denominators only have factors of 2.
How do you identify the repeating block in a decimal?
The repeating block is the digit or group of digits that cycles infinitely. It is written with a bar over the repeating digits. For example, 0.142857 with a bar over 142857 means those six digits repeat forever.