Repeating decimals, also known as recurring decimals, are numbers that have a pattern of digits that repeats infinitely. For example, 0.3333… (where the 3 repeats forever) and 1.252525… (where 25 repeats forever) are repeating decimals. Converting these decimals to fractions is a useful skill in mathematics and can be done using a few simple steps.
Understanding Repeating Decimals
Before diving into the conversion process, let’s understand the concept of repeating decimals better.
- Repeating Block: The repeating pattern of digits in a repeating decimal is called the repeating block. In 0.3333…, the repeating block is ‘3’. In 1.252525…, the repeating block is ’25’.
- Repeating Bar: To represent repeating decimals, we use a bar over the repeating block. For example, 0.3333… is written as 0.3̅ and 1.252525… is written as 1.25̅.
Steps to Convert Repeating Decimals to Fractions
Here’s a step-by-step guide to convert repeating decimals to fractions:
- Set up an equation: Let the decimal be equal to a variable, say ‘x’. For example, if the decimal is 0.3̅, we write:
x = 0.3̅
- Multiply to shift the decimal: Multiply both sides of the equation by 10 raised to the power of the number of digits in the repeating block. In our example, the repeating block has one digit (3), so we multiply by 10:
10x = 3.3̅
- Subtract the original equation: Subtract the original equation from the multiplied equation. This will eliminate the repeating part of the decimal:
10x = 3.3̅
x = 0.3̅
$9x = 3$
- Solve for ‘x’: Solve the resulting equation for ‘x’. In our example:
$9x = 3$
$x = frac{3}{9}$
- Simplify the fraction: Simplify the fraction to its lowest terms. In our example, both 3 and 9 are divisible by 3, so the simplified fraction is:
$x = frac{1}{3}$
Therefore, 0.3̅ is equivalent to the fraction 1/3.
Examples
Let’s work through a few more examples to solidify the process:
Example 1: Converting 0.6̅ to a fraction
Set up the equation:
x = 0.6̅Multiply to shift the decimal:
10x = 6.6̅Subtract the original equation:
10x = 6.6̅
x = 0.6̅
$9x = 6$
Solve for ‘x’:
$9x = 6$
$x = frac{6}{9}$Simplify the fraction:
$x = frac{2}{3}$
Therefore, 0.6̅ is equivalent to the fraction 2/3.
Example 2: Converting 1.25̅ to a fraction
Set up the equation:
x = 1.25̅Multiply to shift the decimal: Since the repeating block has two digits (25), we multiply by 100:
100x = 125.25̅Subtract the original equation:
100x = 125.25̅
x = 1.25̅
$99x = 124$
- Solve for ‘x’:
$99x = 124$
$x = frac{124}{99}$
Therefore, 1.25̅ is equivalent to the fraction 124/99.
Dealing with Mixed Repeating Decimals
Sometimes, the repeating block doesn’t start immediately after the decimal point. These are called mixed repeating decimals. For example, 0.123̅. To convert these, we need to modify the steps slightly:
Set up the equation:
x = 0.123̅Multiply to shift the decimal: Multiply by 1000 (10 raised to the power of 3, the number of digits in the repeating block):
1000x = 123.123̅Adjust for the non-repeating part: Multiply the original equation by 100 (10 raised to the power of 2, the number of digits before the repeating block):
100x = 12.123̅Subtract the adjusted equations:
1000x = 123.123̅
100x = 12.123̅
$900x = 111$
Solve for ‘x’:
$900x = 111$
$x = frac{111}{900}$Simplify the fraction:
$x = frac{37}{300}$
Therefore, 0.123̅ is equivalent to the fraction 37/300.
Conclusion
Converting repeating decimals to fractions is a valuable skill in mathematics. It allows us to express these decimals in a more compact and precise form, making them easier to work with in various mathematical operations and applications. By understanding the steps and applying them systematically, you can confidently convert any repeating decimal to its equivalent fraction.