Repeating decimal numbers, also known as recurring decimal numbers, are those that have digits that repeat infinitely after a certain point. These numbers are commonly encountered in everyday calculations and can be quite difficult to work with. However, with a few simple steps, it is possible to convert repeating decimal numbers into fractions.

Step 1: Identify the repeating pattern

The first step in converting a repeating decimal number into a fraction is to identify the repeating pattern. For example, consider the decimal number 0.33333… In this case, the repeating pattern is ‘3.’ Once you have identified the repeating pattern, you can move on to the next step.

Step 2: Write an equation

The next step is to write an equation that represents the repeating decimal number as a fraction. To do this, you can use the following formula:

x = a.abcabcabc…

where x is the decimal number you want to convert into a fraction, a is the non-repeating part of the number (if any), and b is the repeating pattern.

For example, let’s use the decimal number 0.33333… as an example. Here, a = 0 and b = 3. So, we can write the following equation:

x = 0.33333…
x = 0.33 + 0.0033 + 0.000033 + …

Step 3: Solve the equation

Now that you have written an equation, it’s time to solve it to get the fraction. To do this, we’ll use a little algebraic manipulation. First, multiply both sides of the equation by 100 to get rid of the decimal point:

100x = 33.3333…

Next, let’s subtract the first equation from the second equation. This will help us isolate the repeating pattern:

100x – x = 33.3333… – 0.3333…
99x = 33
x = 33/99

Simplifying this fraction, we get:

x = 1/3

So, the repeating decimal number 0.33333… is equivalent to the fraction 1/3.

Step 4: Simplify the fraction (if necessary)

In some cases, the fraction you get from converting a repeating decimal number may not be in its simplest form. If this happens, you should simplify the fraction. For example, if we convert the repeating decimal number 0.666666… using the same steps as above, we get:

x = 0.666666…
x = 0.6 + 0.060 + 0.006 + …

100x = 66.6666…

100x – x = 66.6666… – 0.6666…

99x = 66

x = 2/3

So, the repeating decimal number 0.666666… is equivalent to the fraction 2/3. However, this fraction can be simplified by dividing both the numerator and denominator by 2:

2/3 = 1/2

So, the repeating decimal number 0.666666… is equivalent to the fraction 1/2.

In conclusion, converting repeating decimal numbers into fractions is not as difficult as it may seem. All you need to do is follow these simple steps: identify the repeating pattern, write an equation, solve the equation, and simplify the fraction if necessary. By doing so, you can easily convert any repeating decimal number into a fraction and make your calculations much easier.

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