Recurring decimals, also known as recurring decimals, are decimal numbers that have one or more digits that repeat indefinitely at regular intervals. Working with repeating decimals can sometimes be confusing, but you can convert them to fractions. Repeating decimals are sometimes represented by a line above the repeating digits. The number 3, 7777 where 7 is repeated, for example, can also be written as 3, 7. To convert a number like this into a fraction you must write it as an equation, multiply, subtract to remove the repeating decimal part and finally solve The equation.
Part 1 of 2: Convert Basic Repeating Decimals
Step 1. Identify the repeating decimal
For example, in the number 0, 4444, the repeating decimal is the
Step 4.. It is a basic repeating decimal in the sense that there is no part of the decimal number that does not repeat. Count how many repeating digits are in the pattern.
- Once you write the equation, you must multiply it by 10and, where and equals the number of digits that repeat in the pattern.
- In the example of 0, 4444 there is only one digit that is repeated, therefore, you must multiply the equation by 101.
- If the repeating decimal is 0, 4545For example, then there are two repeating digits. In that case you must multiply the equation by 102.
- If the repeating digits are three, then you must multiply by 103; and so on.
Step 2. Rewrite the decimal number as an equation
Write it so that "x" equals the original number. In this case, the equation is x = 0.4444. Since there is only one repeating decimal digit, multiply the equation by 101 (which is equal to 10).
- In the example where x = 0.4444, 10x = 4.4444.
- In the example where x = 0.4545, there are two repeating digits. Therefore, you must multiply both sides of the equation by 102 (which is equal to 100). Then: 100x = 45, 4545.
Step 3. Eliminate the repeating decimals
To do this, you just have to subtract x from 10x. Remember that everything you do on one side of the equation you must do on the other as well. Therefore:
- 10x - 1x = 4.4444 - 0.4444
- On the left side, you have 10x - 1x = 9x. On the right side, you have 4.4444 - 0.4444 = 4.
- Therefore, 9x = 4.
Step 4. Solve the equation to find the value of x
Once you know what 9x equals, you can determine the value of "x" by dividing by 9 on both sides of the equation:
- On the left side of the equation you have 9x ÷ 9 = x. On the right side of the equation you have 4/9.
- Therefore, x = 4/9 and the repeating decimal number 0, 4444 can be written as the fraction 4/9.
Step 5. Reduce the fraction
Express the fraction in its simplest form (if necessary) by dividing both the numerator and the denominator by the greatest common divisor between the two.
In the 4/9 example, the fraction is already expressed in its simplest form
Part 2 of 2: Recurring numbers with non-periodic figures
Step 1. Identify the repeating digits
It is nothing out of the ordinary to find numbers that have non-repeating digits before the repeating decimals. These can also be converted to fractions.
For example, take the number 6, 215151. In this case, the non-recurring digits are the 6, 2 and the repeating digits are the
- Again, look at how many repeating digits are in the pattern since you need to multiply by 10and based on that number.
- In this case, there are two repeating digits, so you must multiply the equation by 102.
Step 2. Write the problem as an equation and subtract the repeating decimals
One more time yeah x = 6, 215151 then 100x = 621, 5151. To eliminate repeating decimals, subtract them from both sides of the equation:
- 100x - x (= 99x) = 621, 5151 - 6, 215151 (= 615, 3)
- Therefore, 99x = 615.3
Step 3. Solve the equation to find the value of x
Since 99x = 615, 3, divide both sides of the equation by 99. That returns x = 615, 3/99.
Step 4. Remove the decimals from the numerator
To do this, multiply the numerator and denominator by 10z, where z equals the number of decimal places you must move to remove the decimal. In 615, 3 you have to move the comma just one place, which means that you will have to multiply the numerator and denominator by 101:
- 615, 3x10 / 99 x 10 = 6153/990
- Reduce the fraction by dividing the numerator and denominator by the greatest common divisor, which in this case is 3. Thus, the final result is x = 2051/330.