A periodic decimal number is a value expressed in decimal notation with a finite string of digits that from a certain point on is repeated indefinitely. It is not easy to work with these numbers, but they can be converted into fractions. Sometimes, the periodic decimal places are marked with a hyphen; for example, the number 3, 7777 with 7 periodic can also be reported as 3, 7. To turn a number like this into a fraction, you have to set up an equation, do some multiplication and subtraction to remove the periodic digit and finally solve the equation itself.
Steps
Part 1 of 2: Converting Elementary Periodic Decimal Numbers
Step 1. Find the periodic digits
For example, the number 0, 4444 has as a periodic figure
Step 4.. It is an elementary number, because there is no non-periodic decimal portion. Count how many periodic digits there are.
- Once the equation is written, you need to multiply it by 10 ^ y, where is it y corresponds to the number of digits present in the periodic portion.
- In the example of 0.44444, there is only one repeated digit, so you can multiply the equation by 10 ^ 1.
- If you take into account the number 0, 4545, the periodic portion consists of two digits; accordingly, you multiply the equation by 10 ^ 2.
- If there were three digits, the factor would be 10 ^ 3 and so on.
Step 2. Rewrite the decimal number as an equation
Express it so that "x" is equal to the original number. In the example considered, the equation is x = 0.44444; since there is only one periodic digit, multiply it by 10 ^ 1 (which corresponds to 10).
- In the example: x = 0.44444, so 10x = 4.44444.
- If you consider x = 0.4545 where there are two periodic digits, you have to multiply both terms by 10 ^ 2 (i.e. 100) to get 100x = 45, 4545.
Step 3. Remove the periodic portion
You can do this by subtracting x from 10x. Remember that any operation performed on the right term of the equation must also be reported on the left one:
- 10x - 1x = 4.44444 - 0.44444;
- On the left side you get 10x - 1x = 9x; on the right one 4, 4444 - 0, 4444 = 4;
- Consequently: 9x = 4.
Step 4. Solve for x
When you know what 9x equals, you can find the value of x by dividing both terms of the equation by 9:
- On the right side you have 9x ÷ 9 = x, while on the left you get 4/9;
- You can therefore state that x = 4/9 and that therefore the periodic decimal number 0, 4444 can be rewritten as a fraction 4/9.
Step 5. Reduce the fraction
Simplify it to a minimum (if possible), dividing both the numerator and the denominator by the greatest common factor.
In the example described above, 4/9 is already at its lowest
Part 2 of 2: Converting Numbers with Periodic and Non-Periodic Decimals
Step 1. Determine the periodic digits
It's not uncommon to find a number with a non-periodic portion before the repeating sequence, but even then you can convert to a fraction.
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For example, consider the number 6, 215151; in this case, 6, 2 it is not periodic while
Step 15. it is.
- Again you have to note how many digits the repeating portion is made up of, because you have to multiply by 10 ^ y, where "y" is just the quantity of those digits.
- In this example, there are two repeating digits, so you need to multiply the equation by 10 ^ 2.
Step 2. Write the problem as an equation, then subtract the periodic part
Again, if x = 6.25151, it follows that 100x = 621.5151. To remove repeating digits, subtract from both terms of the equation:
- 100x - x (= 99x) = 621, 5151 – 6, 215151 (= 615, 3);
- So 99x = 615, 3.
Step 3. Solve for x
Since 99x = 615, 3 divide both terms by 99; by doing so, you earn x = 615, 3/99.
Step 4. Remove the decimal place from the numerator
To do this, simply multiply both the numerator and the denominator by 10 ^ z, where is it z corresponds to the number of decimal places you need to delete. In 615, 3 you only have to move the decimal one place, which means you have to multiply by 10 ^ 1:
- 615.3 x 10 / 99 x 10 = 6153/990;
- Simplify the fraction by dividing the numerator and denominator by the greatest common factor, which in this case is 3: x = 2051/330.
