And for the case of 0.818181…, that meant that we had to multiply by 100. The General Rule for Dealing with Repeating Decimalsīut why did we multiply by 100 in this new problem dealing with a repeating pattern of two digits instead of by 10 as we did earlier when dealing with a single repeating digit? Well, the important thing to realize is that in order for this technique to work, we had to multiply whatever the repeating decimal was by some number so that the repeating part after the decimal point cancelled out when we subtracted the original number from the new multiplied number. As it turns out, you can divide both the top and bottom of this fraction by 9, which means that 0.818181… = 81/99 = 9/11. So this something, which is actually our repeating decimal 0.818181…, must be equal to the fraction 81/99. That means we’ve found that 99 of something is equal to 81 in this problem. In this case, what we’ve really done is to subtract 1 of something from 100 of something (before it was from 10 of something), which is just equal to 99 of something. Just like before, let’s now subtract these two numbers to get 81.818181… – 0.818181… = 81. When we do that, we get the new repeating decimal 81.818181…. The twist is that this time we’re not going to multiply the number by 10, we’re going to multiply it by 100. If you need a more thorough reminder about how this all works, you can go back and take a look at the last article where we explain it in much more detail.īelieve it or not, this is exactly the same technique that we need to use to convert the repeating pattern of digits 0.818181… into a fraction-with one small twist. So this something, which is actually our repeating decimal 0.777…, is just equal to 7/9. And that means that we’ve found that 9 of something is equal to the number 7 in this problem. But what we’ve really done here is to subtract 1 of something from 10 of something, which leaves us with 9 of something. If we now subtract the original 0.777… from this, we’re left with 7 since the repeating decimal part subtracts away. Namely, if we take the repeating decimal 0.777… and multiply it by 10, we get the new repeating decimal 7.777…. To figure out how to convert a decimal like 0.818181… that has a repeating pattern, let’s first recall why the quick and dirty method for converting simple repeating decimals to fractions that we talked about last time works. How to Turn Repeating Decimal Patterns Into Fractions So 0.111… = 1/9, 0.444… = 4/9, 0.777… = 7/9, and so on.īut what about decimals that have a whole pattern of repeating digits instead of one single repeating number-something like 0.818181…? Or what if the digits don’t start repeating right away-something like 0.7222… where there’s an extra 7 in there before 2 starts repeating forever? Well, these situations require a little extra explanation-so let’s start by looking at dealing with repeating patterns. This gives you the following: = Subtract the equation from step 1 from the equation in step = 7.The quick and dirty rule we discovered is that these types of repeating decimals are equivalent to the fraction that has the number doing the repeating in its numerator and the number 9 in its denominator. To move the decimal to the right of the 7, you need to multiply by 10. Begin by writing = the repeating = Multiply both sides of the equation by a power of 10 which will move the decimal to the right of the repeating number.
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