Transforming Simple Repeating Decimals to Fractions and Vice Versa 03:41 minutes

Video Transcript

Transcript Transforming Simple Repeating Decimals to Fractions and Vice Versa

The brothers Zooey, Louie and Phooey formed a band called the Musical Triplets and entered their school’s annual Battle of the Bands contest. Look at that?! The Triplets beat last year's winner, the Math Bros. and win first place! The grand prize is a one hundred dollar bill! They plan to divide the prize money evenly amongst themselves.

They go to their Uncle Huge, who's as good as a bank, to exchange their one hundred dollar bill for 10 ten dollar bills. After dividing the money evenly, they're left with one ten dollar bill. They then ask their uncle to change the last ten dollar bill into ten, one-dollar bills. Each boy now gets three dollars, leaving one one-dollar bill. This goes on and on...they exchange dollars for dimes and dimes for pennies until...

Using long division

They divided all the money, except for one last penny. What if they want to split that one remaining penny? We know how to split one cent into three even parts with math. We write this as a fraction one-third but how much is this, exactly? You already know the fraction bar indicates division, so one third is the same as one divided by three.

Do you remember how we calculate the quotient? That's right! We use long division! Now do the math. Do you see a pattern? Since we'll always have a remainder, we call this number, and numbers like this, a repeating decimal. Instead of writing the repeating part again and again, we can use a horizontal bar to indicate the digits that repeat.One third is equal to one divided by three, which we rewrote in long division form. While evaluating this problem, we get zero point three, three, three, three, three, three. Oh, sorry.

Converting a repeating decimal into a fraction

Okay, so you can use division to see if a fraction has a repeating decimal, but how do you convert a repeating decimal into a fraction? Changing a repeating decimal in which all numbers after the decimal repeat, such as zero point one repeating, zero point two repeating, etc. and zero point 73 repeating, all the way up to zero point 123456789 repeating, into a fraction, might seem daunting at first. But I'll show you a little trick that'll amaze your friends or maybe just your math teacher. Use place value to determine the denominator. To write these numbers as fractions, first find out how many numbers repeat after the decimal. Here, there are one, two and three digits to the right of the decimal that repeat, respectively.

Next, you write the repeating part in the numerator like so. For each of the denominators we need to write the same number of 9s as there are numbers in the numerator. So one 9 here, two 9s here and three 9s here. Now all we have to do is simplify! Two-ninths is already in its reduced form, so we don't have to do anything to it. The greatest common factor of 36 and 99 is 9. So we can divide the numerator and denominator by 9. Doing so leaves us with four-elevenths. The greatest common factor of 459 and 999 is 27. So we can divide the numerator and denominator by 27. Doing so leaves us with seventeen-thirty-sevenths. Let's see if the Triplets have figured out the trick to splitting the last penny. Uh-oh, it looks like Uncle Huge is gonna keep that penny! Well, a penny saved is a penny earned!