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## 8th grade

### Unit 1: Lesson 1

Repeating decimals- Converting a fraction to a repeating decimal
- Writing fractions as repeating decimals
- Converting repeating decimals to fractions (part 1 of 2)
- Converting repeating decimals to fractions
- Converting repeating decimals to fractions (part 2 of 2)
- Converting multi-digit repeating decimals to fractions
- Writing repeating decimals as fractions review
- Writing fractions as repeating decimals review

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# Converting a fraction to a repeating decimal

CCSS.Math:

Learn how to rewrite 19/27 as a repeating decimal. What's a repeating decimal? THAT is a great question. This video explains. Created by Sal Khan.

## Want to join the conversation?

- is 8th grade math important in college?(17 votes)
- Unlike some other subjects, Math builds on itself, so the concepts in 8th grade will be very important to do 9th grade Algebra and 10th grade Geometry which is needed to move to Algebra II, pre-calculus and calculus in college. For most degrees, you have to take some math classes.(31 votes)

- Why is a repeating decimal has only 6 digits?(0 votes)
- They have an infinite number of digits. But since it would take a while to write that many, he stops when he has written enough for the pattern to become visible.(43 votes)

- how do you know how many 0s to put down?(8 votes)
- Supposed you've reached the stage of putting down only zeros after the decimal point from the dividend (inside number).

Continue putting down zeros either until the remainders begin to repeat, or until a remainder is 0.

In the first case, look at the repeating group of remainders. Consider the digit in the answer that is computed just after each remainder in the repeating group is found. These digits form the repeating group in the answer; the answer becomes a repeating decimal.

In the second case, there are no more decimal digits to put in the answer; the answer is a terminating decimal.

Have a blessed, wonderful day!(2 votes)

- does it change if you get a three digit number as the denominator, or numerator(6 votes)
- No- you still have to divide the denominator into the numerator no matter what. You’d have a harder time doing that, though(3 votes)

- What is the difference between repeating decimals and a decimal that terminates?(4 votes)
- A repeating decimal has a group of digits that repeats infinitely many times, but a terminating decimal has only a finite number of digits.(6 votes)

- I'm really confused where you got those 0's from and why it's such a big equation(2 votes)
- No were, hes using it to make the number bigger so he can divide it.(8 votes)

- why does it always have to be the smaller number in the box and the bigger number outside the box?(0 votes)
- You can't think in terms of the size of the numbers to figure out what goes where.

It is always numerator (number at the top of the fraction) goes inside the box and denominator (number on the bottom of the fraction) goes outside the box. There are times where the numerator will be larger than the denominator like 17/3 = 5.66666....(12 votes)

- I'm confused. Why can we just bring down the 0?(3 votes)
- we could but then we should add another 0 to the answer anyways doing 27x0 helps us understand how we did the porblem(1 vote)

- Thank you this helps at lot(3 votes)
- 0.13333...= 13/99

0.99999...= ?(2 votes)- Interesting question!

In the real number system, 0.99999... is equal to 1! Amazing!

We can see this by considering the difference 1-0.99999... . This difference is greater than or equal to 0, but less than**every**decimal in the**infinite**sequence 0.1, 0.01, 0.001, 0.0001, 0.00001,... . The only real number that meets the conditions in the previous sentence is 0, so 1-0.99999... = 0. Therefore, 0.99999... = 1 in the real number system.(3 votes)

## Video transcript

PROBLEM: "Express the rational number 19/27 (or 19 27ths) as a terminating decimal or a decimal that eventually repeats. Include only the first six digits of the decimal in your answer." Let me give this a shot. So we want to express 19/27 – which is the same thing as 19 ÷ 27 – as a decimal. So let's divide 27 into 19. So 27 going into 19. And we know it's going to involve some decimals over here, because 27 is larger than 19, and it doesn't divide perfectly. So let's get into this. So 27 doesn't go into 1. It doesn't go into 19. It does go into 190. And it looks like 27 is roughly 30. It's a little less than 30. 30 times 6 would be 180. So let's go with it going 6 times. Let's see if that works out. Well, 6 × 7 is 42. 6 × 2 is 12, + 4 is 16. And when we subtract, 190 - 162 is going to get us – Actually, we could've had another 27 in there. Because when we subtract – So we get a 10 from the 10's place. So that becomes 8 10's. This became 28. So we could have put one more 27 in there. So let's do that. So let's put one more 27 in there. So 7 27's. 7 × 7 is 49. 7 × 2 is 14, + 4 is 18. And now our remainder is 1. We can bring down another 0. 27 goes into 10 0 times. 0 × 27 is 0. [Not "10," as Sal states by mistake.] Subtract – we have a remainder of 10. But now, we have to bring down another 0. So let's bring down this 0 right over here. So now, 27 goes into 100 3 times. 3 × 27 is 60 + 21, is 81. And then we subtract: 100 - 81. Well, we could take 100 from the 100's place, and make it 10 10's. And then we could take 1 of those 10's from the 10's place and turn it into 10 1's. And so 9 10's minus 8 10's is equal to 1 10. And then 10 -1 is 9. So it's equal to 19. You probably – You might have been able to do that in your head. And then we have – And I see something interesting here – because when we bring down our next 0, we see 190 again. We saw 190 up here. But let's just keep going. So 27 goes into 190 – And we already played this game. It goes into it 7 times. 7 × 27 – we already figured out – was 189. We subtracted. We had a remainder of 1. Then we brought down another 0. We said 27 goes into 10 0 times. 0 × 27 is 0. Subtract. Then you have – We still have the 10, but we've got to bring down another 0. So you have 27, which goes into 100 – (We've already done this.) –3 times. So you see something happening here. It's 0.703703. And we're just going to keep repeating 703. This is going to be equal to 0.703703703703 – on and on and on forever. So the notation for representing a repeating decimal like this is to write the numbers that repeat – in this case 7, 0, and 3 – and then you put a line over all of the repeating decimal numbers to indicate that they repeat. So you put a line over the 7, the 0, and the 3, which means that the 703 will keep repeating on and on and on. So let's actually input it into the answer box now. So it's 0.703703. And they tell us to include only the first six digits of the decimal in your answer. And they don't tell us to round or approximate – because, obviously, if they said to round to that smallest, sixth decimal place, then you would round up because the next digit is a 7. But they don't ask us to round. They just say, "Include only the first six digits of the decimal in your answer." So that should do the trick. And it did.