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PROBLEM: "Express the rational number 19/27

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(or 19 27ths) as a terminating decimal

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or a decimal that eventually repeats.

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Include only the first six digits

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of the decimal in your answer."

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Let me give this a shot.

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So we want to express 19/27 –

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which is the same thing as 19 ÷ 27 – as a decimal.

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So let's divide 27 into 19.

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So 27 going into 19.

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And we know it's going to involve some decimals

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over here, because 27 is larger than 19,

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and it doesn't divide perfectly.

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So let's get into this.

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So 27 doesn't go into 1.

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It doesn't go into 19.

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It does go into 190.

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And it looks like 27 is roughly 30.

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It's a little less than 30.

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30 times 6 would be 180.

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So let's go with it going 6 times.

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Let's see if that works out.

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Well, 6 × 7 is 42.

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6 × 2 is 12, + 4 is 16.

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And when we subtract, 190 - 162 is going to get us –

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Actually, we could've had another 27 in there.

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Because when we subtract –

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So we get a 10 from the 10's place.

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So that becomes 8 10's.

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This became 28.

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So we could have put one more 27 in there.

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So let's do that.

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So let's put one more 27 in there.

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So 7 27's.

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7 × 7 is 49.

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7 × 2 is 14, + 4 is 18.

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And now our remainder is 1.

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We can bring down another 0.

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27 goes into 10 0 times.

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0 × 27 is 0. [Not "10," as Sal states by mistake.]

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Subtract – we have a remainder of 10.

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But now, we have to bring down another 0.

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So let's bring down this 0 right over here.

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So now, 27 goes into 100 3 times.

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3 × 27 is 60 + 21, is 81.

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And then we subtract: 100 - 81.

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Well, we could take 100 from

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the 100's place, and make it 10 10's.

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And then we could take 1 of those 10's from

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the 10's place and turn it into 10 1's.

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And so 9 10's minus 8 10's is equal to 1 10.

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And then 10 -1 is 9.

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So it's equal to 19.

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You probably –

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You might have been able to do that in your head.

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And then we have –

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And I see something interesting here –

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because when we bring down our next 0,

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we see 190 again.

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We saw 190 up here.

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But let's just keep going.

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So 27 goes into 190 –

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And we already played this game.

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It goes into it 7 times.

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7 × 27 – we already figured out – was 189.

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We subtracted.

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We had a remainder of 1.

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Then we brought down another 0.

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We said 27 goes into 10 0 times.

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0 × 27 is 0.

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Subtract.

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Then you have –

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We still have the 10,

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but we've got to bring down another 0.

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So you have 27, which goes into 100 –

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(We've already done this.)

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–3 times.

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So you see something happening here.

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It's 0.703703.

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And we're just going to keep repeating 703.

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This is going to be equal to 0.703703703703 –

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on and on and on forever.

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So the notation for representing

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a repeating decimal like this

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is to write the numbers that repeat –

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in this case 7, 0, and 3 –

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and then you put a line over all of

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the repeating decimal numbers

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to indicate that they repeat.

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So you put a line over the 7, the 0, and the 3,

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which means that the 703 will keep

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repeating on and on and on.

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So let's actually input it into the answer box now.

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So it's 0.703703.

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And they tell us to include only

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the first six digits of the decimal in your answer.

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And they don't tell us to round or approximate –

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because, obviously, if they said to round

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that final sixth decimal place,

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then you would round it up from 6 to 7,

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because the next digit is a 7.

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But they don't ask us to round.

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They just say, "Include only the first six digits

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of the decimal in your answer."

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So that should do the trick.

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And it did.