Decimal Representation of 8/27 (Step-by-Step Long Division + Why It Repeats)

The decimal representation of 8/27 is 0.296296296…, and the block 296 repeats forever.

You write it as 0.296 with a bar over all three digits, or as 0.(296) if a bar is awkward to type.

Property	Value
Fraction	8/27
Decimal form	0.296296…
Repeating block	296
Length of the block	3 digits
Type of decimal	Pure repeating (recurring)
Rounded to 2 decimal places	0.30
Rounded to 3 decimal places	0.296
As a percentage	about 29.63%

If that’s all you needed, you’re set. If you have to show your working for homework or an exam, keep reading. We’ll go through the long division, work out why the decimal repeats, and then turn it back into a fraction as a final check.

☰ Table of Contents

What Does "Decimal Representation of a Fraction" Mean?

A fraction is just a division that hasn’t been done yet. The line in 8/27 means “8 divided by 27.” So the decimal representation of 8/27 is whatever you get when you actually carry out that division.

Fractions turn into one of two kinds of decimals:

Terminating decimals stop after a few digits, like 1/4 = 0.25.
Repeating decimals carry on forever in a pattern, like 1/3 = 0.333…

8/27 is the second kind. And because 8 is smaller than 27, the answer is below 1, so it starts with 0 before the decimal point.

Step-by-Step Long Division of 8 ÷ 27

Let’s do it slowly. Once you see the pattern, it’s actually quite satisfying.

Setup: Write 8 as 8.000000 so you have plenty of zeros to bring down.

Step 1: Start with 8 ÷ 27. 27 doesn’t go into 8, so write 0 in the answer, then a decimal point. Add a zero to 8 to make 80.

Step 2: Divide 80 by 27. 27 × 2 = 54, and 27 × 3 = 81 is just too big. So the first digit after the decimal point is 2. Subtract: 80 − 54 = 26.

Step 3: Bring down a zero to make 260. 27 × 9 = 243, and 27 × 10 = 270 is too big. The next digit is 9. Subtract: 260 − 243 = 17.

Step 4: Bring down a zero to make 170. 27 × 6 = 162, and 27 × 7 = 189 is too big. The next digit is 6. Subtract: 170 − 162 = 8.

Step 5: Spot the repeat. The remainder is 8, which is exactly where we began. So the next step is 80 ÷ 27 all over again, and the whole cycle starts again: 2, 9, 6, 2, 9, 6, and on it goes.

Here’s the working on paper:

0 . 2 9 6 2 9 6 …

______________________

27 | 8 . 0 0 0 0 0 0

8 0

5 4

—-

2 6 0

2 4 3

—–

1 7 0

1 6 2

—–

8 0 <- back where we began

8 ÷ 27 = 0.296296… ✓

The Remainder Table (This Is the Key to Everything)

Long division for any fraction is really a story about remainders. Here’s the story for 8/27:

Step	Dividend	Digit	Remainder
1	80	2	26
2	260	9	17
3	170	6	8
4	80	2	26
5	260	9	17
…	…	…	…

The remainders go 26, 17, 8, then 26, 17, 8 again. The moment a remainder comes back, the digits repeat too, because the next step starts from the exact same place as before. That’s the whole reason repeating decimals exist.

Why Does 8/27 Repeat Forever?

There are two ways to explain it: a simple one and a more precise one.

The simple reason: there are only so many remainders

When you divide by 27, every remainder has to be somewhere between 0 and 26. That’s only 27 possibilities. So sooner or later you’re going to land on a remainder you’ve already seen, and from that point the pattern loops. The only way out is hitting a remainder of 0, which ends the decimal. With 8/27, we never get there.

The precise reason: 27 has a factor of 3

A fraction in lowest terms gives a terminating decimal only if its denominator has no prime factors other than 2 and 5. That’s because we count in base 10, and 10 = 2 × 5.

Let’s test 8/27:

Is it in lowest terms? 8 = 2³ and 27 = 3³. They share no factors, so yes.
What are the prime factors of the denominator? 27 = 3 × 3 × 3.
Are they only 2s and 5s? No, they’re 3s.

So 8/27 can’t terminate. It has to repeat.

Why a block of exactly 3 digits?

Here’s a neat bit of number magic. 10³ − 1 = 999, and 999 = 27 × 37. Because 27 divides evenly into 999, the decimal repeats every 3 digits. (For a denominator like 7, you’d need 999999 before it divides evenly, which is why 1/7 has a 6-digit block.)

This gives us a handy shortcut too: 1/27 = 37/999 = 0.037037…, and we’ll use that below.

How to Turn 0.296296... Back Into a Fraction

This is a great way to check your answer. Let x = 0.296296…

Step 1: The repeating block has 3 digits, so multiply by 10³ = 1000:
1000x = 296.296296…

Step 2: Subtract the original x from it:
1000x − x = 296.296296… − 0.296296…
999x = 296

Step 3: Solve for x:
x = 296/999

Step 4: Simplify. Both numbers divide by 37, since 296 = 8 × 37 and 999 = 27 × 37:
x = 8/27 ✓

We’re right back where we started, so the decimal is correct.

A Handy Shortcut: The "27 Pattern"

27 is a friendly denominator, because 1/27 = 0.037037… and every other fraction over 27 is just a multiple of that. Take a look:

Fraction	Decimal
1/27	0.037037…
2/27	0.074074…
4/27	0.148148…
5/27	0.185185…
7/27	0.259259…
8/27	0.296296…
10/27	0.370370…
13/27	0.481481…
26/27	0.962962…

Spot the trick? The repeating block is just the numerator times 37. For 8/27, that’s 8 × 37 = 296. It’s a quick way to check your long division in a few seconds (as long as the numerator is below 27, so the block stays within 3 digits).

Rounding 8/27 to Different Decimal Places

Since the decimal never ends, you’ll often be asked to round it:

Rounded to	Answer
1 decimal place	0.3
2 decimal places	0.30
3 decimal places	0.296
4 decimal places	0.2963
5 decimal places	0.29630

Watch the 4 decimal place case. The digits run 0.2962 96…, and the next digit is 9, so the last 2 rounds up to a 3. This is a spot where lots of people slip.

As a percentage, 8/27 is about 29.63%.

Is 8/27 a Rational Number?

Yes. A rational number is any number that can be written as a fraction of two whole numbers, and 8/27 is exactly that. Every repeating decimal is rational. Irrational numbers like π or √2 are the ones whose digits go on forever with no repeating pattern at all.

So although 0.296296… never ends, there’s nothing mysterious about it. It’s predictable, and it always converts back to a neat fraction.

Mistakes Almost Everyone Makes

Writing 0.296 and stopping. That’s only an approximation. The exact value is 0.296296…, with the bar over the whole block.
Putting the bar over the wrong digits. The bar goes over all of 296, not just the 6.
Forgetting the starting zero. Because 8 is smaller than 27, the answer begins 0. and your first step is dividing 80 by 27.
Waiting for a zero remainder. You’ll be waiting forever. Instead, watch for a remainder you’ve seen before.
Mixing up 1/27 and 8/27. 1/27 = 0.037037… and 8/27 = 0.296296… Always check the numerator.
Rounding the 4th decimal place wrongly. 0.2962963… becomes 0.2963, not 0.2962.

Try It Yourself

Want to see if it’s sunk in? Try these with long division:

5/27 (answer: 0.185185…)
4/27 (answer: 0.148148…)
2/9 (answer: 0.222…)

Find the remainder that comes back around, and you’ll find the repeating block every time.

Conclusion

- 8/27 = 0.296296…, with a bar over 296
- The repeating block is 296, and it has 3 digits
- Long division gives remainders 26, 17 and 8, and when 8 comes back, the digits repeat
- It repeats because 27 = 3³ contains a factor other than 2 or 5
- Shortcut: 1/27 = 0.037037…, and the block for k/27 is k × 37
- Convert it back with 999x = 296, which gives 296/999 = 8/27

Once you’ve worked through this one by hand, repeating decimals stop feeling scary. They’re just a reminder coming back around.

Decimal Representation of 8/27 (Step-by-Step Long Division + Why It Repeats)

Decimal Representation of 8/27 (Step-by-Step Long Division + Why It Repeats)

What Does "Decimal Representation of a Fraction" Mean?

Step-by-Step Long Division of 8 ÷ 27

The Remainder Table (This Is the Key to Everything)

Why Does 8/27 Repeat Forever?

How to Turn 0.296296... Back Into a Fraction

A Handy Shortcut: The "27 Pattern"

Rounding 8/27 to Different Decimal Places

Is 8/27 a Rational Number?

Mistakes Almost Everyone Makes

Try It Yourself

Conclusion

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Decimal Representation of 8/27 (Step-by-Step Long Division + Why It Repeats)

What Does "Decimal Representation of a Fraction" Mean?

Step-by-Step Long Division of 8 ÷ 27

The Remainder Table (This Is the Key to Everything)

Why Does 8/27 Repeat Forever?

How to Turn 0.296296... Back Into a Fraction

A Handy Shortcut: The "27 Pattern"

Rounding 8/27 to Different Decimal Places

Is 8/27 a Rational Number?

Mistakes Almost Everyone Makes

Try It Yourself

Conclusion

Related Posts

Leave a Comment Cancel Reply

Start Your Child's Learning Journey

Thank You! 🎉

Book Your Free Demo