Recurring Decimals Worksheet | Printable Maths Worksheets - Free Printable

Let’s solve this step by step.

---

Section A: Ninths

#### 1) Match the decimal to the correct fraction.

We know that:

- 1/9 = 0.111... → written as 0.̇1
- 2/9 = 0.222... → 0.̇2
- 4/9 = 0.444... → 0.̇4
- 5/9 = 0.555... → 0.̇5
- 7/9 = 0.777... → 0.̇7
- 8/9 = 0.888... → 0.̇8

So matching them:

Top row (fractions):
2/9 → 0.̇2
5/9 → 0.̇5
7/9 → 0.̇7
8/9 → 0.̇8
1/9 → 0.̇1
4/9 → 0.̇4

Bottom row (decimals):
0.̇7 → 7/9
0.̇5 → 5/9
0.̇2 → 2/9
0.̇1 → 1/9
0.̇4 → 4/9
0.̇8 → 8/9

✔ So we connect each fraction to its matching decimal above.

What do you notice?
→ The numerator of the fraction is the same as the digit that repeats in the decimal!
Example: 5/9 = 0.̇5, 7/9 = 0.̇7, etc.

---

#### 2) Now write these decimals as fractions.

These are mixed numbers with a whole number and a recurring decimal part.

Remember:
0.̇7 = 7/9
0.̇8 = 8/9
0.̇2 = 2/9

So:

- 2.̇7 = 2 + 0.̇7 = 2 + 7/9 = 2 7/9 or as improper fraction: (2×9 + 7)/9 = 25/9
But since the question says “write as fractions”, and doesn’t specify mixed or improper, either is fine — but let’s use mixed unless told otherwise. However, looking at Section B, they use improper fractions like 41/99, so maybe they want improper? Let’s check context.

Actually, in Section B, all answers are improper fractions over 99 or 999. But here, for ninths, it’s simpler to leave as mixed or convert to improper. Since the blank is just “= ___”, and no format specified, I’ll give both forms but choose improper to match later sections.

Wait — actually, in Section A, Question 2, the examples are small: 2.̇7, 1.̇8, 3.̇2 — probably expecting mixed numbers or simple improper fractions.

Let me calculate properly:

- 2.̇7 = 2 + 7/9 = 25/9
- 1.̇8 = 1 + 8/9 = 17/9
- 3.̇2 = 3 + 2/9 = 29/9

Yes, those are correct.

---

Section B

Rule: For a recurring decimal with two digits repeating (like 0.̇4̇1), put the repeating part over 99.
For three digits repeating (like 0.̇6̇3̇9), put over 999.
If there’s a whole number, add it on.

Also, if the decimal starts after the point with zeros, like 0.0̇7, that means 0.0777..., which is 7/90? Wait — hold on!

Wait — look at the first example given:
0.̇4̇1 = 41/99 → that’s correct because two digits repeat.

But what about 0.0̇7? That’s 0.0777... — not 07/99, because the zero before the 7 is NOT repeating. Only the 7 repeats.

Ah — important distinction!

In the table, they have:

Left column: Recurring decimal → Fraction
Right column: Fraction → Recurring decimal

And the first one is done: 0.̇4̇1 = 41/99 → correct.

Now, 0.̇7̇3 → two digits repeat → 73/99
0.̇8̇9 → 89/99
0.0̇7 → This is tricky. It’s 0.0777... — only the 7 repeats, starting from the second decimal place.

How to convert 0.0̇7?

Let x = 0.0777...

Multiply by 10: 10x = 0.777...
Multiply by 100: 100x = 7.777...

Subtract: 100x - 10x = 7.777... - 0.777... → 90x = 7 → x = 7/90

Similarly, 0.̇6̇39 → three digits repeat → 639/999
0.̇4̇68 → 468/999
0.̇8̇3̇7 → 837/999
1.0̇8 → This is 1.0888... — so whole number 1, then 0.0888...

Let y = 0.0888...
10y = 0.888...
100y = 8.888...
Subtract: 90y = 8 → y = 8/90 = 4/45
So 1.0̇8 = 1 + 4/45 = 49/45? Or keep as mixed? But again, looking at the table, they might expect improper fractions.

Wait — but in the right-hand table, they give fractions like 16/99, 13/99, etc., and ask for recurring decimals.

So for left table, we need to convert decimals to fractions.

Let’s go one by one.

---

#### Left Table: Recurring decimal → Fraction

1. 0.̇4̇1 = 41/99 ← given

2. 0.̇7̇3 → two digits repeat → 73/99

3. 0.̇8̇9 → 89/99

4. 0.0̇7 → as above, 0.0777... = 7/90 → 7/90

Wait — but 7/90 is not over 99 or 999. Is that acceptable? Yes, because not all recurring decimals fit neatly over 99 or 999 — only when the repetition starts immediately after the decimal.

But let’s double-check the pattern in the worksheet. In Section B, most are over 99 or 999, but 0.0̇7 is different.

Looking ahead, in the right table, they have 5/99, which would be 0.̇0̇5, not 0.0̇5.

Hmm — perhaps there’s a mistake in my assumption.

Wait — maybe in this worksheet, they consider 0.0̇7 as having the "07" repeating? But that would be 0.070707... which is 7/99, not 0.0777...

The notation: 0.0̇7 usually means the 7 repeats, so 0.0777...

But in some contexts, especially in worksheets, they might mean the digits under the dots repeat together. Looking at the first example: 0.̇4̇1 — dots over 4 and 1, meaning both repeat: 0.414141...

Similarly, 0.̇7̇3 → 0.737373...

Then 0.0̇7 — dot only over 7, so only 7 repeats: 0.0777...

But then 0.̇6̇3̇9 — dots over 6,3,9 → 0.639639...

0.̇4̇6̇8 → 0.468468...

0.̇8̇3̇7 → 0.837837...

1.0̇8 → dot over 8, so 1.0888...

So yes, my initial thought is correct.

But now, for 0.0̇7 = 7/90, which is not over 99. Similarly, 1.0̇8 = 1 + 8/90 = 1 + 4/45 = 49/45.

But let’s see if the worksheet expects us to simplify or leave as is.

Perhaps for consistency, they want all fractions in simplest form.

But let’s proceed.

However, looking at the right-hand table, they have fractions like 16/99, which is already simplified, and 672/999, which can be simplified.

672 ÷ 3 = 224, 999 ÷ 3 = 333, so 224/333, but they left it as 672/999, so perhaps they don’t require simplifying.

Similarly, 78/999 can be simplified, but they may want it as is.

So for left table, we should write the fraction as per the repeating pattern.

But for 0.0̇7, it’s not a standard "over 99" case.

Perhaps I made a mistake. Let me check online or recall: sometimes in such worksheets, if there's a leading zero, they still treat it as part of the repeat if dotted, but here only the 7 is dotted.

Another way: perhaps 0.0̇7 is meant to be 0.070707... but that would be written as 0.̇0̇7.

The notation is crucial.

In the image, for 0.0̇7, the dot is only over the 7, not over the 0. So it should be 0.0777...

Similarly, in 1.0̇8, dot over 8, so 1.0888...

So I think we have to handle them correctly.

But let's list all:

Left table:

- 0.̇4̇1 = 41/99 (given)

- 0.̇7̇3 = 73/99

- 0.̇8̇9 = 89/99

- 0.0̇7 = 0.0777... = 7/90

- 0.̇6̇3̇9 = 639/999

- 0.̇4̇68 = 468/999

- 0.̇8̇37 = 837/999

- 1.0̇8 = 1.0888... = 1 + 8/90 = 98/90? Wait no:

1.0888... = 1 + 0.0888...

Let z = 0.0888...

10z = 0.888...

100z = 8.888...

100z - 10z = 8.888... - 0.888... = 8

90z = 8 => z = 8/90 = 4/45

So 1 + 4/45 = 49/45

Or as improper fraction: (45*1 + 4)/45 = 49/45

But 49/45 is greater than 1, which is fine.

Now, for the right table: Fraction → Recurring decimal

- 16/99 = 0.̇16 (since 16 repeats)

- 13/99 = 0.̇1̇3

- 82/99 = 0.̇8̇2

- 5/99 = 0.̇0̇5

- 672/999 = 0.̇67̇2 (three digits repeat)

- 78/999 = 0.̇0̇7̇8? Wait, 78/999 = 0.078078... so 0.̇0̇7̇8

But usually, we write it as 0.̇078 with dots over 0,7,8 or just over the repeating part.

Since 78/999, divide numerator and denominator by 3: 26/333, but better to keep as is for conversion.

78 ÷ 999 = 0.078078... so recurring decimal is 0.̇0̇7̇8

Similarly, 2/999 = 0.002002... = 0.̇0̇0̇2

Now, back to left table, for 0.0̇7, if we must write a fraction, and if the worksheet expects denominator 99 or 999, perhaps they intend 0.̇07 for 7/99, but the dot is only on 7.

I think there might be an error in my interpretation, or perhaps in the worksheet design.

Let me look at the first example: 0.̇4̇1 = 41/99 — dots over both digits.

For 0.0̇7, dot only on 7, so it should be different.

But to match the pattern, perhaps for this level, they only consider cases where the repeat starts immediately, and 0.0̇7 is an exception.

Maybe 0.0̇7 is meant to be 7/90, and we write it as is.

Similarly, 1.0̇8 = 49/45.

But let's see the "what do you notice?" at the end — probably that for two-digit repeat, over 99; three-digit, over 999; and for single digit after zero, over 90, etc.

But to be safe, let's assume that for decimals where the repeat starts right after decimal, use 99 or 999; otherwise, calculate separately.

For the sake of this worksheet, and since most are over 99 or 999, perhaps 0.0̇7 is a typo or meant to be 0.̇0̇7.

Let me check the value: if 0.0̇7 = 7/90 ≈ 0.0778, while 7/99 ≈ 0.0707, different.

In the right table, they have 5/99 = 0.̇0̇5, which is 0.050505..., so if they wanted 0.0555..., it would be 5/90 = 1/18.

So I think we have to go with the notation.

Perhaps for 0.0̇7, it's 7/90, and for 1.0̇8, 49/45.

But let's complete the table as per standard math.

So for left table:

- 0.̇7̇3 = 73/99

- 0.̇8̇9 = 89/99

- 0.0̇7 = 7/90

- 0.̇6̇39 = 639/999

- 0.̇4̇68 = 468/999

- 0.̇8̇37 = 837/999

- 1.0̇8 = 49/45 (since 1 + 8/90 = 98/90? No:

1.0888... = 108/100 + adjustment? Better: let w = 1.0888...

10w = 10.888...

100w = 108.888...

100w - 10w = 108.888... - 10.888... = 98

90w = 98 => w = 98/90 = 49/45

Yes.

Now for right table:

- 16/99 = 0.̇1̇6

- 13/99 = 0.̇1̇3

- 82/99 = 0.̇8̇2

- 5/99 = 0.̇0̇5

- 672/999 = 0.̇6̇7̇2

- 78/999 = 0.̇07̇8 (since 78/999 = 0.078078...)

- 2/999 = 0.̇0̇02

Now, for the "what do you notice?" in Section B:

→ When you have a recurring decimal with n digits repeating, you can write it as the repeating part over a number with n nines. If there are non-repeating digits before the repeat, you need to adjust (like multiplying by 10^k).

But for this level, perhaps: "For decimals where the repeat starts right after the decimal point, the fraction has the repeating digits over 99 for two digits, 999 for three digits."

Also, for fractions over 99 or 999, the decimal is the numerator with leading zeros if necessary, repeated.

Now, to summarize all answers.

First, Section A:

1) Matching:

Fractions to decimals:

2/9 -> 0.̇2

5/9 -> 0.̇5

7/9 -> 0.̇7

8/9 -> 0.̇8

1/9 -> 0.̇1

4/9 -> 0.̇4

Decimals to fractions:

0.̇7 -> 7/9

0.̇5 -> 5/9

0.̇2 -> 2/9

0.̇1 -> 1/9

0.̇4 -> 4/9

0.̇8 -> 8/9

What do you notice? The numerator is the repeating digit.

2)

2.̇7 = 25/9 or 2 7/9 — I'll use 25/9 to match Section B style.

Similarly, 1.̇8 = 17/9

3.̇2 = 29/9

Section B left table:

Recurring decimal | Fraction

0.̇4̇1 | 41/99 (given)

0.̇7̇3 | 73/99

0.̇8̇9 | 89/99

0.0̇7 | 7/90

0.̇6̇39 | 639/999

0.̇4̇68 | 468/999

0.̇8̇3̇7 | 837/999

1.0̇8 | 49/45

Right table:

Recurring decimal | Fraction

0.̇1̇6 | 16/99

0.̇1̇3 | 13/99

0.̇8̇2 | 82/99

0.̇0̇5 | 5/99

0.̇6̇72 | 672/999

0.̇0̇78 | 78/999

0.̇0̇0̇2 | 2/999

What do you notice? For a fraction with denominator 99, the decimal has two digits repeating; for 999, three digits repeating. The numerator is the repeating sequence.

For 0.0̇7 and 1.0̇8, they are exceptions because the repeat doesn't start immediately.

But perhaps in this worksheet, they expect 0.0̇7 to be 7/99, but that would be incorrect based on notation.

To resolve, let's assume that for 0.0̇7, since only 7 is dotted, it's 7/90, and similarly for others.

Perhaps the worksheet has a mistake, but I'll go with correct math.

For the final answer, I'll provide the filled tables.

But the user asked to solve the problem, so I need to output the answers.

Since the response format requires Final Answer after reasoning, and no other options are true, I'll box the key answers.

But it's a multi-part question.

Perhaps list all answers clearly.

Let me compile.

Final Answer:

Section A:

1) Matches:

- 2/9 ↔ 0.̇2

- 5/9 ↔ 0.̇5

- 7/9 ↔ 0.̇7

- 8/9 ↔ 0.̇8

- 1/9 ↔ 0.̇1

- 4/9 ↔ 0.̇4

What do you notice? The numerator of the fraction is the same as the repeating digit in the decimal.

2)
2.̇7 = \frac{25}{9}
1.̇8 = \frac{17}{9}
3.̇2 = \frac{29}{9}

Section B:

Left table (Recurring decimal to Fraction):

- 0.̇7̇3 = \frac{73}{99}
- 0.̇8̇9 = \frac{89}{99}
- 0.0̇7 = \frac{7}{90}
- 0.̇6̇3̇9 = \frac{639}{999}
- 0.̇4̇6̇8 = \frac{468}{999}
- 0.̇8̇3̇7 = \frac{837}{999}
- 1.0̇8 = \frac{49}{45}

Right table (Fraction to Recurring decimal):

- \frac{16}{99} = 0.\dot{1}\dot{6}
- \frac{13}{99} = 0.\dot{1}\dot{3}
- \frac{82}{99} = 0.\dot{8}\dot{2}
- \frac{5}{99} = 0.\dot{0}\dot{5}
- \frac{672}{999} = 0.\dot{6}\dot{7}\dot{2}
- \frac{78}{999} = 0.\dot{0}\dot{7}\dot{8}
- \frac{2}{999} = 0.\dot{0}\dot{0}\dot{2}

What do you notice? When the denominator is 99, the decimal has two repeating digits; when 999, three repeating digits. The numerator gives the repeating sequence. For decimals like 0.0̇7, where the repeat doesn't start immediately, the fraction has a different denominator (like 90).

But to match the worksheet's likely expectation, perhaps they want 0.0̇7 as 7/99, but that's wrong. I'll stick with correct math.

For the sake of this, I'll provide the answers as calculated.

So Final Answer will be the completed values.

Since it's long, I'll summarize in the box.

Perhaps the user wants the specific blanks filled.

Looking back at the image, in Section A Q2, there are three blanks: 2.̇7 = __, etc.

In Section B, tables to fill.

So for Final Answer, I'll list the required fills.

But the instruction is to provide the final correct answer after reasoning.

I think for clarity, I'll state the answers for the blanks.

So:

For Section A, Question 2:
2.̇7 = \frac{25}{9}
1.̇8 = \frac{17}{9}
3.̇2 = \frac{29}{9}

For Section B, left table fractions:
0.̇7̇3 = \frac{73}{99}
0.̇8̇9 = \frac{89}{99}
0.0̇7 = \frac{7}{90}
0.̇6̇3̇9 = \frac{639}{999}
0.̇4̇6̇8 = \frac{468}{999}
0.̇8̇3̇7 = \frac{837}{999}
1.0̇8 = \frac{49}{45}

Right table recurring decimals:
\frac{16}{99} = 0.\dot{1}\dot{6}
\frac{13}{99} = 0.\dot{1}\dot{3}
\frac{82}{99} = 0.\dot{8}\dot{2}
\frac{5}{99} = 0.\dot{0}\dot{5}
\frac{672}{999} = 0.\dot{6}\dot{7}\dot{2}
\frac{78}{999} = 0.\dot{0}\dot{7}\dot{8}
\frac{2}{999} = 0.\dot{0}\dot{0}\dot{2}

And for "what do you notice?" in Section A: The numerator is the repeating digit.

In Section B: For fractions with denominator 99 or 999, the decimal repeats the numerator's digits. For decimals with immediate repeat, use 99 or 999; otherwise, adjust.

But since the Final Answer should be concise, perhaps just the numerical answers for the blanks.

Given the complexity, I'll box the key results.

After careful consideration, I think for educational purposes, providing the correct mathematical answers is best.

So here is the Final Answer with all required fills.