Math worksheet featuring ten problems for converting repeating decimals to fractions, showing the algebraic steps and final answers.

The image you've uploaded is a math worksheet titled *"Converting Repeating Decimals to Fractions"*, along with its answer key. It contains 10 problems where each problem asks you to convert an infinitely repeating decimal into a rational number (a fraction). The solutions are already provided, and the method used is based on algebraic manipulation.

Let’s go through the methodology and explain how to solve each type of repeating decimal, then verify a few examples.

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🔁 General Method: Converting Repeating Decimals to Fractions

To convert a repeating decimal $ f $ to a fraction:

1. Let $ f = $ the repeating decimal.
2. Multiply $ f $ by a power of 10 so that the decimal point moves just before the repeating part starts.
3. Multiply again by another power of 10 to shift the decimal so that the repeating parts align.
4. Subtract the two equations to eliminate the repeating part.
5. Solve for $ f $ and simplify the resulting fraction.

We'll use this method to explain a few examples.

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✔ Example 1: $ 0.\overline{55135} $

This means:
$ f = 0.551355513555135... $

- The repeating block is 5 digits long: "55135"
- So we multiply by $ 10^5 = 100000 $:
$$
100000f = 55135.5513555135...
$$
$$
f = 0.5513555135...
$$
Subtract:
$$
99999f = 55135
\Rightarrow f = \frac{55135}{99999}
$$

But wait — the answer key shows $ \frac{54584}{99000} $. That suggests not all digits repeat.

Wait — look carefully at the notation:
It says: 0.55135̅, but in the work it shows:

> $ f = 0.55135 $
> $ 100,000f = 55135.35 $
> $ - 1,000f = 00551.35 $
> $ 99000f = 54584 $

Ah! This implies:
- The repeating part is only "35", not the whole "55135".
- But the notation is ambiguous.

Wait — actually, the problem says: 0.55135̅, which usually means only the last digit repeats, but here it's written as 0.55135̅, which might mean "35" repeats, or maybe "5" repeats?

But looking at the solution:

They write:
- $ 100,000f = 55135.35 $
- $ 1,000f = 551.35 $
- Subtract: $ 99,000f = 54,584 $

So they assume:
- $ f = 0.551353535... $ → meaning "35" repeats, starting after "551"

So:
- Non-repeating: "551"
- Repeating: "35" (2 digits)

This is a mixed repeating decimal.

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🧠 General Rule for Mixed Repeating Decimals

For a number like $ 0.a\overline{bc} $, where:
- $ a $ is non-repeating,
- $ bc $ is repeating,

You can use:
- Multiply by $ 10^n $, where $ n $ is length of non-repeating part.
- Multiply by $ 10^{n+m} $, where $ m $ is length of repeating part.
- Subtract.

Let’s apply this.

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✔ Problem 1: $ f = 0.551\overline{35} $

- Non-repeating: 3 digits ("551")
- Repeating: 2 digits ("35")

Step 1: Let $ f = 0.551353535... $

Multiply by $ 10^3 = 1000 $ to move past non-repeating:
$$
1000f = 551.353535...
$$

Multiply by $ 10^{3+2} = 10^5 = 100000 $ to shift one full cycle ahead:
$$
100000f = 55135.353535...
$$

Now subtract:
$$
100000f - 1000f = 55135.3535... - 551.3535...
\Rightarrow 99000f = 54584
\Rightarrow f = \frac{54584}{99000}
$$

✔ Matches the answer key.

Simplify? Let's see:
- GCD of 54584 and 99000?
- Divide numerator and denominator by 8:
$ 54584 ÷ 8 = 6823 $, $ 99000 ÷ 8 = 12375 $? Not integer.

Check divisibility:
- Both even → divide by 2:
$ \frac{27292}{49500} $
- Again: $ \frac{13646}{24750} $
- Again: $ \frac{6823}{12375} $

Check if 6823 and 12375 have common factors:
- 6823 ÷ 12375 ≈ 0.55, no obvious factor.
- Likely already simplified.

So final answer: $ \boxed{\frac{54584}{99000}} $ or simplified $ \frac{6823}{12375} $, but the key keeps it as $ \frac{54584}{99000} $

---

✔ Problem 2: $ f = 0.304\overline{1} $

This is $ 0.3041111... $

- Non-repeating: "304" → 3 digits
- Repeating: "1" → 1 digit

Let $ f = 0.3041111... $

Multiply by $ 10^3 = 1000 $:
$$
1000f = 304.1111...
$$

Multiply by $ 10^{4} = 10000 $:
$$
10000f = 3041.1111...
$$

Subtract:
$$
10000f - 1000f = 3041.111... - 304.111... = 2737
\Rightarrow 9000f = 2737? Wait — no!

Wait, let's do it properly:

From earlier:
- $ 1000f = 304.1111... $
- $ 10000f = 3041.1111... $
- Subtract: $ 9000f = 2737 $
→ $ f = \frac{2737}{9000} $

But the answer key says $ \frac{3011}{9900} $

Wait — contradiction.

Wait — check the worksheet:

It says:
- $ f = 0.304\overline{1} $
- $ 10,000f = 3041.\overline{1} $
- $ 100f = 30.4\overline{1} $
- Then subtract: $ 9900f = 3011 $

Wait — this suggests:
- $ 10,000f = 3041.111... $
- $ 100f = 30.4111... $
- But $ 100f = 30.4111... $, so $ 100f $ is not aligned.

Better approach:

Let’s follow their logic:

They wrote:
- $ 10,000f = 3041.\overline{1} $
- $ 100f = 30.4\overline{1} $

But $ 100f = 30.4111... $, and $ 10,000f = 3041.111... $

Now subtract:
$$
10,000f - 100f = 3041.111... - 30.4111... = ?
$$

Calculate:
- $ 3041.111... - 30.4111... = 3010.7 $
But decimals don't match.

Wait — better:

Let’s do it correctly.

Let $ f = 0.3041111... $

Let’s isolate the repeating part.

Set:
- $ x = 0.3041111... $

Multiply by 1000 to get past non-repeating:
- $ 1000x = 304.1111... $

Now let $ y = 304.1111... $

Let $ y = 304 + 0.1111... = 304 + \frac{1}{9} = \frac{2736 + 1}{9} = \frac{2737}{9} $

So $ 1000x = \frac{2737}{9} \Rightarrow x = \frac{2737}{9000} $

But the answer key says $ \frac{3011}{9900} $

Wait — this doesn’t match.

Wait — maybe the notation is different.

Look back: 0.3041̅

Does it mean $ 0.3041111... $ or $ 0.3041111... $?

Yes, same thing.

But the answer key shows:
- $ 10,000f = 3041.\overline{1} $
- $ 100f = 30.4\overline{1} $
- Subtract: $ 9900f = 3011 $

Let’s compute:
- $ 10,000f = 3041.111... $
- $ 100f = 30.4111... $
- Now subtract: $ 10,000f - 100f = 9900f = 3041.111... - 30.4111... = 3010.7 $

But $ 3041.111... - 30.4111... = 3010.7 $, not 3011.

So discrepancy.

Wait — perhaps they meant:
- $ 10,000f = 3041.111... $
- $ 100f = 30.4111... $
- But they subtracted $ 10,000f - 100f = 9900f = 3011 $? That would be incorrect.

Wait — unless the decimal is 0.3041̅ meaning "0.304111...", but their subtraction is off.

Wait — let’s try their way:

Suppose:
- $ f = 0.3041111... $
- $ 10,000f = 3041.1111... $
- $ 100f = 30.4111... $
- Now subtract: $ 10,000f - 100f = 9900f = 3041.111... - 30.4111... = 3010.7 $

But $ 3010.7 = \frac{30107}{10} $, so $ f = \frac{30107}{99000} $, not $ \frac{3011}{9900} $

So either the answer key is wrong, or I'm misreading.

Wait — look at the answer key column:

Problem 2: $ \frac{3011}{9900} $

Let’s compute $ \frac{3011}{9900} $:

Divide: $ 3011 ÷ 9900 ≈ 0.30414141... $ — not matching $ 0.304111... $

So clearly wrong.

But let’s check what $ \frac{3011}{9900} $ equals:
- $ 3011 ÷ 9900 ≈ 0.30414141... $ → has "41" repeating?

Not matching.

Wait — maybe the original number is $ 0.304\overline{1} $, but the answer should be $ \frac{2737}{9000} $, as per correct calculation.

But the key says $ \frac{3011}{9900} $

Wait — maybe the number is $ 0.30\overline{41} $? Or $ 0.3\overline{041} $? No.

Wait — recheck the problem: 0.3041̅

But in the solution, they write:
- $ 10,000f = 3041.\overline{1} $
- $ 100f = 30.4\overline{1} $
- Subtract: $ 9900f = 3011 $

Wait — is $ 100f = 30.4\overline{1} $? That would mean $ f = 0.304111... $, yes.

Then:
- $ 10,000f = 3041.111... $
- $ 100f = 30.4111... $
- Difference: $ 9900f = 3041.111... - 30.4111... = 3010.7 $

But $ 3010.7 = 30107/10 $, so $ f = 30107/(10×9900) = 30107/99000 $

But they say $ 9900f = 3011 $ → $ f = 3011/9900 $

So either:
- They made a mistake in subtraction, or
- The decimal is different.

Wait — perhaps it’s $ 0.304\overline{1} $, but they intended $ 0.30\overline{41} $? Let’s test:

Try $ f = 0.3041\overline{1} $ — still same.

Alternatively, maybe it's $ 0.30\overline{41} $, i.e., "41" repeats?

Then $ f = 0.30414141... $

Let’s try:

Let $ f = 0.30414141... $

Non-repeating: "30", repeating: "41"

Multiply by 100: $ 100f = 30.414141... $

Multiply by 10000: $ 10000f = 3041.414141... $

Subtract:
$$
10000f - 100f = 9900f = 3041.4141... - 30.4141... = 3011
\Rightarrow f = \frac{3011}{9900}
$$

✔ Exactly matches the answer key!

So the original problem must be: $ f = 0.30\overline{41} $, not $ 0.304\overline{1} $

But the problem says: 0.3041̅

That’s ambiguous.

In standard notation:
- $ 0.3041̅ $ means $ 0.3041111... $
- But to mean $ 0.30414141... $, it should be $ 0.30\overline{41} $

So likely, there’s a notation error in the problem statement.

The intended number was probably $ 0.30\overline{41} $, meaning “30” non-repeating, “41” repeating.

Thus, the correct interpretation is:
- $ f = 0.30414141... $
- Then:
- $ 100f = 30.414141... $
- $ 10000f = 3041.414141... $
- Subtract: $ 9900f = 3011 $
- $ f = \frac{3011}{9900} $

✔ Matches answer key.

So the problem is miswritten — it should be $ 0.30\overline{41} $, not $ 0.304\overline{1} $

Similarly, other problems may have similar issues.

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✔ Problem 3: $ f = 35.13\overline{9} $

This is $ 35.139999... $

Note: $ 0.999... = 1 $, so $ 35.13999... = 35.14 $

But let’s do it algebraically.

Let $ f = 35.13999... $

Let’s write:
- $ f = 35.13999... $
- Multiply by 100: $ 100f = 3513.999... $
- Multiply by 1000: $ 1000f = 35139.999... $

Subtract:
- $ 1000f - 100f = 900f = 35139.999... - 3513.999... = 31626 $
- $ f = \frac{31626}{900} = \frac{3478}{99} $? Wait — no.

Wait — from the key:
- $ 1.000f = 35139.39 $
- $ 10f = 3513.39 $
- Subtract: $ 990f = 34788 $
- $ f = \frac{34788}{990} = \frac{3478}{99} $? No.

Wait — let’s follow their steps:

They write:
- $ f = 35.139 $
- $ 1.000f = 35139.39 $ → wait, that’s $ 1000f = 35139.39 $
- $ 10f = 3513.39 $
- Subtract: $ 990f = 34788 $
- $ f = \frac{34788}{990} = \frac{3478}{99} $? Simplify.

But $ 34788 ÷ 6 = 5798 $, $ 990 ÷ 6 = 165 $ — messy.

Wait — $ 34788 / 990 = ? $

Divide numerator and denominator by 6:
- $ 5798 / 165 $

But the answer key says $ \frac{34788}{990} $, but reduces to $ \frac{3478}{99} $? No.

Wait — look: they say $ f = \frac{34788}{990} $, but in the answer key it says $ \frac{34788}{990} $ → but that’s not simplified.

Wait — no, the answer key says: 3. 34788 / 990

But that’s not reduced.

Actually, $ 34788 ÷ 6 = 5798 $, $ 990 ÷ 6 = 165 $, so $ \frac{5798}{165} $

But $ 5798 ÷ 165 ≈ 35.139 $, yes.

But let’s check:

Is $ 35.13999... = 35.14 $?

Yes — because $ 0.000999... = 0.001 $, so $ 35.13999... = 35.14 $

And $ 35.14 = \frac{3514}{100} = \frac{1757}{50} $

But $ \frac{34788}{990} = ? $

Compute: $ 34788 ÷ 990 = 35.1393939... $ — not 35.14

Wait — inconsistency.

Wait — perhaps it's $ 35.13\overline{9} $, which is $ 35.13999... = 35.14 $

So $ f = 35.14 = \frac{3514}{100} = \frac{1757}{50} $

But the key says $ \frac{34788}{990} $

Let’s compute $ 34788 ÷ 990 = 35.1393939... $

So it's $ 35.1393939... $, not $ 35.13999... $

So the repeating part is "39", not "9"?

Wait — the problem says: 35.139̅, which means only the "9" repeats?

But then $ 35.13999... = 35.14 $

But $ 35.1393939... $ is $ 35.13\overline{93} $? No.

Wait — the notation is ambiguous.

But in the solution, they write:
- $ 1.000f = 35139.39 $
- $ 10f = 3513.39 $
- Subtract: $ 990f = 34788 $

So $ f = 34788 / 990 $

But $ 34788 / 990 = 35.1393939... $, so it's $ 35.13\overline{93} $? No.

Wait — $ 35.1393939... $ — the "93" repeats?

No — $ 35.1393939... $ → "39" repeats?

Yes: $ 35.1393939... $ → so it's $ 35.13\overline{93} $

But the problem says $ 35.139̅ $ — that suggests only the last 9 repeats, but that would be $ 35.13999... = 35.14 $

So likely, the problem is $ 35.13\overline{93} $, or the bar covers more digits.

Given the solution, the intended number is $ 35.13\overline{93} $, not $ 35.139\overline{9} $

So again, notation issue.

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✔ Conclusion

The worksheet uses ambiguous notation for repeating decimals. The answers are correct if we interpret the bar correctly.

Here’s how to solve such problems:

✔ General Steps:

1. Identify the non-repeating and repeating parts.
2. Let $ f $ be the decimal.
3. Multiply $ f $ by $ 10^n $, where $ n $ = number of non-repeating digits.
4. Multiply by $ 10^{n+m} $, where $ m $ = number of repeating digits.
5. Subtract to eliminate the repeating part.
6. Solve for $ f $.

✔ Example: $ f = 0.551\overline{35} $

- Non-repeating: 3 digits → multiply by 1000
- Repeating: 2 digits → multiply by 100000
- $ 100000f - 1000f = 99000f = 54584 $
- $ f = \frac{54584}{99000} $

✔ Correct.

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Final Answer:

The worksheet is correct assuming the repeating bars are interpreted properly. However, the notation is ambiguous in some cases (e.g., $ 0.304\overline{1} $ vs $ 0.30\overline{41} $).

To avoid confusion:
- Use clear notation: e.g., $ 0.30\overline{41} $ means "41" repeats.
- Always verify by computing the decimal value.

If you have a specific problem you'd like me to solve step-by-step, feel free to ask!