Rewrite as a simplified fraction 2.67 repeating
The correct answer and explanation is:
✅ Correct Answer:
The decimal 2.67 repeating (written as 2.6̅7̅) can be written as the simplified fraction
267100\frac{267}{100}
✅ Explanation (300 words):
To convert a repeating decimal like 2.67 repeating into a fraction, follow a systematic method. First, understand the notation:
- 2.67 repeating means that both 6 and 7 repeat forever: 2.67676767…2.67676767\ldots
We can write this repeating decimal as a fraction using algebra:
Step 1: Let x=2.676767…x = 2.676767\ldots
Step 2: Multiply x by 100 (since two digits repeat): 100x=267.676767…100x = 267.676767\ldots
Step 3: Subtract the original x from this equation: 100x−x=267.676767…−2.676767…⇒99x=265100x – x = 267.676767\ldots – 2.676767\ldots \Rightarrow 99x = 265
Step 4: Solve for x: x=26599x = \frac{265}{99}
So the repeating decimal 2.676767… equals 26599\frac{265}{99}
✅ Now simplify if possible. But in this case, 265 and 99 have no common factor other than 1, so this is already simplified.
🔁 Alternate Method Using Long Decimal:
You can also convert by rewriting the decimal:
- 2.676767… = 2 + 0.676767…
- Convert 0.676767… to a fraction first:
- Let y = 0.676767…
- Then 100y = 67.676767…
- Subtract: 100y − y = 67.676767… − 0.676767… = 67
- So 99y = 67 → y = 67/99
Now add back the whole number:
- 2 + 67/99 = (2 × 99 + 67)/99 = (198 + 67)/99 = 265/99
✅ Final Answer:
26599\boxed{\frac{265}{99}}