Answer
\( 2.\overline{718} = \frac{2716}{999} \) and \( 10.\overline{213} = \frac{3401}{333} \).
Solution
Certainly! Let's tackle both parts of the problem step-by-step.
---
### **Problem (a): Convert the Repeating Decimal \( 2.\overline{718} \) to an Improper Fraction**
**Step 1: Let \( x \) represent the repeating decimal.**
\[
x = 2.\overline{718} = 2.718718718\ldots
\]
**Step 2: Multiply \( x \) by \( 1000 \) to shift the decimal point three places to the right (since the repeating block has three digits).**
\[
1000x = 2718.\overline{718} = 2718.718718\ldots
\]
**Step 3: Subtract the original \( x \) from this new equation to eliminate the repeating part.**
\[
1000x - x = 2718.\overline{718} - 2.\overline{718}
\]
\[
999x = 2716
\]
**Step 4: Solve for \( x \).**
\[
x = \frac{2716}{999}
\]
**Step 5: Simplify the fraction (if possible).**
To check if the fraction can be simplified, find the greatest common divisor (GCD) of 2716 and 999.
- Using the Euclidean algorithm:
- \( 2716 \div 999 = 2 \) with a remainder of 718
- \( 999 \div 718 = 1 \) with a remainder of 281
- \( 718 \div 281 = 2 \) with a remainder of 156
- \( 281 \div 156 = 1 \) with a remainder of 125
- \( 156 \div 125 = 1 \) with a remainder of 31
- \( 125 \div 31 = 4 \) with a remainder of 1
- \( 31 \div 1 = 31 \) with a remainder of 0
The GCD is **1**, so the fraction is already in its simplest form.
**Final Answer for (a):**
\[
2.\overline{718} = \frac{2716}{999}
\]
---
### **Problem (b): Convert the Repeating Decimal \( 10.\overline{213} \) to an Improper Fraction**
**Step 1: Let \( y \) represent the repeating decimal.**
\[
y = 10.\overline{213} = 10.213213213\ldots
\]
**Step 2: Multiply \( y \) by \( 1000 \) to shift the decimal point three places to the right (since the repeating block has three digits).**
\[
1000y = 10213.\overline{213} = 10213.213213\ldots
\]
**Step 3: Subtract the original \( y \) from this new equation to eliminate the repeating part.**
\[
1000y - y = 10213.\overline{213} - 10.\overline{213}
\]
\[
999y = 10203
\]
**Step 4: Solve for \( y \).**
\[
y = \frac{10203}{999}
\]
**Step 5: Simplify the fraction (if possible).**
First, divide both numerator and denominator by **3**:
\[
\frac{10203 \div 3}{999 \div 3} = \frac{3401}{333}
\]
Next, check if the fraction can be simplified further by finding the GCD of 3401 and 333.
- Using the Euclidean algorithm:
- \( 3401 \div 333 = 10 \) with a remainder of 71
- \( 333 \div 71 = 4 \) with a remainder of 49
- \( 71 \div 49 = 1 \) with a remainder of 22
- \( 49 \div 22 = 2 \) with a remainder of 5
- \( 22 \div 5 = 4 \) with a remainder of 2
- \( 5 \div 2 = 2 \) with a remainder of 1
- \( 2 \div 1 = 2 \) with a remainder of 0
The GCD is **1**, so the fraction is already in its simplest form.
**Final Answer for (b):**
\[
10.\overline{213} = \frac{3401}{333}
\]
---
**Summary of Solutions:**
- **(a)** \( 2.\overline{718} = \dfrac{2716}{999} \)
- **(b)** \( 10.\overline{213} = \dfrac{3401}{333} \)
Reviewed and approved by the UpStudy tutoring team
Explain
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