Divide 3 by 13 in decimal expansion and show the division

To divide \( 3 \div 13 \) and express the result in decimal expansion, we can perform long division. Here's how the process works step by step:

1. Set up the division: \( 3 \div 13 \).

2. Since \( 3 \) is smaller than \( 13 \), we'll add a decimal point and proceed with division by introducing zeros after the decimal point.

### Long Division:

```
    0.2307692...
   ______________
13 | 3.000000000
     - 26
    ______
      40
     - 39
    ______
       10
      - 0
    ______
      100
     - 91
    ______
        90
      - 78
    ______
       120
     - 117
    ______
         30
      - 26
    ______
         40
```

#### Explanation of the steps:
1. \( 13 \) into \( 30 \) goes 2 times (since \( 13 \times 2 = 26 \)), subtract \( 26 \), remainder \( 4 \).
2. Bring down a 0 to make 40. \( 13 \) into \( 40 \) goes 3 times (since \( 13 \times 3 = 39 \)), subtract \( 39 \), remainder \( 1 \).
3. Bring down a 0 to make 10. \( 13 \) into \( 10 \) goes 0 times, so continue.
4. Bring down another 0 to make 100. \( 13 \) into \( 100 \) goes 7 times (since \( 13 \times 7 = 91 \)), subtract \( 91 \), remainder \( 9 \).
5. Bring down another 0 to make 90. \( 13 \) into \( 90 \) goes 6 times (since \( 13 \times 6 = 78 \)), subtract \( 78 \), remainder \( 12 \).
6. Bring down another 0 to make 120. \( 13 \) into \( 120 \) goes 9 times (since \( 13 \times 9 = 117 \)), subtract \( 117 \), remainder \( 3 \).
7. The remainder brings us back to 30, which starts the cycle over again.

### Final Decimal Expansion:
The decimal expansion of \( 3 \div 13 \) is a repeating decimal:
\[
0.\overline{230769}
\]
So, \( 3 \div 13 = 0.230769230769... \) with the repeating block of "230769."

Would you like more details, or do you have any questions?

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Here are 5 related questions for further understanding:
1. How would you divide other numbers by 13 using long division?
2. Can you express other fractions as repeating decimals, such as \( 5 \div 7 \)?
3. What is the length of the repeating block in the decimal expansion of \( 1 \div 17 \)?
4. How do you convert repeating decimals back to fractions?
5. Why do some fractions result in repeating decimals while others terminate?

**Tip:** Repeating decimals can always be converted into fractions! The length of the repeating cycle helps determine the denominator in such cases.

Divide 3 by 13 in decimal expansion and show the division.

Learn how to divide 3 by 13 using long division, with a step-by-step guide that reveals the repeating decimal expansion 0.230769. This method demonstrates the process of handling division with a non-terminating repeating decimal. Suitable for students in Grades 5-7.

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