Question

Which of the following rational numbers can be expressed as a repeating decimal:7/8, 5/12, 17/25, or 31/40?

Answer 1

Express 7/13 as a repeating decimal.Mark the repeating digits with a bar.
Note:you will find the option to add a bar above the tes in the Geometry

Answer 2

ur wrong

Answer 3

Convert 0.8 repeating to a rational number

Answer 4

To convert 0.82 both repeating to a rational number, you would set x equal to 0.82 and then multiply both sides of the equation by what number?

Answer 5

To determine which of the given rational numbers can be expressed as a repeating decimal, we need to look at their denominators in simplest form. A rational number can be expressed as a repeating decimal if, after simplifying the fraction, its denominator has prime factors other than 2 and 5.

Let's analyze each fraction one by one:

1. **\( \frac{7}{8} \)**:
- The denominator \( 8 \) factors as \( 2^3 \).
- Since it only contains the prime factor 2, \( \frac{7}{8} \) can be expressed as a terminating decimal, specifically \( 0.875 \).

2. **\( \frac{5}{12} \)**:
- The denominator \( 12 \) factors as \( 2^2 \times 3 \).
- Since it contains the prime factor 3 (in addition to 2), \( \frac{5}{12} \) can be expressed as a repeating decimal.

3. **\( \frac{17}{25} \)**:
- The denominator \( 25 \) is \( 5^2 \).
- Since it only contains the prime factor 5, \( \frac{17}{25} \) can be expressed as a terminating decimal, specifically \( 0.68 \).

4. **\( \frac{31}{40} \)**:
- The denominator \( 40 \) factors as \( 2^3 \times 5 \).
- Since it only contains the prime factors 2 and 5, \( \frac{31}{40} \) can be expressed as a terminating decimal, specifically \( 0.775 \).

Based on this analysis, the only rational number from the list that can be expressed as a repeating decimal is **\( \frac{5}{12} \)**.

Answer 6

To express \( \frac{7}{13} \) as a repeating decimal, we can perform long division.

1. **Divide 7 by 13**:
- \( 7.000000 \) divided by 13 gives:
- 13 goes into 70 five times (5), which gives 65.
- Subtract 65 from 70, leaving a remainder of 5.
- Bring down the next 0 (making it 50).

2. **Continue dividing**:
- 13 goes into 50 three times (3), which gives 39.
- Subtract 39 from 50, leaving a remainder of 11.
- Bring down the next 0 (making it 110).

3. **Continue dividing**:
- 13 goes into 110 eight times (8), which gives 104.
- Subtract 104 from 110, leaving a remainder of 6.
- Bring down the next 0 (making it 60).

4. **Continue dividing**:
- 13 goes into 60 four times (4), which gives 52.
- Subtract 52 from 60, leaving a remainder of 8.
- Bring down the next 0 (making it 80).

5. **Continue dividing**:
- 13 goes into 80 six times (6), which gives 78.
- Subtract 78 from 80, leaving a remainder of 2.
- Bring down the next 0 (making it 20).

6. **Continue dividing**:
- 13 goes into 20 once (1), which gives 13.
- Subtract 13 from 20, leaving a remainder of 7.
- Bring down the next 0 (making it 70).

At this point, we see that we have returned to a previous remainder of 7, which indicates that the decimal will start repeating.

Putting this all together, we get:
\[
\frac{7}{13} = 0.538461 \, \overline{538461}
\]

Therefore, the decimal representation of \( \frac{7}{13} \) is:

\[
0.\overline{538461}
\]

Where the digits **538461** are the repeating part, marked with a bar above them.

Answer 7

I apologize for the confusion in my previous response. Let’s redo the long division for \( \frac{7}{13} \) correctly and find its decimal representation with the repeating digits.

### Long Division of \( 7 \div 13 \)

1. **7 divided by 13**:
- 13 goes into 7 **0** times.
- So, we place **0.** and multiply \( 0 \times 13 = 0 \).
- Subtract 0 from 7, we still have 7.

2. Now bring down a 0 (here, we consider 7 as 7.000):
- Now, 70 divided by 13:
- 13 goes into 70 **5** times.
- \( 5 \times 13 = 65 \).
- Subtract: \( 70 - 65 = 5 \).

3. Bring down another 0 (making it 50):
- 13 goes into 50 **3** times.
- \( 3 \times 13 = 39 \).
- Subtract: \( 50 - 39 = 11 \).

4. Bring down another 0 (making it 110):
- 13 goes into 110 **8** times.
- \( 8 \times 13 = 104 \).
- Subtract: \( 110 - 104 = 6 \).

5. Bring down another 0 (making it 60):
- 13 goes into 60 **4** times.
- \( 4 \times 13 = 52 \).
- Subtract: \( 60 - 52 = 8 \).

6. Bring down another 0 (making it 80):
- 13 goes into 80 **6** times.
- \( 6 \times 13 = 78 \).
- Subtract: \( 80 - 78 = 2 \).

7. Bring down another 0 (making it 20):
- 13 goes into 20 **1** time.
- \( 1 \times 13 = 13 \).
- Subtract: \( 20 - 13 = 7 \).

Now we've reached a remainder of **7** again, which means that the digits we've calculated will start repeating.

### Final Result

Combining the results from the division:

\[
\frac{7}{13} = 0.538461 \overline{538461}
\]

Thus, the correct decimal representation of \( \frac{7}{13} \) is:

\[
0.\overline{538461}
\]

Thank you for your patience, and I apologize for the earlier error!

Answer 8

To convert the repeating decimal \(0.8\overline{8}\) (which can be written as \(0.8888...\)) into a rational number, we can use the following approach:

Let \(x = 0.8888...\).

### Step 1: Set up the equation
\[
x = 0.8888...
\]

### Step 2: Multiply by 10 to move the decimal
Now, multiply both sides of the equation by 10:
\[
10x = 8.8888...
\]

### Step 3: Set up a new equation
Now, we can write:
\[
10x = 8 + 0.8888...
\]
Notice that \(0.8888...\) is just \(x\):
\[
10x = 8 + x
\]

### Step 4: Solve for \(x\)
Now, subtract \(x\) from both sides:
\[
10x - x = 8
\]
\[
9x = 8
\]
Now, divide both sides by 9:
\[
x = \frac{8}{9}
\]

### Conclusion
Therefore, the repeating decimal \(0.8\overline{8}\) can be expressed as the rational number:
\[
\frac{8}{9}
\]

Answer 9

To convert the repeating decimal \(0.82\overline{2}\) (which means \(0.822222...\)) to a rational number, we will follow these steps:

1. **Set up the equation**:
Let \(x = 0.82\overline{2}\).

2. **Multiply by 100** to shift the decimal point two places to the right (to account for the non-repeating part "82"):
\[
100x = 82.2222...
\]

3. **Multiply by 10** to shift the decimal point one place to the right for the repeating part. This gives:
\[
10x = 8.2222...
\]

4. **Set up the equations**:
From the first multiplication:
\[
100x = 82.2222...
\]
From the second multiplication:
\[
10x = 8.2222...
\]

5. **Subtract the two equations**:
Subtract the equation \(10x = 8.2222...\) from \(100x = 82.2222...\):
\[
100x - 10x = 82.2222... - 8.2222...
\]
Simplifying this gives:
\[
90x = 74
\]

6. **Solve for \(x\)**:
Divide both sides by 90:
\[
x = \frac{74}{90}
\]
Now simplify \(\frac{74}{90}\):
\[
x = \frac{37}{45}
\]

### Conclusion
So, \(0.82\overline{2}\) can be expressed as the rational number \(\frac{37}{45}\).

### Summary
To convert \(0.82\overline{2}\) into a rational number, you would initially set \(x = 0.82\overline{2}\) and then multiply the equation by **100**. Therefore, the first step includes multiplying by **100** to shift the decimal to the second place, allowing for isolation of the repeating part effectively.