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

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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 \( \frac{5}{12} \).

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

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.

ur wrong

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!

Convert 0.8 repeating to a rational number

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: \[ \frac{8}{9} \]

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?

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.