Consider the following decimal expansions.

Question

1/99 = 0. overline 01
68/99 = 0. overline 68
3 82/99 = 3. overline 82

Use the pattern above to determine which fraction is correctly converted into its decimal equivalent.
A. 78/99 = 07 overline 6

B. 3/99 = 0. overline 3

C. 8/99 = 0. over line 08

D. 8 47/99 = 0.8 overline 47

Answer 1

To convert these fractions correctly into decimal form, we need to analyze each option:

1. A. 78/99 \[ \frac{78}{99} = \frac{78 \div 3}{99 \div 3} = \frac{26}{33} \] To express \( \frac{26}{33} \): \[ 26 \div 33 = 0.7878\ldots = 0.\overline{78} \] Thus, \( 78/99 \) actually equals \( 0.\overline{78} \), not \( 07\overline{6} \).

2. B. 3/99 \[ \frac{3}{99} = \frac{1}{33} \] To express \( \frac{1}{33} \): \[ 1 \div 33 = 0.030303\ldots = 0.\overline{03} \] So, \( 3/99 \) is not equal to \( 0.\overline{3} \).

3. C. 8/99 \[ \frac{8}{99} \] A direct division yields: \[ 8 \div 99 = 0.080808\ldots = 0.\overline{08} \] Thus, \( 8/99 = 0.\overline{08} \) is correct.

4. D. 847/99 \[ 847/99 = 8 + \frac{47}{99} = 8 + 0.\overline{47} \] Thus, \( 847/99 = 8.0\overline{47} \), which means it is also correctly expressed as \( 0.8\overline{47} \).

In conclusion, the options that are correctly converted into their decimal equivalents are:

• C. 8/99 = 0.\overline{08}
• D. 847/99 = 0.8\overline{47}

However, between these options:

• C. is fully stated as correct without any transformation.
• D. while technically correct, it should ideally be interpreted as \( 8.\overline{47} \).

Overall: The best fit for the question only asks about clear representation in decimal form without any ambiguity is C.

Answer 2

What is the decimal expansion of the following fraction?
1/18

A. 1.18

B. 0.18

C. 0. overline 06

D. 0.0 overline 5

Answer 3

To find the decimal expansion of \(\frac{1}{18}\), we can perform long division:

1. Long Division of 1 by 18:
   • 18 goes into 1, 0 times. So, we write 0. and move to the next decimal place.
   • Bring down the next digit to get 10. 18 goes into 10, 0 times again.
   • Bring down another 0 to make it 100. 18 goes into 100 by 5 (since \(18 \times 5 = 90\)).
   • Subtract 90 from 100 to get 10.
   • Bring down another 0 to make it 100 again, and repeat the previous steps.
   • This results in the same cycle since \(100\) divided by \(18\) gives another 5 remainder \(10\), thus continuing indefinitely.

From this, we find: \[ \frac{1}{18} = 0.055555\ldots = 0.0\overline{5} \]

So, the decimal expansion of \(\frac{1}{18}\) is:

• D. \(0.0\overline{5}\)

Thus, the correct answer is D.

Answer 4

Use the following number to complete the statements below.
0.1 overline 6
The decimal above is a/an ____ number because it is a ______ decimal.
The fraction equivalent of this number ________.

Answer 5

The number \( 0.1\overline{6} \) can be analyzed as follows:

1. Type of Number:

   • The decimal above is a/an irrational number because it is a repeating decimal.

   Correction: Actually, \( 0.1\overline{6} \) is typically considered a rational number because repeating decimals can be expressed as fractions.

2. Fraction Equivalent: To convert \( 0.1\overline{6} \) to a fraction, we represent it as:

   • Let \( x = 0.1\overline{6} \).

   We can express this as: \[ x = 0.166666\ldots \] To separate the non-repeating and repeating parts: \[ 10x = 1.66666\ldots \] Now, subtract the original \( x = 0.16666\ldots \) from the multiplied equation: \[ 10x - x = 1.6666\ldots - 0.1666\ldots \] This simplifies to: \[ 9x = 1.5 \] Now, isolate \( x \): \[ x = \frac{1.5}{9} = \frac{15}{90} = \frac{1}{6} \]

   So, the fraction equivalent of this number is \( \frac{1}{6} \).

Final Completed Statements:

• The decimal above is a/an rational number because it is a repeating decimal.
• The fraction equivalent of this number is \( \frac{1}{6} \).

Answer 6

What is the decimal expansion of the following fraction?
1/4

A. 0.14

B. 1.4

C. 0.25

D. 0. overline 25

Answer 7

To find the decimal expansion of the fraction \(\frac{1}{4}\), we can perform the division:

1. Long Division of 1 by 4:
   • 4 goes into 1, 0 times. So we write \(0.\) and move to the next decimal place.
   • Bring down a 0 to make 10. 4 goes into 10, 2 times (since \(4 \times 2 = 8\)).
   • Subtract 8 from 10, leaving a remainder of 2.
   • Bring down another 0 to make 20. 4 goes into 20, 5 times (since \(4 \times 5 = 20\)).
   • Subtract 20 from 20, leaving a remainder of 0.

So, \(\frac{1}{4} = 0.25\).

Now, let's look at the options:

• A. \(0.14\) — Incorrect.
• B. \(1.4\) — Incorrect.
• C. \(0.25\) — Correct.
• D. \(0.\overline{25}\) — Incorrect (this signifies a repeating decimal, which doesn't apply here).

Correct Answer:

The decimal expansion of \(\frac{1}{4}\) is C. \(0.25\).

Answer 8

What is the fraction form of the following decimal expansion?

1. overline 7
A. 15/9

B. 14/9

C. 17/9

D. 16/9

Answer 9

To convert the decimal \(1.\overline{7}\) (which means \(1.7777\ldots\)) into a fraction, we can use the following steps:

1. Let \(x = 1.\overline{7}\): \[ x = 1.7777\ldots \]

2. Multiply both sides by 10 (to shift the decimal point one place to the right): \[ 10x = 17.7777\ldots \]

3. Now subtract the original \(x\) from this equation: \[ 10x - x = 17.7777\ldots - 1.7777\ldots \] This simplifies to: \[ 9x = 16 \]

4. Now solve for \(x\): \[ x = \frac{16}{9} \]

Thus, the fraction form of \(1.\overline{7}\) is \(\frac{16}{9}\).

Final Answer:

The correct option is D. \(\frac{16}{9}\).

Answer 10

Which phrase describes the decimal expansion of the following number?
square root of -36

A.
does not have a decimal expansion
B.
a non-terminating, non-repeating decimal
C.
a repeating decimal
D.
a terminating decimal

Answer 11

The expression \(\sqrt{-36}\) is not a real number, as the square root of a negative number involves imaginary numbers.

In fact: \[ \sqrt{-36} = 6i \] where \(i\) is the imaginary unit, defined as \(i = \sqrt{-1}\).

Since the square root of \(-36\) results in an imaginary number, it does not have a decimal expansion in the traditional sense, which applies only to real numbers.

Final Answer:

The correct phrase that describes the decimal expansion of \(\sqrt{-36}\) is: A. does not have a decimal expansion.

Answer 12

What is the decimal expansion of the following fraction?

1/10
A. 0. overline 1

B. 0.1

C. 0.0 overline 1

D.

Answer 13

To find the decimal expansion of the fraction \(\frac{1}{10}\), we can perform the division:

1. Division of 1 by 10:
   • When we divide 1 by 10, we can express it as: \[ 1 \div 10 = 0.1 \]

So, the decimal expansion of \(\frac{1}{10}\) is \(0.1\).

Now, let's look at the options:

• A. \(0.\overline{1}\) — This represents a repeating decimal \(0.1111\ldots\), which is not correct.
• B. \(0.1\) — Correct.
• C. \(0.0\overline{1}\) — This represents a repeating decimal that starts after one zero, which is also not correct.

Final Answer:

The decimal expansion of \(\frac{1}{10}\) is B. \(0.1\).

Answer 14

Drag each number to the correct location on the mixed number. Not all numbers will be used.

What is the fraction form of the following decimal expansion?

12.5 overline 18

500 57 259 9 110 12 50

Answer 15

To convert the decimal expansion \(12.5\overline{18}\) into fraction form, we can break it down into its integer and fractional parts.

1. Separate the Whole Number: \[ 12.5\overline{18} = 12 + 0.5\overline{18} \]

2. Convert the Non-Repeating Decimal \(0.5\overline{18}\) into a Fraction: Let's represent \(0.5\overline{18}\) as \(x\): \[ x = 0.5\overline{18} \]

3. Isolate the Repeating Part: We can express \(x\) as: \[ x = 0.5 + 0.\overline{18} \]

4. Convert \(0.\overline{18}\) into a Fraction: Let \(y = 0.\overline{18}\).

   • Multiply by 100 to shift the decimal point two places to the right: \[ 100y = 18.\overline{18} \]
   • Now subtract the original \(y\) from this equation: \[ 100y - y = 18.\overline{18} - 0.\overline{18} \] \[ 99y = 18 \] \[ y = \frac{18}{99} = \frac{2}{11} \quad (\text{after simplification}) \]

5. Add \(0.5\) to \(0.\overline{18}\): Now substitute back: \[ 0.5 = \frac{1}{2} \] So: \[ x = \frac{1}{2} + \frac{2}{11} \] To add these fractions, find a common denominator (which would be 22): \[ x = \frac{11}{22} + \frac{4}{22} = \frac{15}{22} \]

6. Combine the Whole and Fractional Parts: Now compile everything: \[ 12.5\overline{18} = 12 + 0.5\overline{18} = 12 + \frac{15}{22} = \frac{12 \times 22 + 15}{22} = \frac{264 + 15}{22} = \frac{279}{22} \]

Final Fraction Form:

The fraction form of the decimal expansion \(12.5\overline{18}\) is: \[ \frac{279}{22} \]

From the Given Options:

From the numbers provided (500, 57, 259, 9, 110, 12, 50), it appears there might be a misunderstanding as they do not seem to relate directly to the final fraction obtained.

However, if the question requires you to use these numbers to fit into a mixed number format (e.g., numerator and denominator), you'd need to place \(279\) as the numerator and \(22\) as the denominator if approximate options existed. In that case:

Numerator: 279 (this isn't listed in your numbers) Denominator: 22 (this isn't listed either)

Please clarify if there are specific ways you would like those numbers to be used, or how you intended for them to relate to the mixed number format!

Answer 16

Drag each number to the correct location on the mixed number. Not all numbers will be used.

What is the fraction form of the following decimal expansion?
_?__
12.5 overline 18. _?__
__?_
A. 500
B. 57
C. 259
D. 9
E. 110
F. 12
G. 50

Answer 17

To convert the decimal expansion \(12.5\overline{18}\) into fraction form, we previously calculated (as detailed earlier) that it can be expressed as:

\[ 12.5\overline{18} = \frac{279}{22} \]

When forming a mixed number, it typically consists of an integer part and a fractional part.

1. Identify the Integer Part: The integer part of \(12.5\overline{18}\) is \(12\).

2. Identify the Fractional Part: The fractional part corresponds to \(0.5\overline{18}\), which we found to be \(\frac{15}{22}\) (not included in the options).

Since we need to express this as a combination of the mixed number \( \frac{?}{?} \):

Mixed Number Format:

The mixed number can be expressed as: \[ 12 \frac{15}{22} \]

Numbers to Drag:

From the options provided, we can fill them into the format of \(\text{Integer} , \frac{\text{Numerator}}{\text{Denominator}}\).

• Integer Part: 12 (from options F)
• Numerator: 15 does not directly match any provided number.
• Denominator: 22 does not directly match any provided number.

Since \(15\) and \(22\) do not appear in your list, we can look for the closest relation they may have. However, considering just the components of the mixed number, we focus solely on those which are:

• For a fraction representing \(0.5 \overline{18}\) equating to \(\frac{15}{22}\):
  • If interpreting \(0.5\) separately as a fraction, it would come from \(50/100\) or similar.

Summary of Best Options Based on Provided Choices:

To match what is closest:

• Integer: 12 (from F)
• If we were interpreting to use something for possible fractions, we cannot find direct matching numerators or denominators from the specified options.

If it allows those closest options to actually fit in context:

• The appropriate fill-ins based on the number formats would be \(\underline{12} , \frac{?}{?}\) where \(?/?\) cannot be explicitly filled as per your listed values.

So the filled mixed number would look like:

For the Actual Drag:

1. 12 (For the integer part)
2. Fill the fraction with the most suitable if needed, but nothing directly available from your numbers for that fractional representation as numerator and denominator. Based on traditional approach:

If you must drag, perhaps place:

• \(F. 12\) in the integer part,
• The numerator and denominator would remain unfilled realistically based on options unless re-evaluated if broken down specifically into components relatable.

Please let me know if further breakdown or a different approach is needed to prioritize any of those other listed numbers!