Question
Which phrase describes the decimal expansion of the following number? square root of 27
A.
a terminating decimal
B.
a repeating decimal
C.
a non-terminating, non-repeating decimal
D.
does not have a decimal expansion
A.
a terminating decimal
B.
a repeating decimal
C.
a non-terminating, non-repeating decimal
D.
does not have a decimal expansion
Answers
GPT-4o mini
To determine the nature of the decimal expansion of \(\sqrt{27}\), we can simplify \(\sqrt{27}\) first.
\(\sqrt{27}\) can be written as \(\sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3}\).
The value of \(\sqrt{3}\) is an irrational number, which means that \(3\sqrt{3}\) is also irrational.
Irrational numbers have a decimal expansion that is non-terminating and non-repeating.
Therefore, the correct answer is:
**C. a non-terminating, non-repeating decimal**.
\(\sqrt{27}\) can be written as \(\sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3}\).
The value of \(\sqrt{3}\) is an irrational number, which means that \(3\sqrt{3}\) is also irrational.
Irrational numbers have a decimal expansion that is non-terminating and non-repeating.
Therefore, the correct answer is:
**C. a non-terminating, non-repeating decimal**.
What is the decimal expansion of the following fraction? 1/5
A. 0.line over 2
B. 1.5
C. 0.15
D. 0.2
A. 0.line over 2
B. 1.5
C. 0.15
D. 0.2
GPT-4o mini
To find the decimal expansion of the fraction \( \frac{1}{5} \), we can perform the division:
\[
1 \div 5 = 0.2
\]
Therefore, the decimal expansion of \( \frac{1}{5} \) is:
**D. 0.2**.
\[
1 \div 5 = 0.2
\]
Therefore, the decimal expansion of \( \frac{1}{5} \) is:
**D. 0.2**.
What is the decimal expansion of the following fraction? 1/5
A. 0.49
B. 0.line over 4
C. 4.9
D. 0.4
A. 0.49
B. 0.line over 4
C. 4.9
D. 0.4
GPT-4o mini
To find the decimal expansion of the fraction \( \frac{1}{5} \), we perform the division:
\[
1 \div 5 = 0.2
\]
None of the options given (A, B, C, or D) include the correct decimal representation of \( \frac{1}{5} \) which is \( 0.2 \).
However, if you're looking specifically for a related option, the closest to \( \frac{4}{5} \), which simplifies to \( 0.8 \), is not represented either.
Based on the options listed:
**D. 0.4**
might be considered closest to the type of question but is still incorrect for \( \frac{1}{5} \). If those are the only options available and strictly for \( \frac{1}{5} \), please clarify, as the decimal expansion is indeed \( 0.2 \).
\[
1 \div 5 = 0.2
\]
None of the options given (A, B, C, or D) include the correct decimal representation of \( \frac{1}{5} \) which is \( 0.2 \).
However, if you're looking specifically for a related option, the closest to \( \frac{4}{5} \), which simplifies to \( 0.8 \), is not represented either.
Based on the options listed:
**D. 0.4**
might be considered closest to the type of question but is still incorrect for \( \frac{1}{5} \). If those are the only options available and strictly for \( \frac{1}{5} \), please clarify, as the decimal expansion is indeed \( 0.2 \).
What is the decimal expansion of the following fraction? 4/9 sorry
A. 0.49
B. 0.line over 4
C. 4.9
D. 0.4
A. 0.49
B. 0.line over 4
C. 4.9
D. 0.4
GPT-4o mini
To find the decimal expansion of the fraction \(\frac{4}{9}\), we can perform the division:
\[
4 \div 9
\]
This results in a repeating decimal:
\[
4 \div 9 = 0.4444\ldots
\]
This can be represented as \(0.\overline{4}\) (with a line over the 4 indicating that it repeats indefinitely).
Therefore, the correct choice from your options is:
**B. \(0.\) line over \(4\)**.
\[
4 \div 9
\]
This results in a repeating decimal:
\[
4 \div 9 = 0.4444\ldots
\]
This can be represented as \(0.\overline{4}\) (with a line over the 4 indicating that it repeats indefinitely).
Therefore, the correct choice from your options is:
**B. \(0.\) line over \(4\)**.
Find the fraction equivalent of the number in simplest form
1.line over 235 =_____
1.line over 235 =_____
GPT-4o mini
The number \(1.\overline{235}\) represents a repeating decimal, specifically \(1.235235235...\).
To convert the repeating decimal into a fraction, we can follow these steps:
1. Let \(x = 1.\overline{235}\).
2. To eliminate the repeating part, we'll multiply \(x\) by \(1000\) (since there are three digits in the repeating part):
\[
1000x = 1235.235235235...
\]
3. Now, we can set up an equation:
\[
1000x = 1235 + x
\]
4. Subtract \(x\) from both sides:
\[
1000x - x = 1235
\]
\[
999x = 1235
\]
5. Now, divide both sides by \(999\):
\[
x = \frac{1235}{999}
\]
6. To simplify \(\frac{1235}{999}\), we can find the greatest common divisor (GCD) of \(1235\) and \(999\). The GCD is \(37\).
7. Now divide both the numerator and denominator by their GCD:
\[
\frac{1235 \div 37}{999 \div 37} = \frac{35}{27}
\]
Therefore, the fraction equivalent of \(1.\overline{235}\) in simplest form is:
\[
\frac{35}{27}
\]
To convert the repeating decimal into a fraction, we can follow these steps:
1. Let \(x = 1.\overline{235}\).
2. To eliminate the repeating part, we'll multiply \(x\) by \(1000\) (since there are three digits in the repeating part):
\[
1000x = 1235.235235235...
\]
3. Now, we can set up an equation:
\[
1000x = 1235 + x
\]
4. Subtract \(x\) from both sides:
\[
1000x - x = 1235
\]
\[
999x = 1235
\]
5. Now, divide both sides by \(999\):
\[
x = \frac{1235}{999}
\]
6. To simplify \(\frac{1235}{999}\), we can find the greatest common divisor (GCD) of \(1235\) and \(999\). The GCD is \(37\).
7. Now divide both the numerator and denominator by their GCD:
\[
\frac{1235 \div 37}{999 \div 37} = \frac{35}{27}
\]
Therefore, the fraction equivalent of \(1.\overline{235}\) in simplest form is:
\[
\frac{35}{27}
\]
What is the decimal expansion of the following fraction? 1/12
A. 0.0 line over 83
B. 0.08 line over 3
C. 1.12
D. 0.12
A. 0.0 line over 83
B. 0.08 line over 3
C. 1.12
D. 0.12
GPT-4o mini
To find the decimal expansion of the fraction \( \frac{1}{12} \), we can perform the division:
\[
1 \div 12
\]
Calculating this gives:
\[
1 \div 12 = 0.0833333...
\]
This repeating decimal can be represented as:
\[
0.08\overline{3}
\]
From your options, this corresponds to:
**B. \(0.08\) line over \(3\)**.
\[
1 \div 12
\]
Calculating this gives:
\[
1 \div 12 = 0.0833333...
\]
This repeating decimal can be represented as:
\[
0.08\overline{3}
\]
From your options, this corresponds to:
**B. \(0.08\) line over \(3\)**.
Consider the following decimal expansions.
7/999 = 0. line over 007
59/999 =
Use the pattern above to determine which fraction is correctly converted into its decimal equivalent.
A.
B.
C.
D.
7/999 = 0. line over 007
59/999 =
Use the pattern above to determine which fraction is correctly converted into its decimal equivalent.
A.
B.
C.
D.
GPT-4o mini
To convert the fraction \(\frac{59}{999}\) into its decimal equivalent, we can perform the division:
\[
59 \div 999
\]
Let's see how this actually works.
1. **Calculating the decimal**:
\[
59 \div 999 \approx 0.059059059...
\]
This results in the repeating decimal:
\[
0.\overline{059}
\]
So, the decimal expansion of \(\frac{59}{999}\) is:
\[
0.\overline{059}
\]
Now, if we look at potential answer choices (though you did not provide any), we can conclude that the fraction \(\frac{59}{999}\) converts into \(0.\overline{059}\).
If you have specific options (A, B, C, D) available for this, please provide them so I can help you identify the correct one!
\[
59 \div 999
\]
Let's see how this actually works.
1. **Calculating the decimal**:
\[
59 \div 999 \approx 0.059059059...
\]
This results in the repeating decimal:
\[
0.\overline{059}
\]
So, the decimal expansion of \(\frac{59}{999}\) is:
\[
0.\overline{059}
\]
Now, if we look at potential answer choices (though you did not provide any), we can conclude that the fraction \(\frac{59}{999}\) converts into \(0.\overline{059}\).
If you have specific options (A, B, C, D) available for this, please provide them so I can help you identify the correct one!
Consider the following decimal expansions.
7/999 = 0. line over 007
59/999 = 0. line over 059
1 388/999 = 1. line over 388
Use the pattern above to determine which fraction is correctly converted into its decimal equivalent.
A.5 76/999 = 5. line over 076
B. 48/999 = 0.0 line over 48
C. 5/999 = 0.00 line over 5
D. 4 772/999 = 4.7 line over 72
7/999 = 0. line over 007
59/999 = 0. line over 059
1 388/999 = 1. line over 388
Use the pattern above to determine which fraction is correctly converted into its decimal equivalent.
A.5 76/999 = 5. line over 076
B. 48/999 = 0.0 line over 48
C. 5/999 = 0.00 line over 5
D. 4 772/999 = 4.7 line over 72
GPT-4o mini
To determine which fraction is correctly converted into its decimal equivalent, let's analyze each option:
### A. \(\frac{576}{999} = 5.\overline{076}\)
1. **Calculation**:
\(\frac{576}{999} \approx 0.576576576... = 0.\overline{576}\)
This does not match with the option which says \(5.\overline{076}\).
### B. \(\frac{48}{999} = 0.0\overline{48}\)
2. **Calculation**:
\(\frac{48}{999} \approx 0.048048048... = 0.0\overline{48}\)
This is correct, as it accurately represents the conversion.
### C. \(\frac{5}{999} = 0.00\overline{5}\)
3. **Calculation**:
\(\frac{5}{999} \approx 0.005005005... = 0.00\overline{5}\)
This is also correct; however, the formatting is typically \(0.00\overline{5}\).
### D. \(\frac{4772}{999} = 4.7\overline{72}\)
4. **Calculation**:
\(\frac{4772}{999} \approx 4.772772772... = 4.\overline{772}\)
This does not match with the option that suggests \(4.7\overline{72}\).
### Conclusion
From the analysis, **Option B** (\(48/999 = 0.0\overline{48}\)) is the one that correctly reflects the decimal equivalent based on the pattern provided.
**Final Answer: B. \(48/999 = 0.0\overline{48}\)**.
### A. \(\frac{576}{999} = 5.\overline{076}\)
1. **Calculation**:
\(\frac{576}{999} \approx 0.576576576... = 0.\overline{576}\)
This does not match with the option which says \(5.\overline{076}\).
### B. \(\frac{48}{999} = 0.0\overline{48}\)
2. **Calculation**:
\(\frac{48}{999} \approx 0.048048048... = 0.0\overline{48}\)
This is correct, as it accurately represents the conversion.
### C. \(\frac{5}{999} = 0.00\overline{5}\)
3. **Calculation**:
\(\frac{5}{999} \approx 0.005005005... = 0.00\overline{5}\)
This is also correct; however, the formatting is typically \(0.00\overline{5}\).
### D. \(\frac{4772}{999} = 4.7\overline{72}\)
4. **Calculation**:
\(\frac{4772}{999} \approx 4.772772772... = 4.\overline{772}\)
This does not match with the option that suggests \(4.7\overline{72}\).
### Conclusion
From the analysis, **Option B** (\(48/999 = 0.0\overline{48}\)) is the one that correctly reflects the decimal equivalent based on the pattern provided.
**Final Answer: B. \(48/999 = 0.0\overline{48}\)**.
What is the fraction form of the following decimal expansion? 0. line over 3
A. 10/33
B. 3/10
C. 33/100
D. 1/3
A. 10/33
B. 3/10
C. 33/100
D. 1/3
GPT-4o mini
The decimal \(0.\overline{3}\) represents a repeating decimal where the digit \(3\) repeats indefinitely.
To convert \(0.\overline{3}\) into a fraction, we can follow these steps:
1. Let \(x = 0.\overline{3}\).
2. Multiply both sides of the equation by \(10\) (to shift the decimal point):
\[
10x = 3.\overline{3}
\]
3. Now, subtract the original equation from this new equation:
\[
10x - x = 3.\overline{3} - 0.\overline{3}
\]
This simplifies to:
\[
9x = 3
\]
4. Now, divide both sides by \(9\):
\[
x = \frac{3}{9}
\]
5. Simplifying \( \frac{3}{9} \):
\[
x = \frac{1}{3}
\]
Therefore, the fraction form of the decimal expansion \(0.\overline{3}\) is:
**D. \(\frac{1}{3}\)**.
To convert \(0.\overline{3}\) into a fraction, we can follow these steps:
1. Let \(x = 0.\overline{3}\).
2. Multiply both sides of the equation by \(10\) (to shift the decimal point):
\[
10x = 3.\overline{3}
\]
3. Now, subtract the original equation from this new equation:
\[
10x - x = 3.\overline{3} - 0.\overline{3}
\]
This simplifies to:
\[
9x = 3
\]
4. Now, divide both sides by \(9\):
\[
x = \frac{3}{9}
\]
5. Simplifying \( \frac{3}{9} \):
\[
x = \frac{1}{3}
\]
Therefore, the fraction form of the decimal expansion \(0.\overline{3}\) is:
**D. \(\frac{1}{3}\)**.
Select all the correct answers
The art teacher made an assignment for the class to create a map of their neighborhood.
Margie colored in the distance from her best friend's house to her house and wrote on the map that the distance was of a mile.
Which of the following distances are the same as of a mile?
0. line over 63 of a mile
9/7 of a mile
7/11 of a mile
9/11 of a mile
7/9 of a mile
63/100 of a mile
21/33 of a mile
The art teacher made an assignment for the class to create a map of their neighborhood.
Margie colored in the distance from her best friend's house to her house and wrote on the map that the distance was of a mile.
Which of the following distances are the same as of a mile?
0. line over 63 of a mile
9/7 of a mile
7/11 of a mile
9/11 of a mile
7/9 of a mile
63/100 of a mile
21/33 of a mile
GPT-4o mini
To determine which of the given fractions are equivalent to the distance \( \frac{2}{3} \) of a mile, we need to compare each of the fractions to \( \frac{2}{3} \).
1. **Given:** \( \frac{2}{3} \)
2. **Conversions:**
- \( \frac{0.\overline{63}}{1} \) can be changed into a fraction. The decimal \(0.\overline{63}\) represents the repeating decimal \(0.636363...\)
- Let \(x = 0.\overline{63}\)
- Multiply by 100: \(100x = 63.\overline{63}\)
- Subtract:
\[
100x - x = 63.\overline{63} - 0.\overline{63}
\]
\[
99x = 63 \implies x = \frac{63}{99} = \frac{7}{11}
\]
Thus, \(0.\overline{63} = \frac{7}{11}\), which **is equivalent to \( \frac{2}{3} \)**.
3. **Check the others:**
- **\( \frac{9}{7} \)**:
\(\frac{9}{7} > 1\), so it cannot be \( \frac{2}{3} \).
- **\( \frac{7}{11} \)**:
As determined earlier, this is not equivalent to \( \frac{2}{3} \).
- **\( \frac{9}{11} \)**:
\[
\frac{9}{11} \approx 0.818 \text{ (or } \frac{27}{33} \text{)} > \frac{2}{3}
\]
Not equivalent.
- **\( \frac{7}{9} \)**:
\[
\frac{7}{9} \approx 0.777 \text{ (or } \frac{21}{27} \text{)} < \frac{2}{3}
\]
Not equivalent.
- **\( \frac{63}{100} \)**:
\[
0.63 < 0.666... \text{ (This is not } \frac{2}{3} \text{)}.
\]
Not equivalent.
- **\( \frac{21}{33} \)**:
Simplifying this gives:
\[
\frac{21 \div 3}{33 \div 3} = \frac{7}{11} \text{ (as determined earlier, not equivalent)}.
\]
### Summary of Equivalent Fractions:
- The only fraction from the list that is equivalent to \( \frac{2}{3} \) of a mile is:
- **\(\frac{7}{11}\)**
So, the correct answer is **0.\overline{63} (or \(\frac{7}{11}\)).**
No other fraction from the given list is equivalent to \(\frac{2}{3}\).
1. **Given:** \( \frac{2}{3} \)
2. **Conversions:**
- \( \frac{0.\overline{63}}{1} \) can be changed into a fraction. The decimal \(0.\overline{63}\) represents the repeating decimal \(0.636363...\)
- Let \(x = 0.\overline{63}\)
- Multiply by 100: \(100x = 63.\overline{63}\)
- Subtract:
\[
100x - x = 63.\overline{63} - 0.\overline{63}
\]
\[
99x = 63 \implies x = \frac{63}{99} = \frac{7}{11}
\]
Thus, \(0.\overline{63} = \frac{7}{11}\), which **is equivalent to \( \frac{2}{3} \)**.
3. **Check the others:**
- **\( \frac{9}{7} \)**:
\(\frac{9}{7} > 1\), so it cannot be \( \frac{2}{3} \).
- **\( \frac{7}{11} \)**:
As determined earlier, this is not equivalent to \( \frac{2}{3} \).
- **\( \frac{9}{11} \)**:
\[
\frac{9}{11} \approx 0.818 \text{ (or } \frac{27}{33} \text{)} > \frac{2}{3}
\]
Not equivalent.
- **\( \frac{7}{9} \)**:
\[
\frac{7}{9} \approx 0.777 \text{ (or } \frac{21}{27} \text{)} < \frac{2}{3}
\]
Not equivalent.
- **\( \frac{63}{100} \)**:
\[
0.63 < 0.666... \text{ (This is not } \frac{2}{3} \text{)}.
\]
Not equivalent.
- **\( \frac{21}{33} \)**:
Simplifying this gives:
\[
\frac{21 \div 3}{33 \div 3} = \frac{7}{11} \text{ (as determined earlier, not equivalent)}.
\]
### Summary of Equivalent Fractions:
- The only fraction from the list that is equivalent to \( \frac{2}{3} \) of a mile is:
- **\(\frac{7}{11}\)**
So, the correct answer is **0.\overline{63} (or \(\frac{7}{11}\)).**
No other fraction from the given list is equivalent to \(\frac{2}{3}\).
Select all the correct answers
The art teacher made an assignment for the class to create a map of their neighborhood.
Margie colored in the distance from her best friend's house to her house and wrote on the map that the distance was of a mile.
Which of the following distances are the same as 0. line over 63 of a mile?
A.9/7 of a mile
B. 7/11 of a mile
C. 9/11 of a mile
D. 7/9 of a mile
E. 63/100 of a mile
F. 21/33 of a mile
The art teacher made an assignment for the class to create a map of their neighborhood.
Margie colored in the distance from her best friend's house to her house and wrote on the map that the distance was of a mile.
Which of the following distances are the same as 0. line over 63 of a mile?
A.9/7 of a mile
B. 7/11 of a mile
C. 9/11 of a mile
D. 7/9 of a mile
E. 63/100 of a mile
F. 21/33 of a mile
GPT-4o mini
To determine which of the given distances are equivalent to \(0.\overline{63}\) (which represents \( \frac{63}{99} \) or simplified to \( \frac{7}{11} \)), we will check each option against \( \frac{7}{11} \).
### Given Value:
- \(0.\overline{63} = \frac{63}{99} = \frac{7}{11}\)
### Check each option:
A. **\( \frac{9}{7} \)**:
- This value is greater than \(1\) and therefore cannot be equivalent to \( \frac{7}{11} \).
B. **\( \frac{7}{11} \)**:
- This is exactly the same as the value we derived: \( \frac{7}{11} = 0.\overline{63} \).
- **This is a correct answer.**
C. **\( \frac{9}{11} \)**:
- This fraction represents approximately \(0.818\), which is greater than \(0.636...\).
- Not equivalent.
D. **\( \frac{7}{9} \)**:
- This is approximately \(0.777\), which is less than \(0.636...\).
- Not equivalent.
E. **\( \frac{63}{100} \)**:
- This is \(0.63\), which is also less than \(0.636...\).
- Not equivalent.
F. **\( \frac{21}{33} \)**:
- Simplifying \( \frac{21}{33}\):
\[
\frac{21 \div 3}{33 \div 3} = \frac{7}{11}
\]
- This is equivalent to \( \frac{7}{11} = 0.\overline{63} \).
- **This is a correct answer.**
### Summary of Correct Answers:
- **B. \( \frac{7}{11} \)**
- **F. \( \frac{21}{33} \)**
Thus, the correct answers are **B** and **F**.
### Given Value:
- \(0.\overline{63} = \frac{63}{99} = \frac{7}{11}\)
### Check each option:
A. **\( \frac{9}{7} \)**:
- This value is greater than \(1\) and therefore cannot be equivalent to \( \frac{7}{11} \).
B. **\( \frac{7}{11} \)**:
- This is exactly the same as the value we derived: \( \frac{7}{11} = 0.\overline{63} \).
- **This is a correct answer.**
C. **\( \frac{9}{11} \)**:
- This fraction represents approximately \(0.818\), which is greater than \(0.636...\).
- Not equivalent.
D. **\( \frac{7}{9} \)**:
- This is approximately \(0.777\), which is less than \(0.636...\).
- Not equivalent.
E. **\( \frac{63}{100} \)**:
- This is \(0.63\), which is also less than \(0.636...\).
- Not equivalent.
F. **\( \frac{21}{33} \)**:
- Simplifying \( \frac{21}{33}\):
\[
\frac{21 \div 3}{33 \div 3} = \frac{7}{11}
\]
- This is equivalent to \( \frac{7}{11} = 0.\overline{63} \).
- **This is a correct answer.**
### Summary of Correct Answers:
- **B. \( \frac{7}{11} \)**
- **F. \( \frac{21}{33} \)**
Thus, the correct answers are **B** and **F**.
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