A.
a terminating decimal
B.
a repeating decimal
C.
a non-terminating, non-repeating decimal
D.
does not have a decimal expansion
21 answers
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.
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.
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 \( 0.2 \).
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 \(4\).
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:
Let \(x = 1.\overline{235}\).
To eliminate the repeating part, we'll multiply \(x\) by \(1000\) (since there are three digits in the repeating part):
\[ 1000x = 1235.235235235... \]
Now, we can set up an equation:
\[ 1000x = 1235 + x \]
Subtract \(x\) from both sides:
\[ 1000x - x = 1235 \] \[ 999x = 1235 \]
Now, divide both sides by \(999\):
\[ x = \frac{1235}{999} \]
To simplify \(\frac{1235}{999}\), we can find the greatest common divisor (GCD) of \(1235\) and \(999\). The GCD is \(37\).
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 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 \(3\).
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.
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!
To determine which fraction is correctly converted into its decimal equivalent, let's analyze each option:
Calculation: \(\frac{576}{999} \approx 0.576576576... = 0.\overline{576}\)
This does not match with the option which says \(5.\overline{076}\).
Calculation: \(\frac{48}{999} \approx 0.048048048... = 0.0\overline{48}\)
This is correct, as it accurately represents the conversion.
Calculation: \(\frac{5}{999} \approx 0.005005005... = 0.00\overline{5}\)
This is also correct; however, the formatting is typically \(0.00\overline{5}\).
Calculation: \(\frac{4772}{999} \approx 4.772772772... = 4.\overline{772}\)
This does not match with the option that suggests \(4.7\overline{72}\).
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. \(48/999 = 0.0\overline{48}\).
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:
Let \(x = 0.\overline{3}\).
Multiply both sides of the equation by \(10\) (to shift the decimal point):
\[ 10x = 3.\overline{3} \]
Now, subtract the original equation from this new equation:
\[ 10x - x = 3.\overline{3} - 0.\overline{3} \]
This simplifies to:
\[ 9x = 3 \]
Now, divide both sides by \(9\):
\[ x = \frac{3}{9} \]
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 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 \( \frac{2}{3} \).
Given: \( \frac{2}{3} \)
Conversions:
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: \[ \frac{21 \div 3}{33 \div 3} = \frac{7}{11} \text{ (as determined earlier, not equivalent)}. \]
So, the correct answer is 0.\overline{63} (or \(\frac{7}{11}\)). No other fraction from the given list is equivalent to \(\frac{2}{3}\).
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 \( \frac{7}{11} \).
A. \( \frac{9}{7} \):
B. \( \frac{7}{11} \):
C. \( \frac{9}{11} \):
D. \( \frac{7}{9} \):
E. \( \frac{63}{100} \):
F. \( \frac{21}{33} \):
Thus, the correct answers are B and F.
