Select all the correct answers

Question

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

Answer 1

To determine which of the provided options is equivalent to \( \frac{0.\overline{63}}{1} \) of a mile, we first need to convert \( 0.\overline{63} \) into a fraction.

Let \( x = 0.\overline{63} \).

To eliminate the repeating decimal, we can multiply \( x \) by 100:

\[ 100x = 63.\overline{63} \]

Now, we can subtract the first equation from the second:

\[ 100x - x = 63.\overline{63} - 0.\overline{63} \]

This simplifies to:

\[ 99x = 63 \]

Now, solving for \( x \):

\[ x = \frac{63}{99} \]

We can simplify \( \frac{63}{99} \) by finding the greatest common divisor (GCD) of 63 and 99, which is 9:

\[ x = \frac{63 \div 9}{99 \div 9} = \frac{7}{11} \]

Now we know \( 0.\overline{63} = \frac{7}{11} \).

Let’s compare this value to the options given:

A. \( \frac{9}{7} \) — Not equal to \( \frac{7}{11} \)

B. \( \frac{7}{11} \) — Equal to \( \frac{7}{11} \)

C. \( \frac{9}{11} \) — Not equal to \( \frac{7}{11} \)

D. \( \frac{7}{9} \) — Not equal to \( \frac{7}{11} \)

E. \( \frac{63}{100} \) — Not equal to \( \frac{7}{11} \)

F. \( \frac{21}{33} \) — Can be simplified:

\[ \frac{21 \div 3}{33 \div 3} = \frac{7}{11} \]

Thus, the correct answers are:

B. \( \frac{7}{11} \)

F. \( \frac{21}{33} \)

Answer 2

What is the fraction form of the following decimal expansion? 0.84 line over 12

A. 695/824

B. 694/824

C. 694/825

D. 695/826

Answer 3

To convert the decimal \(0.84\overline{12}\) into a fraction, we can break it down as follows:

Let:

• \(x = 0.84\overline{12}\).

To express \(x\) mathematically, we first separate the non-repeating part and the repeating part:

• The non-repeating part is \(0.84\).
• The repeating part is \(0.\overline{12}\).

1. Convert the repeating decimal part

Let:

• \(y = 0.\overline{12}\).

To convert \(y\) into a fraction:

Multiply \(y\) by 100 (to move two decimal places):

\[ 100y = 12.\overline{12} \]

Now subtract the original \(y\) from this equation:

\[ 100y - y = 12.\overline{12} - 0.\overline{12} \]

This simplifies to:

\[ 99y = 12 \]

So,

\[ y = \frac{12}{99} = \frac{4}{33} \quad \text{(after simplification)} \]

2. Combine both parts

Now we can express \(x\) as:

\[ x = 0.84 + 0.\overline{12} \]

We convert \(0.84\) into a fraction:

\[ 0.84 = \frac{84}{100} = \frac{21}{25} \quad \text{(after simplification)} \]

Now we add both fractions:

\[ x = \frac{21}{25} + \frac{4}{33} \]

3. Find a common denominator

The least common multiple (LCM) of \(25\) and \(33\) is \(825\):

Now convert both fractions:

\[ \frac{21}{25} = \frac{21 \times 33}{25 \times 33} = \frac{693}{825} \] \[ \frac{4}{33} = \frac{4 \times 25}{33 \times 25} = \frac{100}{825} \]

4. Add the fractions

Now add them:

\[ x = \frac{693}{825} + \frac{100}{825} = \frac{793}{825} \]

Final Fraction

So, the complete conversion of \(0.84\overline{12}\) gives us:

\[ x = \frac{793}{825} \]

Thus, the simplified fraction form of \(0.84\overline{12}\) is:

\[ \frac{793}{825} \]

Now looking back at your answer options, none of them match \( \frac{793}{825} \). Please double-check the options or provide any additional context for verifying the numbers.