Question

1. nolan wrote ( 1.overline{63} ) and ( 0.overline{2} ) as fractions in simplified form. use numbers from the box to write the fraction.

box with numbers: 2, 7, 9, 90, 11, 18
( 1.overline{63} = \frac{square}{square} )
( 0.overline{2} = \frac{square}{square} )

1. which of the following is a rational number? circle all that apply.

a. ( pi )
b. ( 21.082 )
c. ( sqrt{9} )
d. ( 1.4382768ldots )
e. ( \frac{2}{7} )
f. ( 9 )

1. write the number ( 12.overline{1} ) as a fraction.

1. select true or false for each statement.

a. ( sqrt{19} ) is approximately 4.358. ( circ ) true ( circ ) false
b. ( 3.overline{6} ) is equal to ( \frac{11}{3} ). ( circ ) true ( circ ) false
c. ( sqrt{44} ) is a rational number. ( circ ) true ( circ ) false
d. ( \frac{9}{12} ) is a rational number. ( circ ) true ( circ ) false
e. ( 0.529639ldots ) is an irrational number. ( circ ) true ( circ ) false

Explanation:

Problem 1: Converting Repeating Decimals to Fractions

For \( 1.\overline{63} \)

Step 1: Let \( x = 1.\overline{63} \)

Since the repeating part has 2 digits, multiply \( x \) by 100:
\( 100x = 163.\overline{63} \)

Step 2: Subtract the original equation from the new one

\( 100x - x = 163.\overline{63} - 1.\overline{63} \)
\( 99x = 162 \)

Step 3: Solve for \( x \)

\( x = \frac{162}{99} \)
Simplify by dividing numerator and denominator by 9:
\( x = \frac{18}{11} \) (using numbers 18 and 11 from the box)

For \( 0.\overline{2} \)

Step 1: Let \( y = 0.\overline{2} \)

Since the repeating part has 1 digit, multiply \( y \) by 10:
\( 10y = 2.\overline{2} \)

Step 2: Subtract the original equation from the new one

\( 10y - y = 2.\overline{2} - 0.\overline{2} \)
\( 9y = 2 \)

Step 3: Solve for \( y \)

\( y = \frac{2}{9} \) (using numbers 2 and 9 from the box)

Problem 2: Identifying Rational Numbers

A rational number can be expressed as a fraction \( \frac{p}{q} \) ( \( q
eq 0 \) ), terminating, or repeating decimal.

• A. \( \pi \): Irrational (non - repeating, non - terminating).
• B. \( 21.082 \): Terminating decimal, so rational.
• C. \( \sqrt{9}=3 \): Integer (can be written as \( \frac{3}{1} \) ), rational.
• D. \( 1.4382768 \): Terminating decimal, rational.
• E. \( \frac{2}{7} \): Fraction, rational.
• F. \( 9 \): Integer ( \( \frac{9}{1} \) ), rational.

So circle B, C, D, E, F.

Problem 3: Converting \( 12.\overline{1} \) to a Fraction

Step 1: Let \( z = 12.\overline{1} \)

Multiply \( z \) by 10 (since repeating part has 1 digit):
\( 10z = 121.\overline{1} \)

Step 2: Subtract the original equation from the new one

\( 10z - z = 121.\overline{1} - 12.\overline{1} \)
\( 9z = 109 \)

Step 3: Solve for \( z \)

\( z=\frac{109}{9} \) (or \( 12\frac{1}{9} \))

Problem 4: True/False Statements

A. \( \sqrt{19}\approx4.358 \)

Calculate \( 4.358^2 = 4.358\times4.358\approx19 \), so True.

B. \( 3.\overline{6}=\frac{11}{3} \)

\( \frac{11}{3}=3.666\cdots = 3.\overline{6} \), so True.

C. \( \sqrt{44} \) is rational

\( \sqrt{44} = 2\sqrt{11} \), \( \sqrt{11} \) is irrational, so \( \sqrt{44} \) is irrational. False.

D. \( \frac{9}{12} \) is rational

\( \frac{9}{12}=\frac{3}{4} \), which is a fraction, so rational. True.

E. \( 0.529639\cdots \) is irrational

If it is a non - repeating, non - terminating decimal, it is irrational. But the notation here is a bit unclear. If it is a non - repeating decimal, it is irrational. But if it is a repeating decimal (the “_” is unclear), assuming it is non - repeating, it is irrational. But usually, if it is written as \( 0.529639\ldots \) without a repeat bar, and non - repeating, it is irrational. But maybe a typo. However, if we assume it is a non - repeating decimal, it is irrational. But if it was a repeating decimal (e.g., \( 0.529639\overline{529639} \)), it would be rational. Given the problem, we assume it is non - repeating, so True (that it is irrational). But this is a bit ambiguous.

Answer:

s:

1. \( 1.\overline{63}=\boldsymbol{\frac{18}{11}} \), \( 0.\overline{2}=\boldsymbol{\frac{2}{9}} \)
2. Circle B. \( 21.082 \), C. \( \sqrt{9} \), D. \( 1.4382768 \), E. \( \frac{2}{7} \), F. \( 9 \)
3. \( 12.\overline{1}=\boldsymbol{\frac{109}{9}} \) (or \( 12\frac{1}{9} \))
4. A. True; B. True; C. False; D. True; E. True (assuming non - repeating)