Solved on Feb 14, 2024

Convert the repeating decimal $3.\overline{2}$ to a mixed number.

STEP 1

Assumptions
1. $3 . \overline{2}$ represents the number 3.2222..., where the digit 2 repeats indefinitely.
2. A mixed number is a number consisting of an integer part and a proper fraction part.
3. To express the repeating decimal as a fraction, we will use algebraic manipulation.

STEP 2

Let's represent the repeating decimal $3 . \overline{2}$ as $x$ .
$x = 3 . \overline{2}$

STEP 3

Multiply $x$ by 10 to shift the decimal point one place to the right, which will help us eliminate the repeating part.
$10x = 32 . \overline{2}$

STEP 4

Subtract the original number $x$ from the result of the multiplication to get rid of the repeating decimal.
$10x - x = 32 . \overline{2} - 3 . \overline{2}$

STEP 5

Perform the subtraction on the left side of the equation.
$9x = 32 . \overline{2} - 3 . \overline{2}$

STEP 6

Perform the subtraction on the right side of the equation, noting that the repeating decimals cancel each other out.
$9x = 32 - 3$

STEP 7

Calculate the result of the subtraction on the right side of the equation.
$9x = 29$

STEP 8

Divide both sides of the equation by 9 to solve for $x$ .
$x = \frac{29}{9}$

STEP 9

Now, we will convert the improper fraction $\frac{29}{9}$ to a mixed number.
The integer part of the mixed number is the quotient of the division of 29 by 9, and the fractional part is the remainder over the divisor.

STEP 10

Divide 29 by 9 to find the quotient and remainder.
$29 \div 9 = 3 \text{ remainder } 2$

STEP 11

Express the quotient and remainder as a mixed number.
$x = 3 \frac{2}{9}$
So, $3 . \overline{2}$ expressed as a mixed number in simplest form is $3 \frac{2}{9}$ .

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Convert the repeating decimal $3.\overline{2}$ to a mixed number.

STEP 1

Assumptions
1. $3 . \overline{2}$ represents the number 3.2222..., where the digit 2 repeats indefinitely.
2. A mixed number is a number consisting of an integer part and a proper fraction part.
3. To express the repeating decimal as a fraction, we will use algebraic manipulation.

STEP 2

Let's represent the repeating decimal $3 . \overline{2}$ as $x$ .
$x = 3 . \overline{2}$

STEP 3

Multiply $x$ by 10 to shift the decimal point one place to the right, which will help us eliminate the repeating part.
$10x = 32 . \overline{2}$

STEP 4

Subtract the original number $x$ from the result of the multiplication to get rid of the repeating decimal.
$10x - x = 32 . \overline{2} - 3 . \overline{2}$

STEP 5

Perform the subtraction on the left side of the equation.
$9x = 32 . \overline{2} - 3 . \overline{2}$

STEP 6

Perform the subtraction on the right side of the equation, noting that the repeating decimals cancel each other out.
$9x = 32 - 3$

STEP 7

Calculate the result of the subtraction on the right side of the equation.
$9x = 29$

STEP 8

Divide both sides of the equation by 9 to solve for $x$ .
$x = \frac{29}{9}$

STEP 9

Now, we will convert the improper fraction $\frac{29}{9}$ to a mixed number.
The integer part of the mixed number is the quotient of the division of 29 by 9, and the fractional part is the remainder over the divisor.

STEP 10

Divide 29 by 9 to find the quotient and remainder.
$29 \div 9 = 3 \text{ remainder } 2$

STEP 11

Express the quotient and remainder as a mixed number.
$x = 3 \frac{2}{9}$
So, $3 . \overline{2}$ expressed as a mixed number in simplest form is $3 \frac{2}{9}$ .

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Convert the repeating decimal 3.2‾3.\overline{2}3.2 to a mixed number.

STEP 1

Assumptions1. 3.2‾3 . \overline{2}3.2 represents the number 3.2222..., where the digit 2 repeats indefinitely.2. A mixed number is a number consisting of an integer part and a proper fraction part.3. To express the repeating decimal as a fraction, we will use algebraic manipulation.

STEP 2

Let's represent the repeating decimal 3.2‾3 . \overline{2}3.2 as xxx.x=3.2‾x = 3 . \overline{2}x=3.2

STEP 3

Multiply xxx by 10 to shift the decimal point one place to the right, which will help us eliminate the repeating part.10x=32.2‾10x = 32 . \overline{2}10x=32.2

STEP 4

Subtract the original number xxx from the result of the multiplication to get rid of the repeating decimal.10x−x=32.2‾−3.2‾10x - x = 32 . \overline{2} - 3 . \overline{2}10x−x=32.2−3.2

STEP 5

Perform the subtraction on the left side of the equation.9x=32.2‾−3.2‾9x = 32 . \overline{2} - 3 . \overline{2}9x=32.2−3.2

STEP 6

Perform the subtraction on the right side of the equation, noting that the repeating decimals cancel each other out.9x=32−39x = 32 - 39x=32−3

STEP 7

Calculate the result of the subtraction on the right side of the equation.9x=299x = 299x=29

STEP 8

Divide both sides of the equation by 9 to solve for xxx.x=299x = \frac{29}{9}x=929​

STEP 9

Now, we will convert the improper fraction 299\frac{29}{9}929​ to a mixed number.The integer part of the mixed number is the quotient of the division of 29 by 9, and the fractional part is the remainder over the divisor.

STEP 10

Divide 29 by 9 to find the quotient and remainder.29÷9=3 remainder 229 \div 9 = 3 \text{ remainder } 229÷9=3 remainder 2

STEP 11

Express the quotient and remainder as a mixed number.x=329x = 3 \frac{2}{9}x=392​So, 3.2‾3 . \overline{2}3.2 expressed as a mixed number in simplest form is 3293 \frac{2}{9}392​.

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Convert the repeating decimal $3.\overline{2}$ to a mixed number.

Assumptions
1. $3 . \overline{2}$ represents the number 3.2222..., where the digit 2 repeats indefinitely.
2. A mixed number is a number consisting of an integer part and a proper fraction part.
3. To express the repeating decimal as a fraction, we will use algebraic manipulation.

Let's represent the repeating decimal $3 . \overline{2}$ as $x$ .
$x = 3 . \overline{2}$

Multiply $x$ by 10 to shift the decimal point one place to the right, which will help us eliminate the repeating part.
$10x = 32 . \overline{2}$

Subtract the original number $x$ from the result of the multiplication to get rid of the repeating decimal.
$10x - x = 32 . \overline{2} - 3 . \overline{2}$

Perform the subtraction on the left side of the equation.
$9x = 32 . \overline{2} - 3 . \overline{2}$

Perform the subtraction on the right side of the equation, noting that the repeating decimals cancel each other out.
$9x = 32 - 3$

Calculate the result of the subtraction on the right side of the equation.
$9x = 29$

Divide both sides of the equation by 9 to solve for $x$ .
$x = \frac{29}{9}$

Now, we will convert the improper fraction $\frac{29}{9}$ to a mixed number.
The integer part of the mixed number is the quotient of the division of 29 by 9, and the fractional part is the remainder over the divisor.

Divide 29 by 9 to find the quotient and remainder.
$29 \div 9 = 3 \text{ remainder } 2$

Express the quotient and remainder as a mixed number.
$x = 3 \frac{2}{9}$
So, $3 . \overline{2}$ expressed as a mixed number in simplest form is $3 \frac{2}{9}$ .