Math / Numbers & Operations

Question\text{Write the recurring decimal } 99.9\overline{9} \text{ as a fraction in its lowest terms.}

Studdy Solution

STEP 1

1. The recurring decimal $99.9\overline{9}$ can be expressed as a fraction.
2. We aim to convert the recurring decimal into a fraction in its simplest form.

STEP 2

1. Define the variable.
2. Set up an equation to represent the recurring decimal.
3. Eliminate the recurring part by manipulating the equation.
4. Solve for the fraction.
5. Simplify the fraction to its lowest terms.

STEP 3

Let $x = 99.9\overline{9}$ .

STEP 4

Multiply $x$ by $10$ to shift the decimal point one place to the right.
$10x = 999.9\overline{9}$

STEP 5

Subtract the original $x = 99.9\overline{9}$ from the equation $10x = 999.9\overline{9}$ .
$10x - x = 999.9\overline{9} - 99.9\overline{9}$
This simplifies to:
$9x = 900$

STEP 6

Solve for $x$ by dividing both sides by $9$ .
$x = \frac{900}{9} = 100$

STEP 7

Since $100$ is already in its simplest form, the fraction representing $99.9\overline{9}$ is:
$\boxed{100}$
The recurring decimal $99.9\overline{9}$ as a fraction in its lowest terms is:
$\boxed{100}$

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Question\text{Write the recurring decimal } 99.9\overline{9} \text{ as a fraction in its lowest terms.}

Studdy Solution

STEP 1

1. The recurring decimal $99.9\overline{9}$ can be expressed as a fraction.
2. We aim to convert the recurring decimal into a fraction in its simplest form.

STEP 2

1. Define the variable.
2. Set up an equation to represent the recurring decimal.
3. Eliminate the recurring part by manipulating the equation.
4. Solve for the fraction.
5. Simplify the fraction to its lowest terms.

STEP 3

Let $x = 99.9\overline{9}$ .

STEP 4

Multiply $x$ by $10$ to shift the decimal point one place to the right.
$10x = 999.9\overline{9}$

STEP 5

Subtract the original $x = 99.9\overline{9}$ from the equation $10x = 999.9\overline{9}$ .
$10x - x = 999.9\overline{9} - 99.9\overline{9}$
This simplifies to:
$9x = 900$

STEP 6

Solve for $x$ by dividing both sides by $9$ .
$x = \frac{900}{9} = 100$

STEP 7

Since $100$ is already in its simplest form, the fraction representing $99.9\overline{9}$ is:
$\boxed{100}$
The recurring decimal $99.9\overline{9}$ as a fraction in its lowest terms is:
$\boxed{100}$

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Question\text{Write the recurring decimal } 99.9\overline{9} \text{ as a fraction in its lowest terms.}

Studdy Solution

STEP 1

1. The recurring decimal 99.99‾ 99.9\overline{9} 99.99 can be expressed as a fraction.2. We aim to convert the recurring decimal into a fraction in its simplest form.

STEP 2

1. Define the variable.2. Set up an equation to represent the recurring decimal.3. Eliminate the recurring part by manipulating the equation.4. Solve for the fraction.5. Simplify the fraction to its lowest terms.

STEP 3

Let x=99.99‾ x = 99.9\overline{9} x=99.99.

STEP 4

Multiply x x x by 10 10 10 to shift the decimal point one place to the right.10x=999.99‾ 10x = 999.9\overline{9} 10x=999.99

STEP 5

Subtract the original x=99.99‾ x = 99.9\overline{9} x=99.99 from the equation 10x=999.99‾ 10x = 999.9\overline{9} 10x=999.99.10x−x=999.99‾−99.99‾10x - x = 999.9\overline{9} - 99.9\overline{9}10x−x=999.99−99.99This simplifies to:9x=9009x = 9009x=900

STEP 6

Solve for x x x by dividing both sides by 9 9 9.x=9009=100x = \frac{900}{9} = 100x=9900​=100

STEP 7

Since 100 100 100 is already in its simplest form, the fraction representing 99.99‾ 99.9\overline{9} 99.99 is:100\boxed{100}100​ The recurring decimal 99.99‾ 99.9\overline{9} 99.99 as a fraction in its lowest terms is:100\boxed{100}100​

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1. The recurring decimal $99.9\overline{9}$ can be expressed as a fraction.
2. We aim to convert the recurring decimal into a fraction in its simplest form.

1. Define the variable.
2. Set up an equation to represent the recurring decimal.
3. Eliminate the recurring part by manipulating the equation.
4. Solve for the fraction.
5. Simplify the fraction to its lowest terms.

Let $x = 99.9\overline{9}$ .

Multiply $x$ by $10$ to shift the decimal point one place to the right.
$10x = 999.9\overline{9}$

Subtract the original $x = 99.9\overline{9}$ from the equation $10x = 999.9\overline{9}$ .
$10x - x = 999.9\overline{9} - 99.9\overline{9}$
This simplifies to:
$9x = 900$

Solve for $x$ by dividing both sides by $9$ .
$x = \frac{900}{9} = 100$

Since $100$ is already in its simplest form, the fraction representing $99.9\overline{9}$ is:
$\boxed{100}$
The recurring decimal $99.9\overline{9}$ as a fraction in its lowest terms is:
$\boxed{100}$