Express the repeating decimal 2.1 repeating as a fraction.
The Correct Answer and Explanation is :
To express the repeating decimal ( 2.\overline{1} ) (which means ( 2.1111… )) as a fraction, follow these steps:
Step 1: Set up the equation
Let the repeating decimal be represented by ( x ):
[
x = 2.1111\ldots
]
Step 2: Eliminate the repeating decimal
To remove the repeating part, multiply both sides of the equation by 10 (since the repeating block is one digit long):
[
10x = 21.1111\ldots
]
Step 3: Subtract the original equation from the new equation
Now subtract the original equation ( x = 2.1111\ldots ) from the equation ( 10x = 21.1111\ldots ):
[
10x – x = 21.1111\ldots – 2.1111\ldots
]
Simplifying both sides:
[
9x = 19
]
Step 4: Solve for ( x )
To solve for ( x ), divide both sides by 9:
[
x = \frac{19}{9}
]
Step 5: Final answer
Therefore, the repeating decimal ( 2.\overline{1} ) can be expressed as the fraction:
[
2.\overline{1} = \frac{19}{9}
]
Explanation
A repeating decimal occurs when a number has a portion that repeats indefinitely. In this case, the decimal ( 2.\overline{1} ) consists of the digit “1” repeating indefinitely. To convert such a repeating decimal into a fraction, we use algebraic manipulation.
First, we assign the repeating decimal to a variable, ( x ), and then multiply both sides of the equation by 10 to shift the decimal point. By subtracting the original equation from the new one, the repeating decimals cancel out, leaving us with an equation we can solve. This method works for any repeating decimal, regardless of the number of repeating digits.
The final fraction, ( \frac{19}{9} ), is an improper fraction, which means the numerator is greater than the denominator. If needed, it can also be expressed as a mixed number:
[
\frac{19}{9} = 2 \frac{1}{9}
]