What is 1.21 repeating as a fraction?
The correct answer is: (\frac{40}{33})
The Correct Answer and Explanation is :
To express the repeating decimal (1.21\overline{21}) as a fraction, we can use a method that involves setting up an equation.
Step 1: Define the Decimal
Let:
[
x = 1.212121\ldots
]
Step 2: Multiply by a Power of 10
To eliminate the repeating part, we multiply (x) by (100) (because the repeating part consists of two digits):
[
100x = 121.212121\ldots
]
Step 3: Set Up the Equation
Now we have two equations:
- (x = 1.212121\ldots)
- (100x = 121.212121\ldots)
Next, we subtract the first equation from the second:
[
100x – x = 121.212121\ldots – 1.212121\ldots
]
This simplifies to:
[
99x = 120
]
Step 4: Solve for (x)
Now, solve for (x) by dividing both sides by (99):
[
x = \frac{120}{99}
]
Step 5: Simplify the Fraction
To simplify (\frac{120}{99}), we need to find the greatest common divisor (GCD) of (120) and (99).
The GCD can be found using the Euclidean algorithm:
- (120) divided by (99) gives a remainder of (21).
- (99) divided by (21) gives a remainder of (15).
- (21) divided by (15) gives a remainder of (6).
- (15) divided by (6) gives a remainder of (3).
- (6) divided by (3) gives a remainder of (0).
Thus, the GCD is (3). Now, divide both the numerator and denominator by their GCD:
[
\frac{120 \div 3}{99 \div 3} = \frac{40}{33}
]
Conclusion
So, the repeating decimal (1.21\overline{21}) as a fraction is:
[
\frac{40}{33}
]
This fraction cannot be simplified further, as (40) and (33) have no common factors other than (1). Thus, the final result is (\frac{40}{33}).