What is the recurring decimal 0.1 as a fraction in its simplest form?
The Correct Answer and Explanation is :
To express the recurring decimal ( 0.1 ) (specifically, ( 0.111…) where the digit “1” repeats indefinitely) as a fraction, we can follow a systematic approach.
Let’s denote the recurring decimal as ( x ):
[
x = 0.111…\text{ (where the digit “1” repeats)}
]
To eliminate the decimal, we can multiply both sides of the equation by ( 10 ):
[
10x = 1.111…\text{ (the “1” still repeats)}
]
Now, we have two equations:
- ( x = 0.111…)
- ( 10x = 1.111…)
Next, we can subtract the first equation from the second:
[
10x – x = 1.111… – 0.111…
]
This simplifies to:
[
9x = 1
]
To isolate ( x ), we divide both sides by ( 9 ):
[
x = \frac{1}{9}
]
Thus, the recurring decimal ( 0.111…) can be expressed as the fraction ( \frac{1}{9} ).
To verify that this fraction is in its simplest form, we need to check for any common factors between the numerator and the denominator. The number ( 1 ) is a prime number and only has factors of ( 1 ) and ( 1 ). The number ( 9 ) factors into ( 3 \times 3 ). Since ( 1 ) and ( 9 ) share no common factors other than ( 1 ), ( \frac{1}{9} ) is indeed in its simplest form.
In conclusion, the recurring decimal ( 0.111…) can be expressed as the fraction ( \frac{1}{9} ). This process illustrates how to convert a repeating decimal into a fraction by using algebraic manipulation, making it clear and straightforward to find the simplest form of the fraction.