Converting 0.17777 Recurring to a Fraction
The decimal 0.17777 recurring is a repeating decimal that follows a consistent pattern. To convert it to a fraction, we need to identify the repeating part and then use some algebraic manipulation.
Identifying the Repeating Part
The repeating part of 0.17777 is 77. Let's isolate this part by multiplying the decimal by an appropriate power of 10.
0.17777 × 100 = 17.777
Now, we can see that the repeating part is 77, and it starts after the decimal point.
Converting to a Fraction
Let's use the variable x to represent the decimal 0.17777 recurring.
x = 0.17777
Multiply both sides of the equation by 100 to get:
100x = 17.777
Since the repeating part is 77, we can subtract x from 100x to eliminate the decimal part:
100x - x = 17.777 - 0.17777
This simplifies to:
99x = 17.6
Now, divide both sides by 99:
x = 17.6/99
Simplify the fraction by dividing both numerator and denominator by their greatest common divisor (GCD), which is 11:
x = 176/990
Simplifying the Fraction
We can simplify the fraction further by dividing both numerator and denominator by their GCD, which is 10:
x = 176/990 = 44/165 = 4/15
Therefore, the fraction equivalent to 0.17777 recurring is 4/15.
