Convert the following repeating decimal to a fraction in simplest form.
point, 2, start overline, 1, end overline
.2
1
To convert the repeating decimal to a fraction, we need to set up an equation. Let x be the repeating decimal.
First, we can multiply the decimal by a power of 10 that moves the repeating part to the left of the decimal point. In this case, we can multiply x by 10 to get:
10x = 2.12121212...
Next, we can subtract the original decimal from the multiplied decimal to eliminate the repeating part:
10x - x = 2.12121212... - 0.2
This simplifies to:
9x = 1.92121212...
Now, we want to eliminate the repeating part again. Since there are two digits repeating, we multiply the equation by 100 to move the repeating part to the left of the decimal point:
100(9x) = 100(1.92121212...)
900x = 192.12121212...
We can subtract the previous equation from this equation to eliminate the repeating part:
900x - 9x = 192.12121212... - 1.92121212...
This simplifies to:
891x = 190
Now we can solve for x by dividing both sides of the equation by 891:
x = 190/891
To simplify the fraction, we can find the greatest common divisor (GCD) of 190 and 891, which is 1. Dividing both the numerator and denominator by 1, we get:
x = 190/891 = 190/891
Therefore, the fraction in simplest form for the repeating decimal .2 1 is 190/891.