Question

Convert 0.09¯¯¯¯¯
to a rational number in simplest form.(1 point)
Responses

9100
9 over 100

910
9 tenths

19
1 ninth

111

Answer 1

The notation "0.09¯¯¯¯¯" represents a repeating decimal where the digit 9 repeats infinitely after the decimal point. To convert this repeating decimal to a fraction, we can use the following approach:

Let x = 0.09999...

Then, multiplying both sides of this equation by 100 to shift the decimal two places to the right gives us:

100x = 9.9999...

Now we have two expressions for x, one multiplied by 100 and the other unchanged. Subtracting the original expression (x) from the multiplied expression (100x) allows us to remove the repeating decimal:

100x = 9.9999...
- x = 0.09999...
----------------
99x = 9.9

Dividing both sides of the equation by 99 to solve for x yields:

x = 9.9 / 99

We can simplify this fraction:

x = 9.9 / 99
x = (9 * 1.1) / (9 * 11)
x = 1.1 / 11

Now we remove the decimal point by multiplying both the numerator and denominator by 10:

x = (1.1 * 10) / (11 * 10)
x = 11 / 110

Since 11 is a common factor of both the numerator and denominator, we can simplify further:

x = 11 / 110
x = 1 / 10

Thus, the repeating decimal 0.09¯¯¯¯¯ corresponds to the rational number 1/10 in its simplest form.