What is the product of the numerator and the denominator when $0.\overline{09}$ is expressed as a fraction in lowest terms?

Guest Apr 3, 2021

#1**0 **

I'm not sure how to explain this, but when you have a line, the fraction denominator is the number of digits and 9 amount of that.

So under this line, there are 2 digits, so the denominator is 2 nines, 99.

The numberator is the number under the line, 9.

So our fraction is 9/99 = 1/11.

That would make the product 11.

Sorry for the sad explanation.

I hope this helped. :))

=^._.^=

catmg Apr 3, 2021

#2**0 **

There is an algebraic way through manipulation:

Let $x=0.\overline{09}$

Multiply by 100 to get $100x=9.\overline{09}$

Subtract x from 100x to get $100x-x=9.\overline{09}-0.{09}$ which cancels out the infinite decimals leaving $99x=9$

Dividing both sides by 99 gives $x=\frac{9}{99}$, which simplifies to $x=\frac{1}{11}$. The product of the numerator and denominator is $1*11=\boxed{11}$

If you are familiar with geometric series there is also another way:

x=0*10^-1+9*10^-2+0*10^-3+9*10^-4+0*10^-5+...

simplifying gives x=9*10^-2+9*10^-4+9*10^-6+... (you got lucky the 0's canceled out, there is a trick to dealing with them when there is no 0's to cancel out)

We can see this is an infinite geometric series with first term 9*10^-2 and common ration 10^-2. Using the formula for the sum of an infinite geometric series gives:

$S=\frac{9*10^-2}{1-10^-2}$

$S=\frac{\frac{9}{100}}{\frac{99}{100}}$

$S=\frac{9}{99}$

$S=\frac{1}{11}$

The product is 11 :) These are the two algebraic solutions

RiemannIntegralzzz Apr 3, 2021