Rational numbers?

Question

I need help with this problem? I don't get it at all. 

Compare 0.1 and 0.111..., 0.13 and 0.131313... And 0.157 and 0.157157157... When written as fractions. Make a conjecture about expressing repeating decimals like these as fractions.

Answer 1

Kim, remember that a fraction means we are dividing the numerator by the denominator, which then results in a rational number.   How do you write 0.1 as a fraction? Now, what do you have to divide 1 by to get .1111...? Get out your calculator and play with different denominators to figure it out.  It'll help with the rest.

Answer 2

I know how to change these repeating decimals into fractions but like you I have no idea what point the question is making.

 

here's the scoop.

 

let x = your repeating decimal

 

if you decimal has 1 repeating digit, multiply by 10, 2 repeating digits, multiply by one hundred, 3 repeating digits, multiply by one thousand.

 

so for .111111.....

find the value of 10x. This means move the decimal 1 place to the right

 

for .13131313.....

find the value of 100x. This means move the decimal 2 places to the right

 

for .157157157.......

find value of 1000x. This means move decimal 3 places to the right

 

So let's go back to the first one.

 

10x = 1.11111....

   x =   .11111....

 

Subtract the second equation from the first

 

10x - x = 1

9x = 1

x = 1/9

But x = .11111...

This means .11111..... and the fraction 1/9 have the same value

in other words, the fraction that equals this repeating decimal is 1/9

 

let's quickly do the other two examples

 

100x = 13.131313...

     x =     .131313...

 

99x = 13

x = 13/99

 

1000x = 157.157157.....

       x =       .157157....

 

999x = 157

x = 157/999