Question

0.19494949494.....in a fraction???

Answer 1

It might be easier to write 1.9494949494... As a fraction.

1 + 94/99 = (99+94)/99 = 193/99

Done

Okay, now how do we answer the question?

Answer 2

you can multiply this by the period

Answer 3

Example:
This is my favorite way:
Say we wanted to write .331331331331331... as a fraction.
\[\text{ Let } x= .331331331... \\ \text{ so } 1000x=331.331331... \\ 1000x-x=331.331331...-.331331331 \\ 999x=331 \\ x=\frac{331}{999} \\ \]

Answer 4

@perl are you saying 194/99?right

Answer 5

not exactly

Answer 6

i dont think you have to find the exact rational number

the question says, is this a rational number

It is a mathematical fact that repeating decimals are rational numbers.

Answer 7

eventually repeating decimals*

Answer 8

mathematical fact = theorem (which means we can prove it)

Answer 9

I'll just finish teh argument

if x = 0.19494949494....
then 10x = 1.949494...
1000x = 194.949494...

1000x - 10x = 193
990x = 193
x = 193/990

Answer 10

ok i see my mistake earlier, there is a nice algorithm


.19494949 = .1 + .0949494.... = 1/10 + 94/990

Answer 11

so if you have an eventually repeating decimal

cut out the part which is not repeating and add it the repeating decimal.

Answer 12

lets do another example

consider the decimal
.13678678678... we see that '678' repeats

.13678678678... = .13 + .00678678... = 13/100 + 678/99900