Question

Express the given repeating decimal .7 as a quotiant of integers and reduce if possible?

Answer 1

What if it isn't possible? Does there exist a and b such that a/b = 0.77777777777...?

Answer 2

integers a and b*

Answer 3

well according to my book it said to let a given number be n and to multiply each side by 100 ending with 100n = 70.7 then subtract one n and .7 and you have 99n=70/99 but it doesn't seem right

Answer 4

I found one, but it wasn't easy. :(
\[\frac{7}{9}\]

Answer 5

I wish I could explain it how I got there...

Answer 6

Brute force?

Answer 7

No, I started analyzing repeating decimals which I already knew, such as 10/3 and played around with them to get a repeating 7.

Answer 8

well. if you multiply 0.77777777... 9 times you get a whole number

Answer 9

There is a method to these. I just don't remember

Answer 10

Let x=.77777777...

Then 10x=7.777777....

But 7.777777...= 7+.777777... = 7 + x

So we have 7 + x = 10x

which gives 7 = 9x

which gives x= 7/9.

Answer 11

In general (just as a short cut), if you have a repeating decimal, say:

0.abcabcabcabc....

Then the fraction will be abc/999

Its one complete cycle over the same number of 9s.

Examples:
.4545454545... = 45/99= 5/11

.1234123412341234... = 1234/9999

Answer 12

You should get 10 or more medals for that outstanding explanation