express the repeating decimal as ratio of two integers :

- 1018

express the repeating decimal as ratio of two integers :

- katieb

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- 1018

\[5.215=5.2151515...\]

- 1018

15 is the repeating

- campbell_st

multiply you number by 10, what do you get..?

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## More answers

- campbell_st

then multiply the number by 1000 what do you get...?

- campbell_st

then its let the number be x
1000x = number
- 10x = number
------------------ do the subtraction and solve for x

- 1018

@phi so should i separate the 5 first and solve for the decimals only?

- 1018

@campbell_st wait is this correct 5.215 x 10?

- phi

0.2151515...
the trick is to give that infinitely repeating expansion a name like x
x= 0.2151515...
then multiply both sides by 10
10x = 2.151515...
now do 1000x
1000x = 215.151515...

- 1018

but how bout the two? its not repeating is that ok? @phi

- campbell_st

yes so 10x = 52.151515.......
and 1000x = 5215.15151515...
the subtraction eliminates the recurring part of the decimal...

- phi

we multiply by 10 to move the "2" to the left, leaving 0.151515..
then we multiply the original by 1000 to get another number with 0.1515...

- 1018

@campbell_st so that leaves me with 990x =5163? then x right?

- campbell_st

thats correct... so now solve for x, you'll have an improper fraction

- 1018

it gave me the same but the 15 repeated 4 times only

- 1018

1721/330

- campbell_st

then change the improper fraction to a mixed number
and that's the answer

- phi

**it gave me the same but the 15 repeated 4 times only***
on your calculator? it has limited number of digits to show

- campbell_st

an altenate method is to treat the recurring decimal as a geometric series... the common ratio is < 1 so you can find the infinite sum
but the multiplication by multiples of 10 is the easiest

- 1018

@phi no sorry, what i meant was it gave me almost the same but this time the number 15 only showed up 4 times, and yeah its limited
@campbell_st is this correct? 5(71/330)

- phi

the original question wants a ratio of two integers, so leave the answer
1721/330

- phi

see
http://www.wolframalpha.com/input/?i=1000+digits+1721%2F330%3D

- campbell_st

well that is 1650 + 71 = 721 so that seems correct...
so I'm unsure if you only need the faction 71/330 or the improper fraction as the answer.
I'd probably give the mixed number answer

- 1018

oh wait now what. hahaha. so just remove the 5?

- 1018

how bout you @phi you still think its the original one?

- phi

***express the repeating decimal as ratio of two integers :
\[ 5.2\overline{15} = \frac{1721}{330}\]

- 1018

ok thanks ill see what to put as the answer. haha. thanks guys!

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