1018
  • 1018
express the repeating decimal as ratio of two integers :
Mathematics
katieb
  • katieb
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1018
  • 1018
\[5.215=5.2151515...\]
1018
  • 1018
15 is the repeating
campbell_st
  • campbell_st
multiply you number by 10, what do you get..?

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campbell_st
  • campbell_st
then multiply the number by 1000 what do you get...?
campbell_st
  • campbell_st
then its let the number be x 1000x = number - 10x = number ------------------ do the subtraction and solve for x
1018
  • 1018
@phi so should i separate the 5 first and solve for the decimals only?
1018
  • 1018
@campbell_st wait is this correct 5.215 x 10?
phi
  • 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
  • 1018
but how bout the two? its not repeating is that ok? @phi
campbell_st
  • campbell_st
yes so 10x = 52.151515....... and 1000x = 5215.15151515... the subtraction eliminates the recurring part of the decimal...
phi
  • 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
  • 1018
@campbell_st so that leaves me with 990x =5163? then x right?
campbell_st
  • campbell_st
thats correct... so now solve for x, you'll have an improper fraction
1018
  • 1018
it gave me the same but the 15 repeated 4 times only
1018
  • 1018
1721/330
campbell_st
  • campbell_st
then change the improper fraction to a mixed number and that's the answer
phi
  • 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
  • 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
  • 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
  • phi
the original question wants a ratio of two integers, so leave the answer 1721/330
phi
  • phi
see http://www.wolframalpha.com/input/?i=1000+digits+1721%2F330%3D
campbell_st
  • 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
  • 1018
oh wait now what. hahaha. so just remove the 5?
1018
  • 1018
how bout you @phi you still think its the original one?
phi
  • phi
***express the repeating decimal as ratio of two integers : \[ 5.2\overline{15} = \frac{1721}{330}\]
1018
  • 1018
ok thanks ill see what to put as the answer. haha. thanks guys!

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