### Warm-ups:

1. What do you get when you multiply 5.

You are watching: Express the repeating decimal as a quotient of two integers

23 through 10? by 100? by 1000?2. What"s the difference between 5.232323 and 5.232323...?3. What carry out you acquire when you multiply 5.232323... by 10? by 100? by 1000?4. What execute you have to multiply 2.343434 ... through to acquire 234.343434... ?Answersto warm-ups

### The Idea:

We"d choose to take it a repeating decisnucongo.orgl and express it together a quotient of twowhole numbers. The idea is to multiply by some number (10, 100, 1000, etc.)so that as soon as we subtract the original number indigenous the multiple, the repeatingpart cancels out.Examples: 1. X = .
444... 10 x = 4.444 ... x = .444...So when we subtract, the .444... will cancel out on theright, and also we"ll it is in left with 9x = 4,which we deserve to solve to obtain x = 3/42. X = .232323...If us multiply through 10, we gain 10x = 2.32323... If we shot to subtract x, we have10x = 2.32323... x = .232323...This isn"t any good -- the two"s room lined increase under the 3"s and the 3"s underthe 2"s, so when we subtract, us won"t have actually things cancelling out. What"sthe problem? The pattern the repeats is 2 digits, so we require to relocate thedecisnucongo.orgl suggest two number to snucongo.orgke the repeating part cancel out. That meanswe need to multiply by 100:100x = 23 .232323... x = .232323...Now things line up, for this reason we deserve to subtract and also get99x = 23, then solve to get x = 23/993. Here"s a variation: x = 2.4232323 ...The repeating pattern is two digits, therefore we shot multiplying by 100:100x = 242.32323 ... x = 2.42323 ...Now the repeating patterns 23line up, but only after ~ the first decisnucongo.orgl place.If we subtract, we get99x =239 .9 (go with the detailsof the subtraction closely yourself)then resolve to getx = 239.9/99But we"re no done, since we want a entirety number end a totality number, andthis doesn"t have actually a entirety number on top. But if us multiply the optisnucongo.orgl by 10we will gain a totality number. If we multiply both numerator and denominatorby ten, we won"t readjust the worth of the number. Law this offers our finalanswer:x = 2399/990

### Exercises to do:

For the offered number x, figure out what you must multiply through toget the desired cancelling, and why. Then express the number together a rationalnumber.1. X = 3.
777...2. X = 3.7666 ...3. X = 3. 464646 ...4. X = 3.5464646 ...5. X = 4.123123... 6. X = 4.

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