Does 0.33333….. really equal 1/3?

Hello.  What rain yesterday, here in Campbell River!  Well, we sure needed it.  It’s nice to have more seasonal temperatures after the oven that was last week.

A math tutor often encounters the topic of converting decimals to fractions.  Terminating decimals are easy:  for instance, 0.9 is 9/10.  Then, 0.31 is just 31/100.  As well, 0.222 is 222/1000, which reduces to 111/500.

What about repeating decimals, such as 0.333333…….?

Well, there’s an algebraic trick for that:

Let x=0.3333…..(Note that x=1x; we just don’t usually write the one.)



1x =0.33333……(2)

Subtracting (2) from (1) gives 9x=3.00000

Of course, 3.00000…. = 3, so 9x=3

Next, divide both sides by 9 to isolate x:


Recall, we began by defining x as 0.333333…..Now, since we see x is also equal to 1/3, we know that it must be true:


Have a great day, and come back for more hints.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

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