Repeating decimals

[jonathanb875]

Hi,

In the first strategy guide (FDPs), I quote "It can also be helpful to know that if the denominator is 9, 99, 999 or another number equal to a power of 10 minus 1, then the numerator tells you the repeating digits. for example : 23 / 99 = 0,232323..."

So I tried with another fractions to see if this rule works every time. But I came across this one : 58 / 999 = 0.058058058... In that case, the repeating sequence is "058", not "58". Can anyone please explain me why ?

Thanks a lot !

[RonPurewal]

did you actually DO the long division that produces this result? if so, it should be clear what's going on here.

when you divide by 9, you get single-digit repetitions.
when you divide by 99, you get double-digit repetitions.
when you divide by 999, you get triple-digit repetitions.
and so on.

so, for instance, 7/999 is going to be the same as 007/999 = 0.007007007007...

again, it should be absolutely clear WHY this stuff happens if you actually do the long division that produces the relevant results.
if you genuinely have no idea why these decimals end up the way they end up, then i suspect you're just plugging them into a calculator, rather than working them out by hand and seeing why they turn out the way they do.

[RonPurewal]

also—
make sure you apprehend the results of these calculations with a bit of common-sense intuition!

if you are dividing 58 by 999, then obviously you should not expect 0.5858585858...—since that would be more than 0.5, and the fraction 58/999 is very clearly nowhere close to as big as 1/2.

you should be able to look at 58/999 and see that it's way smaller than 1/10 (= 0.1)... so, you should already know that the result is going to be 0.0xxxxx before you even do the work.

[jonathanb875]

also—
make sure you apprehend the results of these calculations with a bit of common-sense intuition!

if you are dividing 58 by 999, then obviously you should not expect 0.5858585858...—since that would be more than 0.5, and the fraction 58/999 is very clearly nowhere close to as big as 1/2.

you should be able to look at 58/999 and see that it's way smaller than 1/10 (= 0.1)... so, you should already know that the result is going to be 0.0xxxxx before you even do the work.

Wow thanks a lot ! You're right : common sense and doing the long division help a lot !

[RonPurewal]

no problem. you're welcome.