Find a fraction with greater than five but less than 10 digits in the repeating part of its decimal. Write the fraction and its equivalent decimal expansion.

Question

 this is hard

Answer 1

This is not hard!  This problem expands your mind so you can put more wonderful information into it.

"The object of opening the mind, as with opening the mouth, is to shut it again on something solid."  – G K Chesterton

Let’s start at the end and get to the beginning.

I’ll just pick any set of repeating digits (greater than five of them and less than 10 of them), say 0478.    (You should pick some others and try this .. using mine is plagiarism; note: this is my Social Security Number!).

Now, a couple rules of algebra:

 A fraction will have a terminating decimal only if the factors of the denominator are 2 or 5 or any multiples of powers of 2 or 5.  If there are any other factors in the denominator, the decimal equivalent will repeat.    (learn this later)

To change a repeating decimal to a fraction, place the repeating digits over (the next larger power of 10) - 1 then reduce the fraction.

    So, with 0478, I put that over 10000-1  (1000-1 won’t work even though it’s bigger than 478 – those are not my repeating digits):

         0478 / 9999   (sorry, no common factors, so it can’t be reduced) (don't use this, it's my Social Security Number!)

Try another one:

      The repeating digits are 108108108…  (note: only three, so don't use this example either)

      108 / 999  = 12 / 111 = 4/37  (answer)

This has got to be fun, because my little sister can’t do it!

 

 

Answer 2

Let a, b, c, d, e, f be any integers such that the string abcdef is not formed of repeating parts.

 

Then X=0.abcdefabcdefabcdefabcdef.......is a repeating 6 digit decimal.

 

1000000X - X= abcdef.abcdefabcdefabcdef - 0.abcdefabcdef......= abcdef

 

999999X=abcdef

 

X=abcdef/999999

 

Choose a=1 b=2 c=3 d=4 e=5 f=6

 

X=123456/999999=41152/333333=.123456123456123456.....   etc.

 

Response question: Can you find a fraction whose denominator is fewer than 6 digits but whose decimal representation is a 6 digit repeater?

 

 

Answer 3

One way to do this is trial and error.  Using a numerator of 1, start with 1 in the denominator and work your way through until you find one:

 

1/1 = 1.000...   NO, only 1 repeating digit

1/2 = 0.500...  NO

1/3 = 0.333...  NO

1/4 = 0.250000...  NO

1/5 = 0.2000...   NO

1/6 = 0.1666...   NO

1/7 = 0.142857142857142857142857142857...   YES!!  The repeating section is 142857, which has 6 digits.