# Thread: changing fractions to decimals

1. ## changing fractions to decimals

Hello.
i have a problem with changing fractions to decimals. For instance if i have the fraction....121/239 and i have to write it as a repeating decimal i am not sure how to go about it. in my example it says...(121*4649*9)/(239*4649*9)
Which equals 5062761/9999999 and can then be written as a decimal. However i am not sure where the 4649 and the 9 come from. if anybody could help me on this topic it would be good thanks.

2. the key is to change the denominator so that all of its digits are nines. The approach that your book used was to first change it so that all the digits were one, and then multiply by nine. So we have the denominator 239, and we need to change it (via multiplication) to a number whose digits are all ones.

$1111\div 239 \approx 4.6$

$11111\div 239 \approx 46.4$

$111111\div 239 \approx 464.9$

$1111111\div 239 = 4649$

so we find 1111111 is a multiple of 239, hence:

$\frac{121}{239} * \frac{4649}{4649} = \frac{562529}{1111111}$

now, in order to rewrite the fraction with a denominator of nines (so that we can write it as a repeating decimal) we multiply by 9/9:

$\frac{562529}{1111111} * \frac{9}{9} = \frac{5062761}{9999999}$

finally, we convert to the repeating decimal which will follow the usual pattern:

$\frac{5062761}{9999999} = 0.\overline{5062761}$

3. The repeating decimal is $.5062761 5062761 5062761 ...$

If you were given this decimal expansion and asked to find the rational number is represents you could do the following.

Let
$x = .5062761 5062761 5062761 ...$
so
$10000000x = 5062761.5062761 5062761 ...$
so
$10000000x - x = 5062761.5062761 5062761 ... - .5062761 5062761 ... = 5062761$
which is really
$9999999x = 5062761$
and therefore
$x = 5062761/9999999$

4. thanks for the help. the first method seems to be working well except on one of my sums. I have to work out the repeating decimal of 284/2151. Using that method i have got to the sum 284/2151*5165556072/5165556072. The answer i get does not seem right. the example says....284/2151*4649/4649= 1320316/9999999 being the answer. Where have i gone wrong?

5. try skipping the step involving the ones, and go straight to a denominator of nines...

$99999 \div 2151 \approx 46.4$

$999999\div 2151 \approx 464.8$

$9999999\div 2151 = 4649$

and you should see how to get the answer from there. The reason why it didn't work the same way as the first one is because 11111...111 being divisible by a number guarantees that 99999...999 is also divisible by the number. However, the reverse is not necessarily true. The reason why the first example stopped in "oneville" before calculating the final denominator is probably for numerical (ease of computation and accuracy) reasons. You might have another problem where you could first change the denominator to 33333...333 and then multiply by 3/3. If you favor a direct approach you can always go straight for the nines as was done above.