If you add the denominator to both the numerator and denominator of a fraction, you get a new fraction that sometimes is larger than the original fraction.

Question

Find a fraction where the new fraction is four times as large as the original fraction after you add the denominator to both the numerator and denominator.

Extra: In order for the new fraction to be n times greater than the original fraction, how many times larger than the original numerator must the original denominator be? Express your answer in terms of n.

Please help me its going in my grade!!!

Answer 1

(n+d)/(d+d) = 4n/d
nd+d^2 = 8nd
d^2-7nd = 0
d-7n = 0
d = 7n

so, you can have
1/7: 8/14 = 4/7
3/21: 24/42 = 4/7
and so on

Now, replace 4 with k and see what you get to have the new fraction k times as large.

Answer 2

original fraction -- n/d

new fraction formed = (n+d)/2d

you want: (n+d)/2d = 4(n/d)
8nd = nd + d^2
7nd = d^2
7n = d

it looks like we can find an infinite number of cases like that.
e.g. let n = 1, then d = 7
then the original fraction was 1/7 and
the fraction formed by your rule = 8/14, which is 4/7 and it is 4 times as large

let n = 13 , then the original fraction is
13/91 and the new fraction is 104/182
now 13/91 = 1/7, and 104/182 = 4/7

etc

notice it works even for fractions not in lowest terms, after your reduce them they all reduce to 1/7 and 4/7