help with Algebra!!!

Question

The denominator of a fraction in its simplest form is greater than the numerator by 5 . If 3 is added to the numerator, and 2 added to the denominator, then the fraction itself is increased by 1/3 . Find the original fraction.

Answer 1

n/(n + 5)

(n + 3)/(n + 7) = n/(n + 5) + 1/3

Once we apply the LCD, 3(n+ 5)(n + 7), we only deal with the numerators. It's too long to explain why.

3(n + 3)(n + 5) = 3n(n + 7) + 1(n + 5)(n + 7)

3n² + 24n + 45 = 3n² + 21n + n² + 12n + 35

n² + 9n - 10 = 0

(n + 10)(n - 1) = 0

n + 10 = 0

n = -10

n - 1 = 0

n = 1

-10/-5 = 2, not a "fraction", so this is out.

1/6

Let's check.

4/8 = 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2

Check!

Answer 2

d= n+5

n/d = n/(n+5)

(n+3)/(n+5+2) = n/(n+5) + (1/3)[n/(n+5)]

(n+3)/(n+7) = (4/3)[n/(n+5)]

(n+3)/(n+7) = 4n/(3n+15)

cross multiply

4n(n+7) = (n+3)(3n+15)

4n^2 + 28n = 3n^2 + 24n + 45

n^2 +4n -45 = 0

(n-9)(n+5) = 0

n=9 or -5

the fraction is 9/14

-5/0 is not a proper fraction

9/14

add 3 to numerator, 2 to denominator to get

12/16 = 3/4 which is

9/14 plus 1/3 (9/14)

= 4/3 times 9/14 = 36/52 = 3/4

The fraction is 9/14