how to split a fraction into two fractions on python?

Question

from the user, we are required to ask for (1) the number of test cases, and (2) two integers that make up the fraction (i did it as a:numerator and b:denominator). then we are required to print that fraction in this form (1/2 + 1/y). also, we have to use the gcd function in writing our code

we were given a couple of sample inputs to check our work:
- input 2 and 3 must output '1/2+1/6'
- input 1 and 4 must output '1/2+1/-4'
- input 7 and 10 must output '1/2+1/5'
- input 1 and 1 must output '1/2+1/2'

here is my code:

N = int(input())
index = 1

while index<=N:
  a = int(input())
  b = int(input())
  num = 2*a -b
  den = 2*b
  def gcd(x,y): #this calculates for the greatest common divisor
    if x>y:
      smaller=y
    else:
      smaller=x 
    for factor in range(1,smaller+1):
      if (x%factor==0) and (y%factor==0):
        hcf=factor #this is the gcd
        newNum=num//hcf
        newDen=den//hcf
        newFrac=str(newNum)+'/'+str(newDen) #to simplify the fraction, both numerator and denominator are divided by the gcd
        print('1/2+'+newFrac)
  index = index + 1
  gcd(num,den)

when trying to run the code, there are some inputs that give me the result I want, but there are some that don't...what's wrong with my code? should I put a broken line somewhere or something?

sample input # and what they output (in comments)
4 # number of test cases
2
3
#outputs 1/2+1/6 ---> correct
1
4
#outputs nothing ---> incorrect
7
10
#outputs 1/2+4/20
#        1/2+2/10
#        1/2+1/5 ---> outputs too much ---> incorrect
1
1
#outputs 1/2+1/2 ---> correct

thanks in advance! :)

Answer 1

You can try this:

N = int(input())

for __ in range(N):

    a = int(input())
    b = int(input())

    a_b_str = ('%i/%i' %(a,b)).ljust(5) # for printout purposes; optional

    num = 2*a - b
    den = 2*b

    if num == 0:
        print(a_b_str + ' -->  1/2')

    elif den % num == 0:
        y = den / num
        print(a_b_str + ' -->  1/2+1/%i' %y)

    else:
        print(a_b_str + ' CAN NOT BE REPRESENTED IN THE FORM: "1/2+1/y"') 

Results: (for the respective a,b)

2/3   -->  1/2+1/6
1/4   -->  1/2+1/-4
7/10  -->  1/2+1/5
1/1   -->  1/2+1/2

1/2   -->  1/2
1/5   CAN NOT BE REPRESENTED IN THE FORM: "1/2+1/y"

3/2   -->  1/2+1/1
5/2   CAN NOT BE REPRESENTED IN THE FORM: "1/2+1/y"

-1/2  -->  1/2+1/-1
-1/4  CAN NOT BE REPRESENTED IN THE FORM: "1/2+1/y"

---

Actually, you don't need to do much stuff for your task since:

a/b = 1/2 + 1/y 
=> 1/y = a/b - 1/2
=> 1/y = (2*a - b) / 2*b
=> y = 2*b / (2*a - b) = den / num

---

UPD.

The alternative solution - with gcd():

# gcd based on Euclid's algorithm
def gcd(b, a): 
    while a:
        b, a = a, b%a
    return abs(b)

N = int(input())

for __ in range(N):

    a = int(input())
    b = int(input())

    a_b_str = ('%i/%i' %(a,b)).ljust(5) 

    num = 2*a - b
    den = 2*b

    if num == 0:
        print(a_b_str + ' -->  1/2')

    else:
        gcd_temp = gcd(den, num)

        if gcd_temp == abs(num):
            y = (num/gcd_temp) * (den/gcd_temp)
            print(a_b_str + ' -->  1/2+1/%i' %y) 

        else:
            print(a_b_str + ' CAN NOT BE REPRESENTED IN THE FORM: "1/2+1/y"')