# Fraction Sum Divisors

Given a pair of reduced fractions, e.g., 1/30 and 1/42, or 3/10 and 1/5, we add them with the LCD Algorithm.

If the sum reduces, then it will reduce by a factor of the GCD of the denominators.

#### Example 1/30+1/42

• Find the LCM
30=2 * 3 * 5 and 42=2 * 3 * 7
so the LCM =2 * 3 * 5 * 7 = 210 and the GCD =2* 3=6.
• Modify the first fraction
210/30=7 so we rewrite 1/30 = (7*1)/(7*30) = 7/210
• Modify the second fraction
210/42=5 so 1/42 = (5*1)/(5*42) = 5/210
• Combine the fractions with the common denominator
1/30+1/42 = 7/210+5/210 = 12/210
• Notice if they reduce
12/210= (12 -: 6)/(210 -: 6) = 2/35

#### Example 11/30+1/42

11/30+1/42 = 77/210+5/210 = 82/210= (82 -: 2)/(210 -: 2) = 41/105

#### Proof

Applying the LCD algorithm a/c+b/d=(au+bv)/lcm

lcm=LCM(c,d)
Note that the LCM of two numbers is the product of the highest power of each prime from the prime factorizations of the two numbers.

gcd=GCD(c,d)
Note that the GCD is the lowest power of the primes common to the two numbers.

These sets of prime factors are disjoint and complementary to the product of the two numbers, i.e.,

c* d = lcm * gcd (1)

Alternatively,

lcm/c=d/gcd(2)

To apply the LCD Algorithm we need to findĀ u and v, the complementary factors of the lcm for c and d respectively.

u=lcm/c and v=lcm/d(3)

Since u=lcm/c=d/gcd, then u is made from factors of d that are not common to c. Similarly, v is made from factors of c that are not common to d. Hence u and v have no common factors, i.e., GCD(u,v)=1.

Notice that u*v = lcm/c * lcm/d and gcd=(c*d)/lcm

Hence, lcm = gcd * u * v

so a/c+b/d=(au+bv)/(gcd * u * v)

If this sum is reducible, then there exists an Integer n such that

(au+bv)/n and (gcd * u * v)/n

If n divides u then n must divide bv. However, we know that u is relatively prime to v. Additionally, since b is relatively prime to d, b is also relatively prime to u. Hence n can’t divide bv and then must not divide u,

So if n is to be the GCD of the numerator and denominator of the sum, it must divide the GCD(c,d).