A way to work this is to

- find the least common multiple (LCM) of the denominators

- express each fraction using that LCM as the denominator

- add the numerators

- reduce, by dividing numerator and denominator by their greatest common divisor (GCD).

Denominators: 3, 8, 12. One way to find the LCM is to look at the prime factors.

3=3

8=2*2*2

12=2*2*3, so the least common multiple is 2*2*2*3 = 24.

2/3 = ?/24

(2/3)*24 = ? = 2*24/3 = 2*8 = 16, so our first fraction is 16/24

3/8 = ?/24

(3/8)*24 = ? = 3*24/8 = 3*3 = 9, so our second fraction is 9/24

5/12 = ?/24

(5/12)*24 = ? = 5*24/12 = 5*2 = 10, so our last fraction is 10/24

16/24 + 9/24 + 10/24 = (16+9+10)/24 = 35/24

35=5*7; 24=2*2*2*3. They share no common factors except 1, so this is the "lowest terms."

Expressed as a mixed fraction, 35/24 = 1 11/24.

______

Changing the denominator of a fraction can be considered as multiplication by 1.

2/3 = (2/3)*1 = (2/3)*(24/24)

= (2*24)/(3*24) = (2*3*8)/(3*24) = ((2*8)/(24))*(3/3)

= (16/24)*1 = 16/24

Note the fraction at the end is still equal to the fraction at the beginning. It is just expressed using a different denominator.

- find the least common multiple (LCM) of the denominators

- express each fraction using that LCM as the denominator

- add the numerators

- reduce, by dividing numerator and denominator by their greatest common divisor (GCD).

Denominators: 3, 8, 12. One way to find the LCM is to look at the prime factors.

3=3

8=2*2*2

12=2*2*3, so the least common multiple is 2*2*2*3 = 24.

2/3 = ?/24

(2/3)*24 = ? = 2*24/3 = 2*8 = 16, so our first fraction is 16/24

3/8 = ?/24

(3/8)*24 = ? = 3*24/8 = 3*3 = 9, so our second fraction is 9/24

5/12 = ?/24

(5/12)*24 = ? = 5*24/12 = 5*2 = 10, so our last fraction is 10/24

16/24 + 9/24 + 10/24 = (16+9+10)/24 = 35/24

35=5*7; 24=2*2*2*3. They share no common factors except 1, so this is the "lowest terms."

Expressed as a mixed fraction, 35/24 = 1 11/24.

______

Changing the denominator of a fraction can be considered as multiplication by 1.

2/3 = (2/3)*1 = (2/3)*(24/24)

= (2*24)/(3*24) = (2*3*8)/(3*24) = ((2*8)/(24))*(3/3)

= (16/24)*1 = 16/24

Note the fraction at the end is still equal to the fraction at the beginning. It is just expressed using a different denominator.