let

p = the number of apples pudge has

a = the number of apples ace has

c = the number of apples christi has

pudge has three times as many apples as ace

p = 3a

ace has twice as many as christi

a = 2c

plug a = 2c into p = 3a

p = 3*2c

p = 6c

pudge has 12 more than both ace and christi together

p = 12 + a + c

plug a = 2c and p = 6c into the last equation

6c = 12 + 2c + c

6c = 12 + 3c

6c - 3c = 12

3c = 12

c = 4 apples

a = 2*4 = 8 apples

p = 6c = 6*4 = 24 apples

ace has 8 apples, pudge has 24 apples and christi has 4 apples.