A bag holds red and blue marbles in the ratio 3 : 5. Four more red marbles are dropped in, and a marble now drawn at random is equally likely to be red or blue. How many marbles were in the bag at first?

Write the counts as 3k red and 5k blue. After the four reds go in there are 3k + 4 red among 8k + 4 marbles, and that must be one half: 2(3k + 4) = 8k + 4, so 6k + 8 = 8k + 4 and k = 2. The bag began with 6 red and 10 blue, that is 16 marbles — and afterwards 10 red against 10 blue, as required. The value 20 is the number of marbles AFTER the four were added, and 8 comes from taking k = 1 without testing it: 3 red and 5 blue would become 7 red against 5 blue, which is not an even chance.