Allen started out with x marbles. He doubled his number of marbles by buying an additional x marbles from Betty. Then he gave away one-third of all his marbles to his friend Clarence. At the end of the day, he had 16 marbles. How many marbles did he start out with

start --- x

after Betty deal --- 2x
after give-away ---- (2/3)(2x)

(4/3)x = 16
4x = 48
x = 12

check:
start: 12
after doubling : 24
gave (1/3) of 24 or 8 away
leaving him with 16

SO HE STARTED OUT WITH 12?

To solve this problem, let's break it down step by step:

1. Allen started with x marbles.
2. He doubled his number of marbles by buying an additional x marbles from Betty. So, his total number of marbles became x + x = 2x.
3. Then he gave away one-third of all his marbles to his friend Clarence. One-third of his marbles is (1/3) * 2x = 2x/3.
4. At the end of the day, he had 16 marbles. So, the equation becomes:
2x - 2x/3 = 16

To solve this equation, you can follow these steps:

1. Multiply through by 3 to eliminate the fraction:
3 * (2x) - 3 * (2x/3) = 3 * 16
6x - 2x = 48

2. Combine like terms:
4x = 48

3. Divide both sides by 4:
x = 48/4
x = 12

Therefore, Allen initially started with 12 marbles.