Tim has a bag full of marbles He takes out half of the marbles present in the bag and puts back one marble into the bag. He repeats this process four more times. In the end, only three marbles are left in the bag.
How many marbles were there in the bag originally?
working backwards,
x/2 + 1 = 3, so x=4
x/2 + 1 = 4, so x=6
now finish it off in two more steps
hi oobleck please help me on my work
english folktales
Define x:
Let the number of marbles in the bag be x
Take out half and put back 1 (1st Time):
⇒1/2 x + 1
Take out half and put back 1 (2nd Time):
⇒ 1/2 (1/2 x + 1) + 1
⇒ 1/4x + 1/2 + 1
⇒ 1/4x + 3/2
Take out half and put back 1 (3rd Time):
⇒ 1/2 (1/4x + 3/2) + 1
⇒ 1/8x + 3/4 + 1
⇒ 1/8 x + 7/4
Take out half and put back 1 (4th Time):
⇒ 1/2 (1/8 x + 7/4) + 1
⇒ 1/16 x + 7/8 + 1
⇒ 1/16 x + 15/8
Solve x:
There were 3 marbles left
1/16 x + 15/8 = 3
1/16 x = 3 - 15/8
1/16 x = 9/8
x = 9/8 ÷ 1/16
x = 18
Answer: There were 18 marbles in the bag.
To find out the original number of marbles in the bag, we can work backward from the given information. Let's break down the process step by step and keep track of the number of marbles at each stage.
1. At the end of the process, we're told that only three marbles are left in the bag.
2. Before the last repetition, there must have been six marbles (3 initially plus 1 added back during each repetition).
3. Prior to the fifth repetition, there must have been 12 marbles (6 initially plus 1 added back for each repetition).
4. Prior to the fourth repetition, there must have been 24 marbles (12 initially plus 1 added back each time).
5. Prior to the third repetition, there must have been 48 marbles.
6. Prior to the second repetition, there must have been 96 marbles.
7. Finally, before the first repetition, the bag must have contained 192 marbles (96 initially plus 1 added back each time).
Thus, the original number of marbles in the bag was 192.