Albert and Zelda agreed to play a game. If heads appeared on the toss
of an ordinary coin, Zelda had to double the amount of money that Albert
had. If the result was tails, then Albert had to pay Zelda $24. As it
turned out, the coin came up heads, tails, heads, tails, heads, and tails.
Then Albert was broke. How much money did Albert start with?
There are two ways to do the problem. The fun way is to "show the movie backwards".
We'll do that way first and then we'll do it by algebra.
Albert ended up broke.
The 6th (last) toss was tails so he gave Zelda $24. That left him broke so before the 6th toss he had $24.
The 5th toss was heads so Zelda doubled his money, making him have $24. So before the 5th toss he had $12.
The 4th toss was tails so he gave Zelda $24. That left him with $12 so before the 4th toss he had $36.
The 3rd toss was heads so Zelda doubled his money, making him have $36. So before the 3rd toss he had $18.
The 2nd toss was tails so he gave Zelda $24. That left him with $18 so before the 2nd toss he had $42.
The 1st toss was heads so Zelda doubled his money, making him have $42. So before the 1st toss he had $21.
Answer: Albert started out with $21.
Now we'll do it the algebra way, which is the way your teacher
probably wanted you to do it, although the backwards way is more fun!
Suppose Albert started out with $X
1. The coin came up heads, so Zelda doubled Albert's money, so he then had $2X
2. The coin came up tails, so Albert paid Zelda $24, so he then had $(2X-24)
3. The coin came up heads, so Zelda doubled Albert's money, so he then had $2(2X-24) or $(4X-48)
4. The coin came up tails, so Albert paid Zelda $24, so he then had $(4X-48)-$24 or $(4X-48-24) or $(4X-72)
5. The coin came up heads, so Zelda doubled Albert's money, so he then had $2(4X-72) or $(8X-144)
6. The coin came up tails, so Albert paid Zelda $24, so he then had $(8X-144)-$24 or $(8X-144-24) or $(8X-168)
The Albert was broke, so we set that final amount = 0
8X - 168 = 0
8X = 168
X = 21
So Albert started with $21.
Let's check:
Albert started out with $21
1. The coin came up heads, so Zelda doubled Albert's money, so he then had $42
2. The coin came up tails, so Albert paid Zelda $24, so he then had $42-$24 or $18.
3. The coin came up heads, so Zelda doubled Albert's money, so he then had $36
4. The coin came up tails, so Albert paid Zelda $24, so he then had $36-$24 or $12.
5. The coin came up heads, so Zelda doubled Albert's money, so he then had $24
6. The coin came up tails, so Albert paid Zelda $24, so he then he was broke.
Edwin