During lunchtime today at school, Criss, a junior, shared a problem that was posed to him by an adult friend over the weekend. This friend considered the problem difficult that he didn't think anybody could solve. He allegedly said to Criss that he would give him a car as a prize if he was able to solve it. Unfortunately, the young man failed to solve it. Now he is anxious to know how.
"I know how to do it. It's easy." I said.
"Really? His face brightened up. He may have just found someone who could possibly solve his friend's intriguing problem!
Here's my modified version of the problem: Criss pays a dollar to start playing Game 1. He spends half of his money on this game and pays a dollar to log out. He then proceeds to pay a dollar to start playing Game 2. As in Game 1, he spends half of his money on this game and pays a dollar to log out. He decides to pay a dollar for Game 3, too. As in each of the previous two games, he spends half of his money on this game. When he pays a dollar to log out, he finds out that he has no money left. How much money did he have before he played the first game?
Criss couldn't wait for my algebraic solution. Could you?
Why don't you try it yourself first? Once you're done, scroll down to compare your answer with mine.
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Solution:
Let x be the amount of money Criss had before he started playing the first game. Then:
(x - 1) equals money left after paying a dollar to start Game 1.
(x - 1) - 1/2(x - 1) equals money left after spending half his money on Game 1.
(x - 1) - 1/2(x - 1) - 1 -1 equals money left after paying a dollar to log out and another dollar to start Game 2.
(x - 1) - 1/2(x - 1) - 1 -1 - 1/2[(x - 1) - 1/2(x - 1) - 1 - 1] equals money left after spending half his money on Game 2.
(x - 1) - 1/2(x - 1) - 1 -1 - 1/2[(x - 1) - 1/2(x - 1) - 1 - 1] - 1 - 1 equals money left after paying a dollar to log out and another dollar to start Game 3.
(x - 1) - 1/2(x - 1) - 1 -1 - 1/2[(x - 1) - 1/2(x - 1) - 1 - 1] - 1 - 1 - 1/2{(x - 1) - 1/2(x - 1) - 1 - 1 - 1/2[(x - 1) - 1/2(x - 1) - 1 - 1] - 1 - 1} equals money left after spending half his money on Game 3.
(x - 1) - 1/2(x - 1) - 1 -1 - 1/2[(x - 1) - 1/2(x - 1) - 1 - 1] - 1 - 1 - 1/2{(x - 1) - 1/2(x - 1) - 1 - 1 - 1/2[(x - 1) - 1/2(x - 1) - 1 - 1] - 1 - 1} - 1 = 0, no more money left.
The equation above may be simplified as 0.125x - 2.625 = 0, which gives us x = 21.
Criss, therefore, had $21.00 before he started playing Game 1.
Check for reasonableness. Do you agree with my solution?
Are there other ways to solve this problem? Criss would be happy to know. :D
I did mine so much faster! I got my dollar too!:) u should write about it haha
ReplyDeleteOMG, this is Tania. I was there with Criss, Itzel, and Bekka when this was told to us. First we got offered a car, then 100 dollars, and we never got it!! :( Now I understand why math is important; it can get us cars or money!! ;p haha :)
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