The Egg Problem

[joey101]

Looking for some assistance please...

Chris has some eggs to sell and Maria, Ai-Ling and Jermaine want to buy them. Chris sells half the eggs plus half an egg to Maria, then sells half the remaining eggs plus half an egg to Ai-Ling, and finally sells half the remaining eggs plus half an egg to Jermaine.

At the end of the three sales, Chris is out of eggs. The strange thing is that Chris never had to break an egg. How many eggs did Chris have to begin with?

Does only one solution exist? If so, why?

How many eggs would Chris have started with if the same scenario had taken place with different numbers of people buying the eggs? For example, what if there had been four buyers?

Can you find a pattern that will determine the number of eggs for any number of buyers? Describe the pattern.

 

[Deleted member 4993]

Looking for some assistance please...

Chris has some eggs to sell and Maria, Ai-Ling and Jermaine want to buy them. Chris sells half the eggs plus half an egg to Maria, then sells half the remaining eggs plus half an egg to Ai-Ling, and finally sells half the remaining eggs plus half an egg to Jermaine.

At the end of the three sales, Chris is out of eggs. The strange thing is that Chris never had to break an egg. How many eggs did Chris have to begin with?

Does only one solution exist? If so, why?

How many eggs would Chris have started with if the same scenario had taken place with different numbers of people buying the eggs? For example, what if there had been four buyers?

Can you find a pattern that will determine the number of eggs for any number of buyers? Describe the pattern.

"sells half the remaining eggs plus half an egg to Jermaine....At the end of the three sales, Chris is out of eggs." - This tells you that Chris sold 1 egg to Jermaine - Why??

Now your turn....

Please share your work with us .

If you are stuck at the beginning tell us and we'll start with the definitions e.g. define retained earnings.

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[joey101]

Attempt

How do we know that Chris sold one egg to Jermaine? He sold 1/2 of what he had left over, plus 1/2.

I'm looking at different ways to solve this problem. I guess I could write it out mathematically:

1. x/2 + 1/2 =

2. x/2 + 1/2 =

3. x/2 + 1/2 =

and try to figure out how many he started out with and hope it fits the pattern.

Let's say he had 7 eggs... 7/2 is 3.5 plus 1/2 = 4; 7 - 4 = 3 eggs left
3/2 = 1.5 plus 1/2 = 2; 7 - 4 - 2 = 1
1/2 + 1/2 = 1; 7 - 4 - 3 - 1 = 0 eggs left over...

 

[joey101]

Pattern

Does only one solution exist? If so, why?

Is there a pattern that I am missing that will determine the number of eggs for any number of buyers?

 

[HallsofIvy]

Does only one solution exist? If so, why?

Is there a pattern that I am missing that will determine the number of eggs for any number of buyers?

What have you done to look for a pattern?

Let X be the number of eggs Chris has to start with. "Chris sells half the eggs plus half an egg to Maria". So he sells X/2+ 1/2= (X+ 1)/2 to Maria which leaves Chris with X- (X+1)/2= (X- 1)/2 eggs.

"then sells half the remaining eggs plus half an egg to Ai-Ling". So he sells ((X- 1)/2)/2+ 1/2= X/4- 1/4+ 1/2= (X+1)/4 eggs which leaves Chris with (X- 1)/2- (X+ 1)/4= X/2- X/4- 1/2- 1/4= (X- 3)/4 eggs.

"and finally sells half the remaining eggs plus half an egg to Jermaine." So he sells ((X- 3)/4)/2+ 1/2 = X/8- 3/8+ 1/2= X/8+ 1/8= (X+ 1)/8 eggs which leaves Chris with (X- 3)/4- (X+1)/8= (X- 7)/8 eggs.

The number of eggs given each time are (X+ 1)/2, (X+ 1)/4, (X+ 1)/8 and the number of eggs he has left are (X- 1)/2, (X- 3)/4, (X- 7)/8. Do you not see a pattern? Powers of 2 perhaps?

After selling those eggs to 3 people, Chris has none left: he has sold all X of his eggs. (X- 7)/8= 0 so \(\displaystyle X= 7= 2^3- 1\)
See a pattern?

 

[joey101]

What have you done to look for a pattern?

Let X be the number of eggs Chris has to start with. "Chris sells half the eggs plus half an egg to Maria". So he sells X/2+ 1/2= (X+ 1)/2 to Maria which leaves Chris with X- (X+1)/2= (X- 1)/2 eggs.

"then sells half the remaining eggs plus half an egg to Ai-Ling". So he sells ((X- 1)/2)/2+ 1/2= X/4- 1/4+ 1/2= (X+1)/4 eggs which leaves Chris with (X- 1)/2- (X+ 1)/4= X/2- X/4- 1/2- 1/4= (X- 3)/4 eggs.

"and finally sells half the remaining eggs plus half an egg to Jermaine." So he sells ((X- 3)/4)/2+ 1/2 = X/8- 3/8+ 1/2= X/8+ 1/8= (X+ 1)/8 eggs which leaves Chris with (X- 3)/4- (X+1)/8= (X- 7)/8 eggs.

The number of eggs given each time are (X+ 1)/2, (X+ 1)/4, (X+ 1)/8 and the number of eggs he has left are (X- 1)/2, (X- 3)/4, (X- 7)/8. Do you not see a pattern? Powers of 2 perhaps?

After selling those eggs to 3 people, Chris has none left: he has sold all X of his eggs. (X- 7)/8= 0 so \(\displaystyle X= 7= 2^3- 1\)
See a pattern?

So let's say that there are 7 people. In order for Chris to have none left and follow a similar patten, I would go: 2^7-1?

= 127?

 

[Deleted member 4993]

So let's say that there are 7 people. In order for Chris to have none left and follow a similar patten, I would go: 2^7-1?

= 127?

Try it out:

first person gets 64 eggs - 63 left

second person gets 32 eggs - 31 left

and so on......