PLEASE HELP. No one knows this, please help!

Question

A farmer has to make 8 stops in delivering eggs. He begins with exactly the number of eggs he needs for these 8 deliveries. At the first stop he delivers half of the eggs he had plus 1/2 of an egg. At each of these next 7 stops, he delivers half of the remaining eggs plus 1/2 of an egg. When he is finished he has no eggs left, and none have been broken when making the deliveries. How many eggs did he have when he began his deliveries?

Answer 1

Hi Marlene - I agree with your method except that the problem states that deliveries are made at all 8 stops. Your  table shows no deliveries at the first stop. 

 

If you assume that deliveries are made at all 8 stops per the problem description, and using the same approach as described by Marlene, you come up with a total of 256 eggs. The table would look like this.

 

Stop Number                   Number of Eggs Delivered       Numer of Eggs Left

 

0                                          0                                       256

1                                          128.5                                 127.5

2                                          64.25                                 63.25

3                                          32.12                                 31.12

4                                          16.06                                 15.06

5                                          8.03                                   7.03

6                                          4.01                                   3.01

7                                          2.01                                   1.01

8                                          1.01                                    0

Answer 2

Farmer starts with X eggs

 

First stop he delivers (1/2x + 1/2) eggs and has x - (1/2x + 1/2) left.  This can be reduced to (x-1)/2.

2nd stop he delivers 1/2[(x-1)/2] + 1/2 eggs and has (x-1)/2-{1/2[(x-1)/2] + 1/2} left.  This can be reduced to (x-3)/4.

3rd stop he delivers 1/2[(x-3)/4] + 1/2 eggs and has (x-3)/4-{1/2[(x-3)/4] + 1/2 } left.  This can be reduced to (x-7)/8.

4th stop he delivers 1/2[(x-5)/8] + 1/2 eggs and has (x-7)/8 - {1/2[(x-7)/8] + 1/2} left, reduced to (x-15)/16

 

Let's stop here and see the pattern.

at the 4th stop, we can restate the eggs left as

[x - (2^t-1)]/2^t

where t = the stop number

 

So after the 7th stop, there are [x - (2⁷-1)]/2⁷ left.

We're told that he has no eggs left so let's set that equal to 0

[x - (2⁷-1)]/2⁷ = 0

[x - (2⁷-1)] = 0

x = 2⁷ - 1 = 127

 

To check this look at the chart below

 

Stop number     # eggs delivered     # eggs left

0                                                        127

1                               64                       63

2                               32                       31

3                               16                       15

4                                8                         7

5                                4                         3

6                                2                         1

7                                1                         0

 

Hope this has helped.