Fatuma writes down in her book list of positive integers from 1-1000.Amani then erases all even numbers from list and replaces them with their halves

Question

Fatuma writes down in her book list of positive integers from 1-1000.Amani then erases all even numbers from list and replaces them with their halves.How many different numbers will be formed

Answer 1

Start to list the numbers, and see if you can see the pattern.

 

1  2  3  4  5  6   7   8  9   10   11  12  13  14  15  16  17  18   ...      996   997   998    999    1000

 

 

Now replace every even number with its half:

 

1  1  3  2  5  3  7  4  9  5  11  6  13  7  15  8  17  9  ...  498  997  499  999  500

 

Notice, each original ODD number repeats one time. 

Every even number that is only divisible by 2 (not 4) results in an odd number, so it is a repeat.

 

It is only the even numbers that were divisible by 4 that gave us even numbers and there for non-repeats (because not odd).

 

 

So we have 500 odd numbers to start with.   500 even numbers to start with.   After dividing the even numbers by 2, we only have 250 that are unique (even)

 

So there are 500 + 250 different numbers.

 

Or looking at it another way, look at the new list and see that only every 4th number is even.   So in a group of 4 numbers, only 3 will be unique.  (two will be a repeated odd number from somewhere in the list so they only count as one unique number, 1 will be a different odd, 1 will be even) - this is a 3:4 ratio of unique numbers : total numbers.   1000*3/4 = 750 unique numbers.