Annie loves giving away puppies. When her friend Bobby walks by, she instantly gives him 50% of her puppy stash. Bobby, stunned with this shower of puppies,

Let the original number of puppies=x
.5x Annie's total after giving 50% to Bobby
.5x Bobby's total of puppies
.6x Annie's total after getting back 20% of her puppies
.4x Bobby's total after giving back 20% of the puppies
.3x Annie's total after giving Bobby 50% of her puppies
.7x Bobby's total after getting 50% of Annie's puppies.
So, we have:
.7x =7 divide both sides by .7
x=10 original number of puppies that Annie started with.

Annie loves giving away puppies. When her friend Bobby walks by, she instantly gives him 50% of her puppy stash. Bobby, stunned with this shower of puppies, gives back 20% of his total. Annie counters by giving Bobby 50% of her new puppy total. Bobby responds by giving Annie back 50% of the puppies he now has and then running off, having gone from 0 to 7 puppies in just a few seconds! How many puppies did Annie begin with?

Let x  be the number that Annie starts with

When she gives Bobby 50%  of her puppies she has  (.50)x  left and he has (.50)x

When Bobby gives back 20%  of his stash, he has .80 (.50)x  = .40x and Annie has (.50)x + .20 (.50) x  =  .60x

When Annie gives Bobby 50%  of her new total, she has (.50)(.60)x = .30x  left   and Bobby now has  .40x + (.50)(.60)x   =  .70x

And lastly.....he gives her 50% of his total back to Annie and he is left with (.50) (.70)x  =  .35x  and she is left with .30x + .35x  = .65x

And his total = .35x   = 7     so    7/.35   =  x   = 20....and this is the number of puppies that Annie started with

Proof :  Annie gives Bobby 50%.....she has 10 and Bobby has 10

Bobby gives he 20% of his total back to Annie  ....he has 8   and she has .60(20) = 12

Annie gives Bobby 50%  of her total.....she now has 6 and he has 8 + 6 = 14

Bobby finally gives 50% of his total back to Annie....so he has 7 and she has 6 + 7  = 13

Note that, at each step, the number of puppies between them always = 20