Apple picking problem

Question

A woman was picking apples in a garden. To get out of the garden, she had to go through 4 doors, and at each door she had to leave half of her apples for the guards. She brought 10 apples home. How many apples did the guards keep?

Candy Clicker

Answer 1

Let x = #apples she started with. That means she arrives at Door 1 with x apples.

x -> Door 1: -> (1/2)x she leaves with

(1/2)x -> Door 2: -> (1/4)x she leaves the guard 1/2 of what she started with so has 1/2 of those apples in her basket.

(1/4)x -> Door 3: -> (1/8)x (this is 1/2 of what she came to the door with)

(1/8)x -> Door 4: -> (1/16)x -- finally this is what remains

But we know she remained with 10 apples so:

(1/16)x = 10

x = 160; the woman started with 160 apples

This is actually an exponential decay problem with base (1/2).

A = initial(1/2)^t

We know that A = 10, when t = 4:

10 = x(1/2)⁴

(1/16)x = 10

x = 160

So the exponential decay function is:

A(t) = 160(1/2)^t, where t is number of doors she has gone through.

A(0) = 160

A(1) = 80

A(2) = 40

A(3) = 20

A(4) = 10