SOLUTION: A box contains some apples. Andree takes 1/2 of them along with one extra apple. Beatrice takes 1/3 of the remaining apples along but put 2 apples back in thebox & finally, corrine

A box contains some apples. Andree takes 1/2 of them along
with one extra apple. Beatrice takes 1/3 of the remaining
apples along but put 2 apples back in thebox & finally,
Corrine takes 5/6 of the remaining apples along with one more
apple. There are now 7 apples left in the box. How many
apples were in the box before Andree took her share?

Two ways to solve it.  The FUN way and the ALGEBRA way. ;-)

The FUN way:

Think of showing the movie of that backward and tell what you 
see at each step, then when you figure how many were in would
the basket have at the first of that movie, that will be the 
answer.

Showing the movie backwards:

 "7 apples left in the box."

So in the backwards movie we start out seeing a box with 7
apples in it.

"Corrine takes 5/6 of the remaining apples along with one
more apple"

So in the backwards movie we see Corrine walking backwards
carrying a bunch of apples up to the box which contains 7
apples.  Then she puts one apple in the box.  Now in order
for her to have been able (in the forward movie) to have taken
away 5/6 and to have left those 8, then those 8 would have
been the other 1/6 of what was there before she came along.
Therefore there had to have been 6x8 or 48 when Corrine first
walked up. Then she took 5/6 or 40 away, and left the other
8.

So, back to the backwards movie, after seeing Corrine put one
in the box, she then proceeds to put in 40 apples.  So we see
her walk backwards to the box with an armload of 41 apples.
She first puts one, pauses, and then dumps the other 40 in
the box.  Now she walks backwards away appleless leaving the
box containing 48 apples.  

So in the backwards movie we now see a box with 48 apples in
it. Then we see Beatrice walking backwards carrying a bunch
of apples up to the box which contains 48 apples. 

  "(Beatrice)...put 2 apples back in the box."

So in the backwards movie she takes 2 apples from the 48 in
the box, and puts them in her arms along with her other
apples, that leaves 46 in the box.  

  "Beatrice takes 1/3 of the remaining apples..."

Now in order for her to have been able (in the forward movie)
to have taken away 1/3 and to have left those 46, then those
46 would have been the other 2/3 of what was there before she
came.  Half of 2/3 is 1/3, and half of 46 is 23. So Beatrice
took away 23, before she put back those 2.      
 
So, back to the backwards movie, after seeing Beatrice take
two from the box, leaving 46 in the box, she then proceeds to
put 23 more in the box.  So we see her walk backwards to the
box with an armload of 21 apples. She takes 2 out of the box,
pauses, and then dumps the 23 in the box.  Now she walks 
backwards away appleless, leaving the box containing 46+23
or 69 apples.  

So in the backwards movie we now see a box with 69 apples in
it. Then we see Andree walking backwards carrying a bunch of
apples up to the box which contains 69 apples.

 "Andree takes 1/2 of them along with one extra apple."

Let's interpret that as 

"Andree takes 1/2 of what's there and then takes one more
apple.  

So in the backwards movie, Andree first puts 1 apple in the
box.  That makes 70 apples in the box.  In the forward movie,
he took half and left half, so in the backwards movie he put
another 70 in the box.  So the box contained 140 apples
originally.

--------------------------------------------------------

Now for the ALGEBRA way:

A box contains N apples. Andree takes (1/2)N plus one extra
apple. So he takes away (1/2)N + 1 from N.

Subtracting, N - [(1/2)N + 1] = N - (1/2)N - 1 = (1/2)N - 1
in the box.

Beatrice takes 1/3 of the remaining (1/2)N - 1 
So Beatrice takes away (1/3)[(1/2)N - 1] from the remaining
(1/2)N-1.  Subtracting,

(1/2)N - 1 - (1/3)[(1/2)N - 1] = (1/2)N - 1 - (1/6)N + 1/3 =
(1/3)N - 2/3 

That's how many are left in the box  

But puts 2 apples back in the box

Now there are (1/3)N - 2/3 + 2 = (1/3)N + 4/3 apples in the
box.

& finally, Corrine takes 5/6 of the remaining (1/3)N + 4/3.
So multiplying:

(5/6)[(1/3)N + 4/3] = (5/18)N + 10/9

Now we subtract these (5/18)N + 10/9 from the (1/3)N + 4/3

(1/3)N + 4/3 - [(5/18)N + 10/9] = (1/3)N + 4/3 - (5/18)N - 10/9 =
(1/18)N + 2/9 left in the box 

" along with one more apple"

So now we subtract one more and get 

(1/18)N + 2/9 - 1 = (1/18)N - 7/9 left in the box.

Since there are now 7 apples left in the box, we have the
equation:

(1/18)N - 7/9 = 7

Multiply through by 18

N - 14 = 126

N = 140

Edwin McCravy
AnlytcPhil@aol.com