SOLUTION: The Knave of Hearts stole some tarts. He ate half of them, and half a tart more. The Knave of Diamonds ate half of what was left, and half a tart more. Then the Knave of Clubs

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Question 1183541: The Knave of Hearts stole some tarts. He ate half of them, and half a tart more.
The Knave of Diamonds ate half of what was left, and half a tart more.
Then the Knave of Clubs ate half of what remained, and half a tart more.
This left just one tart for the Knave of Spades.
How many tarts did the Knave of Hearts steal?
Found 2 solutions by ikleyn, MathTherapy:
Answer by ikleyn(52790)   (Show Source): You can put this solution on YOUR website!
.
The Knave of Hearts stole some tarts. He ate half of them, and half a tart more.
The Knave of Diamonds ate half of what was left, and half a tart more.
Then the Knave of Clubs ate half of what remained, and half a tart more.
This left just one tart for the Knave of Spades.
How many tarts did the Knave of Hearts steal?
~~~~~~~~~~~~~~~ 

I will solve the problem  from the ending state to the starting step. 


I will organize my logic, my solution and my calculations using this table below.


    Name           starting       consumed         ending
   (step)           amount         amount          amount


  H (Heart)           x           0.5x+0.5        h = x - (0.5x+0.5) = 0.5x-0.5     (1)


  D (diamond)         h           0.5h+0.5        d = h - (0.5h+0.5) = 0.5h-0.5     (2)


  C (club)            d           0.5d+0.5        c = d - (0.5d+0.5) = 0.5d-0.5     (3)


  S (spade)           c = 1                                                         (4)



        So, I have 4 equations  (1),  (2),  (3)  and  (4)  for  4  unknowns  x, h,  d  and  c.

        Now I will move backward from (4) to (1) to determine the unknowns one after another.



The last equation  c = 1  (4)  in the table gives me the opportunity to find "d" from (3)

    1 = 0.5d-0.5  --->  1 + 0.5 = 0.5d  --->  1.5 = 0.5d  --->  d = 1.5/0.5 = 3;  d = 3.       (5)



Now, knowing d= 3, I substitute it into equation (2) and find h

    3 = 0.5h-0.5  --->  3 + 0.5 = 0.5h  --->  3.5 = 0.5h  --->  h = 3.5/0.5 = 7;  h = 7.       (6)



Finally, knowing h= 7, I substitute it into equation (1) and find x

    7 = 0.5x-0.5  --->  7 + 0.5 = 0.5x  --->  7.5 = 0.5x  --->  x = 7.5/0.5 = 15;  x = 15.     (7)



Thus the problem is just solved, and the ANSWER is


      +-----------------------------------------+
      |   The Knave of Hearts stole 15 tarts.   |
      +-----------------------------------------+

Solved.

This method is one of standard methods of solving such problems.

What you will meet rarely in other solutions, is perfect organization of logic and data.

This problem is good for a Math Circle session.


////////////


Do not forget to post your  "THANKS"  to me for my teaching.



Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!

The Knave of Hearts stole some tarts. He ate half of them, and half a tart more.
The Knave of Diamonds ate half of what was left, and half a tart more.
Then the Knave of Clubs ate half of what remained, and half a tart more.
This left just one tart for the Knave of Spades.
How many tarts did the Knave of Hearts steal?
Let original number of tarts be T

1st time around: Ate  of the tarts, and  more gives us a remainder of: 

2nd time around: Ate  of REMAINING tarts, plus  more gives us a remainder of: 

3rd time around: Ate  of REMAINING tarts, plus  more gives us a remainder of: 

After 3rd time around, 1 tart remained. so we get: 

T - 7 = 8 ----- Multiplying by LCD, 8

Original number of tarts, or  

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