Lesson One more lesson on solving problems by the backward method

One more lesson on solving problems by the backward method

Problem 1

             (a)  Analyzing the ending situation from the problem



The stamps were circulated inside the team and did not go out or go into from outside.

Therefore, as they started with 6750 stamps at the beginning, so they ended with the same 
6750 stamps at the end.


      From it, we can analyze the ending situation from the problem.


The problem says that at the end 

    "Cheryl had twice the total number of stamps that Andy and Berlin had."


It means that at the end Cheryl had 2/3 of 6750, i.e. 4500 stamps, while Andy and Berlin
had 1/3 of 6750, or  2250 stamps, together.


Also, the problem says that at the end 

    "the ratio of the number of Andy’s stamps to the number of Berlin’s stamps became 4:5".


It means that at the end Andy had 4/9 of 2250, or 1000 stamps,
while Berlin had at the end 5/9 of 2250, or 1250 stamps.


So, at the end, Andy had 1000 stamps;  Berlin had 1250 stamps, and Cheryl had 4500 stamps.



             (b)  Making steps back from the end to the beginning



                     (b32)  From step (3) to step (2)



At step (3), Cheryl gave 1/6 of her stamps to Andy. 

So, we can write this equation

    C -  = 4500,  or   = 4500,  giving  C =  = 6*900 = 5400.


So, immediately before step (3), Cheryl had 5400 stamps.

Next, at step 3, Cheryl gave  = 900 stamps to Andy; so, immediately before step (3), Andy had 1000-900 = 100 stamps.


Thus, immediately after step 2, Andy had 100 stamps;  Berlin had 1250 stamps; Cheryl had 5400 stamps.



                     (b21)  From step (2) to step (1)



At step (2), Berlin gave 1/3 of her stamps to Cheryl.

So, we can write this equation

    B -  = 1250,  or   = 1250,  giving  B =  = 1875.


So, immediately before step (2), Berlin had 1875 stamps.

Next, at step 2, Berlin gave  = 625 stamps to Cheryl; so, immediately before step (2), Cheryl had 5400-625 = 4775 stamps.


Thus, immediately after step 1, Andy had 100 stamps;  Berlin had 1875 stamps; Cheryl had 4775 stamps.



                     (b10)  From step (1) to the beginning



At step (1), Andy gave 1/2 of his stamps to Berlin. 

So, we can write this equation

    A -  = 100,  or   = 100,  giving A = 200.


So, at the beginning, Andy had 200 stamps.

Next, at step (1), Andy gave  = 100 stamps to Berlin; so, at the beginning, Berlin had 1875-100 = 1775 stamps.


Thus, at the beginning, Andy had 200 stamps; Berlin had 1775 stamps, and Cheryl had 4775 stamps.


Finally, the problem asks "How many stamps did Berlin and Cheryl have in total at first ?"


The ANSWER  is  1775 + 4775 = 6550.