2. Alice, Ben and Carl collect stamps. They exchange stamps among themselves according to the following scheme: Alice gives Ben as many stamps as Ben has and Carl as many stamps as Carl has. After that, Ben gives Alice and Carl as many stamps as each of them has, and then Carl gives Alice and Ben as many stamps as each has. If each finally has 64 stamps, with how many stamps does Alice start?
Answers
Step-by-step explanation:
ALICE HAVE 11 STAMPS WHEN HE START
Answer:
Alice has 104 stamps
Step-by-step explanation:
At the start Alice, Ben, and Carl have A,B,and C stamps, respectively.
On the first transaction, Alice gives B stamps to Ben and C stamps to Carl.
So after the first transaction
Alice has A-B-C
Ben has 2B
Carl has 2C
On the second transaction, Ben gives A-B-C stamps to Alice and 2C stamps to Carl.
Alice has 2(A-B-C) = 2A-2B-2C
Ben has 2B - (A-B-C)-2C = 2B - A +B+C = -A + 3B -C
Carl has 4C
On the final transaction Carl gives 2A-2B-2C stamps to Alice and -A+3B-C stamps to Ben
Alice has 4A-4B-4C stamps.
Ben has -2A+6B-2C stamps
Carl has 4C - (2A-2B-2C)-(-A+3B-C) =
4C -2A + 2B + 2C + A - 3B + C =
-A - B+7C
Each of these is 64 stamps, so....
Alice:
4A-4B-4C =64
A-B-C = 16 <--- this is equation #1
Ben:
-2A+6B-2C = 64
-A + 3B - C = 32<--- divides by 2; this is equation #2
Carl: -A - B+7C = 64 <--- this is equation #3
The 3x3 system of equations is:
A-B-C = 16
-A + 3B -C = 32
-A - B+7C = 64
Pairing up the first to of these 3 equations, A is eliminated by adding together equations #1 and #2:
2B - 2C = 48
B - C = 24
B = C+24 <---- please label this equation ALPHA
Next SUBTRACTS Equation #3 MINUS equation #2, which also eliminates A:
8c - 4b = 32
2c - b = 8
2c - (C+24) = 8
2c - c - 24 = 8
c - 24 = 8
c = 32
By equation ALPHA in bold above, B = 32+24 = 56
Finally, using the ORIGINAL first equation A-b-c = 16
a - 56 - 32 = 16
a - 88 = 16
a = 104
So altogether , they starts with A+B+C = 104+56+32 = 192 stamps.
At the end, there are also 3 x 64 = 192 stamps.
Finally, reviewing the transactions:
A=104
B=56
C=32
Alice gives 56 stamps to Ben and 32 stamps to Carl:
A=16
B=112
C=64
Ben gives 16 stamps to Alice and 64 stamps to Carl:
A=32
B=32
C=128
Finally Carl gives 32 stamps to both Alice and Ben:
A=64
B=64
C=64
Which shows the correct answer of:
A=104
B=56
C=32