Water is contained in the three tanks A, B, and C.
First, after moving 1/3 of the water of A to B, 1/3 of the water remaining in A was moved to C.
Next, after moving 1/3 of the water of B to C, 1/3 of the water remaining in B was moved to A.
As a result, the volume ratio of water remaining in A, B, and C was 2 : 3 : 4.
Find the volume ratio of the water which was contained in the tank of A, B, and C at first by the least integer ratio.
First, after moving 1/3 of the water of A to B, 1/3 of the water remaining in A was moved to C.
Next, after moving 1/3 of the water of B to C, 1/3 of the water remaining in B was moved to A.
As a result, the volume ratio of water remaining in A, B, and C was 2 : 3 : 4.
Find the volume ratio of the water which was contained in the tank of A, B, and C at first by the least integer ratio.
Answer
Since the volume of water in B became 3 after moving 1/3 of the water remaining in B to A, the volume in B before moving is 3 / (1 - 1/3) = 9/2, and A is 2 - (9/2 × 1/3) = 1/2.
At the previous operation, 1/3 of the water in B was moved to C, the volume in B before moving is 9/2 / (1 - 1/3) = 27/4 and C is 4 - (27/4 × 1/3) = 7/4.
At the previous operation, 1/3 of the water in A was moved to C, the volume in A before moving is 1/2 / (1 - 1/3) = 3/4 and C is 7/4 - (3/4 × 1/3) = 3/2.
At the previous operation, 1/3 of the water in A was moved to B, the volume in A before moving is 3/4 / (1 - 1/3) = 9/8 and B is 27/4 - (9/8 × 1/3) = 51/8.
Therefore, the volume ratio of the water at first is 9/8 : 51/8 : 3/2 = 9 : 51 : 12 = 3 : 17 : 4.
3 : 17 : 4
Solution
Assuming the volume of the water remaining in A, B, and C is 2, 3, and 4 respectively as a result of moving operation, it is traced back in order.
Since the volume of water in B became 3 after moving 1/3 of the water remaining in B to A, the volume in B before moving is 3 / (1 - 1/3) = 9/2, and A is 2 - (9/2 × 1/3) = 1/2.
At the previous operation, 1/3 of the water in B was moved to C, the volume in B before moving is 9/2 / (1 - 1/3) = 27/4 and C is 4 - (27/4 × 1/3) = 7/4.
At the previous operation, 1/3 of the water in A was moved to C, the volume in A before moving is 1/2 / (1 - 1/3) = 3/4 and C is 7/4 - (3/4 × 1/3) = 3/2.
At the previous operation, 1/3 of the water in A was moved to B, the volume in A before moving is 3/4 / (1 - 1/3) = 9/8 and B is 27/4 - (9/8 × 1/3) = 51/8.
Therefore, the volume ratio of the water at first is 9/8 : 51/8 : 3/2 = 9 : 51 : 12 = 3 : 17 : 4.
A
|
2
|
1/2
|
1/2
|
3/4
|
9/8
|
B
|
3
|
9/2
|
27/4
|
27/4
|
51/8
|
C
|
4
|
4
|
7/4
|
3/2
|
3/2
|
