He originally had a total of 54 gallons of milk in three churns, and he wanted to make sure each churn contained 18 gallons of milk.
In order to do this, he did the following:
First, he poured 1/4 of the first churn into the second churn. He then poured 1/2 of the second churn into the third churn. Finally, he poured 1/3 of the third churn into the first churn.
How many gallons did each churn contain before Farmer Stone started pouring?
Hint
Try working backwards with each churn containing 18 gallons.
Answer
12, 33, and 9 gallons respectively for churns 1, 2, and 3.
Reasoning
Working backwards, at the end after Pour 3, the churns (C1, C2, C3) contained:
C1 C2 C3
————————————
18 18 18 after Pour 3
Pour3 was 1/3 of C3 into C1, the remaining 2/3 has to be the 18 gallons left in C3 after the pour, which means that 1/3 is 9 gallons. So 9 gallons was poured from C3 into C1. Before Pour3, C1 must have contained 9 gallons, and C3 contained 27 gallons.
C1 C2 C3
————————————
9 18 27 after Pour 2
18 18 18 after Pour 3
Pour2 was 1/2 of C2 into C3, the remaining 1/2 has to be the 18 gallons left in C2 after the pour, which means that 1/2 is 18 gallons. So 18 gallons was poured from C2 into C3. Before Pour2, C2 must have contained 36 gallons, and C3 contained 9 gallons.
C1 C2 C3
————————————
9 36 9 after Pour 1
9 18 27 after Pour 2
18 18 18 after Pour 3
Pour1 was 1/4 of C1 into C2, the remaining 3/4 has to be the 9 gallons left in C1 after the pour, which means that 1/4 is 3 gallons. So 3 gallons was poured from C1 into C2. Before Pour2, C1 must have contained 12 gallons, and C2 contained 33 gallons.
C1 C2 C3
————————————
12 33 9 at the start
9 36 9 after Pour 1
9 18 27 after Pour 2
18 18 18 after Pour 3
Answer
The missing sum is 40.
Circle = 12
Star = 3
Square = 7
Triangle = 9
Reasoning
Looking at Column 1 and Column 3 we have:
[1] Square + Star + Circle + Square = 29
[2] Square + Square + Triangle + Star = 26
If we subtract [1] – [2] we get:
[3] Circle – Triangle = 3
And Row 3 is:
[4] Circle + Circle + Triangle + Circle = 45
If we add [3] + [4] we get:
Circle + Circle + Circle + Circle = 48
So, Circle = 12.
Using [3] we find that Triangle = 9.
Using Column 2 we find that Star = 3.
Using Row 1 we find that Square = 7.
Double-Checking
The rows add to: 35 + 26 + 45 + 25 = 131.
The columns add to: 29 + 36 + 26 + 40 = 131.
The rows and columns must add to the same number, so this could have been used a shortcut to the missing sum.
In total, there are:
5 Squares = 5 x 7 = 35
5 Circles = 5 x 12 = 60
3 Stars = 3 x 3 = 9
3 Triangles = 3 x 9 = 27
Which is also a total of 131.
So all of the totals are the same, as expected.
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