# Simple Math Problem 4 [closed]

## How much liquor did they originally have?

 Three alcoholic adults Takilu, Whisku and Rumu got some liquor.
Takilu gave Whisku and Rumu as much liquor as they already had.
Then Whisku gave Takilu and Rumu as much liquor as they already had.
Finally, Rumu gave Whisku and Takilu as much liquor as they already had.

Now each of them have 24 bottles of liquor.

• Easy. They originally had 72 bottles of liquor plus how many they drank during this exchange. Commented Dec 15, 2017 at 14:17

Solution:

Explanation:

You can get to this by just going through the instructions backwards and dividing the number of bottles of the people that received liquor by 2, then add the value that's left for these people to the one giving away the bottles.

You can write it as a matrix system :

They start with $\pmatrix{ A \\ B \\ C}$ bottles each.

Then the first action looks like : $\pmatrix{ A \\ B \\ C} \to \pmatrix{1 & -1 & -1 \\ 0 & 2 & 0 \\ 0 & 0 & 2}\pmatrix{ A \\ B \\ C}$. Then $\pmatrix{1 & -1 & -1 \\ 0 & 2 & 0 \\ 0 & 0 & 2}\pmatrix{ A \\ B \\ C} \to \pmatrix{2 & 0 & 0 \\ -1 & 1 & -1 \\ 0 & 0 & 2}\pmatrix{1 & -1 & -1 \\ 0 & 2 & 0 \\ 0 & 0 & 2}\pmatrix{ A \\ B \\ C}$. Then the third action : $\pmatrix{2 & 0 & 0 \\ -1 & 1 & -1 \\ 0 & 0 & 2}\pmatrix{1 & -1 & -1 \\ 0 & 2 & 0 \\ 0 & 0 & 2}\pmatrix{ A \\ B \\ C} \to \pmatrix{2 & 0 & 0 \\ 0 & 2 & 0 \\ -1 & -1 & 1}\pmatrix{2 & 0 & 0 \\ -1 & 1 & -1 \\ 0 & 0 & 2}\pmatrix{1 & -1 & -1 \\ 0 & 2 & 0 \\ 0 & 0 & 2}\pmatrix{ A \\ B \\ C}$

But this must equal $\pmatrix{24 \\ 24 \\ 24}$.

Invert the matrix system however you prefer, matrix by matrix then multiply or multiply then invert. $\pmatrix{ 4 & -4 & -4 \\ -2 & 6 & -2 \\ -1 & -1 & 7 }\pmatrix{ A \\ B \\ C} = \pmatrix{24 \\ 24 \\ 24 }$ gives $\pmatrix{ A \\ B \\ C } = \pmatrix{ 39 \\ 21 \\ 12 }$

(I'm sure this can be formatted better, hiding the equations is a pain)

Takilu started with 39; Whisku started with 21; Rumu started with 12

How I did it:

If W and T are doubled at the end and ended up with 24, they had 12 each before that. R ended with 24 and gave 24 so had 48. Using the same logic, before W gave the liquor, T had 6, W had 42, and R had 24. And at the start, before T gave the liquor, T had 39, W had 21, and R had 12.