Juan has two machines: one exchanges 1 white token into 4 red tokens, while the other exchanges 1 red token into 3 white one. juan has 4 white tokens. after exactly 11 exchanges , he has 31 tokens. how many of those are red?

14 are red

Step-by-step explanation:

Let's identify the 2 types of transactions Juan machines do:

w : changes 1 white token into 4 red tokens (gain: 3 tokens overall)

r : changes 1 red token into 3 white tokens (gain: 2 tokens overall)

We know that Juan does 11 trades, so we can say:

w + r = 11

Overall, Juan started with 4 tokens and ends with 31... so, gain of 27 tokens.

We can model this as:

3w + 2r = 27

(3w because each w transaction, he gains 3 tokens, 2r because every r transaction, he gains 2 tokens).

From first equation we get r = 11 - w, which we place in the second equation:

3w + 2 (11 - w) = 27

3w + 22 - 2w = 27

w = 5, so r = 6 (from first equation).

Now, we know Juan did 5 w transactions and 6 r transactions.  But how many token of each color does he have?

Start: 4W (4 White chips)

He does 4 w transactions, he now has 16R tokens

He does 6 r transactions, he now has 10 (16-6) R tokens, plus 18 W tokens

He does another w transaction, he now has 14 R (10 + 4), plus 17 W(18-1) tokens

He did 11 transactions, and he has 31 tokens at the end, out of which 14 are red.