tmua.me Extreme Set C Paper 2 · Question 6

Initially, tank $A$ contains $1$ litre of pure dye, and tank $B$ contains $2$ litres of water. A cycle consists of transferring $t$ litres from $A$ to $B$ , mixing thoroughly, then transferring $t$ litres from $B$ back to $A$ and mixing thoroughly again. The same value of $t$ is used in each cycle, with $0<t<1$ .

Let $c_n$ be the proportion by volume of dye in tank $A$ after $n$ complete cycles. It is given that

\frac{c_1}{c_3}=\frac85.

What is the sum of all the possible values of $t$ ?

\dfrac25

2-\dfrac{4\sqrt7}{7}

\dfrac45

\dfrac{1+\sqrt7}{4}

\dfrac{34}{3}-4\sqrt7

\dfrac{12}{5}-\dfrac{4\sqrt7}{7}

examday

tmua.me Extreme Set C Paper 2 · Q6