How to calculate the ratio of rate constants of competing reactions? There are two compounds $\ce{A}$ and $\ce{B}$ that react with one common reagent $\ce{C}$ following first order kinetics. 
  Suppose 99.9% of $\ce{A}$ must react before 0.1% of $\ce{B}$ has reacted then what is the minimum ratio of their respective rate constants? 


My Attempt at the question:

We know that the reactions follow first order kinetics, hence I tried to find the time at which the given percentage of $\ce{A}$ and $\ce{B}$ is left using the formula 
$$ t = \frac{2.303}{k} \cdot \log\left(\frac{C_0}{C_1}\right)$$ 
where $t$ is the time, $k$ is the rate constant for each reaction and $C_0$ and $C_1$ are initial and final concentration respectively.

After this I tried to find the ratio of rate constants directly but couldn't arrive at the answer. 
I derived the above formula using the integrated rate law.
$$k_1t = 2.303 \log\left(\frac{1}{(1-0.999)}\right)$$
$$k_2t = 2.303 \log\left(\frac{1}{(1-0.001)}\right)$$