Answer
4.0 kg.
Work Step by Step
When the combination begins to sink, the weight of the wood and lead equals the buoyant force on the objects.
$$m_{wood}g+m_{lead}g=V_{wood}\rho_{water}g+ V_{lead}\rho_{water}g $$
$$m_{wood}+m_{lead}=\frac{ m_{wood}}{ \rho_{wood}}\rho_{water}+ \frac{ m_{lead}}{ \rho_{lead}}\rho_{water} $$
Solve for the mass of the lead.
$$m_{lead}(1-\frac{\rho_{water}}{\rho_{lead}})= m_{wood}(\frac{\rho_{water}}{\rho_{wood}}-1)$$
$$m_{lead}= m_{wood}\frac{(\frac{\rho_{water}}{\rho_{wood}}-1)}{ (1-\frac{\rho_{water}}{\rho_{lead}})}$$
Putting in numbers, we find a lead mass of 4.0 kg.
