A duck is floating on a lake with 25% of its volume beneath the water. What is the average density of the duck

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the average density of the duck =d, volume =V
average density of water = 1000 kg/m^3
weight of duck = dV * 9.81
= weight of water displaced
= 0.25* V * 1000 * 9.81
dV * 9.81 = 0.25* V * 1000 * 9.81
d= 0.25* 1000
d = 250 kg/m^3
answer
average density of the duck

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Density = mass / volume

Mass of duck = volume of water displaced x density of water

( because it's floating and the gravity force on the duck must be the same as the upthrust from the displaced water - strictly speaking the equation should not be equating masses - it should be forces but that just means putting a "g" on each side which would cancel so why bother with it in the first place?)

Mass of duck = (V/4) . 1.0

where V = volume of duck and 1.0 = density of water ( 1 g/ml)

Density of duck = (V/4 ) / V = 1/4 = 0.25 g/ml

One quarter the density of water.

If it takes 25% of the ducks volume in displaced water, then from Archimedes' principle it directly follows that the average density of the duck equals 25% of the density of water.

It cannot be explained simpler than that, but if you insist on formulaic steps:

Rho_duck * g= m_duck * g / V_duck = m_displacedwater * g / V_duck
= Rho_water * V_displacedwater * g / V_duck
= Rho_water * 25% * g

So Rho_duck = 25% * Rho_water