A sample of metallic frewium weighs 185N on a spring scale in air. When immersed in pure water, the frewium pulls on the scale with a force of 155N. What is the buoyant force provided by the water? What is the density of metallic frewium?

Wow !  This one could have some twists and turns in it.Fasten your seat belt.  It's going to be a boompy ride.-- The buoyant force is precisely the missing 30N .--  In order to calculate the density of the frewium sample, we need to knowits mass and its volume.  Then, density = mass/volume .-- From the weight of the sample in air, we can closely calculate its mass.   Weight = (mass) x (gravity)   185N = (mass) x (9.81 m/s²)   Mass = (185N) / (9.81 m/s²) = 18.858 kilograms of frewium  -- For its volume, we need to calculate the volume of the displaced water.The buoyant force is equal to the weight of displaced water, and the density of water is about 1 gram per cm³.  So the volume of the displaced water (in cm³) is the same as the number of grams in it. The weight of the displaced water is 30N, and weight = (mass) (gravity).           30N = (mass of the displaced water) x (9.81 m/s²)           Mass = (30N) / (9.81 m/s²) = 3.058 kilograms           Volume of displaced water = 3,058 cm³Finally, density of the frewium sample = (mass)/(volume)      Density = (18,858 grams) / (3,058 cm³) = 6.167 gm/cm³ (rounded)================================================I'm thinking that this must  be the hard way to do it, because I noticed that       (weight in air) / (buoyant force) =  185N / 30N = 6.1666... So apparently . . .         (density of a sample) / (density of water) =                                   (weight of the sample in air) / (buoyant force in water) .I never knew that, but it's a good factoid to keep in my tool-box.