Density Practice Problems With Answers

Why Density Problems Make Everyone Overcomplicate Things

Density is mass divided by volume. That is the entire concept. The reason people struggle with it has nothing to do with the math and everything to do with unit conversions and geometry. I have graded enough of these to know that the student who gets the concept perfectly will still lose points because they used centimeters when the answer required cubic meters. The formula itself is trivial. The trap is in the details. Here is the thing nobody tells you about practicing density: if you only work problems where the shape is a perfect cube or cylinder, you are setting yourself up to fail when the actual problem involves an irregular object. Most textbooks start with clean shapes. Real exams and real lab work do not. A rock sample, a piece of jewelry, a plastic component — these things do not come with labeled dimensions. That is where the water displacement method matters more than any formula. I spent a semester as a teaching assistant and watched roughly sixty students fail the same way on the same question. The problem gave the mass of an irregular metal slug and asked for density. Half the class tried to measure it with a ruler. The other half measured it but forgot to convert from grams per cubic centimeter to kilograms per cubic meter before submitting. The correct approach is simple: find the mass on a balance, determine the volume by water displacement in a graduated cylinder, then divide. But you have to do it in the right order and keep your units straight the entire time.

The Method First: Water Displacement and How to Not Mess It Up

When you need the volume of an irregular object, you fill a graduated cylinder partway with water, record that volume, submerge the object completely, and record the new volume. The difference is the object's volume. It sounds stupidly simple and it is. The error comes from three sources that everyone ignores until they lose points. First, the object has to be fully submerged. If it floats, you need to push it down with a thin rod or wire and account for the volume of that rod if it displaces measurable water. I once had a student trying to find the density of a paraffin wax sample. He just pushed it under with his finger and wondered why his result was nowhere near the accepted value. His finger displaced water too. Not a big deal in most lab settings, but on an exam where they want precision, that is a problem worth noting. Second, read the meniscus at eye level. The bottom of the curve is your measurement. If you look from above or below, your volume reading shifts by anywhere from 0.2 to 1 milliliter depending on the cylinder size. That translates directly into a density error, and on close-call problems that error changes your answer grade entirely.

Third, make sure the object is dry before measuring mass but wet enough that you are not losing water during transfer. Weigh first, then displace, or accept that a wet object adds water mass to your reading. I always tell people to weigh the dry object first, then do displacement. It saves you from having to guess how much surface water clings to something when you lift it out.

Unit Conversions: Where People Actually Lose Points

This is the real bottleneck. Density can be expressed in grams per cubic centimeter, kilograms per cubic meter, pounds per cubic foot, or a dozen other combinations. The numerical value changes completely depending on which units you use. Water has a density of 1 g/cm³, which is the same as 1000 kg/m³. If a problem gives you mass in kilograms and volume in cubic centimeters, you cannot just divide and call it a day. You have to convert one side so the units match. The conversion factor between g/cm³ and kg/m³ is exactly 1000. Multiply by 1000 to go from g/cm³ to kg/m³. That is because one kilogram equals one thousand grams and one cubic meter equals one million cubic centimeters, so the ratio works out to a factor of a thousand. You do not need to memorize more than that if you understand the underlying relationship. I remember a specific midterm where the problem involved a block of aluminum with a mass of 2.7 kilograms and a volume of 1000 cubic centimeters. The expected answer in kg/m³ required converting the volume to cubic meters first. 1000 cm³ is 0.001 m³. So 2.7 divided by 0.001 gives 2700 kg/m³. A student who just did 2.7 divided by 1000 got 0.0027 and marked it down thinking they were wrong. They were not wrong about the arithmetic. They were wrong about what the question asked for. These kinds of misreads happen constantly.

Step-by-Step Worked Problems

Problem one: A rectangular block of material measures 4.0 cm by 5.0 cm by 2.0 cm and has a mass of 96 grams. Find the density in g/cm³ and kg/m³. Volume equals length times width times height. Four times five times two is forty cubic centimeters. Density equals mass divided by volume. Ninety-six divided by forty is 2.4 g/cm³. To convert to kg/m³, multiply by one thousand. The answer is 2400 kg/m³. Problem two: A metal bolt has a mass of 48.5 grams. When dropped into a graduated cylinder containing 30.0 mL of water, the water level rises to 36.5 mL. Find the density.

Volume of the bolt is thirty-six point five minus thirty point zero, which is six point five milliliters. One milliliter equals one cubic centimeter, so the volume is 6.5 cm³. Density is forty-eight point five divided by six point five, which equals approximately 7.46 g/cm³. That density is consistent with steel or possibly iron. Not gold, despite what some students assume when the number looks high. Problem three: A solution has a density of 1.05 g/mL. What is the mass of 250 mL of this solution? Rearrange the density formula. Mass equals density times volume. One point zero five times two hundred fifty equals two hundred sixty-two point five grams. This is a straightforward application but people sometimes second-guess whether they should divide instead of multiply. If density is mass over volume, then mass is density times volume. Algebra is not negotiable here.

Problem four: An irregular stone has a mass of 180 grams. It is placed in a overflow can filled to the brim with water. The displaced water is collected and found to have a volume of 60 mL. Find the density in kg/m³. The displaced water volume equals the stone's volume, so the volume is 60 cm³. Density in g/cm³ is one hundred eighty divided by sixty, which is 3.0 g/cm³. Converting to kg/m³ requires multiplying by one thousand, giving 3000 kg/m³. This confirms the stone is likely granite or a similarly dense silicate rock.

Common Pitfalls That Show Up Repeatedly

The most common error is mixing mass and weight. Density uses mass. If a problem gives you weight in newtons, you need to divide by the acceleration due to gravity to get mass in kilograms before applying the density formula. I see this on AP-level exams regularly and it is an easy point to lose if you do not catch it early. Another frequent mistake is assuming density is constant across all conditions. It is not. Density changes with temperature and pressure. A gas like air has a density of about 1.225 kg/m³ at sea level and fifteen degrees Celsius, but that drops to roughly 0.946 kg/m³ at thirty degrees Celsius at the same pressure. Liquids change less dramatically but still measurably. Mercury at twenty degrees Celsius is 13.546 g/cm³ and at fifty degrees it is about 13.470 g/cm³. For most introductory problems this variation is ignored, but if you are doing lab work or working at an advanced level, you need to know whether the given density matches your conditions. There is also the floating versus sinking confusion. An object floats when its density is less than the fluid's density. It sinks when its density is greater. It hovers neutrally when the densities are equal. This is useful for quickly checking whether your calculated density makes physical sense. If you calculate that a wooden block has a density of 1.8 g/cm³, you have made a calculation error because wood floats on water and water's density is 1.0 g/cm³.

What These Practice Problems Cannot Fully Prepare You For

Textbook density problems present idealized conditions. Real measurements have uncertainty. A typical graduated cylinder has an uncertainty of about 0.5 to 1 mL. A top-loading balance might read to 0.01 grams but still carry an uncertainty in that range. When you propagate these uncertainties through your calculation, the final density value has a range, not a single exact number. On introductory courses this is often skipped entirely, but in any real lab setting it matters a lot. For instance, if your mass measurement is 48.5 ± 0.02 grams and your volume from displacement is 6.5 ± 0.5 mL, your density is not simply 7.46 g/cm³. Using basic uncertainty propagation, the volume uncertainty dominates here and your density could reasonably range from about 7.1 to 7.8 g/cm³. Reporting 7.46 implies a precision that your equipment does not support. The correct reported value would be 7.5 ± 0.3 g/cm³ with appropriate significant figures. This kind of rigor is usually reserved for upper-level chemistry or physics labs, but it is worth understanding early. The habit of treating every calculated number as exact instead of as an estimate within a range is something I wish more students developed before they hit college-level coursework.

Where to Find Reliable Density Practice Problems With Answers

There are several solid sources online, but most of them recycle the same basic problems with no variation in difficulty. Khan Academy has a decent set if you work through the full module including the word problems. The textbook resources from open educational projects like OpenStax include problem sets with answers in the back, though the answers sometimes skip steps which makes self-checking harder. A few university physics departments post problem sets with full solutions, and those tend to be the most realistic because the professors who write them actually grade these things. If you are looking for practice that mirrors what shows up on standardized tests, focus on problems that combine density with other concepts. Problems that ask you to find mass from density and volume, or volume from mass and density, are standard. Problems that layer in buoyancy, pressure, or thermal expansion are where the harder questions live. I recommend doing at least five problems of each type: pure density calculation, unit conversion, irregular object displacement, floating and sinking, and multi-concept combined problems.

A Practical Edge Case I Encountered

A student once brought me a problem about a hollow copper sphere. The outer diameter was given, the inner diameter was given, and they needed the average density of the sphere as a whole object, not just the copper material. The accepted answer required calculating the volume of the copper shell by subtracting the inner sphere volume from the outer sphere volume, then dividing the mass by that shell volume. Most students either used the outer volume alone or the inner volume alone and got completely wrong answers. The workaround is straightforward if you think about it correctly. Find the volume of the outer sphere using four-thirds pi r cubed with the outer radius. Find the volume of the inner hollow space the same way with the inner radius. Subtract the inner volume from the outer volume to get the actual copper volume. Then you need the mass, which you either calculate from the known density of copper times the shell volume, or which is given directly in the problem. Divide mass by shell volume and you have your answer. This showed up on a regional science competition exam and roughly forty percent of the students who attempted it got it wrong because they treated the hollow sphere as a solid one.

The Bottom Line

Density problems are not hard because density is hard. They are hard because the problems hide unit mismatches, require geometric calculations you might not have practiced, or involve shapes that do not have clean formulas. The strategy is to slow down, write out every conversion explicitly, and check that your final number makes physical sense. A density of negative five grams per cubic centimeter is impossible. A density of five thousand grams per cubic centimeter is also impossible under normal conditions. If your answer falls outside a reasonable range, something went wrong and you should trace backward through your steps to find where. Practice with varied problems. Include irregular objects, hollow objects, unit conversions in both directions, and problems that combine density with buoyancy or temperature effects. The more types you see, the less surprising the actual exam or lab question will be. The concept itself does not change. Only the packaging does.