You recognize the quantities in a chemistry problem, but the calculator still seems to offer two plausible choices: multiply or divide. Dimensional analysis gives you a way to choose before entering the numbers. Write every quantity with its unit, arrange the relationships so unwanted units cancel, and check that the remaining unit answers the question.
This dimensional analysis practice guide focuses on early college chemistry: metric conversions, density, rates, squared units, and a first grams-to-moles calculation. The worked examples and practice questions are original teaching exercises. Keep your course’s conversion table nearby, try each setup yourself, and use the explanations to diagnose mistakes rather than merely compare final numbers.
What dimensional analysis does
A measurement contains both a number and a unit. Converting 0.425 liters into milliliters changes its numerical expression without changing the volume. The relationship 1 L = 1,000 mL gives you two possible fractions. Choose the one that cancels liters and leaves milliliters.
OpenStax Chemistry 2e explains the factor-label approach and the use of equivalent quantities as conversion factors. The key principle is that units participate in the calculation along with numbers. Use the textbook’s treatment as a reference, then practice with the independent examples below.
Correct units are a necessary check, but they do not prove the answer is correct. A wrong numerical conversion factor can still cancel perfectly. Verify both the relationship and its direction. In chemistry, also keep track of the substance: grams of sodium chloride and grams of water have the same unit but refer to different materials.
A five-step setup for every practice problem
- Identify the given quantity. Copy its value, unit, and substance label when relevant.
- Name the target unit. Put it at the right side of your work before calculating.
- List valid relationships. Use definitions, information supplied in the problem, or an approved reference.
- Arrange the factors. Place each unwanted unit opposite an identical unit so it cancels.
- Calculate and check. Confirm the unit, estimate the size, and round at the end.
For a multistep problem, sketch the unit path first. A path such as micrograms to milligrams to grams makes the missing relationship visible. It also lets you pause productively: instead of saying “I don’t understand any of this,” you can ask which factor connects the two units where you got stuck.
Worked example 1: liters to milliliters
Problem: Express 0.425 L in mL.
0.425 L x (1,000 mL / 1 L) = 425 mL
The L in the starting quantity cancels the L in the denominator. Only mL remains. A milliliter is smaller than a liter, so the same volume requires a larger number of milliliters. That estimate agrees with 425.
If you inverted the factor, the expression would contain L squared per mL. That is not the requested unit, even if a calculator produces a neat decimal. Stop at the unit check and repair the setup before doing arithmetic.
Worked example 2: two metric conversions
Problem: Express 7,250 micrograms in grams. Use scientific notation for the answer if the original measurement’s significant figures are not specified.
7,250 micrograms x (1 mg / 1,000 micrograms) x (1 g / 1,000 mg) = 0.00725 g
Micrograms cancel first, then milligrams. The result is 7.25 x 10-3 g. The conversion itself does not tell you whether the trailing zero in 7,250 was intended to be significant. Use the precision given by your instructor or the measurement context.
You could also convert directly using 1 g = 1,000,000 micrograms. Both routes describe the same relationship. During practice, a longer route is acceptable if you can justify each factor. After you understand the path, shorten it without dropping the unit labels.
Worked example 3: density as a relationship
Problem: A liquid has a density of 1.20 g/mL. What mass corresponds to 35.0 mL?
35.0 mL x (1.20 g / 1 mL) = 42.0 g
Milliliters cancel, leaving grams. The density tells you that each milliliter has a mass of 1.20 g under the stated conditions. The mass should therefore be somewhat greater than 35 g. The calculated 42.0 g is consistent with that estimate and has three significant figures.
Now reverse the question. If the same liquid has a mass of 42.0 g, its volume is found using 1 mL / 1.20 g. Density is not a universal conversion between mass and volume. You need the density of the relevant material under the relevant conditions. Do not use a value remembered from a different substance.
Worked example 4: convert both parts of a rate
Problem: A process moves material at 12.0 cm/s. Express that speed in m/min.
12.0 cm/s x (1 m / 100 cm) x (60 s / 1 min) = 7.20 m/min
Centimeters cancel with centimeters, and seconds cancel with seconds. Notice that seconds began in the denominator of the starting rate. The time factor therefore needs seconds in its numerator. This is why a memorized rule such as “always put the new unit on top” is unreliable for compound units.
For a separate check, 12.0 centimeters each second becomes 720 centimeters in one minute. That is 7.20 meters per minute. Solving a problem by a second simple route can expose a factor that was accidentally inverted.
Worked example 5: squared units
Problem: Convert 3.50 m2 to cm2.
3.50 m^2 x (100 cm / 1 m)^2 = 35,000 cm^2
Write the final value as 3.50 x 104 cm2 to communicate three significant figures clearly. The entire conversion factor is squared. Multiplying by only 100 would convert one length dimension while leaving the other unchanged.
Picture a square that is one meter on each side. Each side is 100 centimeters, so its area is 100 x 100 square centimeters. For cubic units, apply the conversion factor three times. Keep powers visible until you have finished canceling units.
Worked example 6: grams to moles
Problem: Find the amount in moles in 11.7 g of sodium chloride, using the supplied molar mass of 58.44 g/mol.
11.7 g NaCl x (1 mol NaCl / 58.44 g NaCl) = 0.200 mol NaCl
The unrounded quotient is approximately 0.200205 mol. The given mass limits the reported result to three significant figures. Because 11.7 g is about one fifth of 58.44 g, an answer near one fifth of a mole is reasonable.
Use the supplied or correctly calculated molar mass for the compound in the problem. A unit path cannot compensate for selecting the wrong substance. In later reaction problems, balanced-equation coefficients connect amounts in moles of different substances. They do not generally provide direct ratios between their masses in grams.
Try these practice problems before reading the answers
Write the full factor-label setup for each question. For the first attempt, keep a reference table available but cover the worked examples. On a later attempt, mix the question order so the previous problem does not reveal the next method.
- Convert 2.75 km to m.
- Convert 640. mg to g. The decimal point here indicates three significant figures.
- A sample occupies 18.0 mL and has a density of 2.50 g/mL. Find its mass.
- Convert a flow rate of 90.0 mL/min to mL/s.
- Convert 0.0250 m2 to cm2.
- Find the amount in moles in 9.00 g of water using a supplied molar mass of 18.02 g/mol.
Answers and unit paths
- 1: 2.75 km x (1,000 m / 1 km) = 2.75 x 103 m.
- 2: 640. mg x (1 g / 1,000 mg) = 0.640 g.
- 3: 18.0 mL x (2.50 g / 1 mL) = 45.0 g.
- 4: 90.0 mL/min x (1 min / 60 s) = 1.50 mL/s.
- 5: 0.0250 m2 x (100 cm / 1 m)2 = 2.50 x 102 cm2.
- 6: 9.00 g water x (1 mol water / 18.02 g water) = 0.499 mol water.
If your answer differs, first compare the uncanceled units, then the numerical factors, then the arithmetic. This order separates a setup error from a calculator error. Write one sentence explaining the correction and solve a new problem of the same type.
Make practice sessions useful
Use a short error log with three labels: relationship, orientation, and arithmetic. A relationship error means the factor itself was wrong. An orientation error means a valid factor was upside down. An arithmetic error means the setup worked but the numerical calculation did not.
Try a 20-minute session with two familiar conversions, two mixed questions, and one explanation written without looking at the solution. Return to missed question types on another day. The UNC Learning Center’s study guidance supports active engagement and distributing study across sessions. Here, the practical goal is to practice choosing the setup, not just rereading it.
For broader preparation, connect this practice to StudyUpload’s chemistry exam study guide. Build a small collection of representative problems instead of repeatedly solving one easy conversion with different numbers.
Create a conversion sheet worth sharing
A useful sheet includes the relationship, one correct setup, an estimate, and a common mistake. Label exact unit definitions separately from measured values. Keep extra calculator digits during intermediate steps and apply the course’s rounding rules to the final answer.
Explore chemistry study resources or browse student documents to compare explanations. Check any uploaded conversion against your textbook before relying on it. Another student’s notes may use different conventions or contain an error.
Once you have checked your examples, upload your own notes to help other students. Include your original practice questions and explain how units cancel. Share only material you are allowed to distribute, and keep restricted assessments and other people’s personal information out of the file.
Frequently asked questions
How do I know whether to multiply or divide?
Write the relationship as a fraction and orient it so the unwanted unit cancels. The resulting expression determines the arithmetic. Do not choose an operation first and add units afterward.
Does this method work for Celsius to Fahrenheit?
A simple ratio alone does not convert temperature readings between those scales because their zero points differ. Use the appropriate temperature equation, including its offset. Temperature differences are a separate case, so read the question carefully.
Should I memorize every conversion?
Learn the relationships your course expects you to know and practice locating permitted reference values. Memorizing a table without learning to arrange factors will not prepare you for unfamiliar compound-unit questions.
Why can the units be right when my answer is wrong?
You may have used an incorrect factor, selected the wrong substance, or entered the arithmetic incorrectly. Unit cancellation checks the structure of the calculation. A verified relationship and a reasonable size estimate supply additional checks.