ChemGem

How to Do Dimensional Analysis (Step-by-Step, with Practice)

Dimensional analysis (also called the factor-label method) changes units with conversion fractions. Matching units cancel. The unit left over shows whether you reached the answer asked for: 18 grams is not 18 moles.

Spot it, solve it, practice it: 5 steps, 1 worked example, 13 practice problems. Free, no account.

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The route · 5 steps

  1. Write the given amount with its unit, over 1.
  2. Name the wanted unit.
  3. Add factors with each unwanted unit underneath.
  4. Cancel matching units; multiply tops, multiply bottoms, divide.
  5. Check the final unit and the size of the answer.
Try one now See a worked example
Spot it

How to recognize dimensional analysis questions

Look for an amount in one unit, a question asking for a different unit, and an equality like 1000 mL = 1 L to bridge them.

Example wording, with the clue words marked:

A soup recipe calls for 600 mL of broth, but your measuring cup is marked in liters. How many liters do you need? (1000 mL = 1 L)

Not sure it’s this type? Compare the neighbors:

Step-by-step

Worked example: dimensional analysis

The idea

A conversion factor equals 1
1 L = 1000 mL
1 L1000 mL = 1   and   1000 mL1 L = 1
Same amount, new unit. Flip the fraction to go back.
How units cancel
250mL1×1L1000mL=0.250Lgiven, over 1wanted unit on top,old unit underneathonly L is left
Cross out unit labels, never numbers. The circled unit remains.
Reading the layout
  • Write the given amount over 1. That 1 has no unit and changes nothing.
  • Old unit underneath, wanted unit on top.
  • Multiply the tops, multiply the bottoms, then divide.
Two kinds of factor
Unit factor
1 L1000 mL
Same amount, new unit.
Reaction factor
2 mol H2O2 mol H2
A recipe from balanced coefficients. It links moles of different substances.
Cancel rules
  • Keep the substance name with each unit. mol H2 cancels only mol H2, not mol H2O.
  • Cancel factors in multiplication, not terms across + or −.
  • Celsius to kelvin adds 273.15; it is not a ratio.
Squared and cubed units: use the factor twice or three times
UnitFactorValue
Length (cm)100 cm1 m100 cm/m
Area (cm2)(100 cm1 m)2104 cm2/m2
Volume (cm3)(100 cm1 m)3106 cm3/m3
m2 means m × m. Square numbers and units.
Volume link: 1 cm3 = 1 mL exactly
10 cm10 cm10 cm1 L10 × 10 × 10 cm³= 1000 cm³= 1000 mL= 1 L
So 1 L = 1000 cm3 = 1000 mL.
Where this method returns in Chemistry I
ConversionKey factor
g → mol → atoms, molecules or formula unitsMolar mass, then Avogadro’s number
g reactant → mol → mol product → g productCoefficient ratio between two molar masses
mL → L before molarityMolarity is mol/L
Gas mass → mol before PV = nRTn is moles, not grams
J ⇄ kJ1 kJ1000 J or its flip
g → mol → phase-change heatEnthalpy is kJ/mol: reach moles first

On the periodic table: Molar masses come from the decimal number in each box: hydrogen (H) 1.008, oxygen (O) 16.00 g/mol.

Worked example · 2 steps

A water bottle holds 250 mL. How many liters is that?

Given250 mL
FindL
Factor1000 mL = 1 L
  1. 1
    Line up the factor

    mL goes underneath so it cancels.

    250 mL1 × 1 L1000 mL
  2. 2
    Cancel, multiply, divide

    Only L remains.

    250 mL1 × 1 L1000 mL = 2501000 L = 0.250 L
  3. ✓
    Answer

    0.250 L. Liters are bigger, so fewer of them.

Your turn

Dimensional analysis practice problems

Answer first, then check. Hints open one at a time; the answer and full solution open after you try.

Problem 1 of 13 · warm-up

A soup recipe calls for 600 mL of broth, but your measuring cup is marked in liters. How many liters do you need? (1000 mL = 1 L)

Hint 1 · the clue

Put mL in the denominator of your conversion factor.

Hint 2 · the next move

An amount in mL is given and liters are asked for, with 1000 mL = 1 L: that signals a one-step unit conversion.

Your path:

  1. Write the amount in mL
  2. Multiply by 1 L / 1000 mL
  3. mL cancels, leaving L
Answer

Answer: 0.600 L

Full worked solution

What you know: The recipe needs 600 mL of broth, and 1000 mL = 1 L. What you want: liters of broth.

  1. Write the input and put mL underneath in the factor so it cancels: 600 mL × (1 L / 1000 mL).
  2. mL cancels, leaving L. Divide: 600 ÷ 1000 = 0.600. The answer is 0.600 L.

Check: A liter is bigger than a milliliter, so you need fewer of them. 0.600 is smaller than 600, which makes sense.

Watch out: Multiplying by 1000 instead (600000 L) means the factor was upside down and mL would not cancel.

Answer: 0.600 L

Problem 2 of 13

You pour a 1.50 L bottle of sparkling water into a pitcher marked in milliliters. How many milliliters is that? (1000 mL = 1 L)

Hint 1 · the clue

A smaller unit needs a larger numerical count.

Hint 2 · the next move

Liters are given and milliliters are asked for: convert with the 1000 mL = 1 L factor, L underneath.

Your path:

  1. Write the amount in L
  2. Multiply by 1000 mL / 1 L
  3. L cancels, leaving mL
Answer

Answer: 1500 mL

Full worked solution

What you know: The bottle holds 1.50 L, and 1000 mL = 1 L. What you want: milliliters.

  1. Now L is the unit to remove, so L goes underneath: 1.50 L × (1000 mL / 1 L).
  2. L cancels, leaving mL. Multiply: 1.50 × 1000 = 1500. The answer is 1500 mL.

Check: A milliliter is much smaller than a liter, so the count must get bigger. 1500 is bigger than 1.50.

Watch out: This is the flipped version of the mL → L factor. Always put the unit you are getting rid of on the bottom.

Answer: 1500 mL

Problem 3 of 13

A small bag of trail mix contains 4500 mg of protein. How many grams of protein is that? (1000 mg = 1 g)

Hint 1 · the clue

Choose the factor with mg underneath.

Hint 2 · the next move

Milligrams are given and grams are asked for: one conversion factor, 1000 mg = 1 g, does the job.

Your path:

  1. Write the amount in mg
  2. Multiply by 1 g / 1000 mg
  3. mg cancels, leaving g
Answer

Answer: 4.5 g

Full worked solution

What you know: The trail mix has 4500 mg of protein, and 1000 mg = 1 g. What you want: grams of protein.

  1. Put mg underneath so it cancels: 4500 mg × (1 g / 1000 mg).
  2. mg cancels, leaving g. Divide: 4500 ÷ 1000 = 4.5. The answer is 4.5 g.

Check: A gram is bigger than a milligram, so the number shrinks. 4.5 × 1000 = 4500 takes you back to the start.

Watch out: The "milli" prefix means one thousandth. Going from mg to g, divide by 1000; do not multiply.

Answer: 4.5 g

Problem 4 of 13

A lab’s inventory software records a bag of coffee beans as 250000 mg. What is its mass in kilograms? (1000 mg = 1 g, 1000 g = 1 kg)

Hint 1 · the clue

Use two bridges: mg → g → kg.

Hint 2 · the next move

Milligrams in, kilograms out, and two equalities given: no single factor works, so chain two factors through grams.

Your path:

  1. Write the amount in mg
  2. Multiply by 1 g / 1000 mg
  3. Then multiply by 1 kg / 1000 g
  4. mg and g cancel, leaving kg
Answer

Answer: 0.250 kg

Full worked solution

What you know: The bag of beans is 250000 mg, 1000 mg = 1 g, and 1000 g = 1 kg. What you want: kilograms.

  1. No single factor goes from mg to kg, so use two bridges: mg → g → kg.
  2. Set up both factors with the old unit underneath: 250000 mg × (1 g / 1000 mg) × (1 kg / 1000 g).
  3. mg cancels, then g cancels, leaving kg. Multiply the bottoms: 1000 × 1000 = 1000000. Divide: 250000 ÷ 1000000 = 0.250. The answer is 0.250 kg.

Check: A kilogram is a million milligrams, so the number should shrink a lot. 0.250 × 1000000 = 250000.

Watch out: Stopping after the first factor gives 250 g, which is the wrong unit.

Answer: 0.250 kg

Problem 5 of 13

A chocolate bar has a mass of 0.080 kg. Your lab sheet records masses in milligrams. What is the bar’s mass in milligrams? (1000 g = 1 kg, 1000 mg = 1 g)

Hint 1 · the clue

Reverse both factors to leave mg.

Hint 2 · the next move

Kilograms in, milligrams out, with two equalities: bridge kg → g → mg using two factors.

Your path:

  1. Write the amount in kg
  2. Multiply by 1000 g / 1 kg
  3. Then multiply by 1000 mg / 1 g
  4. kg and g cancel, leaving mg
Answer

Answer: 80000 mg

Full worked solution

What you know: The chocolate bar is 0.080 kg, 1000 g = 1 kg, and 1000 mg = 1 g. What you want: milligrams.

  1. Bridge through grams: kg → g → mg. Each factor has the old unit underneath.
  2. 0.080 kg × (1000 g / 1 kg) × (1000 mg / 1 g). kg cancels, then g cancels, leaving mg.
  3. Multiply: 0.080 × 1000 × 1000 = 80000. The answer is 80000 mg.

Check: Milligrams are a million times smaller than kilograms, so the number grows a lot. 80000 ÷ 1000000 = 0.080.

Watch out: 0.080 has two significant figures, so 8.0 × 104 mg shows the precision more clearly than 80000.

Answer: 80000 mg

Problem 6 of 13

A movie runs 90 min. How long is it in hours? (60 min = 1 h)

Hint 1 · the clue

The minute label must cancel.

Hint 2 · the next move

Minutes are given and hours are asked for, with 60 min = 1 h: a one-step conversion.

Your path:

  1. Write the time in min
  2. Multiply by 1 h / 60 min
  3. min cancels, leaving h
Answer

Answer: 1.5 h

Full worked solution

What you know: The movie runs 90 min, and 60 min = 1 h. What you want: hours.

  1. Put min underneath so it cancels: 90 min × (1 h / 60 min).
  2. min cancels, leaving h. Divide: 90 ÷ 60 = 1.5. The answer is 1.5 h.

Check: An hour is longer than a minute, so the number shrinks. 1.5 h × 60 min/h = 90 min.

Watch out: 1.5 h means one and a half hours, not 1 hour 50 minutes. Time is not base 10.

Answer: 1.5 h

Problem 7 of 13

An experiment must stir for 2.0 hours, and the stir plate’s timer is set in seconds. How many seconds should you enter? (60 min = 1 h, 60 s = 1 min)

Hint 1 · the clue

Start with hours; bridge through minutes.

Hint 2 · the next move

Hours in, seconds out, and two time equalities given: chain factors h → min → s.

Your path:

  1. Write the time in h
  2. Multiply by 60 min / 1 h
  3. Then multiply by 60 s / 1 min
  4. h and min cancel, leaving s
Answer

Answer: 7200 s

Full worked solution

What you know: The experiment stirs for 2.0 h, 60 min = 1 h, and 60 s = 1 min. What you want: seconds.

  1. No factor goes straight from h to s, so bridge through minutes: h → min → s.
  2. 2.0 h × (60 min / 1 h) × (60 s / 1 min). h cancels, then min cancels, leaving s.
  3. Multiply: 2.0 × 60 × 60 = 7200. The answer is 7200 s.

Check: One hour is 3600 s, so two hours should be twice that: 2 × 3600 = 7200.

Watch out: Using only one factor of 60 gives 120, which is minutes, not seconds.

Answer: 7200 s

Problem 8 of 13

Your friend finished a 5.0-mile run, and her race app shows distances in kilometers. How many kilometers did she run? (1 mile ≈ 1.609 km)

Hint 1 · the clue

Keep km above and miles below.

Hint 2 · the next move

Miles are given and kilometers are asked for, with 1 mile ≈ 1.609 km: one factor, miles underneath.

Your path:

  1. Write the distance in miles
  2. Multiply by 1.609 km / 1 mile
  3. miles cancels; round to sig figs
Answer

Answer: 8.0 km

Full worked solution

What you know: Your friend ran 5.0 miles, and 1 mile ≈ 1.609 km. What you want: kilometers.

  1. Put miles underneath so it cancels: 5.0 miles × (1.609 km / 1 mile).
  2. miles cancels, leaving km. Multiply: 5.0 × 1.609 = 8.045.
  3. Round: this is multiplication, and 5.0 has two significant figures, the fewest among the measured inputs. So 8.045 becomes 8.0 km.

Check: A kilometer is shorter than a mile, so you need more of them. 8.0 is bigger than 5.0.

Answer: 8.0 km

Problem 9 of 13

Your car’s cruise control is set at 50 miles/h. A friend’s phone app shows speed in km/h. About what speed does the app show? (1 mile ≈ 1.609 km)

Hint 1 · the clue

Do not change hours when the requested unit still uses hours.

Hint 2 · the next move

A speed in miles/h must become km/h: only the distance unit changes, so convert miles and leave hours alone.

Your path:

  1. Write the speed in miles/h
  2. Multiply by 1.609 km / 1 mile
  3. miles cancels; h stays on bottom
Answer

Answer: About 80 km/h

Full worked solution

What you know: The cruise control is set at 50 miles/h, and 1 mile ≈ 1.609 km. What you want: km/h.

  1. Only the distance unit changes. Hours is already in the wanted unit, so leave it alone.
  2. Put miles underneath: 50 miles/h × (1.609 km / 1 mile). miles cancels, leaving km/h.
  3. Multiply: 50 × 1.609 = 80.45. The answer is about 80 km/h.

Check: km numbers are about 1.6 times mile numbers. 50 × 1.6 = 80.

Watch out: Do not convert hours to minutes or seconds. The wanted unit still has h on the bottom.

Answer: About 80 km/h

Problem 10 of 13

A car is moving at 72 km/h. What is that speed in meters per second (m/s)? (1000 m = 1 km, 3600 s = 1 h)

Hint 1 · the clue

Hours starts underneath, so place hours above in its conversion factor.

Hint 2 · the next move

km/h must become m/s: two units change, so use one factor for the top unit and one for the bottom.

Your path:

  1. Write the speed in km/h
  2. Multiply by 1000 m / 1 km
  3. Multiply by 1 h / 3600 s
  4. km and h cancel, leaving m/s
Answer

Answer: 20 m/s

Full worked solution

What you know: The car moves at 72 km/h, 1000 m = 1 km, and 3600 s = 1 h. What you want: m/s.

  1. Two units must change. km is on top, so its factor puts km underneath: (1000 m / 1 km).
  2. h is on the bottom, so its factor puts h on top: (1 h / 3600 s).
  3. 72 km/h × (1000 m / 1 km) × (1 h / 3600 s). km and h cancel, leaving m/s. Calculate: 72 × 1000 = 72000, then 72000 ÷ 3600 = 20. The answer is 20 m/s.

Check: Reverse it: 20 m/s × 3600 s/h = 72000 m/h = 72 km/h.

Watch out: Writing (3600 s / 1 h) leaves h2 on the bottom. A unit on the bottom needs its factor flipped.

Answer: 20 m/s

Problem 11 of 13

A floor tile has an area of 225 cm2. What is its area in m2? (1 m = 100 cm)

Hint 1 · the clue

cm2 is cm × cm, so the factor (1 m / 100 cm) has to be squared. Put cm2 underneath.

Hint 2 · the next move

An area in cm2 must become m2, but only a length factor is given: square the factor before using it.

Your path:

  1. Square the factor: 1 m2 = 10,000 cm2
  2. Multiply by 1 m2 / 10,000 cm2
  3. cm2 cancels, leaving m2
Answer

Answer: 0.0225 m2

Full worked solution

What you know: The tile’s area is 225 cm2, and 1 m = 100 cm. What you want: area in m2.

  1. Square the length factor: (1 m / 100 cm)2 = 1 m2 / 10,000 cm2.
  2. Put cm2 underneath so it cancels: 225 cm2 × (1 m2 / 10,000 cm2).
  3. cm2 cancels, leaving m2. Divide: 225 ÷ 10,000 = 0.0225. The answer is 0.0225 m2.

Check: A square meter is much bigger than a square centimeter, so the number should get much smaller. Reverse it: 0.0225 m2 × 10,000 cm2/m2 = 225 cm2.

Watch out: Dividing by 100 only once gives 2.25 m2, which is 100 times too big. A tile is not over 2 m2.

Answer: 0.0225 m2

Problem 12 of 13

A soda can holds 0.355 L. What is its volume in mL and in cm3? (1 L = 1000 mL, 1 mL = 1 cm3)

Hint 1 · the clue

Change L to mL first, with L underneath. Then use 1 mL = 1 cm3.

Hint 2 · the next move

Liters are given and mL and cm3 are asked for: convert L to mL, then use 1 mL = 1 cm3.

Your path:

  1. Multiply L by 1000 mL / 1 L
  2. L cancels, leaving mL
  3. 1 mL = 1 cm3, so same number
Answer

Answer: 355 mL = 355 cm3

Full worked solution

What you know: The can holds 0.355 L, 1 L = 1000 mL, and 1 mL = 1 cm3. What you want: the volume in mL and in cm3.

  1. Put L underneath so it cancels: 0.355 L × (1000 mL / 1 L).
  2. L cancels, leaving mL. Multiply: 0.355 × 1000 = 355 mL.
  3. 1 mL = 1 cm3, so the number does not change: 355 mL × (1 cm3 / 1 mL) = 355 cm3. The answer is 355 mL = 355 cm3.

Check: A milliliter is smaller than a liter, so the number should get bigger. 355 cm3 ÷ 1000 cm3/L = 0.355 L.

Watch out: The 1000 in 1 L = 1000 cm3 comes from cubing 10 cm (10 × 10 × 10). Do not cube it again; mL and cm3 are the same size.

Answer: 355 mL = 355 cm3

Problem 13 of 13

Convert 750 mL to L using 1000 mL = 1 L.

Hint 1 · the clue

Put mL on the bottom of the conversion factor so it cancels. A liter is bigger than a milliliter, so expect a smaller number.

Hint 2 · the next move

Convert mL to L with the factor given: put mL underneath so it cancels.

Your path:

  1. Write the amount in mL
  2. Multiply by 1 L / 1000 mL
  3. mL cancels, leaving L
Answer

750 × 1/1000 = 0.750 L; mL cancels.

Full worked solution

What you know: 750 mL, and 1000 mL = 1 L. What you want: liters.

  1. Put the unit you are removing, mL, underneath: 750 mL × (1 L / 1000 mL).
  2. mL cancels, leaving L. Divide: 750 ÷ 1000 = 0.750. The answer is 0.750 L.

Check: A liter is bigger than a milliliter, so the number should be smaller. 0.750 L × 1000 mL/L = 750 mL.

Watch out: If you multiply by 1000, the factor is upside down and mL does not cancel.

Answer: 750 × 1/1000 = 0.750 L; mL cancels.

Watch out

Common dimensional analysis mistakes