Skills Beta

Units and Dimensional Analysis

Every measurement is a number with a unit.

Practice 5: Mathematical Routines

Question set for this topic

Part 1 · Hook

Why this matters

In 1999 a space probe sent to Mars was lost because one team worked in pound-force seconds and another in newton seconds. Nobody converted, and the probe flew too close to the planet. In chemistry, every number is a number times a unit, and the unit carries half the meaning.

Part 2 · See it

See it first

3.5 hours times 60 minutes per hour times 60 seconds per minute equals 12,600 seconds; hours cancel, then minutes cancel, leaving seconds.
Dimensional analysis: each conversion factor puts the unit you have on the bottom, so it cancels, until only the unit you want is left. LevlPrep original diagram.

Part 3 · Step by step

How it works, step by step

  1. A measurement is a number and a unit togetherchanging the unit must change the number so the quantity stays the same
  2. A conversion factor is a ratio of two equal quantities, such as 1000 g / 1 kgmultiplying by it is multiplying by 1, so the quantity does not change
  3. Units cancel like numbers when the same unit is on the top and the bottomchoosing which way up each factor goes decides which units are left
  4. When only the wanted unit is leftthe setup is right, and the arithmetic gives the answer in that unit

Part 4 · Key ideas

Key ideas

  • Every measured number needs its unit; "12.6" means nothing until it is 12.6 g or 12.6 mL.
  • Metric prefixes scale a unit: kilo (k) = 1000, centi (c) = 1/100, milli (m) = 1/1000, micro (µ) = 1/1,000,000, nano (n) = 1/1,000,000,000.
  • Dimensional analysis: multiply by conversion factors, each with the unit you have on the bottom, until only the unit you want is left.
  • Density = mass ÷ volume (g/mL or g/cm³), and works as a conversion factor between mass and volume. Kelvin = °C + 273.15.

Part 5 · Misconception

A common mistake

The wrong idea: To change kilometers to meters you divide by 1000, because a kilometer is the bigger unit.

What actually happens: A smaller unit needs a larger number to describe the same distance. 2.45 km × (1000 m / 1 km) = 2450 m. Let the units decide: the factor that cancels km is the right one.

Part 6 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

1. You want to change a volume from milliliters to liters. Which conversion factor do you multiply by?

  1. 1000 mL / 1 L
  2. 1 L / 100 mL
  3. 1 L / 1000 mL
  4. 1000 L / 1 mL
Show the answer

The unit you want goes on top and the unit you have goes on the bottom: 1 L / 1000 mL.

  • 1000 mL / 1 L: This factor is upside down: mL would end up squared instead of canceling.
  • 1 L / 100 mL: Milli means one thousandth, not one hundredth; 100 belongs to centi.
  • Correct: 1 L / 1000 mL: Right: liters on top, milliliters on the bottom, so mL cancels and L is left.
  • 1000 L / 1 mL: A liter is larger than a milliliter, so 1000 L can never equal 1 mL.

2. A hiking trail is 2.45 km long. How long is it in meters?

Type a number in m.

Show the answer

2.45 km × (1000 m / 1 km) = 2450 m. Kilo means 1000, so the number gets 1000 times larger as the unit gets 1000 times smaller.

  • Answer: 2450 m

3. A metal block has a mass of 12.6 g and a volume of 4.65 cm³. What is its density? Include the unit.

Type a number and its unit.

Show the answer

Density = mass ÷ volume = 12.6 g ÷ 4.65 cm³ = 2.71 g/cm³. The unit comes straight from the division: grams per cubic centimeter.

  • Answer: 2.71 g/cm³

4. Normal body temperature is 37.0 °C. What is it in kelvin?

Type a number in K.

Show the answer

K = °C + 273.15 = 37.0 + 273.15 = 310.2 K. A kelvin is the same size as a Celsius degree; the scale just starts at absolute zero.

  • Answer: 310.2 K

5. Which setup converts 3.5 hours into seconds?

  1. 3.5 h × (1 h / 60 min) × (1 min / 60 s)
  2. 3.5 h × (60 min / 1 h) × (1 min / 60 s)
  3. 3.5 h × (60 s / 1 h)
  4. 3.5 h × (60 min / 1 h) × (60 s / 1 min)
Show the answer

Chain one factor per step, each with the unit you have on the bottom: h → min → s. 3.5 × 60 × 60 = 12,600 s.

  • 3.5 h × (1 h / 60 min) × (1 min / 60 s): Both factors are upside down: the units multiply into h²/s instead of canceling.
  • 3.5 h × (60 min / 1 h) × (1 min / 60 s): The first factor works, but the second puts seconds on the bottom, giving min²/s.
  • 3.5 h × (60 s / 1 h): There are 3600 seconds in an hour, not 60, so this answer is 60 times too small.
  • Correct: 3.5 h × (60 min / 1 h) × (60 s / 1 min): Right: hours cancel, then minutes cancel, and seconds are left.

6. A student converts 4250 mg to kilograms and reports 4,250,000,000 kg. What went wrong?

  1. The student used 100 mg per gram in place of 1000 mg per gram
  2. Milligrams and kilograms measure different things
  3. Both factors were upside down, so the number grew instead of shrinking
  4. The student should have added 1000 instead of multiplying
Show the answer

The right answer is 4250 mg × (1 g / 1000 mg) × (1 kg / 1000 g) = 0.00425 kg. A kilogram is a million times larger than a milligram, so the number must shrink.

  • The student used 100 mg per gram in place of 1000 mg per gram: A wrong 100 would give a different error; the reported answer comes from correct factors of 1000 used upside down.
  • Milligrams and kilograms measure different things: Both measure mass, and any mass unit can be converted to any other with the right factors.
  • Correct: Both factors were upside down, so the number grew instead of shrinking: Right: mg → g and g → kg each make the number smaller. Flipping both makes it a million times larger instead of smaller.
  • The student should have added 1000 instead of multiplying: Unit conversions multiply or divide by a factor; adding a number changes the quantity itself.

7. Water has a density of 1.00 g/mL. A 30.0 g plastic bead takes up 25.0 mL. Will the bead float or sink in water, and why?

  1. Float: its density, 0.833 g/mL, is less than that of water
  2. Sink: its density, 1.20 g/mL, is greater than that of water
  3. Sink: its mass, 30.0 g, is greater than 1.00 g
  4. Float: its volume, 25.0 mL, is greater than 1.00 mL
Show the answer

Density = 30.0 g ÷ 25.0 mL = 1.20 g/mL. An object denser than water sinks.

  • Float: its density, 0.833 g/mL, is less than that of water: This divides volume by mass. Density is mass ÷ volume.
  • Correct: Sink: its density, 1.20 g/mL, is greater than that of water: Right: 30.0 g ÷ 25.0 mL = 1.20 g/mL, which is more than 1.00 g/mL.
  • Sink: its mass, 30.0 g, is greater than 1.00 g: Floating depends on mass per volume, not on mass alone; a heavy wooden beam still floats.
  • Float: its volume, 25.0 mL, is greater than 1.00 mL: A large volume alone does not decide it; you need mass divided by volume.

Part 7 · Summary

Summary

Every measurement is a number with a unit. Metric prefixes scale units by factors of ten, and conversion factors (ratios equal to 1) change the unit without changing the quantity. Chain factors so that units cancel; density converts between mass and volume, and K = °C + 273.15.

Part 8 · Up next

What comes next

Part 9 · Connections

Connections