01 Units are just algebra
In algebra, when the same thing sits on the top and the bottom of a fraction, it cancels:
Units do the exact same thing. If cm is on top and cm is on the bottom, they cancel — the same way an x cancels an x.
Your workshop says it directly: "one may treat the unit like an algebraic variable." That's the entire method. You're not memorizing formulas — you're stacking fractions so the unit you don't want cancels itself out, leaving the unit you do want.
02 "To" on top, "from" on the bottom
Here is the whole method in one sentence. When you write a conversion factor as a fraction:
The unit you are converting TO goes on the top.
The unit you are converting FROM goes on the bottom.
Because your starting amount already carries the "from" unit on top, putting "from" on the bottom of the factor makes it cancel automatically — and the "to" unit is what's left standing. You never have to guess which way up the fraction goes.
Name your FROM and TO
What unit do you have (from)? What unit do you want (to)? Write the starting amount — it carries the "from" unit.
Build the factor
Write the conversion fraction with the to unit on top and the from unit on the bottom.
Cancel & solve
The "from" units cancel top-and-bottom. Multiply by the top number, divide by the bottom number.
The rule decides for you. Put to on top and from on the bottom, and the setup is automatically correct. Multiplying by the top and dividing by the bottom just falls out of the fraction.
If the unit didn't cancel, the factor went in upside down — you have from on top by mistake. Flip it so from sits on the bottom, underneath your starting amount.
03 The 1000-staircase
Your workshop's prefix table looks like seven things to memorize. It isn't — it's one staircase plus two in-between steps. Five of the seven prefixes sit exactly 1000 apart, in a row:
Then just two extras squeeze between the base unit and milli — these are the only ones that break the 1000 pattern:
Milliliters are smaller pieces, so you need more of them → the number should get bigger. 0.653 → 653. ✔ If your answer shrank instead, you built the factor upside down.
04 Chaining & powered units
No single conversion factor? Add another fraction — one per step. Each one follows the same rule, so each unit cancels the next until only your target survives. Your workshop calls this doing it "serially."
kg cancels kg, then g cancels g. The only unit left standing is lb — proof the chain is built correctly.
cm² means cm × cm, so the conversion happens twice. cm³ means cm × cm × cm, so it happens three times — and you square or cube the number too, not just the unit.
98.4 in² × (2.54 cm / 1.00 in)² = 98.4 × 6.4516 = 635 cm²
26.2 cm³ × (10 mm / 1 cm)³ = 26.2 × 1000 = 2.62 × 10⁴ mm³
05 Exact vs. measured factors
This is the part that costs points in Lab 1, and almost nobody teaches it. Not all conversion factors limit your significant figures. There are two kinds:
| Kind of factor | Examples | Effect on sig figs |
|---|---|---|
| Exact — a definition, infinitely precise | 1 kg = 1000 g 100 cm = 1 m 1 cm³ = 1 mL 10 mm = 1 cm | None. Never limits your answer. Sig figs come only from the measured value. |
| Measured / rounded — a lab value someone determined | 454 g = 1.00 lb (3 sf) 29.5 mL = 1.00 fl oz (3 sf) 55.85 g/mol (4 sf) 6.02 × 10²³ (3 sf) | Yes. Counts like any other measurement — it can limit your answer. |
2348 mL → L. The factor (1 L = 1000 mL) is exact, so it doesn't limit anything. 2348 has 4 sig figs, so the answer keeps 4: 2.348 L — not 2.35 L.
36.8 lb → g. Here the factor (454 g) is measured with 3 sig figs, and 36.8 also has 3, so the answer gets 3: 1.67 × 10⁴ g — not 16,707 g.
985 in × 2.54 cm/in = 2501.9, which rounds to 3 sig figs. Writing "2500 cm" is ambiguous — a reader can't tell if those zeros count. Write 2.50 × 10³ cm. Scientific notation is the only way to show trailing zeros that are significant.
06 The three power-ups
Density, the periodic table, and Avogadro's number are all just conversion factors in disguise. Each one links two quantities, so each one can be written either way up — and you pick the way that cancels what you're holding.
Your workshop insists on this, and it matters: write g K and mol K, not just "g" and "mol." Every element has a different molar mass, so "grams of K" and "grams of Fe" are genuinely different units that must not cancel each other.
07 The Triangle Toolkit
Every relationship in Workshop 1 with three quantities fits in a triangle. Cover the one you want, and the triangle hands you the formula: letters side-by-side = multiply, letters stacked = divide.
Going atoms → moles means dividing by 6.02 × 10²³, and the powers of ten scare people. They shouldn't. Rewrite your number so it also carries ×10²³ — then the 10²³ on top and bottom cancels exactly like a unit, and you just divide the front numbers.
3.05 × 10²⁰ → 0.305 × 10²¹
3.05 × 10²⁰ → 30.5 × 10¹⁹
Memory hook: “Left = Larger exponent.”
Why it works: sliding the decimal one place left divides the front number by 10, so the exponent must go up by 1 to keep the value the same. You're not changing the number — only how it's dressed.
The triangle is fast recall for a single three-quantity relationship. The railroad tracks win whenever you chain steps or cross units (g → mol → atoms, or in² → cm²) — a triangle holds only three boxes, but the tracks take as many fractions as you need. Triangle for speed, tracks for everything.
08 Tricks & memory hooks
“To” on top, “from” on the bottom
The one rule. The "from" unit cancels automatically, so you never guess multiply vs. divide.
Unit didn't cancel? Flip it.
A wrong leftover unit means the fraction went in upside down. Flip that one factor and run it again.
The 1000-staircase
M · k · base · m · mc · n are each ×1000 apart. Only deci and centi break the pattern, and they live between base and milli.
Smaller unit = bigger number
Converting to a smaller unit? The count grows. To a bigger unit? It shrinks. Catches flipped factors instantly.
Cover it with your thumb
Density, moles, atoms: draw the triangle, cover what you want. Side-by-side = ×, stacked = ÷.
“Left = Larger” for exponents
Decimal left raises the power of ten; right lowers it. Use it to match 10²³ so Avogadro's number cancels.
Exact factors don't cost sig figs
Prefix conversions (1 kg = 1000 g) are definitions — infinitely precise. Only measured factors (454 g/lb) can limit your answer.
Squared once, cubed three times
cm² converts twice, cm³ converts three times — and you square or cube the number, not just the unit.
09 Every factor you need
| Physical property | Metric | English | Conversion factor |
|---|---|---|---|
| Mass | gram (g) | pound (lb) | 454 g = 1.00 lb |
| Volume | mL | fluid ounce | 29.5 mL = 1.00 fl oz |
| Length | cm | inch (in) | 2.54 cm = 1.00 in |
| Volume (exact) | cm³ | mL | 1 cm³ = 1 mL |
| Al | Fe | Cu | Co | Au | K | Na | He |
|---|---|---|---|---|---|---|---|
| 26.98 | 55.85 | 63.55 | 58.93 | 196.97 | 39.10 | 22.99 | 4.003 |
A one-page PDF of this entire cheat sheet — factors, the prefix staircase, all four triangles, and the decimal trick — is provided alongside this guide.
10 Verified answer key
Every practice problem from Workshop 1, with the setup written the way you should write it. Answers use your workshop's own factors (454 g/lb, 29.5 mL/fl oz, 2.54 cm/in) and correct significant figures.
| # | Problem | Setup — "to" on top, "from" on bottom | Answer |
|---|---|---|---|
| Metric ↔ English conversions | |||
| 1 | 36.8 lb → g | 36.8 lb × (454 g / 1.00 lb) | 1.67 × 10⁴ g |
| 2 | 956 g → lb | 956 g × (1.00 lb / 454 g) | 2.11 lb |
| 3 | 156.0 fl oz → mL | 156.0 fl oz × (29.5 mL / 1.00 fl oz) | 4.60 × 10³ mL |
| 4 | 356.0 mL → fl oz | 356.0 mL × (1.00 fl oz / 29.5 mL) | 12.1 fl oz |
| 5 | 16.2 cm → in | 16.2 cm × (1.00 in / 2.54 cm) | 6.38 in |
| 6 | 985 in → cm | 985 in × (2.54 cm / 1.00 in) | 2.50 × 10³ cm |
| 7 | 1450 g → lb | 1450 g × (1.00 lb / 454 g) | 3.19 lb |
| Metric prefixes — factors are exact, so sig figs come from the measurement | |||
| 8 | 2348 mL → L | 2348 mL × (1 L / 1000 mL) | 2.348 L |
| 9 | 55.6 cm → m | 55.6 cm × (1 m / 100 cm) | 0.556 m |
| 10 | 895 g → kg | 895 g × (1 kg / 1000 g) | 0.895 kg |
| 11 | 0.000056 g → mcg | 0.000056 g × (1,000,000 mcg / 1 g) | 56 mcg |
| 12 | 0.0296 L → mL | 0.0296 L × (1000 mL / 1 L) | 29.6 mL |
| Multi-step chains & powered units | |||
| 13 | 63.5 lb → kg | 63.5 lb × (454 g / 1.00 lb) × (1 kg / 1000 g) | 28.8 kg |
| 14 | 32.0 fl oz → L | 32.0 fl oz × (29.5 mL / 1.00 fl oz) × (1 L / 1000 mL) | 0.944 L |
| 15 | 26.2 cm³ → mm³ | 26.2 cm³ × (10 mm / 1 cm)³ → ×1000 | 2.62 × 10⁴ mm³ |
| 16 | 568 mcg → mg | 568 mcg × (1 g / 10⁶ mcg) × (1000 mg / 1 g) | 0.568 mg |
| Density — Triangle 2 | |||
| 17 | d = 10.6 g/cm³, V = 25 cm³ → mass | 25 cm³ × (10.6 g / 1 cm³) · cover m → m = d × V | 265 g 2.7 × 10² g at 2 s.f. |
| 18 | m = 15.6 g, d = 10.6 g/cm³ → volume | 15.6 g × (1 cm³ / 10.6 g) · cover V → V = m ÷ d | 1.47 cm³ |
| Molar mass — Triangle 3 | |||
| 19 | 18.3 g Fe → mol | 18.3 g Fe × (1 mol Fe / 55.85 g Fe) | 0.328 mol |
| 20 | 26.9 g Cu → mol | 26.9 g Cu × (1 mol Cu / 63.55 g Cu) | 0.423 mol |
| 21 | 0.698 mol Co → g | 0.698 mol Co × (58.93 g Co / 1 mol Co) | 41.1 g |
| 22 | 25.9 mol Au → g | 25.9 mol Au × (196.97 g Au / 1 mol Au) | 5.10 × 10³ g |
| Avogadro's number — Triangle 4 | |||
| 23 | 0.250 mol Na → atoms | 0.250 mol × (6.02 × 10²³ atoms / 1 mol) | 1.51 × 10²³ atoms |
| 24 | 3.05 × 10²⁰ atoms He → mol | 3.05 × 10²⁰ atoms × (1 mol / 6.02 × 10²³ atoms) | 5.07 × 10⁻⁴ mol |
| 25 | 56.3 g K → atoms | 56.3 g K × (1 mol / 39.10 g K) × (6.02 × 10²³ atoms / 1 mol) | 8.67 × 10²³ atoms |
10.6 × 25 = 265 exactly. If you treat "25 cm³" as two significant figures, the reported answer is 2.7 × 10² g. If your instructor treats 25 as an exact count, 265 g stands. Ask which convention applies — this is the single most common place students and graders disagree.
11 The 5 things you must know
- Units are variables — the same unit on top and bottom cancels.
- "To" on top, "from" on the bottom — the "from" unit cancels, so you never guess multiply or divide.
- The leftover unit is your answer's unit — and your built-in error check.
- Three quantities? Use a triangle — cover what you want; conversions, density, moles, and atoms all fit.
- Exact factors don't limit sig figs — prefixes are definitions; measured factors like 454 g/lb do limit.
"Converting TO goes on top, converting FROM goes on the bottom. The 'from' cancels — whatever's left is your answer."