Units Carried Through Every Line of a Calculation
Why this matters
The expensive math mistakes in the trades are almost never arithmetic. Nobody multiplies 30 by 40 and gets 1,300. What happens is that 4 inches goes into a formula as if it were 4 feet, or cubic feet get divided by 9 because 9 is the number that lives next to "yards" in someone's memory. The arithmetic is perfect and the answer is off by a factor of 3, 12 or 27, and a concrete truck shows up with the wrong load.
Both of those slips are invisible in a calculation written as bare numbers. Both are impossible to miss in a calculation where every number carries its unit and the units are cancelled on paper. This card owns that technique: write the unit next to every value, treat units like algebra, and let a leftover unit tell you the setup is wrong before the order goes in.
Units are algebra
The core idea: a unit behaves exactly like a variable. Feet times feet is square feet, the same way x times x is x squared. Square feet divided by feet is feet. A unit that appears on the top and bottom of a fraction cancels, the same way x/x cancels.
That gives you a free check on every calculation. Before doing any arithmetic, write the units of the setup and simplify them. If what is left is the unit you wanted, the setup is at least dimensionally right. If what is left is anything else, the setup is wrong, and no amount of careful multiplication will fix it.
A conversion factor is a fraction worth exactly 1. "27 cubic feet per cubic yard" is the fraction (1 cu yd / 27 cu ft), and since 1 cu yd and 27 cu ft are the same amount of concrete, the fraction equals 1. Multiplying by 1 never changes a quantity; it only changes the unit it is written in. The whole skill is choosing which way up to write the fraction, and the units decide that for you: put the unit you want to get rid of on the opposite side from where it currently sits.
The worked example: a 30 by 40 slab, 4 inches thick
Here is the slab calculation laid out with the units carried, then the two common slips laid out the same way so you can see what the units do to them.
SLAB 30 ft x 40 ft x 4 in
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GIVENS L = 30 ft W = 40 ft T = 4 in
STEP 1 thickness to feet
4 in x (1 ft / 12 in) = 0.3333 ft
in cancels -> ft
STEP 2 volume
30 ft x 40 ft x 0.3333 ft = 400 cu ft
ft x ft x ft -> cu ft
STEP 3 cubic feet to cubic yards
400 cu ft x (1 cu yd / 27 cu ft) = 14.81 cu yd
cu ft cancels -> cu yd
ANSWER 14.81 cu yd, about 14.8 yd before waste
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SLIP A divided by 9
400 cu ft x (1 sq yd / 9 sq ft) = 44.4 ???
cu ft / sq ft leaves ft x sq yd -> not a volume
SLIP B inches used as feet
30 ft x 40 ft x 4 in = 4800 ft x ft x in
mixed units left over -> stop, convert first
Step 1 converts before anything else. The thickness is the only given in a different unit, so it is converted first, on its own line. Writing (1 ft / 12 in) with inches on the bottom is what makes the inches cancel. Writing it the other way up, (12 in / 1 ft), would leave inches squared over feet, which is nonsense, and that is your signal to flip it.
Step 2 produces the unit you expected. Feet times feet times feet is cubic feet, which is a volume. The result, 400 cu ft, can be checked by a second route: 30 x 40 = 1,200 sq ft of slab, and a third of a foot deep is a third of 1,200, which is 400.
Step 3 uses 27, and the units say why. A cubic yard is a cube 3 ft on each side, so 3 x 3 x 3 = 27 cubic feet. The factor (1 cu yd / 27 cu ft) has cubic feet on the bottom, so it cancels the cubic feet from step 2 and leaves cubic yards. 400 / 27 = 14.81.
Slip A shows itself. The 9 belongs to square yards (3 ft x 3 ft). Writing the factor with its real units, (1 sq yd / 9 sq ft), and cancelling against cubic feet leaves feet times square yards. That is not a volume, so the setup is wrong. The bare-number version, 400 / 9 = 44.4, looks like a perfectly good yardage and is three times the real order.
Slip B shows itself. Multiplying feet by feet by inches leaves a mixed unit that is not cubic anything. The bare-number version, 30 x 40 x 4 = 4,800, divided by 27 gives 177.8 yards, which is twelve times the real order. Nobody orders 178 yards for a 30 by 40 slab on purpose, but people have phoned in orders off a calculator without looking.
What the order line says. 14.81 cu yd is the neat volume, the volume of a perfect slab on a perfect subgrade. The order adds an allowance for subgrade variation, form bulge and spillage, then rounds up to whatever increment the supplier sells. That allowance is a separate step; see 'The Materials Waste Factor Worth Measuring'. Do not fold it into the volume line, because then nobody can tell which part was geometry and which was allowance.
Squares and cubes of a conversion
The single most common unit error after the two above: converting an area or a volume with the linear factor.
- 1 ft = 12 in, so 1 sq ft = 12 x 12 = 144 sq in, and 1 cu ft = 12 x 12 x 12 = 1,728 cu in.
- 1 yd = 3 ft, so 1 sq yd = 9 sq ft, and 1 cu yd = 27 cu ft.
- 1 m is about 3.281 ft, so 1 sq m is about 10.76 sq ft, and 1 cu m is about 35.31 cu ft.
The units force this if you carry them. To turn 720 sq in into square feet, the factor is (1 ft / 12 in), and it has to appear twice to cancel in-squared:
720 sq in x (1 ft / 12 in) x (1 ft / 12 in) = 720 / 144 = 5.0 sq ft.
Using it once leaves in x ft, which is not an area, and gives the bare-number answer 60, twelve times too big. If you see one factor of 12 cancelling a squared unit, the units are telling you to write it again.
A practical case. A return-air grille opening measures 20 in by 36 in, 720 sq in. Its gross opening in square feet is 5.0, not 60 (the free area, from the grille's data, is smaller still), and the airflow math that follows depends on getting this line right. The units are what make the 144 appear without having to remember it.
Conversion chains: one line, every unit cancelled
Real calculations often need several conversions in a row. Write them as one chain, each factor a fraction equal to 1, and cancel as you go. Here is a coverage problem worked as a chain.
A sealer label gives a coverage rate the tech reads as 250 sq ft per gallon (illustrative; your label governs). The slab above is 1,200 sq ft. How many gallons?
1,200 sq ft x (1 gal / 250 sq ft) = 4.8 gal.
The coverage rate is written with square feet on the bottom so the square feet cancel. If you write it the other way up, 1,200 sq ft x 250 sq ft / gal, you get 300,000 sq ft squared per gallon, which is obviously not gallons. The flip is impossible to miss when the units are on the page; it is very easy to miss as "1,200 times 250" on a calculator.
Longer chains. Converting a flow in gallons per minute to cubic feet per hour is two factors in one line:
8 gal/min x (60 min / 1 hr) x (1 cu ft / 7.48 gal) = 64.2 cu ft/hr.
Minutes cancel against minutes, gallons against gallons, and cubic feet per hour is left. The numbers do not matter to the method; the cancellation does. See related: The Water Constants 7.48, 8.34 and 231 for that 7.48 and its neighbors.
Rule for chains. One factor per line on paper if the chain is long, units written in full, and a final check that exactly the units you wanted are left standing. If anything else survives, a factor is upside down or missing.
The leftover-unit check, as a habit
Before you press equals, do three things:
- Write the units of every value in the setup, including the conversion factors.
- Cancel them. Strike through every matched pair, top and bottom.
- Read what is left. It must be the unit of the thing you are trying to find. Cubic yards for a concrete order, gallons for a fill, pounds for a load, square feet for an area.
This takes about ten seconds. It catches every slip in which a number was used in the wrong unit, every upside-down conversion and every missing conversion. It does not catch a wrong given (a mistaped length) or a wrong formula that happens to have the right units, which is why the rough-check line on the ticket still matters. See related: Write the Calculation Down: The Field Math Habit for that layout.
What it will not save you from. Two formulas can share units and still differ. The area of a circle using diameter in place of radius gives square feet either way, but four times too much. Units guard the setup's dimensions, not its geometry. See related: Circles in the Field: Area, Circumference and the Diameter Trap.
Mixed feet and inches
Dimensions taped in the field come as feet and inches: 12 ft 7-1/2 in. Before they go into a formula, convert every one to a single unit, either all decimal feet or all inches, and write the conversion:
12 ft 7-1/2 in = 12 ft + (7.5 in x 1 ft / 12 in) = 12 ft + 0.625 ft = 12.625 ft.
The classic failure is entering it as 12.75, because 7-1/2 "looks like" three quarters. It is 0.625 of a foot. On a 12 ft dimension that is about 1.5 inches wrong, and on an area it is wrong in both directions at once. Pick one unit for the whole calculation, convert everything into it on its own line, and only then multiply. See related: How to Convert Decimal Feet to Feet, Inches and Sixteenths for converting the answer back.
When the units are fine and the answer is still wrong
Carrying units makes the setup honest, not the inputs. Three things that pass the unit check and still fail:
- A given in the right unit and the wrong value. A slab drawing says 4 in and the forms are set at 5 in. 30 x 40 x (5/12) = 500 cu ft = 18.52 cu yd, a quarter more than the 14.81 ordered. The units cannot know the forms moved.
- Nominal versus actual. A "4 in" pipe or a "2x4" is a name, not a dimension. Use the measured or published actual dimension, and write which one you used.
- Average versus spot thickness. A slab measured 4 in at the forms may be 4.5 in in the middle over a dished subgrade. The volume is the average thickness times the area; a single edge reading is a given that undersells it. That is the job of the waste allowance, but only if somebody knew to look.
References
- See related: Common Unit Conversions + Math Formulas
- The Materials Waste Factor Worth Measuring (library card, for the allowance on top of the neat volume)
- See related: Write the Calculation Down: The Field Math Habit
- See related: How to Figure Cubic Yards for Concrete, Mulch and Gravel
- See related: The Water Constants 7.48, 8.34 and 231