How to Convert Between Units Without a Mistake

Why this matters

Almost every conversion error in the field is the same error: multiplying where you should have divided. The factor was right, the arithmetic was right, and the result is off by the square of the factor in the wrong direction. It happens because a conversion is usually done from memory, in a truck, on a number somebody else wrote down, and nothing in the process is watching the direction. The method below makes the direction impossible to get wrong, because the units themselves cancel or they do not, and it takes no longer than doing it from memory once you have done it twice.

A number gets handled at five points between the instrument and the decision: reading the display, writing it down, converting it, comparing it against a specification, and reporting it. A conversion sits in the middle of that chain, and it is the only link that can change the value while leaving it looking entirely reasonable. Treat the whole chain, not just the arithmetic.

Before you go get the number

The readings in this article come off gas systems, so the hazard leads.

If you smell gas at any point, that is not a measurement situation: everyone leaves the building immediately, no switches are operated, no lights turned on or off, no phone used inside, the exterior shutoff is closed only if it is already on your way out, and the call is made from outside. Do not go back in to take a reading.

Before any appliance fires, run a personal carbon monoxide monitor in the occupied space. If ambient carbon monoxide climbs toward the 50 ppm 8-hour time-weighted average of 29 CFR 1910.1000 Table Z-1, or the 200 ppm ceiling NIOSH publishes, everyone leaves the building immediately, no switch is touched on the way out, and the space is ventilated from outside before anyone re-enters.

Opening a pressure tap breaks a gas joint. NFPA 54, the National Fuel Gas Code, requires the piping system to be checked for leakage before it is placed back in operation, so every tap you open gets leak-checked with a listed leak-detection solution or another approved method before the appliance goes back into service, and never with a flame. Reseal the tap to the manufacturer's instruction, not finger-tight, before you walk away.

Step 1: Write the conversion as a fraction that equals one

Any conversion factor can be written two ways, and both are true statements equal to one:

Since 1 psi is about 27.7 inches of water column (a value that shifts slightly with water temperature, so use the figure your instrument or code reference uses),

• (1 psi) divided by (27.7 in w.c.) equals 1
• (27.7 in w.c.) divided by (1 psi) equals 1

Multiplying anything by 1 does not change its value, only its units. That is the whole basis of the method, and it is why it cannot go wrong: you are not choosing between multiplying and dividing, you are choosing which of two forms of the number 1 to multiply by.

Step 2: Arrange it so the unwanted unit cancels

Put the unit you want to get rid of on the bottom of the fraction so it cancels against the unit on top of your measurement.

Converting 7.0 in w.c. to psi:

7.0 in w.c. x (1 psi / 27.7 in w.c.) = 7.0 / 27.7 = 0.253 psi

The in w.c. on top and the in w.c. on the bottom cancel, and psi is what survives. Had you picked the other form, you would be left holding units of in w.c. squared per psi, which is not a pressure and is not anything. If the units of your answer are not the units you wanted, you picked the wrong form, and you will see it before you see the number. That is the entire safety net.

Step 3: Predict the direction before you look at the result

Do this out loud, in one sentence, before computing: a smaller unit gives a bigger number, a bigger unit gives a smaller number.

A psi is a much larger unit of pressure than an inch of water column, so the number must get much smaller. It went from 7.0 to 0.253, which is smaller by roughly the factor we used. Correct.

Now run the error for comparison. Inverting the factor gives 7.0 x 27.7 = 193.9 psi. The direction test kills it instantly: we converted to a larger unit and the number grew by a factor of nearly thirty. The magnitude test kills it again, because a low-pressure fuel gas system does not operate anywhere near that pressure and the piping is not rated for it. Two independent checks, both free, both catching the dominant conversion error.

Step 4: Chain multiple factors in one line rather than in stages

Where two units change at once, write both factors in a single expression and let all the cancelling happen together. Stopping halfway to write an intermediate result is where transcription errors enter.

Converting 12.0 gal/min to cubic feet per hour, using 1 cubic foot equals about 7.48 gallons:

12.0 gal/min x (1 ft3 / 7.48 gal) x (60 min / 1 h) = (12.0 x 60) / 7.48 = 720 / 7.48 = 96.3 ft3/h

Gallons cancel, minutes cancel, and cubic feet per hour survives. Direction check with two competing factors: the volume unit got bigger, which pushes the number down by about 7.5, and the time unit got bigger, which pushes it up by 60. Net, 60 divided by 7.48 is about 8.0, so the number should end up around eight times larger. It went from 12.0 to 96.3, and 96.3 divided by 12.0 is 8.0. The prediction and the result agree.

Step 5: Know which quantities do not take an offset

This is the step that catches the errors the cancelling method cannot, because the units cancel perfectly and the answer is still wrong.

A temperature and a temperature difference convert differently. A temperature of 20 F converts to Celsius as (20 minus 32) divided by 1.8, which is about -6.7 C. A temperature difference of 20 F degrees converts as 20 divided by 1.8, which is 11.1 C degrees, because the offset between the two scales cancels out when you subtract two temperatures. Using the offset formula on a difference here would report -6.7 where the true answer is 11.1, which is not just wrong in size but wrong in sign.

Any scale with an arbitrary zero has this property: temperature in the everyday scales, and gauge pressure relative to absolute. Ratios and differences on those scales convert by the scale factor alone. Absolute values convert by the scale factor and the offset. Ask which one you are holding before you convert it.

Step 6: Round once, at the end, to the precision you actually had

The pressure reading was 7.0 in w.c., which carries two significant figures. Reporting 0.2527 psi claims four. Round the final answer to the precision of the least precise input, so 0.25 psi, and do the rounding once rather than at each intermediate step, because rounding twice compounds.

Where a conversion feeds another calculation, carry the extra digits internally and round only what you write on the ticket. The accuracy-and-resolution card in this library covers how much precision an instrument actually earns you; the rule here is simply that a conversion never creates precision that was not in the original reading.

Step 7: Reverse it as your final check

Take your answer and convert it back. You must land on the number you started with, and you must have used the opposite operation. If you multiplied going out, you divide coming back.

0.253 psi x (27.7 in w.c. / 1 psi) = 7.01 in w.c., which returns the original 7.0 within rounding. If the reverse conversion needs the same operation as the forward one, the factor was inverted somewhere.

Where the shop-level errors come from

Individual arithmetic is rarely the problem. Two systemic causes account for most of it.

A factor memorized slightly wrong and reused forever. Once a factor is in someone's head it never gets checked, and it propagates to everyone they train. Keep the handful of factors your trade uses in one written place, on the truck and in the shop's records, with the conditions attached where the factor depends on them. The water-column-to-psi factor is a good example: it depends mildly on water temperature, so the written version should say which figure you standardize on rather than leaving each tech to remember a slightly different one.

Conversion at the wrong point in the chain. A number converted before it is written down loses the original, so nobody downstream can check it. Record the raw reading in the unit the instrument displayed, then the converted value beside it. Two entries, one line, and the conversion becomes auditable instead of permanent.

The failure mode that follows from getting this wrong is not usually a dramatic one. It is a fleet of tickets carrying values converted the same wrong way, all internally consistent, all comparing fine against each other, and all disagreeing with the manufacturer's specification by the same factor. The tell is a shop whose readings systematically differ from published figures in one direction on one quantity. That is not a fleet of bad equipment, it is a bad factor in circulation.

How to verify you got this right

Take the conversion you do most often and write it out in the fraction form above, on paper, once. Confirm that the unit you want to remove appears on the bottom, that it cancels, and that the surviving units are the ones you intended. Then state the direction rule for it out loud so it is available next time without the paper.

Then check one ticket from last month. Is the raw reading recorded alongside the converted one? If not, nobody can audit that number now, including you, and the correction is to change the ticket format rather than to try harder.

References

• NFPA 54, National Fuel Gas Code, requirement to check piping for leakage before placing a system back in operation
• 29 CFR 1910.1000 Table Z-1 carbon monoxide permissible exposure limit; NIOSH ceiling value for carbon monoxide
• Instrument and code-reference documentation for the exact conversion factor and the conditions it is stated at
• See related: The Units That Get Confused and What It Costs; The Difference Between Accuracy and Resolution; How to Take a Reading That Means Something