by Matt Parker
Problems can also occur in length in terms of the reference starting point, but these are much rarer. When a bridge was being built between Laufenburg (Germany) and Laufenburg (Switzerland), each side was constructed separately out over the river until they could be joined up in the middle. This required both sides agreeing exactly how high the bridge was going to be, which they defined relative to sea level. The problem was that each country had a different idea of sea level.
The ocean is not a neat, flat surface; it’s constantly sloshing around. And that’s before you get to the Earth’s uneven gravitational field, which alters sea heights. So a country needs to make a decision on its sea level. The UK uses the average height of the water in the English Channel as measured from the town of Newlyn in Cornwall once an hour between 1915 and 1921. Germany uses the height of water in the North Sea, which forms the German coastline. Switzerland is landlocked but, ultimately, it derives its sea level from the Mediterranean.
The problem arose because the German and Swiss definitions of ‘sea level’ differed by 27 centimetres and, without compensating for the difference, the bridge would not match in the middle. But that was not the maths mistake. The engineers realized there would be a sea-level discrepancy, calculated the exact difference of 27 centimetres and then … subtracted it from the wrong side. When the two halves of the 225-metre bridge met in the middle, the German side was 54 centimetres higher than the Swiss side.
This is where the phrase ‘Measure sea level twice, build a 225-metre bridge once’ comes from.
Massive problems
Aircraft fuel is calculated in terms of its mass, not its volume. Temperature changes can cause things to expand and contract; the actual volume fuel takes up depends on its temperature, so it’s an unreliable measurement of quantity. Mass stays the same. So when Air Canada flight 143 was taking off from Montreal on 23 July 1983 to fly to Edmonton, it had been calculated that it required a minimum of 22,300 kilograms worth of fuel (plus an extra 300 kilograms for taxiing, and so on).
There was still some fuel left from the flight in to Montreal, and this was measured to check how much fuel needed to be added for the next flight. Except that both the ground maintenance personnel and the flight crew performed their calculations using pounds instead of kilograms. The amount of fuel required was in kilograms, but they filled the aircraft using pounds and 1 pound equals only 0.45 kilograms. This resulted in the aircraft taking off with approximately half as much fuel as it required to make it to Edmonton. The Boeing 767 was now going to run out of fuel mid-flight.
In an unbelievably lucky twist of fate, the aircraft, flying with a dangerously low amount of fuel, had to make a stopover in Ottawa, where the fuel levels would be double-checked before the plane took off again. The plane landed safely, with the eight crew members and sixty-one passengers unaware how close they had come to running out of fuel mid-flight. It’s a near-miss which reminds us that using the wrong units can put people’s lives in danger.
But then, in an unbelievably unlucky twist of fate, the crew doing the fuel check in Ottawa made exactly the same kilogram/pound unit error and the aircraft was allowed to take off again with barely any fuel left. The fuel then ran out mid-flight.
There should be several alarm bells going off as you read this story. It’s so unbelievable as to strain credulity. Surely a plane will have fuel gauges to indicate how much fuel is left? Cars have such a gauge and, if an automobile runs out of fuel, it merely rolls to a stop and causes a mild inconvenience: you have to walk to the nearest petrol station. If a plane runs out of fuel, it also rolls to a stop – but only after dropping thousands of metres (three thousands of feet) out of the sky. The pilots should have been able to glance up at the fuel gauge and see that they were running low.
This was not some light aircraft with a dodgy fuel gauge either. It was a brand-new Boeing 767 recently acquired by Air Canada. A brand-new Boeing 767 … with a dodgy fuel gauge. The Boeing 767 was one of the first aircraft to be kitted out with all manner of avionics (aviation electronics), so much of the cockpit was electronic displays. And, like most electronics, that is all great … until something goes wrong.
Because of the lack of roadside assistance when you’re thousands of feet up, in aviation redundancy is the name of the game. Aeroplanes need to bring their own spares. So the electronic fuel gauge was linked to sensors in the fuel tanks by two separate channels. If the two numbers coming from each tank agreed, then the fuel gauge could confidently show the current fuel level. The signals from the sensors in the tanks (one in each of the aeroplane’s wings) went into a fuel-level processor which then controlled the gauges. Except this processor was on the blink.
One flight before its disastrous trip, the Boeing 767 was sitting in Edmonton and a certified aircraft technician named Yaremko was trying to work out why the fuel gauges were not working. He found that, if he disabled one of the fuel-sensor channels going into the processor, the gauges started working again. He deactivated the circuit breaker for that channel, labelled it with a piece of tape marked ‘inoperative’ and logged the problem. While waiting for a new processor to replace the faulty one, the aircraft could still be compliant with the Minimum Equipment List (required for the plane to be flown safely), if a manual fuel check was carried out. So now the fuel double-check consisted of the gauge with one sensor channel and someone looking in the tank and physically measuring the amount of fuel before take-off.
This is where everything gets a little bit ‘Swiss Cheese’: the disaster makes it through several checks that could have identified and solved the problem.
The plane was flown from Edmonton to Montreal by Captain Weir, who had misunderstood a conversation with Yaremko and thought the fuel-gauge problem was an ongoing issue and not something that had just happened. So when he handed the aircraft to Captain Pearson in Montreal he explained the fuel gauge had a problem but that a manual fuel check was enough to cover this. Captain Pearson took this to mean that the cockpit fuel gauges were completely inoperative.
While this pilot-to-pilot conversation was happening in Montreal a technician named Ouellet was checking out the aircraft. He did not understand the note Yaremko had logged about the fuel gauge so he tested it himself, which involved reactivating the circuit breaker. This caused all the gauges to go blank and Ouellet went off to order a new processor, forgetting to re-deactivate the circuit breaker. Captain Pearson then got into the cockpit to find all the fuel gauges blank and a label on one channel circuit-breaker saying ‘inoperative’, which is exactly what he expected from his misunderstood conversation with Captain Weir. Because of this unfortunate series of events, a pilot was now prepared to fly an aircraft with no working fuel gauge.
This would of course have been fine, if the fuel calculations had been performed correctly. But it was the early 1980s and Canada was starting the transition from imperial units to metric units. In fact, the new fleet of Boeing 767s were the first aircraft Air Canada had which used metric units. All other Air Canada aeroplanes still measured their fuel in pounds.
To add to the complication, the conversion from volume to mass used the enigmatically titled factor ‘specific gravity’. Had it been called ‘pounds per litre’ or ‘kilograms per litre’, the problem would have been avoided. But it wasn’t. So after measuring the depth of the fuel in the tank in centimetres and successfully converting that to litres, everyone then used a specific gravity of 1.77 to do the conversion: this is the number of pounds per litre for the fuel at that temperature. The correct specific gravity of kilograms per litre would have been around 0.8. And a conversion mistake was made both before take-off in Montreal and again during the stopover in Ottawa.
So, sure enough, in mid-flight after leaving Ottawa the plane ran out of fuel and both engines failed within minutes of each other. This resulted in an error-noise bong! which no one in the cockpit had ever heard before. I get nervous when my laptop makes a noise I’ve never heard before; I can’t imagine what it’s like whe
n you’re flying a plane.
The major problem with both engines failing is that – of course – the plane no longer has any power to fly. A smaller but still important issue is that all the new fancy electronic displays in the cockpit needed power to work and, as they ran directly off a generator attached to the engines, all the avionics went dead. The pilots were left only with the analogue displays: a magnetic compass, a horizon indicator, one airspeed indicator and an altimeter. Oh yeah, and the flaps and slats which would normally control the rate and speed of descent also used the same power, so they were dead as well.
STEP 1: Computation of the fuel on board:
Drip stick readings: 62 and 64 centimetres
Converted to litres: 3758 and 3924 litres
Total litres on board: 3758 + 3924 = 7682 litres.
STEP 2: Conversion of litres on board into kilograms:
7682 litres × 1.77 = 13597
Multiplying by 1.77 gave pounds, but everyone involved thought they were kilograms.
STEP 3: Computation of the fuel to be added:
Minimum fuel required, 22,300 kilograms — fuel on board, 13,597 assumed to be
kilograms = 8703 kilograms.
STEP 4: Conversion into litres of kilograms to be added:
8703 kilograms ÷ 1.77 = 4916 assumed to be litres.
The correct calculation using the minimum required fuel as a base, that is, 22,300 kilograms, as called for by the flight plan, would be as follows:
STEP 1: 3924 + 3758 litres from the first drip stick readings of 64 and 62 centimetres = 7682 litres of fuel on board.
STEP 2: 7682 × 1.77 ÷ 2.2 = 6180 kilograms of fuel on board, prior to fuelling.
STEP 3: 22,300 − 6180 = 16,120 kilograms of fuel to be boarded.
STEP 4: 16,120 ÷ 1.77 × 2.2 = 20,036 litres to be boarded.
Breakdown of how the calculation went wrong from the official Board of Inquiry report into the accident.
In the one stroke of good luck, Captain Pearson was also an experienced glider pilot. This was suddenly super useful. He was able to glide the Boeing 767 over 40 miles to a disused military base airfield in the town of Gimli. It was only a 7,200-foot runway but Captain Pearson was able to hit the ground within 800 feet of the start of it.
In a second stroke of good luck, the front landing gear failed, causing the front of the aircraft to scrape along the ground, providing some much-needed braking friction, and the plane came to a halt before the end of the runway – much to the relief of the people staying in tents and caravans at the far end, which was now used as a drag-racing strip. Here’s the thing about turning off all the engines on a 767: they fly much more silently. Some people had the fright of their life when a jet airliner suddenly appeared on the disused runway, seemingly out of nowhere.
Landing the aircraft as a glider was a phenomenal achievement. When other pilots were given the same scenario in a flight simulator, they ended up crashing. After the Boeing 767 was repaired and returned to service in Canada Air’s fleet, it became known as the Gimli Glider and achieved a reasonable level of fame.
It was eventually retired in 2008 and now lives in an aeroplane scrapyard in California. An enterprising company bought some sections of its fuselage and now sells luggage tags made from the metal skin of the Gimli Glider. I guess the idea is that the aircraft was lucky to survive a dangerous situation, so having a part of the plane should bring good luck. But then again, the vast majority of aeroplanes don’t crash at all so, strictly speaking, this plane was bad luck. I bought a piece of the fuselage and attached it to my laptop, which does not seem to have crashed more, or less, than usual.
And just to add some balance, I found an aviation mistake where the pounds–kilograms mix-up went the other way. In the Gimli Glider case, the fuel calculations were done in kilograms but it was actually fuelled using the smaller pounds: they had too little fuel. On 26 May 1994 a cargo flight travelling from Miami, US, to Maiquetía, Venezuela, was loaded with cargo weighed in kilograms when the flight and ground crews thought it was in pounds – so its cargo was about twice as massive as it should have been.
The roll down the runway was described as ‘very sluggish’, yet the flight still took off. Instead of it taking thirty minutes after take-off to reach cruising height, it took a full hour and five minutes. Then the flight used a suspiciously large amount of fuel. In the resulting court case it was estimated that, when the plane landed in Venezuela, it was 30,000 pounds overweight, which is about 13,600 kilograms (more than the total amount of fuel the Gimli Glider took off with).
It makes me feel a bit better about all the times I think I’ve overpacked my suitcase. But I also feel a lot less safe flying between countries which use different units (which is basically the US and everywhere else). I had better hurry up and determine if my piece of the Gimli Glider is good or bad luck!
Don’t forget about the price tag
It is easy to forget that currencies are units. $1.41 is a very different amount to 1.41c, but because the decimal point is often intuitively taken as a punctuation mark to split dollars from cents, people can consider them to be equivalent. There is an internet-famous phone call from 2006 when US resident George Vaccaro rang his mobile provider Verizon after a trip to Canada. Before the trip they had confirmed that their roaming-data charge in Canada would be 0.002c per kilobyte but then, after the trip, they charged him $0.002 per kilobyte.
Mr Vaccaro’s bill came to $72 for around 36 megabytes, which seems a bit laughable now, with over a decade of technological improvement, but at the time it was about right and the ‘correct’ price of $0.72 would have been laughably small. Verizon had definitely made a mistake when quoting him their rate. But Mr Vaccaro had documented it and was now trying to find out what had changed. The call is a painful recording to listen to, all twenty-seven minutes of it, as Mr Vaccaro is escalated up through several managers. None of them can see the difference between $0.002 and 0.002c and use both numbers interchangeably. I can’t get past the part where one of the managers calls the incorrect calculation ‘obviously a difference of opinion’.
There is an extra complication with money when looking at large amounts of it. Convenient multiples are units in their own right, but when dealing with something like metres and kilometres people tend to take them as different units. Kilometres are actually a combination of the distance unit of a metre with the ‘size unit’ of one thousand. But with money, these size units cause problems.
This was the basis of a meme passed around in 2015 when Obama’s Affordable Care Act was up and running, but not without teething problems (and the ACA Marketplace insurance plans don’t all cover dental work). An easy target for criticism was the cost of setting up Obamacare. A figure of $360 million was passed around as the cost of introducing the program, which is a large amount of money: over a third of a billion dollars. So people on the right of the political spectrum looked for ways to highlight just how much money it was. And this meme was born:
It is easy enough to see what is wrong here. $360 million between 317 million people is not $1 million each, it’s roughly $1 each. No million. Just a single buck.
Despite being fairly easy to debunk by dividing one number by the other, this meme was being passed around as a legit calculation. I appreciate that people are far less critical when it comes to evidence which supports their political beliefs, but I’d like to believe that even the most self-affirming pieces of evidence must at least pass some rudimentary sense-check filter before being promulgated. I cling to the theory that at least the threat of public embarrassment will stop people from endorsing patently implausible claims. Part of me cannot be convinced that anyone arguing for this Obamacare meme is not a troll and in it for the lulz. But to give them the benefit of the doubt, let’s try to work out why this false assertion was so tenacious.
My favourite version of this argument online has the protagonist back up the claim that $360 million divided by 317 million people is a million dollars each (with c
ash to spare) by breaking it down like this:
There are 317 people and you have 360 chairs. Do you have enough chairs for everybody to get one?
Well, yes, you do. The fact that 360 is bigger than 317 seems to be a core part of their argument, and no one is denying that bit of logic. But, for some reason, these people cannot see that this same logic does not hold when you have millions of dollars and millions of people. And I think this statement offers an insight into where their logic is breaking down:
Both units are in millions, so it doesn’t make a difference.
They are dealing with ‘millions’ as a unit and doing subtraction instead of division. Which, in some situations, does work!
Quick question: If I had 127 million sheep and I sold 25 million of them, how many do I have left? That’s right: 102 million. I can guarantee that, in your head, you ‘removed’ the million part of those numbers and did the straightforward calculation of 127 – 25 = 102, then put the million back on to get 102 million. You treated ‘million’ as a unit which could be ignored, as was convenient. But, very importantly, in this case, it works!
To millions of people, though, so it’s the same math, just added zeros.
