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Mental maths - tips, tricks and shortcuts
I thought I'd start a little thread that will hopefully be of some benefit to newbies and "old hats" alike.
Maths is something we use on a daily basis as pilots and the more time you can spend looking out the window as opposed to scribbling on a piece of paper, punching numbers in to calculator or wizzing around your wizz wheel. Be it simple 1-60's, a quick and dirty method of working out drift angles for an instrument approach or converting speeds, weight, units of measurement quickly. |
Percentages can be good for quick mental comparison between choices, or straight calculations.
For example: Planned groundspeed 130 kt. Say time for one leg of a nav is 33 minutes, do groundspeed check from two pinpoints and actually getting 115 kt. That's 15 kts slower than planned, close enough to 10% of planned g/s, so leg will take about 10% longer, or 3.3 min, so you can easily revise ETA and fuel burn. It's not perfectly accurate, but can give you good estimates which you can refine when you get time. |
The first thing I would recommend student pilots do is learn to mentally convert minutes to decimals ie (6 min = 0.1 hr etc). Some cadets I fly with have not seen the practicality of this in an age of an app for everything.
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Learn how to manipulate numbers and groups of numbers in your head.
For example 85 USG of fuel, how many L and KG? Start with 85 x4 so that is 170 plus another 170 which is 340. Now we all know that it's about 3.778 not 4 so that is less 5% roughly. So that is less 10%/2 or 34/2 which is 17. Answer is 323 litres. Now doing it accurately it works out at 321.76, so in flight doing a mental check is 323 close enough?:ok: Now we want kg, well 0.71 kg to the litre, ahhh near enough to 30% less and we had 323, but let's allow a tad mor so use 330 to account for the bit more than 0.70 vs 0.71 and we get 10% of 330 being 33, so three of them is 99 and that means 231kg. Or from 323 it would be 226kg. Now doing it accurately I get 228.45kg, so how close is 231 or 226kg.:ok: So the secret is knowing the formulae or conversion constants, and then just being good at manipulating the numbers in easy chunks. Knowing what to add or subtract and doing so. I am blessed with good mental maths and the ability to also know what is near enough and what is wrong. Too many kids today can't do this and they have no clue if hey come up with a wrong number. That is the scary bit. Mental maths is great, but it must be matched with knowing what looks right and wrong. I hope that helps answer your question. As for the useful tricks like descent profiles such as distance times 3 (in hundreds feet) add 100 etc is something others here will have good tricks for. Chimbu Chuck is a classic one for them. |
Minutes to decimals .6 - I've never had a reason to use this except for sometimes when I have to pay for the aircraft after the flight.
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The one I use a lot is the 'clock face' for approximating crosswinds, drift angles etc:
The clock obviously has 60 minutes and you use the proportion of those minutes to estimate the % of the crosswind etc. I know, crap explanation :) So, I'll try harder. The wind is 30 degs off the runway at 20 knots. Take those 30 degs and pretend they're minutes on our clock face. 30 minutes is 1/2 (50%) of the face, so the crosswind component is about 10 knots. Another: wind is 20 degrees off the runway at 15 kts. 20 degs = 20 minutes = 1/3 of the clock face, therefore 1/3 of 15; crosswind is 5 kts. This works all the way up to around 40-45 degs; thereafter just take the whole lot. So a wind 50 degs off the runway at 10 kts will be 10 kts of crosswind. Add in a 'max drift' calc and we can sort out all our nav problems. Max drift is the wind speed divided by our TAS in miles/min(or IAS if you're low down......these are approximates, not absolutes :)) SO; we're in our Cessna and we're bimbling along at 120 kts. 120 kts/60 = 2 miles/min. We've got a wind speed of 30 kts at our cruising level, so max drift (i.e. if it was all across) would be 15 degs. Now we apply our clock face: wind is 15 degs off our track. 15 degs = 15 minutes = 1/4 of the clock. Max drift is 15 degs (piss poor example eh), so our drift angle is 4ish degs (1/4 of 15), so head 4 degs into wind. Another, I hear you cry: TAS 180 (3 miles/min). Wind is 60 kts, therefore max drift is 60/3 = 20 degs. Wind is 40 degs off our track. 40 degs = 40 mins = 2/3 of the clock. 2/3 of 20......14 degs of drift. We can also take that angle off our track of the wind, subtract it from 90 and apply the same calcs to work out our groundspeed. E.g: we've got 20 kts of wind at 60 degs off our nose. 90-60 = 30. 30 on the clock face = 1/2; so we've got 10 kts of headwind. Wind 30 degs off track at 10 kts: 90-30 = 60; therefore take the whole lot: we've got 10 kts headwind. Apply logically for tailwind. Applying it all: wind is 330/20 at cruise and we're going to track 360 in our 120kt TAS Cessna. So, max drift is 10 degs (120/60 = 2 nms/min. 20/2 = 10). wind is 30 degs off track so hdg reqd is 355 (30 = 1/2 the clock. 1/2 of 10 = 5 degs.....apply in the correct sense!). Groundspeed is 100 kts (90-30 = 60. Apply the whole lot 'cos it's greater than 75% of the clock face, therefore, we've got 20 kts headwind). We land on our conveniently orientated runway 36 and the wind is now 320 at 15. We've got a 10 kts crosswind from the left (40 degs off = 2/3 of the clock = 2/3 of 15). I haven't got a whizzwheel to hand but would be interested to hear how close they are. Now, back to the FMC :ok: |
Another good tip is to
STOP USING FR:mad:GGIN' CALCULATORS IN FLIGHT! It takes your attention away from the aircraft attitude while you concentrate on using tiny buttons and a miniscule display. My students soon learn if a calculator comes out in flight it soon ends up in the baggage area behind the back seat.:E:E:E The more you practise "mental gymnastics" the easier it gets. Learn to use your whiz wheel (not just for time vs distance/fuel, but for multiplication and division as well). The dude who invented this thing was a certifiable genius! Round numbers to make rough calcultions easier, Eg. 4 miles off in 38 would be pretty close to 4:40, hence 6:60 or 6 degrees.:ok: For cross and headwinds, I used 30 deg = 1/2 wind strength is x/wind 45 deg = 3/4 and 60 deg = 90% Reverse the order for head/tail winds 30 deg = 90% head/tail wind 45 deg = 3/4 and 60 deg = 1/2. At 60kts, 1 kt x/w = 1 deg drift At 120 kts = 1/2 deg drift. :ok: |
A good tip I was shown (after 20 years of flying) : A persons finger on a WAC chart covers near enough to 10 nm. Try it. 2 fingers 20nm 4 fingers 40nm etc You'll quickly work out for your own hand where it has to cross the track drawn on the chart to be surprisingly accurate.Very handy on a rough day bouncing around with no auto pilot.Wish I had been shown this for my pre GPS days. I still use for identifying items of interest that we see out the window from time to time.
Cheers RA |
Stuff the mental maths!
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But when MIHC chucks your calculator or whizz wheel (same thing really) into the back seat, maybe you might want to unstuff the brain maths.
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That's why I carry two of them!:E
Doesn't anyone have a sense of humor anymore? |
To nail a 3 degree glideslope, use RoD of 5 x groundspeed! (Because a 3 degree slope is a 5% gradient). Simple, but effective.
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No, I don't mind a whiz wheel, really. You can hold it up, operate it with one hand and keep an eye on what's going on outside at the same time.
(Most students let go of the yoke (haven't quite worked out rudder to keep wings level yet, but that doesn't matter cause they're looking down anyhoo) and concentrate fully on a little half by 3 cm screen for the next two minutes.) Mental maths first, double check with the whiz wheel. Plus, no batteries to go flat.:E:ok: |
A good tip I was shown (after 20 years of flying) : A persons finger on a WAC chart covers near enough to 10 nm. Try it. 2 fingers 20nm 4 fingers 40nm etc You'll quickly work out for your own hand where it has to cross the track drawn on the chart to be surprisingly accurate.Very handy on a rough day bouncing around with no auto pilot.Wish I had been shown this for my pre GPS days. I still use for identifying items of interest that we see out the window from time to time. Cheers RA |
Top of descent for a 500ft/min descent.
Thousands of feet to loose x twice the GS in nm/min. Descent from 9000 to 3000 at 180kts. dropping 6000ft and covering 3nm per minute so 6 x 6 = 36nm. To prove it works. 36nm @ 3nm/min = 12 mins. 12 mins x 500ft/min = 6000ft. 4000ft at 120kts = 4 x 4 = 16nm 16nm @ 2nm/min = 8 mins 8 mins x 500 ft/min = 4000ft. GS is a multiple of 2 so easy enough to use either 120kts, 150kts or 180kts. +/- 10kts, adjust by 3nm as appropriate. |
RTFQ = 1/2TFA
Works for me You must have a ball park figure at all times as man's index finger on the keyboard causes more strife than the middle one used to do. :ok: |
Fratemate's clock code works well.
Rather than dividing and multiplying all over the place having one simple number to remember. I use 0.284 to get litres of Jet per side... ie I need 500 lbs of fuel x 0.284 = 145 litres a side. Time on target can be useful as well... 10 seconds late requires a 10 knot increase for (normal cruise speed) seconds (so if normal cruise speed was 240kts, and you were 10 seconds late, you need to do 250kts for 4 minutes to make up the lost time). Remembering 3x height for profile can be rather hand for night circling as well, not just 60nm out. |
Quick numbers checks using what some have mentioned here are fine under certain conditions but it would be very irresponsible of any pilot not to have a calculator at hand at all times. Under pressure or duress the brain can be very unreliable & simply wrong so always have two options for working out any calculations. Why else do we have such high levels of automation these days even in basic A/C, 'cause it takes the 'guess' work out of a lot of flying & that includes number crunching using the brain only.End result safer!
When I was taught to drive a plane everything revolved around a whiz wheel as well as being able to do simple maths in yr head but a calculator was always there close at hand. Simply being alert at all times & having good situational awareness means brain number crunching is kept to a minimum. Wmk2 |
Here is an easy one. How many feet before leveling off from a climb or descent is required for attaining your target altitude. 10% of ROC eg climbing at 1000fpm you'd need to start leveling off 100' prior to target altitude.
The other one is at what point you require to start your turn onto an DME arc. 1% of GS eg 200kts x 1%=2nm (rate 1 turn) Knox. |
Reciprocal headings, add 200 minus 20 or vise versa or add 2 minus 2 to keep it simple.
150 add 200 = 350 minus 20 = 330 300 minus 200 = 100 plus 20 = 120 |
One I use for percentages (especially multiples of 10%) is to divide by 10 and then multiply by whatever number you need to get the right percentage.
I.e. 20% of 20 = 20/5=4 nice and easy. but 20% of 17.8 I would find too hard to just divide by 5. So I do 17.8/10= 1.78 (10%) x2= 3.56. Works for multiples of 5% as well. you just do the same thing to the 10% level below and then add half of the original figure. I.e. 35% of 56= 5.6x3=16.8 (30%) + 2.8 (half of 10%)=19.6. |
Or you can round a bit to make it easier -
35% of 56: Do 35% of 60, which is 3 and a half times 6, ie 21. Take a little bit off because 56 is a bit less than 60, call the answer 20. Or to be slightly more scientific, 35% of 60 is 21. 56 is 4 less than 60, or about 6% less (two-thirds of 10%). So my answer should also be two-thirds of 10% less, or 21 minus 1.4, which is 19.6. |
Ooooooooooooooh! My head hurts just reading this thread!
I am with the Captain on this one. Dr :8 |
The ideal woman is half your age plus seven years....very useful after getting an airline command and going through divorce..!
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Reciprocal headings, add 200 minus 20 or vise versa or add 2 minus 2 to keep it simple 150 add 200 = 350 minus 20 = 330 300 minus 200 = 100 plus 20 = 120 What's wrong with..... 150+180=330 and 300-180=120! :confused: Some of these work-arounds are more complicated than the original! |
Sorry, Wally.
Gotta disagree here (partially). I am seeing WAY too much of students that can't even do simple addition without a calculator. The overreliance on this automation is the problem. Don't laugh, but I have actually seen finger counting to add two single digit numbers. WHAT THE F:mad:CK??!!??:ugh::mad::bored: Not saying everyone finds multiplying two double digit numbers easy, but practise and using unfcukupable tools are a starting point. Did I mention practise? Does anyone notice how when they first learned to drive a car, they had to really work on clutch & gear coordination, parking, hill starts etc...? Notice how it gets easier with time? Because you practise! Being able to get these answers without having to take your socks off has gotta make things easier when your workload has gone ballistic.:ok: Not trying to drift the thread here, but seeing similar things with many AFR navexs where GPS has taken over. |
Always laughed at the time to decimal place card in the cockpit. Till I used it a few times... Oh how quickly we build reliance on something.
Seriously 0.1= 6 mins IS easy, but after 16 hr split duty have to confess, out comes the app, calculator and socks off... Gone back to refusing to use tools and cards for simple arithmetic and low and behold, I can figure stoof out again. Amazing! Even to the point that ops staff were simply amazed when I figured out the fuel in kg just by taking off 20%... wow no fms, calculator, app, whiz wheel or toes required. So best tip, trick and shortcut? Put down the calculator! Pick up the pen! |
Then pen works fine until you forget to carry a 1 and cause a mess as a result.
Then you get back to technology to fill out the occurrence report which will no doubt be an online form. There is a reason technology has become such a reliance in aviation. Not such a problem calculating a 1:60 in a GA aircraft.. your visual and can probably get close enough to see the destination. Doing the math mentally, getting it wrong and not loading enough fuel in the 767 will land you in Gimli. |
I just love the idea that reading little numbers on an electronic calculator is bad airmanship. Apparently reading smaller numbers on a whizz wheel is not!
Get with the times, encourage common sense, if someone is fast and safe using any technique, then rather than "throwing stuff into the baggage hold" take some notes yourself, you might learn something! Bbbbbbbbbbbbbbbzzzzzzzzbzbzbzzzzzzbzzbzzz |
Good o'l Calculators
Got to agree on mental maths.
Make it Happen Captain, agree with the mental stuff and whizz wheel at student level/flying school ops, however outside of GA, the calculator is your best pal. Example.. If you do 8 sectors there is a "Deck log/ Flight log" to be filled out for each sector and it has to be crunched quickly, as the turn around can be 30 mins.. These compile of converting pounds to kg- litres to kg- minutes to dec. points, along with other calculations. A calculator is a must, No guestimations here. It is also stated in Part B/SOP's, that a calculator is to be used for the Deck logs/MEL's. ANOTHER THING GUYS/GIRLS.....GENERATION Y ARE A DIFFERENT BREED WITH REGARDS TO TECHNOLOGY....CALCULATORS RULE FOR THEM. I notice some here mentioned "batteries"....geez you must be a baby boomer or Gen x....:rolleyes: Solar power and battery combination along with pocket size/ compact units that fit right in your right pocket of your shirt,if you lose it, there is also a iPhone.... Anyhow, as mentioned still got to love mental maths for TOD points... Oops, the FMC does it for me....:ok: |
Jeez, and I thought mental maths was supposed to make things simpler and easier... some of the stuff in this thread makes my brain hurt :ugh:
Clock code for xwind, TOD, RoD etc., sure, but fuel conversions, planning etc., I'll stick with a calculator thanks - why make things harder than they have to be?? |
I like that a thread about mental arithmetic shortcuts has been hijacked into whether you are a hero for liking them or not!
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Years ago when I was in the old CAA, I was checking out a new (young) guy for a rating on a certain sector. One aircraft this guy was responsible for was tracking between two reporting points that were 119 nm apart and the pilot's ETI for the route segment was 60 mins. Casually I ask the guy I was checking what was the pilot's approximate groundspeed.
To my surprise he reaches for a calculator! I still shake my head at that one. Surely there must still be a need for some mental maths ability! |
There'll always be a place for it.
I like new technology that makes life easier, but I don't think I've ever owned or used an electronic device that didn't malfunction in some way at some time. Knowing you can deal with loss of gear (GPS being the most notable one these days I think) without totally losing the plot is good. Plus, it's just good airmanship to be able to check if the automation is telling you bull****, be it because of equipment or operator failure, or both. |
Jeez, and I thought mental maths was supposed to make things simpler and easier... some of the stuff in this thread makes my brain hurt :ugh: Clock code for xwind, TOD, RoD etc., sure, but fuel conversions, planning etc., I'll stick with a calculator thanks - why make things harder than they have to be?? Its not that hard to remember that 100lbs of JetA is almost 60 litres. For example if i want 500lbs ill be ordering about 300 litres, and if its been ordered for me if all of a sudden i saw 400l or 200l I know something somewhere isnt quite right for whatever reason. That's not "harder than they have to be". It's actually quicker than removing calc/phone from pocket and punching in 500 / 1.76 = into it! Sure we arent trying to sin/cos/tan angles or anything like that so some things are irrelevant, but mental maths is very useful to dummy-check figures in all aspects of flying or at least get a rough estimate to get things going and work out an absolute later. |
To my surprise he reaches for a calculator! I still shake my head at that one. I suspect that afterwards, when calmed down, there would have been a forehead slapping moment in private. But, of course, you've never been in that situation have you? I just LOVE empathetic examiners :* |
Sorry-I can't help myself. I always do a rough mental calculation to assure myself that the box of silicon chips probably gave me the correct answer. It's too easy to hit for example the x button when you really meant to divide. So if I'm adding 256 L of Avgas and I need to calculate the added weight I mentally note it should be less than 256kg but more than half of 256kg. I then multiply 256 by 0.7 and arrive at 179.2 so it's probably correct. If I arrive at 365.7 I know I've made a mistake,in this case divided instead of multiplying.
Two more handy rule of thumbs. For 75% power use the rule of 48. MP in Inches plus RPM hundreds will be pretty close to 75% power if they total 48 eg 24 inches MP plus 24 hundred revs is about 75% power so is 23plus 25 or 21 plus 27 For TAS use 2% of the IAS X altitude in thousands and then add this result to the IAS eg Indicating 135 knots @ 9500 feet you get 2% of 135 = 2.7 X 9.5 = 25.65 plus 135 =160.65 knots. This doesn't allow for temperature effects but it's pretty close to right. Cheers RA |
sisemen:
But, of course, you've never been in that situation have you? I just LOVE empathetic examiners But you're right there was a forehead slapping moment after the event. |
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To nail a 3 degree glideslope, use RoD of 5 x groundspeed! (Because a 3 degree slope is a 5% gradient). Simple, but effective. |
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