Math Pages

• One in sixty rule
• Reciprocals
• Get down

Estimating ETA

• Chew on This
• Sixes and sixties
• Two-bit math
• It's in the percentages
• Leading a DME arc

Fun Stuff

• Change?
• Ace on top
• Ferry Pilots
• Pilots and Mechanics
• Add Fast

Chew on this

Last month we memorized the magic numbers 60, 120, and 180. So, if I tell you you are flying 180 knots you immediately tell me we are flying three miles per minutes. This month we will learn how to use a technique called "chewing" to solve what at first seems a difficult mental problem. 

Throughout pilot math I will refer to our speed in miles per minute many times. In an upcoming article I will present the most important rule of thumb in aviation which depends upon your knowing (approximately) your speed in miles per minute. But, that is for a later date. Today we are going to examine how to use "chewing" and our speed in miles per minute to figure ETEs.

If I know I am flying three miles per minute and there are 90 miles to go I simply divide 90 by 3. This is pretty easy to do in my head - so I know it will take 30 minutes. 

If I know I am flying two miles per minute and there are 90 miles to go I simply divide 90 by 2. This is just as easy - it will take 45 minutes.

Later in your career, when you fly faster airplanes you will be going four miles per minute in your King Air, or five miles per minute in your Dash-8 or eight miles per minute in your B767. So, you will eventually need to get good at dividing by other magic numbers. This is accomplished by a technique I call "chewing."

For students training for the instrument rating the most common cruise speed range is between 120 and 180. So, we need to concentrate on being able to divide by 2 or 3. From my experience most people are pretty good at dividing by 2, but find it much harder to divide by three. So, let me start with a bit of advice on how to do that easily. (For those who like to look ahead, in the next editorial I will present a special article devoted exclusively to aircraft that cruise at 150 knots, which is a very common speed for light twins that you will be doing your instrument rating in. So, you should pay particular attention to that article. 150 knots is of course exactly half way between 120 and 180 - so we immediately recognize that it is 2.5 miles per minutes.)

O.K. back to the question at hand - how do I divide by three in my head. The first thing to realize is that no human can do that. The human mind is NOT a computer, it does NOT divide. It just memorizes. So, you need to do what people back in the sixties did - memorize the times table. Memorize the following: (Note: if your airplane flies 4 miles per minute memorize the 4 times table, etc. and then follow along below.)

• 3x1 = 3
• 3x2 = 6
• 3x3 = 9
• 3x4 = 12
• 3x5 = 15
• 3x6 =18
• 3x7 =21
• 3x8 =24
• 3x9 = 27
• 3 x 10 = 30

Once your have the table memorized - and I do mean really memorized, so that you can instantly say 3x7 is 21 or 3x6 is 18, and so on, then you have it made. 

The first thing you should do then is examine the number you are trying to divide to see if it is one of the numbers you have memorized as a multiple of 3. For example if there are 27 miles to go you know the answer is 9 minutes. If there are 21 miles to go the answer is 7 minutes, etc. It is CRITICAL that you can recognize 3,6,9,12,15,18,21,24,27 and know which multiple of three they are - there is no substitute here for memorizing - so as Nike says - JUST DO IT!

You should also look for the above numbers with trailing zeros. So, for example 120 will take 40 minutes - because 12 would take 4 minutes. That is how I could so easily divide 90 earlier - I recognized the number 9 which I know is three minutes, so 90 is 30 minutes (900 miles would be 300 minutes, etc.)

What about distances between the numbers we have memorized. For example how long does 14 miles take? You must immediately realize that the closest of your memorized numbers is 15. So, it takes just under 5 minutes. How long does 25 miles take? The answer is, just over 8 minutes. DON'T go on beyond this point until you can state the time to the nearest minute for any distance up to 30 miles. The nearest minute is close enough - so that's all there is to the matter of distances less than 30 miles.

Now we are ready to tackle distances that are bigger than 30 miles. Here's how a pro does it. Lets say we have to go 78 miles. So we are trying to do 78 divided by 3 - it sounds hard. But, there is a trick to it.

We know from our memorized list that 3 x 10 is 30. So any distance more than 30 miles takes more than 10 minutes. Make sure that this point is clear to you before continuing. 

I mentioned earlier that it is humanly impossible to divide. This is a fact of the human brain - all we can do is add or subtract - so make the problem simpler, subtract 30. Every time you subtract thirty just remember you are subtracting 10 minutes from the answer, so keep track of those 10 minute blocks. In our problem we are starting with 78 miles. If we subtract 30 we have 48. That is still more than 30, so do it again. Now, we have 18. Once the number 18 pops up we recognize it as one of our memorized numbers, that takes 6 minutes. So, the answer is 26 - two 10 minute blocks that we subtracted plus the 6 minutes. 

Another way to think of the above - really the same way, but easier for some people to see it this way  is to start counting by 30s. So, if I have to go 78 miles I say to myself. It will take 10 minutes to go 30 miles, 20 minutes to go 60 miles, then there will be 18 more miles to go, which will take 6 minutes, so the answer is 26.

Lets do another example. Say we have to go 97 miles at three miles per minute. It will take 10 minutes to go 30, 20 minutes, to go 60, 30 minutes to go 90 and just over 2 minutes to go 7. So, the answer is 32 minutes and a bit.

I call this system chewing because we "chew up" the question into to 30 mile segments. Note: that at 4 miles per minute you chew it into 40 mile segments and at 8 miles per minute (jet speeds) you chew it into 80 mile segments.

Summary

Now that we know how to divide by 3 we can easily see that the same technique can be used for dividing by 4,5,6,7,8 etc. I call it "chewing" you just memorize the times tables for the speed of your airplane up to 10 times. Then subtract (chew) out multiples of ten until you have a distance that falls in the range you have memorized. This technique works like crazy - all good pilots use it, although it's amazing how many can't consciously describe how they do the division. Give it a try. Sooner or later you will need to develop the ability, so it might as well be sooner.