The Theory Behind Mental Dead Reckoning

Although it may look complex at first, mental dead reckoning can be easy once you know the shortcuts to make the calculations. Like anything, practice and repetition is the best form of learning to improve at mental dead reckoning.

Reciprocals

Reciprocals are simple to calculate, mainly using addition and subtraction

Example 1 - Addition : 090° + 180° = 270°

Example 2 - Subtraction : 240° - 180° = 060°

Essentially, if you add or subtract 180° to a bearing, you will get the reciprocal bearing. If your bearing is over 180°, the simplest method is the subtraction method. Otherwise, the addition method is the simplest for bearings < 180°.

Fractions of 60 for Integer Numbers

Fractions of 60 can be found for integer numbers by dividing them against 60

Example 1 : What fraction of 60 is 12?
Answer : 12/60 = 1/5

Example 2 : What fraction of 60 is 42?
Answer : 42/60 = 7/10

Example 3 : What fraction of 60 is 30?
Answer : 30/60 = 1/2

Example 4 : What fraction of 60 is 45?
Answer : 45/60 = 3/4

The fractions are found through the reduction of the numerator and denominator to the smallest possible values via their lowest common multiple. For example, 12/60 can be reduced to 1/5 by dividing both the numerator and denominator by 3. 42/60 can be reduced to 7/10 by dividing both the numerator and denominator by 2. 30/60 can be reduced to 1/2 by dividing both the numerator and denominator by 30. 45/60 can be reduced to 3/4 by dividing both the numerator and denominator by 15.

Equivalent Ratios for X : 60

Equivalent ratios of the form a : b = c : d for 60 can be found by finding the difference between the b and d term, or through re-arranging the equation

Method 1 - Difference

Start by finding the smaller of the b and d terms. Next, work out how many times the smallest term goes into the largest term. That number will determine the value of the missing X term.

Example 1 : 1 : 60 = X : 240
60 goes into 240 4 times. Therefore, X = 4

Example 2 : 4 : 60 = X : 30
30 goes into 60 twice. Therefore, X = 2

Method 2 - Re-Arranging the Equation

Isolate the X term by multiplying both sides of the equation by the d (or known) term.

Example 1 : 2.5 : 25 = X : 60
X = 60(2.5⁄25)
X = 150⁄25
∴ X = 6
2.5 : 25 = 6 : 60

Example 2 : 6 : 90 = X : 60
X = 60(6⁄90)
X = 360⁄90
∴ X = 4

Example 3: 4 : 60 = X : 10
X = 10(4⁄60)
X = 40⁄60
∴ X = 4⁄6 == 2⁄3

Finding ratios is similar to finding fractions of 60. Both approaches are useful in certain situations, with the difference method being best used for visual comparison tasks. For more complex ratio equations (specifically where the answer is non-integer), the re-arranging method is best.

Non-Integer Multiples

Non-Integer Multiples can be found by working out the number of times 60 can be divided by a given number, and then finding the fraction of 60 that the remainder represents

Example 1 : 260
60 x 4 = 240, remainder 20
20/60 = 1/3
∴ 260 = 4 + 1⁄3

Example 2 : 640
60 x 10 = 600, remainder 40
40/60 = 2/3
∴ 640 = 10 + 2⁄3

The simplest method to follow is to divide the number by 60, and then find the remainder. The remainder is then divided by 60 to find the fraction of 60 that it represents.

Non-Integer Division

Non-Integer Division can be found through two methods, either by division using the closest useful integer whole that matches the fuel quantity, or by adding value to the numerator to better suit the divisor, whilst also dividing the reduction of that amount

Method 1 - Division using the closest integer whole

Example 1 : Fuel Quantity = 1000lbs, Fuel Flow Rate = 660lbs/hr
Step 1: Calculate the flow rate per minute
660⁄60 = 11lbs/min
Step 2: Using 10 as the closest useful integer whole, find the time interval of sustained operations
1000⁄10 = 100mins

Method 2 - Division through separation and subtraction

Example 1 : Fuel Quantity = 1000lbs, Fuel Flow Rate = 660lbs/hr
Step 1: Calculate the flow rate per minute
660⁄60 = 11lbs/min
Step 2: Add 100 to the total fuel quantity, divide by 11. Subtract the result of 100 divided by 11 (approximates are expected)
1100⁄11 - 100⁄11
∴ 1000lbs = 100mins - 10⁄11mins
∴ 1000lbs = 100mins - 0.909mins ≈ 91 minutes


The first method is the simplest to follow, but least precise of the two. As can be seen in the above example, it is approximately 9 minutes different to the actual result, which would be unacceptable in real world endurance calculations.