Multiplication with reference numbers

You may be familiar with the multiplication of two numbers near a threshold, using their reference number. A typical example is:

98 x 96 = (100 - 2) x (100 - 4) =

The threshold number is 100. You would take one of the numbers, subtract from this the difference of the other number to the threshold of 100, and this would be the first part of the product (e.g. 96 - (100 - 98) = 96 - 2 = 94). Then, you would multiply the two numbers' differences from the threshold with each other (i.e. 2 x 4 = 8) and get the other part of the product. Putting both parts together would give the product of 9408. This method is based on the following relation:

(100 - a) x (100 - b) = 10,000 - 100b -100a +ab = 100 x (100 - b - a) + ab = 100 x ([100 - b] - a) + ab

Here, (100 - b) is the first number and a is the difference of the second number from the threshold 100. At the same time, ab is the product of the two threshold differences.

Now, this method can also be used for other thresholds. A useful application for mental arithmetics is the threshold 50. The only difference to the threshold 100 is that the product of the first number with the threshold difference of the other must be divided by two. Like in this example:

49 x 47 = (50 - 1) x (50 - 3) =

The left part is the difference of 49 and 3 (47 = 50 - 3). 49 - 3 = 46. Divide by two and get 23. The right part is just the product of the differences, i.e. 1 x 3 = 3. The result is 49 x 47 = 2303.

Another example, which needs some more thinking (let me write the left part and right part separated by a '|' character):

49 x 46 = (49 - 4)/2 | 1 x 4 = 45/2 | 04 = 22.5 | 04 = 2254

See the difference? The division resulted in a number ending in '.5'. This number needs to be added to the right part of the answer, as 50.

And for the curious, here is the proof:

(50 - a) x (50 - b) = 2500 - 50a - 50b +ab = (5000 - 100a)/2 - 100b/2 +ab = 100 x ( (50 - a)/2 - b/2 ) + ab = 100 x ( (50 - a) - b)/2 ) + ab

… and this last relation is exactly what we do!