The Trachtenberg method can be used to easily multiply numbers by 11. For example:
11 x 23= 253
11 x 16= 176
11 x 63= 693
11 x 81= 891
Notice that in each example,
a)the first number of the answer is the first number of the multiplicand.
b)The last number of the answer is the last number of the multiplicand.
c)The middle number of the answer is the sum of the multiplicand’s digits.
Note: In cases where the sum of the multiplicand’s digits exceeds 9, use only the last digit of the sum and as a carry figure bump the first number of the multiplicand up by one. [The carry figure is never greater than 1 because the sum of even the two greatest possible digits, 9+9, yields a carry figure less than 2.] Examples:
11 x 75= 825
11 x 84= 924
11 x 99= 1089
To multiply 11 times numbers of more than two digits, do the following. Example:
11 x 7421 = 81631
a) The ones digit of the multiplicand is the ones digit of the answer. In the example, the ones digit of 7421 is 1, and this is the ones digit of the answer. _ _ _ _ 1
b) Add the last two digits of the multiplicand. This yields the tens digit of the answer (there may be a figure to carry). In the example, adding the last two digits of 7421 gives the tens digit of the answer. 2 + 1 = _ _ _3_
c) Add each successive digit pair in the multiplicand (plus any carry figures, if any) to give the next successive number of the answer. In the example, adding the next digit pair of 7421 gives the hundreds digit of answer. 4 + 2 = _ _6_ _
Continue in this manner adding successive digit pairs in the multiplicand for successive digits of the answer (plus any carry figures, if any). In the example, adding the next digit pair of 7421 gives the thousands digit of the answer. 7 + 4 = _1_ _ _ (with a carry of 1)
*d) When digit pairs in the multiplicand are used up, simply use the left-most digit that is unpaired for the first digit of the answer. In the example, the digit left without a pair in 7421 is 7. Adding the carry figure gives the first digit of the answer. 7 + (carry figure of 1) = 8_ _ _ _
*I like to consider that the left-most digit having no available pair is paired with 0. This helps me remember to not forget it- I’m not through until I’ve paired a number with 0. Try it with and without pairing the left-most digit with 0, and see which way you prefer.
See how easy these are:
421 x 11 = 4631 295 x 11 = 3245 230 x 11 = 2530
This is an especially easy way to multiply 11 by two digit numbers! You should be able to do all these easily, most in your head and all by just quickly writing down the answer:
11 x 24= ___ 11 x 15=___
11 x 33= ___ 11 x 78=___
11 x 67= ___ 11 x 111=____
11 x 96= ____ 11 x 654321= _______