Why doesn't 11x10 11x11 and 11x12 follow the pattern of the other 11's in multiplication?

Answer 1

They do in fact follow the same pattern, but once you hit two digits, it gets a little more complicated.

One way to look at it is to consider how the usual technique of simply doubling the digits works. Consider 7x11 - that will give you 77. Why though?

It gives you 77 because you're taking the number seven, multiplying it by ten, and adding 7 more to it. Think of it this way:

7 x 11

is equal to:

7 x (10 + 1)

Well that's easy then, you simply distribute the 7 over both of the terms in the brackets:

7 x (10 + 1)

= 7 x 10 + 7 x 1

= 70 + 7

= 77

When you multiply 11 by another two digit number, it works the same way. Consider 11 times 23:

11 x 23

= 23 x (10 + 1)

= 23 x 10 + 23 x 1

= 230 + 23

= 253

This in fact gives you another neat trick to use. When you're multiplying eleven by a two digit number, you can get the answer quickly by realizing that the first digit of the result will be the first digit of the given number, the last digit will be the last digit of the given number, and the middle digit will be the sum. If that sum is greater than nine, then you carry the addional one over to the hundreds column. For example:

23 x 11

= 2 [2 + 3] 3

= 253

or:

54 x 11

= 5 [5 + 4] 4

= 594

and:

98 x 11

= 9 [ 9 + 8 ] 8

= 9 [17] 8

= 10 [7] 8

= 1078

That same logic can be extended to even larger multiplications, but does become more complicated.