Multiplying by 11

Date: 09/16/97 at 05:55:04
From: Benjamin Silbert
Subject: A problem my class wants to know

My teacher is a mathematician and one day she asked us to give 
her a number times 11. We said: 568 x 11, and after about 5 seconds 
she said the answer was 6248, so we checked on our calculators and she 
was right!

We want to know how she does it so quickly!

Date: 10/31/97 at 11:37:11
From: Doctor Trent
Subject: Re: A problem my class wants to know

Multiply it out on paper, and you'll see that she's just adding 
digits:

       568
     x  11
   -------
       568
      568
   -------
      6248

You can do it like this: in 568, 8 is the last digit, so 8 is the 
last digit in the answer.  6 is the second to last digit, so you 
add it to 8, and 6 + 8 = 14, so 4 is the tens digit; 5 and 6 are 
the first two digits of 568, and 5 + 6 = 11 (plus the carried 1 
from before) = 12, so the next digit is 2 with 1 carried.  5 would 
be the first digit, but we add the carried one to get 6.  

With practice, you can do it in your head quite quickly.

Here's a longer example to help put it together: 7348 x 11

 8 is the last digit
 4 + 8 = 12   2 is the tens digit, carry a 1
 3 + 4 = 7    plus the carried 1 is 8, no carries
 7 + 3 = 10   0 is the next digit, carry the 1
 7  plus the carried 1 is 8, the first digit
so: 7348 x 11 = 80828

-Doctor Trent,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 10/31/97 at 17:50:20
From: Benjamin Silbert
Subject: Re: A problem my class wants to know

Dr. Math,

I asked my teacher and then my dad:

My dad said:

   Multiply it by 10 and then add the number on.

My teacher said:

   use 33 x 11

   33

   put down the first number (3)

   3

   add them together

   6

   put down the last number

   3 

   and your total amount is 363.

Benjamin Silbert

Date: 11/01/97 at 09:10:19
From: Doctor Sonya
Subject: Re: A problem my class wants to know

Dear Benjamin,

Thanks for letting us know the other ways to do this problem. 

Your dad's seems to be the simplest, but could be difficult if the 
numbers were really big. For example, what if we wanted to find 
674 x 11?  We'd have to add 674 + 6740. I like your teacher's way 
best.

Write us if you have any more questions.

-Doctor Sonya,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/