Mental Math Tricks I

Multiples of 11:

11 x 14 = 1 ___ 4 , the middle number is (1 + 4) so the answer is 154

Let's try a few calculations mentally.

11 x 33 = 3 __ 3, the middle number is ( 3 + 3 ) so the answer is 363

11 x 52 = 5 __ 2, the middle number is ( 5+2 ) so the answer is 572

11 x 27 = 2 __ 7 , the middle number is ( 2 + 7 ) so the answer is 297

What about if the sum of the middle number is > 9? Then you carry over the "1" to the digit on its left
as you are doing the addition.

Examples:

11 x 67 = 6 __ 7, the middle number is (6 + 7) = 13 so the answer is 737

11 x 89 = 8 __ 9, the middle number is (8 + 9) = 17 so the answer is 979

11 x 47 = 4 __ 7, the middle number is (4 + 7) = 11 so the answer is 517

Harder trick:

11 x 223 = ?  2 _ _3, Write the two digits on the far left and right. Now add two numbers together,
starting with the unit digit. When the sum is  larger than 9,carry over to the next digit to the left.
2 _5(3+2) 3;  2 4 (2 + 2) 53 so the answer is 2453.

11 x 40532 = 4_ _ _ _ 2; 4_ _ _ 5 (2 +3) 2;  4_ _ 8(5+3)52;  4_ 5(0+5)852;  44(4 +0)5852,
so the answer is 445852  (This is fun!!)

Other interesting pattern:

11 x 11 =  121
111 x 111 = 12321
1111 x 1111 = 1234321
etc… till  111111111 x 111111111 = 12345678987654321

Divisibility rules for 11: 

The difference  of the sum of alternative digits is a multiple of 11, including “0”                                               (0 x 11 = 0, a multiple of 11.)

Example:

61985   (6 + 9 + 5) – (1 + 8) = 11   The number is divisible by 11.

7469     (7 + 6) – (4 + 9) = 0   The number is divisible by 11.

Try these mentally:(Answers below.)

#1: 11 x 23 =

#2: 11 x 72 =

#3: 11 x 97 =

#4: 11 x 55 =

#5: 11 x 76 =

#6: 11 x 60 =

#7: Sum of the first multiples of 11 smaller than 150.

#8:11 x 3421 =

#9: 11 x 452360 =

#10:  11 x 204673=

#11: 11111 x 11111=

#12: 1111111 x 1111111=

#13: What is the sum of the digits of 11111111 x 11111111?

#14: What is n if 45732n is divisible by 11?

#15: How many solutions for distinct numbers A and B if 4A8B is divisible by 11?

#16: How many solutions for distinct numbers A and B if A7B2 is divisible by 11?  What is their sum?

#17: How many solutions for distinct numbers A and B if A3B41 is divisible by 11?  

Answer key:
#1: 253
#2: 792
#3: 1067
#4: 605
#5: 836
#6: 660
#7 1001 The easiest way to do this is to see that this is an arithmetic sequence, starting with 11, 22, 33...
143 (11 x 13). There are 13 terms and the median is 77 so 13 x 77 or 13 x 7 x 11 = 1001
To do Mathcounts well, you need to know 7 x 11 x 13 =1001 by heart.
#8: 37631
#9: 4975960
#10: 2251403
#11:  123454321
#12:  1234567654321
#13:  The number is 123456787654321 so the sum of the digit is (4 x 7) x 2 + 8 = 64
#14:  n = 5
#15:  A + B = 12 (9, 3); (8, 4); (7, 5); skip (6, 6) because A and B are distinct (5, 7); (4, 8), (3, 9)
#16:  90, 81, 72, 63, 54, 45, 36, 27, 18;  A can’t be “0” so there are 9 pairs and the numbers are equally spaced, an arithmetic sequence. Thus the sum is median times how many numbers.                                      54 x 9 = 486
#17: A + B + 1 = 7 so A + B = 6 ; There are (6, 0);(5, 1); (4, 2); (2, 4); (1, 5) 
A + B + 1 = 7 + 11 = 18; A + B = 17; There are (9, 8) and (8, 9) so total 7 pairs