Addition
Technique: Add left to right
326 + 678 + 245 + 567 = 900, 1100, 1600, 1620, 1690, 1730, 1790, 1804,
& 1816
Note: Look for opportunities to combine numbers to reduce the number
of steps to the solution. This was done with 6+8 = 14 and 5+7 = 12 above. Look for opportunities to form
10, 100, 1000, and etc. between numbers that are not necessarily next to each other. Practice!
Multiplication & Squaring
Some useful formulae
(a+b)2 = a2 + 2ab + b2 492 = (40 +
9) 2 = 1600 + 720 + 81 = 2401
(a-b) 2 = a2 –2ab + b2 562 = (60 –
4) 2 = 3600 – 480 + 16 = 3136
(a+b) (a-b) = a2 – b2 64 x 56 = (60 – 4) (60 + 4) = 3600 – 16 = 3584
(a+b) (c+d) = (ac + ad) + (bc + bd) 23 x 34 = (20 + 3) (30 + 4) =
600 + 80 + 90 + 12 = 782
(a+b) (c-d) = (ac – ad) + (bc – bd) 34 x 78 = (30 + 4) (80 – 2) =
2400 –60 + 320 – 8 = 2652
(a-b) (c-d) = (ac – ad) – (bc – bd) 67 x 86 = (70 – 3) (90 – 4) =
6300 – 280 – 270 + 12 = 5762
X = 1 to 9 & Y = Any Number
(X5) 2 = 100X(X+1) + 25 652 = 600(7) + 25 = 4200 + 25 = 4225
25 x Y = (Y x 100)/4 25 x 76 = 7600/4 = 1900
50 x Y = (Y x 100)/2 50 x 67 = 6700/2 = 3350
75 x Y = 3(Y x 100)/4 75 x 58 = (5800 x 3)/4 = 17400/4 = 4350
Square any Two Digit Number (a = 10’s digit & b = 1’s digit)
(ab)2 = 100a2 + 20(a x b) + b2 672 =
100(36) + 20(42) + 49 = 4489
Multiply any Two 2 Digit Numbers (a & c = 10’s digit, b & d = 1’a digit)
ab x cd = 100(a x c) + 10[(b x c) + (a x d)] + (b x d) 53 x 68 =
3000 + 580 + 24 = 3604
Tricks using (X5) 2
(X5 – a) 2 = (X5) 2 – X5(2a) + a2) 632 = (65 –
2) 2 = 4225 – 260 + 4) = 3969
(X5 + a) 2 = (X5) 2 + X5(2a) + a2) 672 = (65 +
2) 2 = 4225 + 260 + 4) = 4489
Squaring Numbers 52 to 99
a2 = [a - (100 – a)]100 + (100 – a) 2 932 = (93 –
7)100 + 72 = 8649
Squaring Numbers 101 to 148
a2 = [a + (a – 100)]100 + (a – 100) 2 1082 = (108
+ 8)100 + 82 = 11664
Squaring Numbers near 1000
a2 = [a – (1000 – a)]1000 + (1000 – a) 2 9942 = (994 –
6)1000 + 6 2 = 988036
a2 = [a + (a – 1000)]1000 + (a – 1000) 2 10072 = (1007
+ 7)1000 + 72 = 1014049
Squaring Numbers that end in 1
a2 = (a – 1) 2 + 2a - 1 61 2 = 60 2 + 122 -
1 = 3600 + 121 = 3721
Squaring Numbers that end in 4
a2 = (a + 1)2 – (2a + 1) 442 = 452 – (88 +
1) = 2025 - 89 = 1936
Squaring Numbers that end in 6
a2 = (a - 1)2 + (2a - 1) 562 = 552 + 112 -
1 = 3025 + 111 = 3136
Squaring Numbers that end in 9
a2 = (a + 1) 2 – (2a + 1) 792 = 802 – (158
+ 1) = 6400 – 159 = 6341
Using Squares to Help Multiply
Two Numbers that Differ by 1
If a > b then a x b = a2 – a 35 x 34 = 1225 – 35 = 1190
If a < b then a x b = a2 + a 35 x 36 = 1225 + 35 = 1260
Two Numbers that Differ by 2
a x b = [(a + b)/2]2 -1 26 x 28 = 272 -1 = 729 – 1 = 728
Two Numbers that Differ by 3 (a < b)
a x b = (a + 1)2 + (a – 1) 26 x 29 = 272 + 25 = 729 + 25 = 754
Two Numbers that Differ by 4
a x b = [(a + b)/2]2 – 4 64 x 68 = 662 – 4 = 4356 - 4 = 4352
Two Numbers that Differ by 6
a x b = [(a + b)/2]2 – 9 51 x 57 = 542 – 9 = 2916 – 9 = 2907
Two Numbers that Differ by an Even Number: a < b and c = (b –
a)/2
a x b = [(a + b)/2]2 – c2 59 x 67 = 632 – 42 = 3969 –
16 = 3953
Two Numbers that Differ by an Odd Number: a < b and c = [1 + (b
– a)]/2
a x b = (a + c)2 – [b + (c –1)2] 79 x 92 = 862 – (92 +
36) = 7396 – 128 = 7268
Other Multiplying Techniques
Multiplying by 11
a x 11 = a + 10a 76 x 11 = 76 + 760 = 836
a x 11 = If a > 9 insert a 0 between digits and
add sum of digits x 10 76 x 11 = 706 + 130 = 836
Multiplying by Other Two Digit Numbers Ending in 1 (X = 1 to 9)
a x X1 = a + X0a 63 x 41 = 63 + (40 x 63) = 63 + 2520 = 2583
Multiplying with Numbers Ending in 5 (X = 1 to 9)
a x X5 = a/2 x 2(X5) 83 x 45 = 41.5 x 90 = 415 x 9 = 3735
Multiplying by 15
a x 15 = (a + a/2) x 10 77 x 15 = (77 + 38.5) x 10 = 1155
Multiplying by 45
a x 45 = 50a – 50a/10 59 x 45 = 2950 – 295 = 2655
Multiplying by 55
a x 55 = 50a + 50a/10 67 x 55 = 3350 + 335 = 3685
Multiplying by Two Digit Numbers that End in 9 (X = 1 to 9)
a x X9 = (X9 + 1)a – a 47 x 29 = (30 x 47) – 47 = 1410 – 47 = 1363
Multiplying by Multiples of 9 (b = multiple of 9 up to 9 x 9)
a x b = round b up to next highest 0 29 x 54 = 29 x 60 – (29 x
60)/10
multiply then subtract 1/10 of result = 1740 – 174 = 1566
Multiplying by Multiples of 99 (b = multiple of 99 up to 99 x 10)
a x b = round up to next highest 0 38 x 396 = 38 x 400 – (38 x
400)/100
multiply and then subtract 1/100 of result = 15200 – 152 = 15048
SUBTRACTION
Techniques:
1) Learn to calculate from left to right: 1427 – 698 = (800 – 100)
+ (30 – 10) + 9 = 729
2) Think in terms of what number added to the smaller equals the
larger: 785 – 342 = 443 (left to right)
3) Add a number to the larger to round to next highest 0; then add
same number to the smaller and subtract:
496 – 279 = (496 + 4) – (279 + 4) = 500 – 283 = 217 (left to right)
4) Add a number to the smaller to round to the next highest 10,
100, 1000 and etc.; then subtract and add
the same number to the result to get the answer: 721 – 587 = 721 –
(587 + 13) = (721 – 600) + 13 = 134
5) Subtract a number from each number and then subtract: 829 – 534
= 795 – 500 = 295
DIVISION
Techniques: Examples
Divide by parts of divisor one at a time: 1344/24 = (1344/6)/4 =
224/4 = 56
Method of Short Division
340 ß Remainders (3, 4, and 0 during calculations)
7)1792
256 ßAnswer
Divide both divisor and dividend by same number to get a short
division problem
10
972/27 divide both by 9 = 3)108
36
Dividing by 5, 50, 500, and etc.: Multiply by 2 and then divide by
10, 100, 1000, and etc.
365/5 = 730/10 = 73
Dividing by 25, 250, 2500, and etc.: Multiply by 4 and divide by
100, 1000, 10000, and etc.
Dividing by 125: Multiply by 8 and then divide by 1000
36125/125 = 289000/1000 = 289
It can be divided evenly by:
2 if the number ends in 0, 2, 4, 6, and 8
3 if the sum of the digits in the number is divisible by 3
4 if the number ends in 00 or a 2 digit number divisible by 4
5 if the number ends in 0 or 5
6 if the number is even and the sum of the digits is divisible by
3
7 sorry, you must just try this one
8 if the last three digits are 000 or divisible by 8
9 if the sum of the digits are divisible by 9
10 if the number ends in 0
11 if the number has an even number of digits that are all the
same: 33, 4444, 777777, and etc.
11 if, beginning from the right, subtracting the smaller of the
sums of the even digits and odd digits
results in a number equal to 0 or divisible by 11:
406857/11 Even = 15 Odd = 15 = 0
1049807/11 Even = 9 Odd = 20 = 11
12 if test for divisibility by 3 & 4 work
15 if test for divisibility by 3 & 5 work
Others by using tests above in different multiplication
combinations
SQUARE ROOTS
Examples
Separate the number into groups of 2 digits or less beginning from
the right
(66049)1/2 6 60 49
What number can be squared and be less than 6 = 2 with a reminder
of 2
Bring down the second group of digits next to the remainder to
give 260
Double the first part of the answer to get 4, divide into 26 of the 260
to get 6 as a trial number
Use 4 & 6 to get 46 and multiply by 6 to equal 276 which is
larger than 260, therefore try 5
Use 4 & 5 to get 45, 45 x 5 = 225, 260 – 225 = 35, bring down
the 49 to get 3549
Double the 25 to get 50, divide 50 into 354 to get 7 as a trial second
part of divisor
Use 50 & 7 to get 507 and multiply 507 x 7 to get 3549 with no
remainder.
See complete calculations below:
6 60 49 (257 = Answer
4
45)260
225
507)3549
3549
0
CUBE ROOTS
Memorize the following:
Cube of 1 = 1, 2 = 8, 3 = 27, 4 = 64, 5 = 125, 6 = 216, 7 = 343, 8 = 512, 9 = 729
Note: no result ends in the same digit
(300763)1/3 Divide in to groups of 3 from right = 300 763
Note: the number ends in a 3. Last digit of cube will be 7 if this is
a cube without a
remainder
Since 7 cubed = 343 and 6 cubed = 216, the left most group of 300
is between them and we must use
the smaller, or 6.
The answer is 67 This method works up
to 1,000,000 for true cubes
Cube Roots the Long Way
(636056)1/3
What number cubed is less than 636 = 8. Put 8 down as first part
of answer
Square 8 for 64 and multiply by 300 = 19200. Divide into 114056 =
6, add the 8 and 6 = 14
Multiply 14 x 30 = 420, add 420 to 19200 = 19620, square the 6 and
add to 19620 = 19656
If 6 x 19656 is less than 124056, then it is not necessary to use
lower number.
636 056 = 863
512
19656)124056
117936
7120
USE ESTIMATES
Use estimates to check your answers. Get in the habit of doing
this for all calculations.
NOTE:
Considerable care has been taken to eliminate errors in this document, but the
author does notguarantee that the document is error free by implication or in
fact
