Mental Math with Tricks and Shortcuts

Mental Math with Tricks and Shortcuts

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                                                 Examples

 

(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 = 1740 – 174 = 1566

multiply then subtract 1/10 of result

 

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 = 15200 – 152 = 15048

multiply and then subtract 1/100 of result

 

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 ————— Remainders (3, 4, and 0 during calculations)

7)1792

256 ———————Answer

 

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

 

USE ESTIMATES
Use estimates to check your answers. Get in the habit of doing this for all calculations.