**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} = a^{2} + 2ab + b^{2} 49^{2} = (40 + 9) ^{2} = 1600 + 720 + 81 = **2401**

(a-b) ^{2} = a^{2} –2ab + b^{2} 56^{2} = (60 – 4) ^{2} = 3600 – 480 + 16 = **3136**

(a+b) (a-b) = a^{2} – b^{2} 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 65^{2} = 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} = 100a^{2} + 20(a x b) + b^{2} 67^{2} = 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) + a^{2}) 63^{2} = (65 – 2) ^{2} = 4225 – 260 + 4) = **3969**

(X5 + a) ^{2} = (X5) ^{2} + X5(2a) + a^{2}) 67^{2} = (65 + 2) ^{2} = 4225 + 260 + 4) = **4489**

Squaring Numbers 52 to 99

a^{2} = [a – (100 – a)]100 + (100 – a) ^{2} 93^{2} = (93 – 7)100 + 7^{2} = **8649**

Squaring Numbers 101 to 148

a^{2} = [a + (a – 100)]100 + (a – 100) ^{2} 108^{2} = (108 + 8)100 + 8^{2} = **11664**

Squaring Numbers near 1000

a^{2} = [a – (1000 – a)]1000 + (1000 – a) ^{2} 994^{2} = (994 – 6)1000 + 6 ^{2} = **988036**

a^{2} = [a + (a – 1000)]1000 + (a – 1000) ^{2} 1007^{2} = (1007 + 7)1000 + 7^{2} = **1014049**

Squaring Numbers that end in 1

a^{2} = (a – 1) ^{2} + 2a – 1 61 ^{2} = 60 ^{2} + 122 – 1 = 3600 + 121 = **3721**

Squaring Numbers that end in 4

a^{2} = (a + 1)^{2} – (2a + 1) 44^{2} = 45^{2} – (88 + 1) = 2025 – 89 = **1936**

Squaring Numbers that end in 6

a^{2} = (a – 1)^{2} + (2a – 1) 56^{2} = 55^{2} + 112 – 1 = 3025 + 111 = **3136**

Squaring Numbers that end in 9

a^{2} = (a + 1) ^{2} – (2a + 1) 79^{2} = 80^{2} – (158 + 1) = 6400 – 159 = **6341**

** **

**Using Squares to Help Multiply**

Two Numbers that Differ by 1

If a > b then a x b = a^{2} – a 35 x 34 = 1225 – 35 = **1190**

If a < b then a x b = a^{2} + a 35 x 36 = 1225 + 35 = **1260**

Two Numbers that Differ by 2

a x b = [(a + b)/2]^{2} -1 26 x 28 = 27^{2} -1 = 729 – 1 = **728**

Two Numbers that Differ by 3 (a < b)

a x b = (a + 1)^{2} + (a – 1) 26 x 29 = 27^{2} + 25 = 729 + 25 = **754**

Two Numbers that Differ by 4

a x b = [(a + b)/2]^{2} – 4 64 x 68 = 66^{2} – 4 = 4356 – 4 = **4352**

Two Numbers that Differ by 6

a x b = [(a + b)/2]^{2} – 9 51 x 57 = 54^{2} – 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} – c^{2} 59 x 67 = 63^{2} – 4^{2} = 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 = 86^{2} – (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.