 # Mental Math with Tricks and Shortcuts

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

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.

## Money and Finance

• \\$1/hour = \\$2000/year** (derivation)

• Earn \\$25/hour? That’s about 50k/year.

• Make 200k/year? That’s about \\$100/hour. This assumes a 40-hour work week.

• \\$20/week = \\$1000/year** (derivation)

• Spend \\$20/week at Starbucks? That’s a cool grand a year.

Rule of 72: Years To Double = 72/Interest Rate (derivation)

• Have an investment growing at 10% interest? It will double in 7.2 years.
• Want your investment to double in 5 years? You need 72/5 or about 15% interest.
• Growing at 2% a week? You’ll double in 72/2 or 36 weeks. You can use this rule for any duration of time, not just years.
• Inflation at 4%? It will halve your money in 72/4 or 18 years.

## Infographic: Mental Math Shortcuts for Poker – Red Chip Poker

https://redchippoker.com/infographic-mental-math-shortcuts-for-poker

Published: Aug 24, 2015Author: Doug HullEstimated Reading Time: 1 min

Infographic: Mental Math Shortcuts for Poker. 6. Poker is a game of math, but not everybody thinks math is a game. Doug Hull set out to change that with Poker Work Book for Math Geeks, a “Sudoku book for poker” that presents the reader with a bevy of poker-specific mathematical exercises to flex their gray matter. If you’re already geeked …

## Arthur T. Benjamin, Ph.D. –

iranischool.com

2017-7-25 · Mental Math: The Mathemagician’s Guide to Lightning Calculation and Amazing Math Tricks. Proli¿ c math and science writer Martin Gardner calls it “the clearest, simplest, most entertaining, and best book yet on the art of calculating in your head.” An avid game player, Professor Benjamin was winner of the American Backgammon Tour in 1997.

Seen 168 times

## The difference between two square numbers is the product of their square roots

If you know the square of a number, you can easily find the square of the next number. Say you know that 10*10=100. So 11*11 = 100+10+11 = 121. So 12*12 = 121+11+12 = 144. So 13*13 = 144+12+13 = 169. And so on.

## Math Trick For Multiplying 2-Digit 3-Digit Numbers

https://www.onlinemathlearning.com/math-trick-multiply-2-digit.html

How to do mental math for multiplying 2-digit numbers and 3-digit numbers quickly and efficiently, teach or reinforce some math concepts and skills, Fun Fast Multiplication Trick, how to multiply 3 digit numbers by 2 digit numbers mentally, …

## 7 Fraction tips and tricks –

www.moomoomath.com

For some people the word "Fractions" is not a happy word. Many people don't like adding fractions,or subtracting fractions because it has always been very confusing or frustrating. Here are 10 proven tips,tricks, or helpful information that can help everyone understand a little bit better.

Seen 115 times

## Convert temperatures

To roughly convert from Celsius to Fahrenheit, multiply by 2 and add 30. From Fahrenheit to Celsius, subtract 30 and divide by 2. (To more precisely convert C to F, multiply by 1.8 and add 32.)

The order is important: The addition/subtraction is always closer to the Fahrenheit side of the conversion. If you forget the order, you know that 32° F = 0° C, so you can test your formula against that.

Or just memorise that room temperature is about 20–22 °C or 68–72 °F, and normal body temperature is around 36-37° C or 97-99° F, depending on several factors.

## Significant Psychological Heuristics

Tversky and Kahneman’s 1974 work, Judgment under Uncertainty: Heuristics and Biases, introduced three key characteristics: representativeness, anchoring and adjustment, and availability.

The representativeness heuristic allows people to judge the likelihood that an object belongs in a general category or class based on how similar the object is to members of that category.

To explain the representativeness heuristic, Tversky and Kahneman provided the example of an individual named Steve, who is “very shy and withdrawn, invariably helpful, but with little interest in people or reality. A meek and tidy soul, he has a need for order and structure, and a passion for detail.” What is the probability that Steve works in a specific occupation (e.g. librarian or doctor)? The researchers concluded that, when asked to judge this probability, individuals would make their judgment based on how similar Steve seemed to the stereotype of the given occupation.

The anchoring and adjustment heuristic allows people to estimate a number by starting at an initial value (the “anchor”) and adjusting that value up or down. However, different initial values lead to different estimates, which are in turn influenced by the initial value.

To demonstrate the anchoring and adjustment heuristic, Tversky and Kahneman asked participants to estimate the percentage of African countries in the UN. They found that, if participants were given an initial estimate as part of the question (for example, is the real percentage higher or lower than 65%?), their answers were rather close to the initial value, thus seeming to be "anchored" to the first value they heard.

The availability heuristic allows people to assess how often an event occurs or how likely it will occur, based on how easily that event can be brought to mind. For example, someone might estimate the percentage of middle-aged people at risk of a heart attack by thinking of the people they know who have had heart attacks.

Tversky and Kahneman's findings led to the development of the heuristics and biases research program. Subsequent works by researchers have introduced a number of other heuristics.

## Who this course is for:

• Those preparing for Math Olympiad, GMAT, GRE, CAT, MAT, SAT, Banking exams and any other Math competitive exam
• Anyone who wants to learn short tricks to crack challenging problems in a target time