Monday, October 15, 2012

Shortcuts in Multiplication, Division, Addition & Subtraction


These are some of the practical secrets in mental math calculation speed. Knowing these can help you figure out answers to most of life's daily activities involving numbers. There are many useful Math techniques, tricks, and secrets that can be valuable in your day to day work or study. Knowing these shortcuts are key to computation speed and accuracy in your Mental Mathematical Aptitude. This can be useful in applications such as IQ tests, aptitude tests, job application tests, military tests, college entrance tests and so many more uses in the office, workplace, or in your own home.


SHORTCUTS IN MULTIPLICATION:

Multiplication using multiples
12 x 15
= 12 x 5 x 3
= 60 x 3
= 180

Multiplication by distribution
12 x 17
= (12 x 10) + (12 x 7) ---> 12 is multiplied to both 10 & 7
= 120 + 84
= 204

Multiplication by "giving and taking"
12 x 47
= 12 x (50 - 3)
= (12 x 50) - (12 x 3)
= 600 - 36
= 564

Multiplication by 5 --> take the half(0.5) then multiply by 10
428 x 5
= (428 x 1/2) x 10 = 428 x 0.5 x 10
= 214 x 10
= 2140

Multiplication by 10  ---> just move the decimal point one place to the right
14 x 10
= 140   ---> added one zero

Multiplication by 50 ---> take the half(0.5) then multiply by 100
18 x 50
= (18/2) x 100 = 18 x 0.5 x 100
= 9 x 100
= 900

Multiplication by 100 ---> move the decimal point two places to the right
42 x 100
= 4200  ---> added two zeroes

Multiplication by 500 ---> take the half(0.5) then multiply by 1000
21 x 500
= 21/2 x 1000
= 10.5 x 1000
= 10500

Multiplication by 25 ---> use the analogy $1 = 4 x 25 cents
25 x 14
= (25 x 10) + (25 x 4) ---> 250 + 100 ---> $2.50 + $1
= 350

Multiplication by 25 ---> divide by 4 then multiply by 100
36 x 25
= (36/4) x 100
= 9 x 100
= 900 

Multiplication by 11 if sum of digits is less than 10
72 x 11
= 7_2  ---> the middle term = 7 + 2 = 9
= place the middle term 9 between 7 & 2
= 792

Multiplication by 11 if sum of digits is greater than 10
87 x 11
= 8_7  ---> the middle term = 8 + 7 = 15
because the middle term is greater than 10, use 5 then
add 1 to the first term 8, which leads to the answer of
= 957

Multiplication of 37 by the 3, 6, 9 until 27 series of numbers --> the "triple effect"           
solve 37 x 3               
multiply 7 by 3 = 21, knowing the last digit (1), just combine two more 1's giving the triple digit answer 111           
               
solve 37 x 9               
multiply 7 by 9 = 63, knowing the last digit (3), just combine two more 3's giving the triple digit answer 333                               
solve 37 x 21               
multiply 7 by 21 = 147, knowing the last digit (7), just combine two more 7's giving the triple digit answer 777                           

Multiplication of the "dozen teens" group of numbers --
(i.e. 12, 13, 14, 15, 16, 17, 18, 19) by ANY of the numbers within the group:
solve 14 x 17               
4 x 7 = 28;  remember 8, carry 2               
14 + 7 = 21               
add 21 to whats is carried (2)               
giving the result 23               
form the answer by combinig 23 to what is remembered (8)               
giving the answer 238   

Multiplication of numbers ending in 5 with difference of 10
45 x 35 
first term = [(4 + 1) x 3] = 15; (4 is the first digit of 45 and 3 is the first digit of 35 --> add 1 to the higher first digit which is 4 in this case, then multiply the result by 3, giving 15)
last term = 75
combining the first term and last term,
= 1575

75 x 85
first term = (8 + 1) x 7 = 63
last term = 75
combining first and last terms,
= 6375

15 x 25
= 375

Multiplication of numbers ending in 5 with the same first terms (square of a number)
25 x 25
first term = (2 + 1) x 2 = 6
last term = 25
answer = 625  ---> square of 25

75 x 75
first term = (7 + 1) x 7 = 56
last term = 25
answer = 5625 ---> 75 squared



SHORTCUTS IN DIVISION:

Division by parts ---> imagine dividing $874 between two persons
874/2
= 800/2 + 74/2
= 400 + 37
= 437

Division using the factors of the divisor: "double division"
70/14
= (70/7)/2 ---> 7 and 2 are the factors of 14
= 10/2
= 5

Division by using fractions:
132/2
= (100/2 + 32/2) ---> break down into two fractions
= (50 + 16)
= 66

Division by 5 ---> divide by 100 then multiply by 20
1400/5
= (1400/100) x 20
= 14 x 20
= 280

Division by 10  ---> move the decimal point one place to the left
0.5/10
= 0.05  ---> 5% is 50% divided by ten

Division by 50 ---> divide by 100 then multiply by 2
2100/50
= (2100/100) x 2
= 21 x 2
= 42

700/50
= (700/100) x 2
= 7 x 2
= 14

Division by 100 ---> move the decimal point two places to the left
25/100
= 0.25

Division by 500 ---> divide by 100 then multiply by 0.2
17/500
= (17/100) x 0.2
= 0.17 x 0.2
= 0.034

Division by 25 ---> divide by 100 then multiply by 4
500/25
= (500/100) x 4
= 5 x 4
= 20

750/25
= (750/100) x 4
= 7.5 x 2 x 2
= 30



SHORTCUTS IN ADDITION:

Addition of numbers close to multiples of ten (e.g. 19, 29, 89, 99 etc.)
116 + 39
= 116 + (40 - 1)
= 116 + 40 - 1  ---> add 40 then subtract 1
= 156 - 1
= 155

116 + 97
= 116 + (100 - 3)
= 116 + 100 - 3   ---> add 100 then subtract 3
= 216 - 3
= 213

Addition of decimals
12.5 + 6.25
= (12 + 0.5) + (6 + 0.25)
= 12 + 6 + 0.5 + 0.25   ---> add the integers then the decimals
= 18 + 0.5 + 0.25
= 18.75



SHORTCUTS IN SUBTRACTION:

Subtraction by numbers close to 100, 200, 300, 400, etc.
250 - 96
= 250 - (100 - 4)
= 250 - 100 + 4    ---> subtract 100 then add 4
= 150 + 4
= 154

250 - 196
= 250 - (200 - 4) 
= 250 - 200 + 4    ---> subtract 200 then add 4
= 50 + 4
= 54

216 - 61
= 216 - (100 - 39)
= 216 - 100 + 39
= 116 + (40 - 1)  ---> now the operation is addition, which is much easier
= 156 - 1
= 155

Subtraction of decimals
47 - 9.9
= 47 - (9 + 0.9) ---> "double subtraction"
= 47 - 9 - 0.9   ---> subtract the integer first then the decimal
= 38 - 0.9
= 37.1

18.3 - 0.8
= 18 + 0.3 - 0.8
= (18 - 0.8) + 0.3  ---> subtract 0.8 from 18 first
= 17.2 + 0.3
= 17.5



WORKING ON PERCENTAGES:

30% of 120
= 10% x 3 x 120 ---> it is much easier working with tens (10%)
= 10% x 120 x 3
= 12 x 3
= 36

five percent of a number: 5%
360 x 5%
= 360 x 10%/2   ---> take the 10% and divide by 2
= 36/2
= 18

360 x 5%
= 360 x 50%/10   ---> take the half(0.5) and divide by 10
= (360/2)/10
= 180/10
= 18

ninety percent of a number: 90%
90% of 700
= (100% - 10%) x 700
= (100% x 700) - (10% x 700)  ---> 100% minus 10% of the number
= 700 - 70
= 630

What is 18 as a percentage of 50?
= 18/50 
= (18/100) x 2  ---> method: division by 50 (explained above)
= 0.18 x 2
= 0.36
= 36%

What is 132 as a percentage of 200?
= 132/200
= (132/2)/100
= [100/2 + 32/2]/100  ---> solution by "double division"
= (50 + 16)/100
= 66/100
= 0.66
= 66%

What is 270 as a percentage of 300?
= 270/300
= [(270/3)/100]  ---> "double division" (using the factors of 300)
= 90/100
= 90%

What is 17 as percentage of 500?
= 17/500
= (17/50)/10
= (17/100) x 2/10   ---> solution using the procedure: division by 50
= (0.17 x 2)/10
= 0.34/10
= 0.034
= 3.4 %

percentages close to 100:
95% of 700
= (100% - 5%) x 700
= (100% x 700) - (5% x 700)
= 700 - (10% x 700/2)  -------> 5% is 10%/2
= 700 - 70/2
= 700 - 35
= 665

percentages less than 10 percent:
3% of 70
= (3/100) x 70
= (70/100) x 3  ---> divide by 100 then multiply the percent value
= 0.7 x 3
= 2.1



DECIMALS:

To convert or express percentages as decimals, divide by 100 or simply just move the decimal point by two places to the left of the given number, thus:

1% = 1/100 = 0.01
2% = 2/100 = 0.02 = 1/50
3% = 3/100 = 0.03
4% = 4/100 = 0.04 = 1/25
5% = 5/100 = 0.05 = 1/20
6.25% = 6.25/100 = 0.0625 = 1/16
7% = 7/100 = 0.07
7.5% = 7.5/100 = 0.075
10% = 10/100 = 0.1 = 1/10
12.5% = 12.5/100 = 0.125 = 1/8
20% = 0.2 = 1/5
21% = 0.21
25% = 0.25 = 1/4
30% = 0.3 = 3/10
33.33% = 33.33/100 = 0.3333 = 1/3
37.5% = 0.375 = 3/8
40% = 0.4 = 2/5
50% = 0.5 = 1/2
60% = 0.6 = 3/5
62.5% = 0.625 = 5/8
66.66% = 66.66/100 = 2/3
75% = 0.75 = 3/4
80% = 0.8 = 4/5
87.5% = 0.875 = 7/8
100% = 1
125% = 1.25 = 1 1/4
150% = 1.5 = 1 1/2
200% = 2



FRACTIONS:

What is three quarters of 80?
= 3/4 x 80
= (80/4) x 3  ---> divide by 4 then multiply by 3
= 20 x 3
= 60

How many quarters in two and a half?
2.5/.25
= 10  ---> there are 10 quarters in $2.50


Improper fractions:

3/2 = 1 1/2 = 1.5 = 150%

4/3 = 1 1/3 = 1.3333 = 133.33%  ---> useful number for volume of sphere, etc.

9/5 = 1 4/5 = 1.8 = 180%  ---> conversion factor for Celsius/Fahrenheit temperatures


V = 4/3 pi * r^3

where:

V = volume of sphere
r = radius of sphere


F = (1.8 C) + 32

where:

F = temperature in Fahrenheit
C = temperature in Celsius

See Also: 

.
GEOMETRY: Perimeter, Circumference, Largest Area, Square, Circle, Fencing, Differential Calculus, Maxima, minima

GEOMETRY: Interior, Exterior angles of any polygon, ratio of the angles

IQ TEST: Math, Mixture, Solutions, Concentration, Coin Problems

IQ TEST: Math, Fractions, Series, Sequence, Military Time, Clock, Days of the week

IQ TEST: Math, Pie, Cake, Divisions, Total, Sum, Parts, Age Problems

IQ TEST: Math, Percentages, Markdowns, Discounts, Original, discounted Price, Ratio, Proportion

IQ TEST: Math, Working together, Job, Work problems at the same rate, inverse proportions

IQ TEST: Math, Probability, Permutations, Repetitions, Ordered Combinations, Exclusive events, Number of digits

MATHEMATICS: Statistics, Motion, Fencing

MATHEMATICS: Ratio, Proportion, Variation - Force of Gravitation, Weight, surface Illumination intensity, current, resistance


MATHEMATICS: Ratio, Proportion, Variation - Conductor wire Resistance, Reactance, Capacitance, Photograph exposure time, F stop of lens

MATHEMATICS: Ratio, Proportion, Variation - Power, Resistance of Conductor, Volume, Turbine Flowrate

MATHEMATICS: Ratio, Proportion, Variation - FREEFALL, DISPLACEMENT, RESISTOR POWER and CURRENT, PENDULUM OSCILLATIONS and LENGTH

Trigonometry - Law of sines, Bearing, Ship, Motorboat, Navy, Marines, Sailor

TRIGONOMETRY: Sine law, Law of cosines, pythagorean identities, trigonometric identities, double angle formulas, reduction formulas, power-reducing formulas, common right triangle combination sides


Conversion of commonly used Fractions into Decimals and Percentage equivalents

Greek and Latin Prefixes, Numerals used in English numerical system, counting, ordering, arrangement, measurement

How to convert binary to decimal and decimal to binary number


 

4 comments:

  1. Thanks for posting these. They were fun to look at. I'll come back again to digest more. I teach kids shortcuts in how to work smarter, not harder. Have you read the book by Bill Nye the Science Guy, Secrets of Mental Math? I'm discovering there are more tricks out there. Anyway, thanks for your posts!

    ReplyDelete
  2. That "double divisor" is sweet! Just what I needed tonight for my students tomorrow! Thanks again.

    ReplyDelete
  3. Now i realize that mathematics is fun activity:-)

    ReplyDelete