Maths Shortcuts

MATHEMATICS: Shortcuts in Multiplication, Division, Addition & Subtraction - secrets in mental mathcalculation speed87

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By eternaltreasures

There are many useful Math techniques, tricks, and secrets that can be valuable in your day to day workor study. Knowing these shortcuts are key to computation speed and accuracy in your MentalMathematical Aptitude. This can be useful in applications such as IQ tests, aptitude tests, job applicationtests, military tests, college entrance tests and so many more uses in the office, workplace, or in yourown home.

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

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= 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 answer111

solve 37 x 9


solve 37 x 21


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 higherfirst 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

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


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


500/25

= (500/100) x 4

= 5 x 4

= 20

750/25

= (750/100) x 4

= 7.5 x 2 x 2

= 30

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

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%


= 132/200

= (132/2)/100

= [100/2 + 32/2]/100 ---> solution by "double division"

= (50 + 16)/100



= 66/100

= 0.66

= 66%


= 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 bytwo 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:

MATHEMATICS:

Calculus - Differential and Integral

CALCULUS: Differentiation and Integration formulas

Electricity, Electrical Circuits, Electronics

MATHEMATICS and ELECTRICAL ENGINEERING: Joint and Combined Variation - Wire resistance, length,area

Fractions, Decimals, Percents

MATHEMATICS: Conversion of commonly used Fractions into Decimals and Percentage equivalents

Geography

GEOGRAPHY, MECHANICAL SHOP THEORY & PRACTICE: Arc length, radius, central angle, Angulardisplacement, velocity

Geometry

GEOMETRY: Area of triangle using Hero's Formula, Structural members, joints, elements

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

GEOMETRY: Perimeter, Circumference, Largest Area, Square, Circular, Rectangular, Fencing, DifferentialCalculus, Maxima



MATHEMATICS: Geometry - Area of Sector of a circle, Radian measure, Arc length, Central angle

MATHEMATICS: Geometry - Centroids of Plane areas and Solids

Ratio, Proportion, Variation

MATHEMATICS: Ratio, Proportion, Variation - Conductor wire Resistance, Reactance, Capacitance,Photograph exposure, Lens

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

MATHEMATICS: Ratio, Proportion, Variation - FREEFALL, DISPLACEMENT, RESISTOR POWER andCURRENT, PENDULUM OSCILLATIONS

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

Statistics

MATHEMATICS: Statistics, Motion, Fencing

Trigonometry

TRIGONOMETRY: Sine law, Cosine law, Pythagorean theorem, Trigonometric identities, Double angleformulas

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

See also:

IQ TESTS - MATH:

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

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

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

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

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

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



IQ TESTS - VOCABULARY APTITUDE:

PREFIXES

IQ TEST: Vocabulary Aptitude, Verbal Ability - Greek, Latin, Old English, German, French PREFIXES ---A-C

IQ TEST: Vocabulary Aptitude, Verbal Ability - Greek, Latin, Old English, German, French PREFIXES ---D-L

IQ TEST: Vocabulary Aptitude, Verbal Ability - Greek, Latin, Old English, German, French PREFIXES ---M-P

IQ TEST: Vocabulary Aptitude, Verbal Ability - Greek, Latin, Old English, German, French PREFIXES ---R-Z

PREFIXES - Botany, Biology

MEDICINE, BOTANY, BIOLOGY, ZOOLOGY: Latin, Greek, Old English, French, German ROOTS, PREFIXES,SUFFIXES

PREFIXES - Chemistry, Chemical Engineering

CHEMISTRY, CHEMICAL ENGINEERING: Periodic Table Elements - Roots, Origin, Etymology of Names

CHEMISTRY, CHEMICAL, MECHANICAL ENGINEERING FUELS & COMBUSTION: Latin, Greek, English Roots,Prefixes, Affixes, Suffixes

PREFIXES - Medicine, Medical Industry, Medical Technology, Hospital


PREFIXES - SI & Metric Number systems, Scientific Notations, Metrology

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

PREFIXES - Zoology, Animal Science


SUFFIXES

IQ TEST: Vocabulary Aptitude, Verbal Ability - Greek, Latin, Old English, German, French SUFFIXES

ROOTS

IQ TEST: Vocabulary Aptitude, Verbal Ability - Greek, Latin, Old English, German, French ROOTS ---A



IQ TEST: Vocabulary Aptitude, Verbal Ability - Greek, Latin, Old English, German, French ROOTS ---B

IQ TEST: Vocabulary Aptitude, Verbal Ability - Greek, Latin, Old English, German, French ROOTS ---Ca-Ci

IQ TEST: Vocabulary Aptitude, Verbal Ability - Greek, Latin, Old English, German, French ROOTS ---Cl-Cy

IQ TEST: Vocabulary Aptitude, Verbal Ability - Greek, Latin, Old English, German, French ROOTS ---D

IQ TEST: Vocabulary Aptitude, Verbal Ability - Greek, Latin, Old English, German, French ROOTS ---E

IQ TEST: Vocabulary Aptitude, Verbal Ability - Greek, Latin, Old English, German, French ROOTS ---F

IQ TEST: Vocabulary Aptitude, Verbal Ability - Greek, Latin, Old English, German, French ROOTS ---G-H

IQ TEST: Vocabulary Aptitude, Verbal Ability - Greek, Latin, Old English, German, French ROOTS ---I, J, K,L

IQ TEST: Vocabulary Aptitude, Verbal Ability - Greek, Latin, Old English, German, French ROOTS ---M

IQ TEST: Vocabulary Aptitude, Verbal Ability - Greek, Latin, Old English, German, French ROOTS ---N-O

IQ TEST: Vocabulary Aptitude, Verbal Ability - Greek, Latin, Old English, German, French ROOTS -----P

IQ TEST: Vocabulary Aptitude, Verbal Ability - Greek, Latin, Old English, German, French ROOTS -----Q-R

IQ TEST: Vocabulary Aptitude, Verbal Ability - Greek, Latin, Old English, German, French ROOTS --- S

IQ TEST: Vocabulary Aptitude, Verbal Ability - Greek, Latin, Old English, German, French ROOTS --- T

IQ TEST: Vocabulary Aptitude, Verbal Ability - Greek, Latin, Old English, German, French ROOTS --- U-Z

See also:

CARS, AUTOMOTIVE MECHANICS:

Troubleshooting and Repair of Internal Combustion Engine (Diesel, Gasoline)

SCIENCE: Energy conversion in an Automobile or car

See also:

SCIENCE: Basic, Fundamental, Elementary Knowledge

EARTH: useful facts, figures, and information

SCIENCE: Energy conversion from hydroelectric power plant to the house



SCIENCE: Energy conversion in an Automobile or car

SCIENCE: Physics - Charles' law, Gay-Lussac's law, gas law

SCIENCE: Physics - Mechanical work (constant force)

SCIENCE: Physics - Mechanical work (variable force)

See also:

GEOGRAPHY, GEOLOGY, NATURE, NATURAL SCIENCE, TRAVEL

GEOGRAPHY, MECHANICAL SHOP THEORY & PRACTICE: Arc length, radius, central angle, Angulardisplacement, velocity

Travel Wonders of the Natural World: list of commonly used terms in Natural Tourism Industry fromGeology & Geography

Continental Travel

NORTH AMERICA: Top 10 Travel Destinations, Tourist Spots and Attractions

EUROPE: Top 10 Travel Destinations, Tourist Spots and Attractions

CARIBBEAN ISLANDS & CENTRAL AMERICA: Top 10 Travel Destinations, Tourist Spots and Attractions

ASIA & MIDDLE EAST: Top 10 Travel Destinations, Tourist Spots and Attractions

SOUTH AMERICA, PACIFIC, ANTARCTICA: Top 10 Travel Destinations, Tourist Spots and Attractions

AFRICA: Top 10 Travel Destinations, Tourist Spots and Attractions

World Travel

WORLD's Top 10 Travel Destinations, Tourist Spots and Attractions

International Country Travel



ARGENTINA: Top 10 Travel Destinations, Tourist Spots and Attractions

ARUBA: Top 10 Travel Destinations, Tourist Spots and Attractions

AUSTRALIA: Top 10 Travel Destinations, Tourist Spots and Attractions

BERMUDA: Top 10 Travel Destinations, Tourist Spots and Attractions

BRAZIL: Top 10 Travel Destinations, Tourist Spots and Attractions

CANADA: Top 10 Travel Destinations, Tourist Spots and Attractions

CANCUN (Quintana Roo, Mexico): Top 10 Travel Destinations, Tourist Spots and Attractions

CHINA: Top 10 Travel Destinations, Tourist Spots and Attractions

CUBA: Top 10 Travel Destinations, Tourist Spots and Attractions

DOMINICAN REPUBLIC: Top 10 Travel Destinations, Tourist Spots and Attractions

FRANCE: Top 10 Travel Destinations, Tourist Spots and Attractions

FORT LAUDERDALE (Florida, USA): Top 10 Travel Destinations, Tourist Spots and Attractions

GERMANY: Top 10 Travel Destinations, Tourist Spots and Attractions

INDIA: Top 10 Travel Destinations, Tourist Spots and Attractions

INDONESIA: Top 10 Travel Destinations, Tourist Spots and Attractions

ITALY: Top 10 Travel Destinations, Tourist Spots and Attractions

MEXICO: Top 10 Travel Destinations, Tourist Spots and Attractions

PHILIPPINES: Top 10 Travel Destinations, Tourist Spots and Attractions

RUSSIA: Top 10 Travel Destinations, Tourist Spots and Attractions

SOUTH KOREA: Top 10 Travel Destinations, Tourist Spots and Attractions

SPAIN: Top 10 Travel Destinations, Tourist Spots and Attractions

UNITED STATES OF AMERICA (USA): Top 10 Travel Destinations, Tourist Spots and Attractions

UNITED KINGDOM (UK): Top 10 Travel Destinations, Tourist Spots and Attractions



See also:

MECHANICAL ENGINEERING:

Design of Machinery, Mechanical Equipment, Machine Elements

MECHANICAL ENGINEERING: Design factor, Uncertainties, Maximum allowable load, Load causing failure

MECHANICAL ENGINEERING: Design of Machine Elements - Rod with a circular cross-section

MECHANICAL ENGINEERING: Reliability of a product, failure, failed products, total manufactured

Fluids, Fluid Mechanics and Dynamics, Fluid Machinery

MECHANICAL ENGINEERING: Fluid Mechanics and Dynamics - Conservation of Mass, mass flow rate,volume flow rate

Fuels and Combustion


General Science

MECHANICAL ENGINEERING: General Science - Atmospheric, Barometric, Gage, Absolute, and Vacuumpressure

Physics, Physical Science

PHYSICS & MACHINE SHOP: Linear speed, Angular velocity - belt, pulley, spindle, motor, rotation,revolution per minute

PHYSICS, PHYSICAL SCIENCE: Atmospheric, Barometric, Gage, Absolute, and Vacuum pressure

Pumps

MECHANICAL ENGINEERING: Pump work - work done by a pump in lifting a fluid

Springs

MECHANICAL ENGINEERING: Springs - connected in series, parallel

Thermodynamics

MECHANICAL ENGINEERING: Thermodynamics - Entropy and the Second Law of Thermodynamics

MECHANICAL ENGINEERING: THERMODYNAMICS - First law of thermodynamics, conservation of energy,internal energy



MECHANICAL ENGINEERING: Thermodynamics - Closed system, Isothermal, piston-cylinder, Boyle's law,Parabolic curve

MECHANICAL ENGINEERING: Thermodynamics - Enthalpy, internal energy, pv work, saturated liquid,vapor, quality

MECHANICAL ENGINEERING: Thermodynamics - PV Work: compression-expansion work

MECHANICAL ENGINEERING: Thermodynamics - Important thermodynamic processes

MECHANICAL ENGINEERING: Thermodynamics - Specific heats at constant pressure, volume, Specificheat ratio, Gas constants

Mechanical Shop Theory & Practice, Machine Shops

MECHANICAL SHOP THEORY & PRACTICE, GEOGRAPHY: Arc length, radius, central angle, Angulardisplacement, velocity

MACHINE SHOP, PHYSICS: Linear speed, Angular velocity - belt, pulley, spindle, motor, rotation,revolution per minute

See also:

CHEMISTRY, CHEMICAL ENGINEERING:

CHEMICAL ENGINEERING: Chemistry - Molecular weight, molar mass, molecular mass, moles,Avogadro's number, molecules

CHEMISTRY, CHEMICAL ENGINEERING: Periodic Table Elements - Roots, Origin, Etymology of Names




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