maths shortcuts
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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
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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
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= 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
multiply 7 by 9 = 63, knowing the last digit (3), just combine two more 3's giving the triple digit answer333
solve 37 x 21
multiply 7 by 21 = 147, knowing the last digit (7), just combine two more 7's giving the triple digit answer777
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
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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
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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:
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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
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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
ADDITION:
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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:
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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
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= 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
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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
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= 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
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= 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
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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:
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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
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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
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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
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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
MEDICINE, BOTANY, BIOLOGY, ZOOLOGY: Latin, Greek, Old English, French, German ROOTS, PREFIXES,SUFFIXES
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
MEDICINE, BOTANY, BIOLOGY, ZOOLOGY: Latin, Greek, Old English, French, German ROOTS, PREFIXES,SUFFIXES
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
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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
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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
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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
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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
CHEMISTRY, CHEMICAL, MECHANICAL ENGINEERING FUELS & COMBUSTION: Latin, Greek, English Roots,Prefixes, Affixes, Suffixes
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
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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
CHEMISTRY, CHEMICAL, MECHANICAL ENGINEERING FUELS & COMBUSTION: Latin, Greek, English Roots,Prefixes, Affixes, Suffixes
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