AMT Math for Mechanics •
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Topics Order of Opera3ons on a Calculator Frac3on to Decimal & Decimal to Frac3ons Add/Subtract/Mul3ply/Divide Frac3ons Numbers – Arabic & Roman Ra3o and Propor3on Percentage Powers and Roots Scien3fic Nota3on Prac3cal Measurements Order of Opera3ons 57 + 14 X 22 + 73 = ? 1635 or 438 or 1387 Order of Opera3ons 438 Order of Opera3ons 1.  () – Parentheses !
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(1 + 2) X 3 – Do (1 + 2) first (1 + 2) X 3 = 3 X 3 2.  Mul3ply and Divide 3.  Addi3on and Subtrac3on 57 + 14 X 22 + 73 = ? 57 + 308 + 73 = 438 Whole Number & Frac3ons •  A number is represented as LeY Side.Right Side LeY Side is the Whole Number !  Right Side is the Frac3on !
•  4 3/8 4 -­‐ Whole Number !  3/8 -­‐ Frac3on !
•  4.375 4 -­‐ Whole Number !  .375 – Frac3on !
•  Convert only Frac3on por3on Frac3on to Decimal •  Frac3on is made up of Top/Bo\om •  Convert frac3on by Top ÷ Bo\om •  3/8 !
3 ÷ 8 = .375 •  Remember / means ÷ •  13/32 !
13 ÷ 32 = .40625 •  4 3/8 Remember not to convert Whole Number !  3 ÷ 8 + 4 !  .375 + 4 = 4.375 !
Decimal to Frac3on •  Frac3on X “Bo\om” •  .25 to 1/4s !
.25 X 4 = 1 or 1/4 •  .375 to 1/8s !
.375 X 8 = 3 or 3/8 •  5.3125 to 1/16s .3125 X 16 = 5 or 5/16 !  5 + 5/16 = 5 5/16 !
Reduce (2/4 Looks Dumb) •  1/2, 1/4, 1/8, 1/16, 1/32, 1/64 Bo\om is a power of 2 !  2 X 2 = 4, 2 X 4 = 8, 2 X 8 = 16, 2 X 16 = 32, 2 X 32 = 64 !
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Divide Top and Bo\om by 2 Stop when Top is an odd number 2/4 = 1/2 20/64 = 10/32 = 5/16 Add/Subtract/Mul3ply/Divide Frac3ons 1.  Convert all numbers to decimal 2.  Use Calculator, follow Order of Opera3on 3.  If needed, convert answer to frac3on !
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1.5 X 3/32 + 0.025 + 0.025 = 1.5 X 0.09375 + 0.025 + 0.025 = 0.140625 + 0.025 + 0.025 = 0.190625 0.190625 X 16 (convert to 1/16ths) = 3.05 or about 3/16 Read a Machinist's Rule Rivet Material Diameter & Radius Diameter & Radius •  Diameter: the length of the line through the center of a circle and touching two points on its edge !
Represents the size of circle •  Size of bolt, rivet, cylinder (bore) and piston !
D or d is used to abbreviate diameter •  Radius: the distance from the center of a circle to a point on the circle !
R or r is used to abbreviate radius Diameter & Radius •  R = ½ D •  D = 2 X R = 2R •  The D (diameter) of a rivet is not to be confused with rivet material D (2017) !
Most of the 3me when we talk about the D of a rivet, we are talking about diameter Rivet Dimensioning •  Diameter is in 1/32’s !
#3 – 3/32, #4-­‐ 4/32 (1/8), #5-­‐ 5/32 •  Length is in 1/16 AN470 – shaY length !  AN426 – full rivet length !
Rivet Designa3on •  (Head Type) (Material) (Diameter in 1/32s) – (Length in 1/16’s) •  Universal head, 2117, 3/32 diameter, 5/16 length !
AN470AD3-­‐5 or MS20470AD3-­‐5 •  Countersunk head, 1100, 1/8 (4/32) diameter, 1/2 (8/16) length !
AN426A4-­‐8 or MS20426A4-­‐8 Shop Head Dimensions •  D – Rivet diameter •  1/2 D – Height •  1 1/2 D – Wide Rivet Diameter •  D – At least 3X the thickest sheet or what the aircraY manufacturer used !  Remember to convert to 1/32s !
•  D ≥ 3 X sheet thickness X 32 (conver(ng to 1/32s) •  Sheet thickness -­‐ 0.025” !
D = 3 X 0.025 X 32 = 2.4 -­‐> #3 •  Sheet thickness – 0.032” !
D = 3 X 0.032 X 32 = 3.072 -­‐> #4 •  Sheet thickness – 0.040” !
D = 3 X 0.040 X 32 = 3.84 -­‐> #4 Rivet Length •  Length = 1.5 X D + Total thickness of all sheets •  Length = (1.5 X D + Sheet 1 + Sheet 2. . .) X 16 X 16 is to convert to 1/16s !  Remember to add all the sheets !
Rivet Length •  2 Sheets of 0.025 – 3/32 rivet (1.5 X 3/32 + 0.025 + 0.025) X 16 = 3.05 -­‐> 4 !  AN470AD3-­‐4 or AN426AD3-­‐4 !
•  2 Sheets of 0.032 – 1/8 rivet (4/32) (1.5 X 1/8+ 0.032 + 0.032) X 16 = 4.024 -­‐> 4 or 5 !  AN470AD4-­‐5 or AN426AD4-­‐5 !
•  2 Sheets of 0.040 – 1/8 rivet (4/32) (1.5 X 1/8+ 0.040 + 0.040) X 16 = 4.28 -­‐> 5 !  AN470AD4-­‐5 or AN426AD4-­‐5 !
Rivet Length •  3 Sheets of 0.025 – 3/32 rivet (1.5 X 3/32 + 0.025 + 0.025 + 0.025) X 16 = 3.45-­‐> 4 !  AN470AD3-­‐4 or AN426AD3-­‐4 !
•  Three sheets -­‐ 0.025, 0.032 and 0.040 sheets What is the required rivet diameter? !  #4 rivet because of 0.040 !  (1.5 X 1/8 + 0.025 + 0.032 + 0.040) X 16 = 4.552-­‐> 5 !  AN470AD4-­‐5 or AN426AD4-­‐5 !
Rivet Dimensions Review •  D – At least 3X the thickest sheet or what the aircraY manufacturer used !  Remember to convert to 1/32s !
•  D ≥ 3 X sheet thickness X 32 (conver3ng to 1/32s) •  Length = 1.5 X D + Total thickness of all sheets !
Remember to convert to 1/16s •  Length = (1.5 X D + Sheet 1 + Sheet 2 . . .) X 16 X 16 is to convert to 1/16s !  Remember to add all the sheets !
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Rivet Problems Available rivets are #3, #4 & #5 Two sheets of 0.032” Two sheets of 0.020” Sheet of 0.025” & 0.032” Sheet of 0.025”, 0.032” & 0.040” Rivet Problems •  Available rivets are #3, #4 & #5 •  Two sheets of 0.032” !
AN470AD4-­‐5 or AN426AD4-­‐5 •  Two sheets of 0.020” !
AN470AD3-­‐3 or AN426AD3-­‐3 •  Sheet of 0.025” & 0.032” !
AN470AD4-­‐4 or AN426AD4-­‐4 •  Sheet of 0.025”, 0.032” & 0.040” !
AN470AD4-­‐5 or AN426AD4-­‐5 Remainders •  5 / 2 = 2 with a remainder of 1 !
Remainder is what is leYover aYer dividing a number •  Remainder = frac3on por3on of the answer X bo\om •
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Remainder Use 100 eggs to fill egg cartons Each carton holds 12 eggs How many eggs are leY (remainder) 100/12 = 8.33333 8 cartons are filled 0.33333 X 12 = 4 (remainder) Check: 8 X 12 + 4 = 100 Numerals •  Arabic Numerals 1, 2, 3, 4, 5, 6, 7, 8, 9 !  Numbers you use every day !
•  Roman Numerals Roman Numerals •  3 – III, 4 – IV, 5 – V, 6 – VI, 7 – VII, 8 – VIII •  9 – IX, 10 – X, 11 -­‐ XI •  2014 – MMXIV !
1000 + 1000 + 10 -­‐1 + 5 Ra3o and Propor3on •  Ra3o -­‐ Special kind of frac3on shows the rela3onship between two values Gear ra3o !  Aspect ra3o for the wing !  Drill press pulley !  Engine compression ra3o !
Ra3os using Algebra Output Speed Input Speed Input Speed X Output Speed Input Speed Input Speed X Output Speed Input Speed = = = Input Teeth Output Teeth Input Teeth X Input Speed X Input Speed Output Teeth Input Teeth Output Teeth Ra3o and Propor3on Output Speed Input Teeth = X Output Teeth Input Speed Ra3o Examples •  Gear reduc3on of 2:1 Reduces the input by a factor of 2 !  Output = Input X 1/2 !
•  Gear reduc3on of 4:3 Reduces the input by a factor of 4:3 !  Output = Input X 3/4 !
•  Gear step up of 2:1 Increases the input by a factor of 2 !  Output = Input X 2 !
Wing Aspect Ra3o Wing Aspect Ra3o •  Wing Aspect Ra3o = the length, or span, of an airplane wing compared to its width, or chord For a rectangular wing !  Non-­‐rectangular wing is based on wing area and span (Span2/Wing Area) !
•  Wing Aspect Ra3o = span / chord !
Rectangular wing •  Remember to use the same unit Convert feet to inches – feet X 12 !  Convert inches to feet – inches / 12 !
Wing Aspect Ra3o •  Span = 30 Y •  Cord = 45 in Convert cord to feet – 1 Y / 12 in !  45 in X 1Y /12 in = 3.75 Y !
•  Wing Aspect Ra3o = Span/Cord !
30 Y /3.75 Y = 8 •  Wing Aspect Ra3o = 8:1 or 8 to 1 •
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Glue Ra3o 5 Parts Resin to 2 Parts Hardener Ra3o is 5:2 Resin = 5/2 X Hardener Hardener = 2/5 X Resin Percentage •  A percentage is another way to represent a number •  100% = 1; 50% = 1/2 = 0.5; 25% = 1/4 = 0.25 •  Conver3ng a frac3on to percentage Convert to decimal !  Mul3ple by 100 !  Add a % symbol to the end !  5/16 = 0.3125; X 100 = 31.25 -­‐> 31.25% !  4 3/8 = 4.375; X 100 = 437.5 -­‐> 437.5% !
Percentage •  Conver3ng a percentage to a form to use on your calculator Percentage / 100 !  75% / 100 = 0.75 !  What is the tax on $80 (9.75% tax rate)? !
•  9.75 / 100 = 0.0975 •  $80 X 0.0975 = $7.8 Percentage •  Find a percentage (First Number / Second Number) X 100 plus % !  What percentage is 4 of 5? !
•  4/5 X 100 = 80 -­‐> 80% !
What percentage is 8 of 4? •  8/4 X 100 = 200 -­‐> 200% Signed Numbers •  Numbers with a “+” are posi3ve Greater than zero !  No sign implies posi3ve !
•  Numbers with a “-­‐” are nega3ve !
Less than zero •  Use your calculator to manipulate numbers Powers •  A power of a base number is the number of 3mes it is mul3plied by itself •  32 = 3 X 3 = 9 •  33 = 3 X 3 X 3 = 27 •  34 = 3 X 3 X 3 X 3 = 81 •  Use your calculator to manipulate numbers •
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Powers Power of 2 – A X A -­‐ square Power of 3 – A X A X A -­‐ cube Power of 4 – A X A X A X A Power of 5 – A X A X A X A X A Power of -­‐2 – 1/(A X A) Power of -­‐3 – 1/(A X A X A) Roots •  The number that must be mul3plied 3mes itself a given number of 3mes to equal a given value •  √ is square root !
Given number of 3mes is 2 •  3 = √9; 4 = √16 ; 15 = √225 !
3 X 3 = 9; 4 X 4 = 16; 15 X 15 = 225 •  3√ is cube root !
Given number of 3mes is 3 •  3 = 3√27 (3 X 3 X 3 = 27) •  Use your calculator to manipulate numbers Scien3fic Nota3on •  Scien3fic nota3on is a mathema3cal procedure in which very large or very small numbers are made more manageable by changing them to numbers between 1 and 10: and raised to a power equal to the number of places the decimal point was moved •  Scien3fic nota3on is also called "powers of ten." Power of Tens Review Names for Power of Ten Fig. 1-33
Scien3fic Nota3on •  Covert from a “+” power Move the decimal point right for each power 3
!  1 X 10 = 1000 4
!  4.56 X 10 = 45,600 !
•  Convert to a “+” power Count the number of 3mes you must move the decimal point to the leY !  Try to get the first part of the number between 1 and 10 4 !  20,000 = 2 X 10
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Scien3fic Nota3on •  Covert from a “-­‐” power Move the decimal point leY for each power -­‐3
!  1 X 10 = 0.001 -­‐4
!  4.56 X 10 = 0.000456 !
•  Convert to a “-­‐” power Count the number of 3mes you must move the decimal point to the right !  Try to get the first part of the number between 1 and 10 -­‐4 !  0.0002 = 2 X 10
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Scien3fic Nota3on •  Use your calculator to manipulate numbers •  If you don’t have a scien3fic nota3on calculator, convert the number and then use your calculator •  Engineering Nota3on is a special form of Scien3fic Nota3on !
Like English, it is powers of 3 •  Thousands, millions and billions •  103, 106 and 109 •  First part of the number between 1 and 1000 •  10,000 (floa3ng point) = 1 X 104 (scien3fic) = 10 X 103 (engineering) TI-­‐30X IIS Nota3on Manipula3on •  Nota3ons Floa3ng Point – FLO !  Scien3fic – SCI !  Engineering – ENG !
•  Do all you calcula3ons and then do the nota3on conversion •  Nota3on conversion (2nd)(SCI/ENG) !  Select nota3on with arrow keys !  (=) !
•  Remember to return to FLO when done Area •  Area is a quan3ty that expresses the extent of a two-­‐dimensional surface or shape in the plane. Size of a wing !  Determine the amount of paint needed !  Determine the amount of carpe3ng needed !  Used in hydraulics calcula3ons !
Area 1X1
1 Square
2X2
4 Squares
4X3
12 Squares
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Area Area is measured in in2, Y2 and cm2 Area for a square = side X side = side2 Area for a rectangle = height X width Area for a triangle = base X height X 1/2 Area for a circle = π X R2 π = 3.1415 !  R is the radius of circle !
•  Distance from the center to the edge of the circle •  Radius = Diameter/2 or R = D/2 !
“Pie R square, Pie R not round” •  1 Y2 = 144 in2 !
12 X 12 •  1 yd2 = 9 Y2 !
3 X 3 Area Trapezoid Area •
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Area = height X (base1 + base2)/2 Area = 5 X (9 + 12)/2 Area = 5 X 21/2 Area = 5 X 10.5 Area = 52.5 Volume Volume •  Volume is the quan3ty of three-­‐dimensional space enclosed by a boundary •  in3; cm3 or cc (cubic cen3meter); Y3 •  Volume of a cube = side X side X side = side3 •  Volume of a box = height X width X depth •  Volume of a cylinder = π X R2 X height Volume •
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Volume 1 Y3 = 1,728 in3 1 liter = 1000 cc 1 in3 = 16.387064 cc 100 in3 = 1.639 liter 1 liter = 61.0237 in³ 1 gallon = 231 in3 Piston Displacement Piston & Engine Displacement •  Engine Displacement and Compression Ra3o •  Cylinder Displacement = π X R2 X height Bore = diameter = D !  R = Bore/2 !  Height = Stroke 2
!  Cylinder displacement = π X (Bore/2) X Stroke !
•  Engine Displacement = Cylinder displacement X # of cylinders •
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Cylinder & Engine Displacement Bore = 4.875” Stroke = 3.875” # of cylinders = 4 Piston displacement = π X (Bore/2)2 X Stroke Piston displacement = π X (4.875”/2)2 X 3.875” 3 !  Piston displacement = 72.329 in
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•  Engine Displacement = Piston displacement X # of cylinders 4 X 72.329 = 289.316 in3 !  Lycoming O-­‐290 !
Engine Compression Ra3o •  Compression ra3o = (Displacement of one cylinder + combus3on chamber space)/
combus3on chamber space Compression Ra3o •  If the volume of a cylinder with the piston at bo\om center is 84 cubic inches and the piston displacement is 70 cubic inches, then the compression ra3o is? •  Displacement of cylinder + combus3on chamber space = 84 •  chamber space = 84 -­‐ displacement of cylinder = 84 – 70 = 14 •  Compression ra3o = 84 / 14 = 6:1 Volume •  How many gallons of fuel will be contained in a rectangular-­‐shaped tank which measures 2 feet in width, 3 feet in length, and 1 foot 8 inches in depth? (7.5 gal = 1 cu Y) •  Tank Volume = 2’ X 3’ X 1’8” •  Tank Volume = 2’ X 3’ X 1 8/12’ •  Tank Volume = 2’ X 3’ X 1.667’ = 10 Y3 •  Fuel = 10 Y3 X (7.5 gallons per 1Y3) •  Fuel = 75 gallons Inverse Percentage •  An engine develops 108 horsepower at 87%. What is the horsepower at 100%? 100 % = Sample value/Sample % !  100% HP = 108 HP / 87% (convert to decimal) !  100% HP = 108 HP / 0.87 = 124 HP !
Ra3o •  An airplane flying a distance of 750 miles used 60 gallons of gasoline. How many gallons will it need to travel 2,500 miles? •  First find the miles to gallon ra3o •  Mul3ple the new distance by the ra3o •  60 gallons / 750 miles = 0.08 gallon/mile •  2,500 miles X 0.08 gallon/mile = 200 gallons Engine Displacement •  The total piston displacement of a specific engine is: a. dependent on the compression ra3o. b. the volume displaced by all the pistons during one revolu3on of the crankshaY. c. the total volume of all the cylinders. Answer: b A four-­‐stoke engine cycles through all cylinders in one revolu3on c is wrong because it includes combus3on chamber Binary Number System •  Binary numbers are used by digital computers !
Only two numbers – 1 & 0 (On and Off) •  0 (bin) = 0 (dec); 1 (bin) = 1 (dec) •  For larger numbers, you must add extra digits 10 (bin) = 2 (dec); 11 (bin) = 3 (dec) !  100 (bin) = 4 (dec); 101 (bin) = 5 (dec) !  110 (bin) = 6 (dec); 111 (bin) = 7 (dec) !  1000 (bin) = 8 (dec); 1001 (bin) = 9 (dec) !  1010 (bin) = 10 (dec); 1011 (bin) = 11 (dec) !  1100 (bin) = 12 (dec); 1101(bin) = 13 (dec) !  1110 (bin) = 14 (dec); 1111(bin) = 15 (dec) !  10000 (bin) = 16 (dec) !
Binary Number System •  Power of 2 1 (bin) = 1 (dec) = 20 1 !  10 (bin) = 2 (dec) = 2
2 !  100 (bin) = 4 (dec) = 2
3 !  1000 (bin) = 8 (dec) = 2
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•  Convert to decimal digit by digit What is 1111 (bin) in decimal? !  1111 = 1000 + 1000 + 100 + 10 + 1 !  1111 (bin) = 8 + 4 + 2 + 1 = 15 (dec) !
•  "There are only 10 types of people in the world: those who understand binary, and those who don't." Profit Margin •  FAA ques3on 8422 in on profit margin •  Memorize the answer