math calculations for hers raters 1 why worry 2
TRANSCRIPT
Math Calculations For HERS Raters
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Why Worry
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Why Worry
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Why Worry
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Calculating Areas
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Calculating Areas
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Calculating Areas
Other Complex Shapes
Insulated Hip Roof
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Develop a Sequence for Problem Solving
1. Convert Measurements to Decimals: 1 foot 3” = 1.25 feet - - 0.5 = 6” etc.2. Simplify Shapes to:
Rectangles or Squares Right Triangles (one angle is 90 degrees) Any Shape where the Formula is Known
3. Carefully Evaluate the Known Information4. Solve the Problem (Answer the Question)5. Convert your answer to feet & inches OR
decimals as the test question requires.
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Make Calculations in DecimalsConvert Inches to Feet by:inches / 12 = decimal feet
Remember: Convert your answer to feet & inches OR decimals as the test
question requires.
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Convert Measurements to Decimals
Common Decimals EquivalenceI inch = 0.0833 inches = 0.254 inches = 0.336 inches = 0.508 inches = 0.679 inches = 0.75
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Convert Measurements to Decimals
Example4 ft 8 inches
8 inches = 1/12 = 0.67
Answer4.67 feet
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Convert Measurements to Decimal Feet
Example6.25 feet
0.25 * 12 = 3 inches
Answer6 ft 3 inches
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Convert Measurements to Feet/Inches
Your Turn- Conversions
Convert to Decimal Feet: Convert to Feet/Inches
One foot- two inches = 3. 33 =
Seven inches = 1. 92 =
One foot – five inches = 4. 67 =
Two feet – nine inches = 6. 08 =
Three feet – ten inches = 5. 50 =
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Simplify The Shape
Hint: Look for Rectangles and Right Triangles
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Hint: Look for Rectangles and Right Triangles
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Simplify The Shape
Hint: Look for Rectangles and Right Triangles
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Simplify The Shape
Hint: Look for Rectangles and Right Triangles
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Simplify The Shape
Your Turn- Simplify This Shape
Hint: Look for Rectangles and Right Triangles
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Hint: Look for Rectangles and Right Triangles
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Your Turn- Simplify This Shape
Math Calculations
Right Triangles• Why Right Triangles
–Calculate Length for Rafters
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Right Triangle- Pythagorean Theorem
90°
AC
B
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90°
AC
B
A2 + B2 = C2
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(A2) 3 X 3 = 9, (B2) 4 X 4 = 16, (C2) 9 + 16 = 25 C = √25 = 5
Right Triangle- Pythagorean Theorem
90°
A C
B
A2 + B2 = C2
Solve for: _____________________________
A = √ C2 - B2 _____________________________
B = √ C2 - A2 ______________________________
C = √ A2 + B2 Watch for change in
Sign !!!!
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Right Triangle- Pythagorean Theorem
B
A2 + B2 = C2
25
(A2) 3 X 3 = 9(B2) 4 X 4 = 16(C2) 9 + 16 = 25
C = √25 = 5
90°
A C
Right Triangle- Pythagorean Theorem
90°
4’ 3”Raft Length ?
15’ 8”
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Right Triangle- Sample Calculation
90°
4’ 3” Raft Length ?
15’ 8”
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Right Triangle- Sample Calculation
3 inches = 3/12 ft = 0.25 ft4’ 3” = 4.25 ft
8 inches = 8/12 ft = 0.67 ft15’ 8’ = 15.67 ft
90°
4’ 3” Raft Length ?
15’ 8”
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Right Triangle- Sample Calculation
A2 = 4.25 x 4.25 = 18.06
B2 = 15.67 x 15.67 = 245.55
C2 = 18.06 + 245.55 = 263.61
C = S263.61 = 16.24 ft
Math Calculations
Ratios• Why Ratios
–Using Roof Pitch in Calculations
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Everyday Use of Ratio’s
• Your going to buy lawn fertilizer– Your lawn is 10,000 ft2
– The fertilizer bag label is:– 1 bag per 2000 ft2
• How many bags do you buy?
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Everyday Use of Ratio’s• How many bags do you buy?
If 1 bag covers 2,000 then 10,000/2,000 = 5 bags
As a Ratio 1 bag = “X” bags Cross multiply
2,000 ft² 10,000 ft²
10,000 ft² x 1 bag = “X” bags x 2,000 ft²
“X” bags = 1 bag * 10,000 ft² Divide
2,000 ft²X bags = 5
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Everyday Use of Ratio’s
• Your going to make chili for 2 people– Recipe is of 4 people– The recipe calls for 3 teaspoons of hot pepper
• How much hot pepper do you put in?– The right amount not fire engine chili
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Everyday Use of Ratio’s• How much hot pepper do you put in?
If 3 teaspoons is for 4 people then 1 ½ teaspoons is for 2 people
As a Ratio 3 teaspoons = “X” teaspoons 4 people 2 people
2 people x 3 teaspoons = “X” teaspoons x 4 people
X teaspoons = 3 teaspoons x 2 people 4 people
X = 1.5 teaspoons or 1 ½ teaspoons
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Units of Ratio’s
They have to be the same on both sides of the =
1 bag = X bags2,000 ft² 10,000 ft²
3 teaspoons = X teaspoons 4 people 2 people
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Roof Pitch
• Roof slope express as a ratio– 4 : 12– 6 : 12– 12 : 12
• Drawn on a Plan as –
• In ratio form = _4_ 12
12
4
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Visualizing Slope
Z12
6
12
6
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Calculating Rise or Run
Slope = 4 : 12 or Rise : Run
On Blueprints, Slope = “X” : 12
”x” = Rise 12 Run
12
4Rise
Z
Run
12
X
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Roof Terms
Z12
6
12
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Roof Run
Roof Span
Roof Span = 2 * Roof Run
or
Roof Run = Roof Span 2
Roof Rise (Pitch)
Roof Run and Roof Span
Roof Run is half of the Roof Span.
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Roof Span is double the Roof Run.
Calculate Run
Z 8Rise16 ft
Run
Example:Pitch 8 : 12
Ratio _8 _ = 16ft 12 Run Cross Multiply & Divide
Run x 8 = 16 x 12
Run = 16 x 12 = 24 ft 8
12
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What is the Span ?
Hint: Run is ½ Span
2 x 24 = 48 ft
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Calculate RiseExample: Pitch 4:12
(Ratio) _4_ = Rise 12 10ft Cross multiply & Divide
4 x 10 = Rise x 12
Rise = 10 * 4 = 3.33 ft 12
Convert to feet – inches 3 ft – 4”
Rise
Run10ft
12
4
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Calculate Pitch
ZRise15 ft
Run18ft
Example:Pitch “X” : 12
Ratio “X” = 15ft 12 18ft Cross Multiply & Divide
“X” x 18 = 15 x 12
“X” = 12 x 15 = 10 18
Pitch 10 : 12
12
“X”
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Roof Pitch Calculations
Your Turn
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Calculating Perimeter, Area and Volume
Two Most Common Shapes:• Rectangles• Triangles
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P = 2 x length + 2 x width
width
length
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Perimeter = Distance around the outside edge
Calculating Perimeter - Rectangle
P = width + length + slope
length
width
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Calculating Perimeter - Triangle
Slope
width
length
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For a RectangleArea equal the length times the width
A = length x width
Calculating Area - Rectangle
Calculating Area - Triangle
A = length x width 2 length
width
Area = ½ width times length
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Volume = length x width x height
height
Calculating Volume - Rectangle
width
length
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V = length x width x height 2
height
Volume - Triangle
width
length
Volume = ½ of Length times Width times Height
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Applying the Calculations
• Floor Area• Wall Area• Conditioned Space Volume
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Area by Component (ft2)
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Area by Component (ft2)
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X Y
Z
Area of a Rectangle Z (ft2)
Area of “Z” = length x width
width
length
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Z
Area of Triangle “X” (ft2)
AX = length x height 2
height
length
X Y
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Area of Triangle Y (ft2)
AY = length x width 2
width
length
X Y
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Total Area (ft2)
AT = AX + AY + AZ
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X Y
Z
Area by Component (ft2)
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Area by Component (ft2)
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Area by Component (ft2)
W
XY Z
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Width W
Area by Component “W”(ft2)
AW = length x width
Length
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X
Area by Component “X”(ft2)
AX= length x width
width
length
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Area by Component “Y”(ft2)
AY = length x width 2
LengthYwidth
length
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Area by Component “Z”(ft2)
AZ = length x width 2
width
length
Z
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Area by Component (ft2)
AT = AW + AX + AY + AZ
W
XY Z
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Calculating Volume (ft3)A Room with a Cathedral Ceiling
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Volume – Cathedral Ceiling
A
B C
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Va = length x width x height
Aheight
Volume by Component “A”(ft3)
width
length
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B
Vb = Rise x Run x length 2
Volume by Component “B” (ft3)
Run(width)
length
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Rise(height)
AB C
C
Vc = Rise x Run x length 2
Run(width)
length
Rise(height)
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Volume by Component “C” (ft3)
AB C
Cathedral Ceiling Volume by Component (ft3)
A
B C
Vt = Va + Vb + Vc
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AB C
Volume - Kneewall
Z
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Volume - Kneewall
Z
A
B CD
Added a Small Cube - DVt = Va + Vb + Vc + Vd
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B
Perimeter (ft)
length
width
P = 2 x length + 2 x width
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Perimeter (ft)
A
B
C
D
E
F
C = ??
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Perimeter (ft)
Y
X
length = e √ X2 + Y2
C
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Perimeter (ft)
A
B
C
D
E
F
P = A + B + C + D + E + F
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-Your Turn-
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1. What is the Slope ?2. What is Height of Peak ?
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23’-4”
6’-8” 5’-0”
10’-0
”6’
-1 1 /
2”
9’-4
1 /2”
Building is 40’ long
1. Floor Area2. Wall Area3. Roof Area4. Volume5. Perimeter
Calculate:
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-Your Turn-
Working with a Circular Shape
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Circumference (c)= Distance around the outside edge of the circle
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Circles
Diameter = Distance across a circle (D) If you divide the distance around the circle (circumference – c ) by the diameter the answer will ALWAYS be = 3.14 It is a constant called “pie”
= 3.14D
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Diameter of a Circle
Radius = Distance from the center of a circle to the edge (r)
r
“r” = ½ diameter
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Radius of a Circle
The area of a circle is equal to times the radius (r)
squared.
r a = r²
Remember “” is a constant = 3.14.
The length of “r” is one half of the diameter (the distance across the circle.)
Take “r” and multiply it by itself to get r².
Now multiply times the product of r² to get the area (a) of the circle.”
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Area of a Circle
Area of a Circle (ft2)
a = D2 4 = 3.14 * Diameter * Diameter 4 ora = r2 = 3.14 * radius * radius
Diameter
radius.
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Volume of a Cylinder (ft3)
v = D2 * h 4 = 3.14 * Diameter * Diameter * height 4 orv = r2 * L = 3.14 * radius * radius * height
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h = height of the cylinder
Area of a Semi-Circle (ft2)
a = r2 2 = 3.14 x radius x radius 2 Ora = D 8 = 3.14 *Diameter * Diameter 8
Diameter
radius
Area (a)= “pie” times the length of the radius squared divided by 2
2
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Volume of 1/2 a Cylinder (ft3)
h = height of the cylinder
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Volume = r2 x h 2 = 3.14 x radius x radius x height 2or using diameter (D)
Volume = D2 x h 8 = 3.14 x Diameter x Diameter x height 8
C = ??
Perimeter of a Semi-Circle (ft)
A
B D
C
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Semi-Circle Perimeter (ft)
Diameter
radius
C = x Diameter 2
C = 3.14 x Diameter 2 or
C = x radius
C = 3.14 x radius
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Area by Component (ft2)
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Area by Component (ft2)
Z
Y
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Area of the Rectangle “Y” (ft2)
AY = length x width
Y
length
width
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Area of the Semi-Circle “Z” (ft2)AZ = r2 2
= 3.14 x radius x radius 2 orAZ = D2 8
= 3.14 x Diameter x Diameter 8
Z
Diameter
radius
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Total Area (ft2)
AT = AY + AZ Z
Y
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Volume (ft3)
Know AY + AZ
VY = AY x L
VZ = AZ x L
VT = VY + VZY
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Z
L = Length
Semi-Circle Calculations
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-Your Turn-
Special Cases
• Ducts• Tray Ceilings
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Duct Surface Area
Rectangular Duct:Surface Area = 2 x (height + width) x length
Round Duct:Surface Area = 3.14 x Duct Diameter x length
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Special Case – Tray Ceiling
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Volume – Tray Ceiling
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Volume – Tray Ceiling
1
2
3
4
5
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Volume – Tray Ceiling
V1 = length x width x height
height
widthlength
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1
Volume – Tray Ceiling
V2 = length x width x height
height
widthlength
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2
Volume – Tray Ceiling
2 Sloped Sides
V3 = Rise x Run x length
Rise
Runlength
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3
Volume – Tray Ceiling
2 Sloped Sides
V4 = Rise x Run x length
Rise
Runlength
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Area – Pyramid
4 Sloped Corners (Pyramid)
a = 2 x length x width x height
lengthwidth
height
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Volume – Tray Ceiling
Sloped Corners = Pyramid
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5
Volume – Pyramid
Pyramid
V5 = 1/3 x length x width x height
lengthwidth
height
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Volume – Tray Ceiling
1
2
3
4
5
VT = V1 + V2 + V3 + V4 + V5
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Area – Tray Ceiling
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Ceiling Area – Tray Ceiling
1
2
3
4
5
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Ceiling Area – Tray Ceiling
Area 1
12
3
4
5
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Ceiling Area – Tray Ceiling
Area 2
12
3
4
5
115
2
Ceiling Area – Tray Ceiling
Areas 3 & 4
12
3
4
5
width ?
length
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Ceiling Area – Tray Ceiling
Areas 3 & 4
12
3
4
5
width ?
width = e X2 + Y2
X
Y
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Area – Tray Ceiling
4 Sloped Corners (Pyramid)
A4 = 2 x length x width x height
lengthwidth
height
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12
3
4
5