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transcript
Workplace Math 10 Updated Jan 2018
Section 2: Ratios and Conversions
This book belongs to: Block:
Section Due Date Questions I Find Difficult Marked Corrections Made and Understood
Self-Assessment Rubric
Learning Targets and Self-Evaluation
Learning Target Description Mark
π β π Understanding how ratios and fractions relate to conversion of units
Using tools and appropriate units to measure computational fluency
π β π Executing conversions with a focus on length to increase computation
Using tools and appropriate units to measure computational fluency
π β π Understanding how ratios relate to converting mass, time, and temperature
Solving multiple step, multiple units conversions with emphasis on distance and time relationships
Category Sub-Category Description
Expert
6 Work meets the objectives; is clear, error free, and demonstrates a mastery of the Learning Targets
βYou could teach this!β
5 Work meets the objectives; is clear, with some minor errors, and demonstrates a clear understanding of the Learning Targets
βAlmost Perfect, one little error.β
Apprentice 4 Work almost meets the objectives; contains errors, and demonstrates sound reasoning and thought
concerning the Learning Targets
βGood understanding with a few errors.β
3 Work is in progress; contains errors, and demonstrates a partial understanding of the
Learning Targets
βYou are on the right track, but key concepts
are missing.β
Novice 2 Work does not meet the objectives; frequent errors, and minimal understanding of the Learning Targets
is demonstrated
βYou have achieved the bare minimum to meet the learning outcome.β
1 Work does not meet the objectives; there is no or minimal effort, and no understanding of the
Learning Targets
βLearning Outcomes not met at this time.β
1
Competency Self-Evaluation
A valuable aspect to the learning process involves self-reflection and efficacy. Research has shown that authentic
self-reflection helps improve performance and effort, and can have a direct impact on the growth mindset of the
individual. In order to grow and be a life-long learner we need to develop the capacity to monitor, evaluate, and
know what and where we need to focus on improvement. Read the following list of Core Competency Outcomes
and reflect on your behaviour, attitude, effort, and actions throughout this unit.
Rank yourself with a check mark: E (Excellent), G (Good), S (Satisfactory), N (Needs Improvement)
E G S N
I listen during instruction period and come to class ready to ask questions
Personal Responsibility
I am fully prepared for Unit Quizzes
I am fully prepared to re-Quizzes
I follow instructions and assist peers
I am on task during work blocks
I complete assignments on time
I keep track of my Learning Targets
Self-Regulation
I take ownership over my goals, learning, and behaviour
I can solve problems myself and know when to ask for help
I can persevere in challenging tasks
I take responsibility to be actively engaged in the lesson and discussions
I only use my phone for school tasks
Classroom
Responsibility and Communication
I am focused on the discussion and lessons
I ask questions during the lesson and class
I give my best effort and encourage others to work well
I am polite and communicate questions and concerns with my peers and teacher
Collaborative Actions
I can work with others to achieve a common goal
I make contributions to my group
I am kind to others, can work collaboratively and build relationships with my peers
I can identify when others need support and provide it
Communication Skills
I present informative clearly, in an organized way
I ask and respond to simple direct questions
I am an active listener, I support and encourage the speaker
I recognize that there are different points of view and can disagree respectfully
Overall
Goal for next Unit β refer to the above criteria. Please select (underline/highlight) two areas you want to focus on
2
Section 2.1 - Ratios
Ratios
What is a Ratio?
It is a numerical relationship between two amounts
Example: 1 βΆ 2 this means
1 ππ’π‘ ππ 2
1 π‘π 2 πππ‘ππ
πΉππ ππ£πππ¦ 1 (πππππ) π‘βπππ πππ 2 (πππππ)
Ratios are specifically important when we get to conversions, because we can use relationships
between units
Ratios are also the SIMPLIFIED representation of a FRACTION
Example:
1
2 πππππ 1 βΆ 2
4
5 πππππ 4 βΆ 5
11
12 πππππ 11 βΆ 12
2
6=
1
3 πππππ 1 βΆ 3
When we see or make recipes ratios between items allow us to reduce or increase the batch.
Below is a recipe for Chocolate Chip Cookies
3
The important question to ask in this case is, what item do I base my ratios on?
Look at the ingredients, any ingredient that is a measurement can be adjusted by the ratio
Concrete ingredients: Eggs in this case, I canβt have one and a half eggs of five eighths of an egg
So since the original recipe calls for 2 eggs and I want 1 egg I use the ratio 1 βΆ 2, so 1
2
everything else
So what I have to do is MULTIPLY (Youβll see with Conversions, we always MULTIPLY) everything by
a half
It is really important to understand one thingβ¦
o You may say from above that we just divide everything by 2. You arenβt wrong.
o But the truth is that division is just the MULTIPLICATION of a FRACTION
o If we always multiply we will be able to cancel units, which means CONVERSIONS
Example: DIVISION IS MULTIPLYING OF THE RECIPROCAL
2 Γ· 2 = 2 β1
2 =
2
2 = 1
1
2Γ· 2 =
1
2β
1
2 =
1
4
1
3Γ· 2 =
1
3β
1
2 =
1
6
1
8Γ· 2 =
1
8β
1
2 =
1
16
Now as we move into Conversions we always want to set them up as
MULTIPLICATION. We do this because units (cm , km, m, etc.) cancel out just
like numbers when they are in the numerator and the denominator.
Reciprocal
4
Conversions
o When we are converting, say from kilometers to meters, there may be an inner
monologue: βDo I multiply or Divide?β o Remember that division is multiplication, it is just the multiplication of the reciprocal
(refliprocal) of the given value o The key to conversions is ALWAYS multiply.
Just multiply by the ratio you are comparing
Remember that multiplying by a fraction is: π»ππ
π©πππππβ
π»ππ
π©πππππ=
π»ππ β π»ππ
π©πππππ β π©πππππ
Example:
π
πβ
π
π=
π
ππ
π
πβ
π
π=
π
π ππππ ππππππππ ππ
π
π
Example: I have six items in my recipe: Half it, Double it and Triple it
2 Cups of Flour 4 eggs 1 Tablespoon of Sugar
1
2 Teaspoon of Salt 1 Teaspoon of Baking
Soda 1
1
2=
3
2 Cups of Milk
Half Triple Quarter
2 πΆπ’ππ β1
2= 1 πΆπ’π 2 πΆπ’ππ β
3
1= 6 πΆπ’ππ 2 πΆπ’ππ β
1
4=
1
2 πΆπ’π
4 πΈπππ β1
2= 2 πΈπππ 4 πΈπππ β
3
1= 12 πΈπππ 4 πΈπππ β
1
4= 1 πΆπ’π
1 πππ π β1
2=
1
2 πππ π 1 πππ π β
3
1= 3 πππ π 1 πππ π β
1
4=
1
4 πππ π
1
2ππ π β
1
2=
1
4 ππ π
1
2ππ π β
3
1=
3
2 ππ π
1
2ππ π β
1
4=
1
8 ππ π
1 ππ π β1
2=
1
2 ππ π 1 ππ π β
3
1= 3 ππ π 1 ππ π β
1
4=
1
4 ππ π
3
2 πΆπ’ππ β
1
2=
3
4 πΆπ’π
3
2 πΆπ’ππ β
3
1=
9
2 πΆπ’ππ
3
2 πΆπ’ππ β
1
4=
3
8 πΆπ’π
It always comes back to fractions!
πΆππβπ‘ π πππππππ¦ π‘βππ
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Section 2.1 β Practice Problems
Simplify the following fractions and write the answer as a ratio.
1. 12
24 2.
14
21 3.
6
15 4.
15
25
Multiply the following proper fractions, simplify the answer and write the result as a ratio.
5. 2
3β
6
7 6.
4
5β
20
40 7.
1
3β
6
11
8. 7
8β
16
35 9.
11
12β
12
22 10.
4
7β
49
56
Multiply the following improper fractions, simplify the answer and write the result as a ratio.
11. 5
3β
9
4 12.
7
5β
60
49 13.
9
3β
22
11
14. 13
8β
16
24 15.
13
12β
48
22 16.
15
7β
56
55
17. Explain why multiplying always works when doing conversions.
18. When you are adjusting a list of measurements by a given ratio, what item should you
base your conversions on and why?
6
19. Find a recipe that you like to cook or would want to cook and list the ingredients and
their quantities below.
Using that recipe as a guide.
i) Triple the batch
ii) Half the batch
7
Section 2.2 β Converting Length using Known Ratios
When we are converting units, there will always be a known ratio that we use
This known ratio will be between to different units
Example: 1ππ = 10ππ or 1ππ βΆ 10ππ β 10ππ βΆ 1ππ
If we know these ratios we can convert anything we are given.
Remember always MULTIPLY
o You just have to follow the following structure every time!
πβππ‘ π¦ππ’ π»ππ£π β π ππ‘ππ = π΄ππ π€ππ
Metric System
The Metric System is used by almost the entire world (all but three countries)
It is easy for the purpose of conversion because it is a BASE 10 system
The Base 10 system makes the conversion quite straight forward
Here is a list of the known Metric Conversion we will use:
Equation Ratio Fraction (Read Top per Bottom)
1ππ = 10ππ
1ππ βΆ 10ππ
10ππ βΆ 1ππ
1ππ
10ππβ
10ππ
1ππ
1π = 100ππ
1π βΆ 100ππ
100ππ βΆ 1π
1π
100ππβ
100ππ
1π
1ππ = 1000π
1ππ βΆ 1000π
1000π βΆ 1ππ
1ππ
1000πβ
1000π
1ππ
Example:
1ππ = 10ππ
1π = 100ππ
1ππ = 1000π
All differ by multiples of 10
BASE 10 SYSTEM
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Example:
How many centimeters are in 123 meters?
Solution:
123π β100ππ
1π
123π β100ππ
1π=
123 β 100ππ
1= πππ πππππ
Example:
How many ππ are there in 15 242 ππππ‘ππππ‘πππ ?
Solution:
Step 1:
15 242ππ β1π
100ππ=
15242π
100= 152.42π
Step 2:
152.42π β1ππ
1000π=
152.42ππ
1000= π. πππππππ
We can do it all in one step, just set up the ratios, continuous multiplication, so the units cancel!
15242ππ β1π
100ππβ
1ππ
1000π=
15242ππ
100 β 1000=
15242ππ
100000= 0.15242ππ
I use the Ratio of ππ βΆ π
I set it up so the meters are on B (since my original is on top)
That way they cancel out
πππ‘πππ cancel with πππ‘πππ
Just left with πΆπππ‘ππππ‘πππ
First I get ππππππ using π: ππ ratio
This time ππ is on the bottom
because I want it to cancel out
Now I get ππππππππππ using ππ: π ratio
This time π is on the bottom because I
want it to cancel out
Meters Cancel Centimeters Cancel
9
Imperial System (Only 3 and a Half Countries use this)
Liberia
Myanmar (Burma)
USA
Canada/UK (use it sometimes)
The conversion ratios for the Imperial System are not Base 10, so they are not as easy to visualize
Here they are:
Equation Ratio Fraction (Read Top per Bottom)
1 ππππ = 1760 π¦ππππ
1ππ βΆ 1760π¦ππ
1760π¦ππ βΆ 1 ππ
1ππ
1760π¦ππ β
1760π¦ππ
1ππ
1 ππππ = 5280 ππ‘
1ππ βΆ 5280ππ‘
5280ππ‘ βΆ 1 ππ
1ππ
5280ππ‘β
5280ππ‘
1ππ
1 π¦ππππ = 3 ππππ‘
1π¦π βΆ 3ππ‘
3ππ‘ βΆ 1π¦π
1π¦π
3ππ‘β
3ππ‘
1π¦π
1 ππππ‘ = 12 πππβππ
1ππ‘ βΆ 12ππ
12ππ βΆ 1ππ‘
1ππ‘
12ππβ
12ππ
1ππ‘
Everything still gets set-up the same way
Make sure the ratios are set-up so that the units still cancel out top and bottom
Example:
How many ππππ‘ are in 64 πππβππ ?
Solution:
64ππ β1ππ‘
12ππ=
64ππ‘
12= π. πππ
Inches cancel
10
Example:
How many inches are there in 3 πππππ ?
Solution:
Multi Step Set-Up
3ππ β1760π¦ππ
1ππ= 5280π¦ππ
5280π¦ππ β3ππ‘
1π¦ππ = 15840ππ‘
15840ππ‘ β12ππ
1ππ‘= πππ πππππ
One Step Set-Up
3ππππ β1760π¦π
1ππβ
3ππ‘
1π¦πβ
12ππ
1ππ‘= πππ πππππ
Example:
How many ππππ‘ in 4.5 πππππ ?
Solution:
4.5ππ β1760π¦ππ
1ππ= 7920π¦ππ
7920π¦ππ β3ππ‘
1π¦π= πππππππ
One Step
4.5ππ β1760π¦ππ
1ππβ
3ππ‘
1π¦π= πππππππ
πΆπππππ πππππ
πΆπππππ π¦ππ
πΆπππππ ππππ‘
Multi-Step πΆπππππ πππππ
πΆπππππ π¦ππ
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Metric to Imperial β Imperial to Metric
Again it is the exact same process
In this case since we are dealing with approximate ratios it is good form to switch
within each individual system before you make the ratio switch to the new system
(Youβll see an example)
Here are the conversions from system to system
Equation Ratio Fraction (Read Top per Bottom)
1 ππ β 1.609ππ
1ππ βΆ 1.609ππ
1.609ππ βΆ 1 ππ
1ππ
1.609ππβ
1.609ππ
1ππ
1 ππ‘ β 0.305 π
1ππ‘ βΆ 0.305 π 0.305 π βΆ 1ππ‘
1ππ‘
0.305 πβ
0.305 π
1ππ‘
1 ππ β 2.54ππ
1 ππ βΆ 2.54ππ
2.54ππ βΆ 1 ππ
1ππ
2.54ππβ
2.54ππ
1ππ
Example:
How many kilometers are in 730ft?
Solution:
Since there is NO DIRECT CONVERSION from ππ to ππππ‘, stay in Imperial first
Switch from ππππ ππ πππππ
Then we can switch from πππππ ππ ππ (a DIRECT CONVERSION)
Multi-Step
730ππ‘ β1ππππ
5280ππ‘= 0.138πππππ
0.138πππππ β1.609ππ
1ππππ= π. ππππ
One Step
730ππ‘ β1ππππ
5280ππ‘β
1.609ππ
1ππππ=
730 β 1.609ππ
5280=
1174.57ππ
5280= π. ππππ
πΆπππππ ππππ‘ πΆπππππ πππππ
12
Example:
How many ππππ‘ππππ‘πππ are there in 42π¦ππ ?
Solution:
Multi-Step
We have a direct conversion from centimeters to inches, so letβs go from yards to inches first
42π¦ππ β3ππ‘
1π¦π= 126ππ‘
126ππ‘ β12ππ
1ππ‘= 1512ππ
1512ππ β2.54ππ
1ππ= 3840.48ππ
One-Step
We have a direct conversion from centimeters to inches, so letβs go from yards to inches first
42π¦ππ β3ππ‘
1π¦πβ
12ππ
1ππ‘β
2.54ππ
1ππ= ππππ. ππππ
Example:
How many ππππ‘ are there in 4ππ
Solution:
Multi-Step
We have a direct conversion from meters to feet, so letβs go from kilometers to meters first
4ππ β1000π
1ππ= 4000π
4000π β1ππ‘
0.305π=
4000
0.305ππ‘
4000
0.305ππ‘ = ππ πππ. ππππ
One-Step
We have a direct conversion from meters to feet, so letβs go from kilometers to meters first
4000ππ β1000π
1ππβ
1ππ‘
0.305π=
4000
0.305ππ‘ = ππ πππ. ππππ
All Conversions get set-up the same way. Make sure the Units Cancel and then
just Multiply Across and Divide the Final Fraction.
πΆπππππ π¦ππππ πΆπππππ ππππ‘ πΆπππππ πππβππ
πΆπππππ πππππππ‘πππ πΆπππππ πππ‘πππ π·ππ£πππ π‘π πππ‘ π‘βπ π΄ππ π€ππ
13
Section 2.2 β Practice Problems
Perform the following conversions and show the ratio being used and the cancelling of units,
dos your answer make sense?
Convert the following measurements to centimeters.
1. 3245 ππ
2. 6.2 πππππ
3. 984 π¦ππππ
4. 784.56 ππ‘
5. 0.003 π¦ππππ
14
Convert the following measurements to feet.
6. 12 690 πππππ
7. 0.567 ππ
8. 1 234 567 ππ
9. 3.4 ππ
Convert the following measurement to miles.
10. 43 567 ππ
11. 3562 ππ
12. 0.392 π
15
Convert the following measurements to meters.
13. 9 πππππ
14. 15 555 ππ
15. 38.76 π¦ππ
16. Come up with three of your own questions, of varying level of difficulty. Solve them, these
will be used in class at a later date.
16
Section 2.3 β Converting Mass, Time, and Temperature
The conversion for MASS is still the exact same set-up
Equation Ratio Fraction (Read Top per Bottom)
Metric
1 π‘ = 1000ππ
1ππ = 1000π
1π = 1000ππ
1π‘ βΆ 1000ππ
1ππ βΆ 1000π
1π βΆ 1000ππ
1π‘
1000ππβ
1000ππ
1π‘
1ππ
1000πβ
1000π
1ππ
1π
1000ππβ
1000ππ
1π
Imperial
1 π = 2000ππ
1ππ = 16ππ§
1π βΆ 2000ππ
1ππ βΆ 16ππ§
1π
2000ππβ
2000ππ
1π
1ππ
16ππ§β
16ππ§
1ππ
1 ππ β 2.54ππ
1 ππ βΆ 2.54ππ
2.54ππ βΆ 1 ππ
1ππ
2.54ππβ
2.54ππ
1ππ
Metric to Imperial
1π = 0.04ππ§
1ππ = 2.21ππ
1π‘ = 1.1π
1π βΆ 0.04ππ§ 1ππ βΆ 2.21ππ
1π‘ βΆ 1.1π
1π
0.04ππ§β
0.04ππ§
1π
1ππ
2.21ππβ
2.21ππ
1ππ
1π‘
1.1πβ
1.1π
1π‘
Make sure the Units Cancel and then just Multiply Across and Divide the Final Fraction.
17
Metric
Example:
How many πππππ are in 12ππ? How many πππππ in 2342ππ? How many πππππππππ in 42 758π?
Solution:
12ππ β1000π
1ππ= ππ ππππ
2342ππ β1π
1000ππ=
2342
1000π
π. ππππ
42 758π β1ππ
1000π=
42 758
1000ππ
ππ. ππππ
Imperial
Example:
How many ππ’ππππ in 4πππ ? How many πππ’πππ in 3π? How many ππ’ππππ in 12.4π?
Solution:
4πππ β16ππ§
1ππ= ππππ
3π β2000πππ
1π= πππππππ
12.4π β2000πππ
1πβ
16ππ§
1ππ=
12.4 β 2000 β 16 = πππ πππππ
Metric β Imperial
Example:
How many πππππ in 17ππ’ππππ ? How many πππ’πππ in 42ππ? How many πππππ in 1.4π
Solution:
17ππ§ β28.35π
1ππ§= πππ. πππ
42ππ β1ππ
0.45ππ=
42
0.45ππ =
ππ. ππππ
1.4π β2000πππ
1πβ
16ππ§
1ππ=
1.4 β 2000 β 16 = 44 800ππ§
44 800ππ§ β28.35π
1ππ§= π πππ ππππ
18
Time
Time conversions work the same, but we need to remember: 60π ππ 60ππππ , πππ‘ 100!
Going forward I will only show multistep examples, you can always do your 1step at a time
Equation Ratio Fraction (Read Top per Bottom)
60π ππ = 1πππ
60π ππ βΆ 1πππ
1πππ βΆ 60π ππ
60π ππ
1πππβ
1πππ
60π ππ
60πππ = 1βπ
60πππ βΆ 1βπ
1βπ βΆ 60ππ
60πππ
1βπβ
1βπ
60πππ
24βπ = 1πππ¦
24βπ βΆ 1πππ¦
1πππ¦ βΆ 24βπ
1πππ¦
24βπβ
24βπ
1πππ¦
7πππ¦ = 1π€πππ
7πππ¦ βΆ 1π€πππ
1π€πππ βΆ 7πππ¦
7πππ¦
1π€πππβ
1π€πππ
7πππ¦
52π€πππ = 1π¦πππ
52π€πππ βΆ 1π¦πππ
1π¦πππ βΆ 52π€πππ
52π€πππ
1π¦πππβ
1π¦πππ
52π€πππ
365πππ¦π = 1π¦πππ
365πππ¦π βΆ 1π¦πππ
1π¦πππ βΆ 365πππ¦π
365πππ¦π
1π¦πππβ
1π¦πππ
365πππ¦π
Example: How ππππ’π‘ππ in a πππ¦?
Solution:
Example:
How many π ππππππ in a π€πππ? How π€ππππ in 40 320 ππππ’π‘ππ ?
1π€πππ β7π·ππ¦
1ππππβ
24βπ
1π·ππ¦β
60ππππ
1βπ=
60π ππ
1πππ
= πππ ππππππ
40 320ππππ β1βπ
60πππβ
1πππ¦
24βπβ
1π€πππ
7πππ¦=
40 320
60 β 24 β 7
=40 320
10 080π€πππ = π πππππ
1πππ¦ β24βπ
1πππ¦β
60ππππ
1βπ= ππππππππ
Solution:
19
Temperature
There are three different temperatures in the books.
Celsius (most countries, Canada), Fahrenheit (some countries, USA)
Kelvin (Mainly used during Scientific Processes, Absolute 0 is 0 Degree Kelvin
We are only going to look at the conversion of Celsius to Fahrenheit and Vice-Versa.
Unlike the other Conversions, this is not about ratios, but there are set equations to
express the difference
Fahrenheit to Celsius Celsius to Fahrenheit
πΉ =9
5πΆ + 32
πΆ =5
9(πΉ β 32)
Example:
What is 32β in Fahrenheit What is 101β in Celsius
Solution:
πΉ =9
5(32) + 32
π = ππ. πβ
πΆ =5
9(πΉ β 32)
πΆ =5
9(101 β 32)
πΆ =5
9(69)
πͺ = ππ. πβ
There is a point where Fahrenheit and Celsius values are equal: βππβ = βππβ
Sub in the 32β
Sub in the 101β
20
Conversions of Multiple Units at the Same Time
This is the most challenging situation, but the ratio work and cancelling of the units
works exactly the same
Example:
How fast in ππππππ/ππππππ is a car travelling at: ππππ/ππ
Solution:
70ππ
1βπβ
1000π
1ππβ
1βπ
60ππππ β
1πππ
60π ππ=
70 β 1000π
60 β 60π ππ=
70 000π
3600π ππ= ππ. π
π
π
Example:
The speed of light is 299 792 458 πππ‘πππ /π πππππ
A light year is a measurement of how far light travels in kilometers in a year. Knowing how
fast light travels we can use our ratios to figure this out!
Solution:
299 792 458π
1π ππβ
1ππ
1000πβ
60π ππ
1πππβ
60πππ
1βπβ
24βπ
1πππ¦β
365πππ¦
1π¦π= π. ππ β ππππππ/ππ
Kilometers cancelled top and bottom
Hours cancelled top and bottom
Minutes cancelled top and bottom
Meters cancelled top and bottom
Seconds cancelled top and bottom
Minutes cancelled top and bottom
Hours cancelled top and bottom
Days cancelled top and bottom
21
Section 2.3 β Practice Problems
Perform the following MASS conversions.
1. Convert 2.3π to ππ’ππππ 2. Convert 23.5πππ to ππππππππππ 3. Convert 13.4ππ to πππ’πππ 4. Convert 13 465ππ§ to π‘πππππ (πππ‘πππ) 5. Convert 3.4π to ππππππππππ
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Perform the following TIME conversions.
6. How many π ππππππ are in 3 πππ¦π ? 7. How many π€ππππ are in 3 πππ π βπππ π¦ππππ ? 8. How many ππππ’π‘ππ in the ππππ‘βπ ππ π½π’ππ¦ πππ π΄π’ππ’π π‘? 9. How many π ππππππ are in the first 6 ππππ‘βπ ππ π‘βπ π¦πππ?
Perform the following TEMPERATURE conversions
10. How hot is 112β in β?
11. What is 7β in β?
12. Prove where Celsius and Fahrenheit are the same.
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Perform the following conversions of MULTIPLE UNITS.
13. If I can run at 8ππ/βπ how fast am I going in π/π ? 14. You watch an ant move 8ππ in 3π ππππππ , how fast is it travelling in ππ/βπ? 15. How long, ππ ππππ’π‘ππ , does it take light to travel 12 πππππππ ππ? 16. If you are strong enough to push an object, with constant acceleration at 2 πππ‘πππ /π ππ,
how far can you push it in 2 π€ππππ ?
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Extra Work Space
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Answer Key
Section 2.1 Section 2.2 Section 2.3
1. 1
2
2. 2
3
3. 2
5
4. 3
5
5. 4
7; 4: 7
6. 2
5; 2: 5
7. 2
11; 2:11
8. 2
5; 2:5
9. 1
2; 1: 2
10. 1
2; 1: 2
11. 15
4; 15: 4
12. 12
7; 12: 7
13. 6
1; 6: 1
14. 13
12; 13: 12
15. 26
11; 26: 11
16. 24
11; 24: 11
17. π΄ππ π€πππ ππππ¦
18. π΄ππ π€πππ ππππ¦
19. π΄ππ π€πππ ππππ¦
1. 324 500 000ππ
2. 997 793.3ππ
3. 89 977.0ππ
4. 23 913.4ππ
5. 0.274ππ
6. 67 003 200ππ‘
7. 1859.0ππ‘
8. 4047.8ππ‘
9. 0.11ππ‘
10. 0.69ππππ
11. 0.02ππππ
12. 0.0002ππππ
13. 14 493.6π
14. 395.4π
15. 35.5π
16. π΄ππ π€ππ ππππ¦
1. 73 600ππ§
2. 9 400 000ππ
3. 29.61πππ
4. 0.34π‘
5. 2 720 000 000ππ
6. 259 200π ππππππ
7. 182π€ππππ
8. 89 280ππππ
9. 15 638 400π πππ
10. 44.4β
11. 44.6β
12. See written Answer
13. 2.2 ππ β
14. 0.095 ππβπβ
15. 0.67ππππ
16. 2 419 200π