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

𝟐 − 𝟏 Understanding how ratios and fractions relate to conversion of units

Using tools and appropriate units to measure computational fluency

𝟐 − 𝟐 Executing conversions with a focus on length to increase computation

Using tools and appropriate units to measure computational fluency

𝟐 − 𝟑 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

“You could teach this!”

5 Work meets the objectives; is clear, with some minor errors, and demonstrates a clear understanding of the Learning Targets

“Almost Perfect, one little error.”

Apprentice 4 Work almost meets the objectives; contains errors, and demonstrates sound reasoning and thought

concerning the Learning Targets

“Good understanding with a few errors.”

3 Work is in progress; contains errors, and demonstrates a partial understanding of the

Learning Targets

“You are on the right track, but key concepts

are missing.”

Novice 2 Work does not meet the objectives; frequent errors, and minimal understanding of the Learning Targets

is demonstrated

“You have achieved the bare minimum to meet the learning outcome.”

1 Work does not meet the objectives; there is no or minimal effort, and no understanding of the

Learning Targets

“Learning Outcomes not met at this time.”

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 – 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 ∶ 2 this means

1 𝑜𝑢𝑡 𝑜𝑓 2

1 𝑡𝑜 2 𝑟𝑎𝑡𝑖𝑜

𝐹𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 1 (𝑏𝑙𝑎𝑛𝑘) 𝑡ℎ𝑒𝑟𝑒 𝑎𝑟𝑒 2 (𝑏𝑙𝑎𝑛𝑘)

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 𝑚𝑒𝑎𝑛𝑠 1 ∶ 2

4

5 𝑚𝑒𝑎𝑛𝑠 4 ∶ 5

11

12 𝑚𝑒𝑎𝑛𝑠 11 ∶ 12

2

6=

1

3 𝑚𝑒𝑎𝑛𝑠 1 ∶ 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 can’t 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 ∶ 2, so 1

2

everything else

So what I have to do is MULTIPLY (You’ll see with Conversions, we always MULTIPLY) everything by

a half

It is really important to understand one thing…

o You may say from above that we just divide everything by 2. You aren’t 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 ÷ 2 = 2 ∙1

2 =

2

2 = 1

1

2÷ 2 =

1

2∙

1

2 =

1

4

1

3÷ 2 =

1

3∙

1

2 =

1

6

1

8÷ 2 =

1

8∙

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: “Do I multiply or Divide?” 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: 𝑻𝒐𝒑

𝑩𝒐𝒕𝒕𝒐𝒎∙

𝑻𝒐𝒑

𝑩𝒐𝒕𝒕𝒐𝒎=

𝑻𝒐𝒑 ∙ 𝑻𝒐𝒑

𝑩𝒐𝒕𝒕𝒐𝒎 ∙ 𝑩𝒐𝒕𝒕𝒐𝒎

Example:

𝟐

𝟓∙

𝟏

𝟑=

𝟐

𝟏𝟓

𝟐

𝟑∙

𝟏

𝟐=

𝟐

𝟔 𝒕𝒉𝒆𝒏 𝒔𝒊𝒎𝒑𝒍𝒊𝒇𝒚 𝒕𝒐

𝟏

𝟑

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 𝐶𝑢𝑝𝑠 ∙1

2= 1 𝐶𝑢𝑝 2 𝐶𝑢𝑝𝑠 ∙

3

1= 6 𝐶𝑢𝑝𝑠 2 𝐶𝑢𝑝𝑠 ∙

1

4=

1

2 𝐶𝑢𝑝

4 𝐸𝑔𝑔𝑠 ∙1

2= 2 𝐸𝑔𝑔𝑠 4 𝐸𝑔𝑔𝑠 ∙

3

1= 12 𝐸𝑔𝑔𝑠 4 𝐸𝑔𝑔𝑠 ∙

1

4= 1 𝐶𝑢𝑝

1 𝑇𝑏𝑠𝑝 ∙1

2=

1

2 𝑇𝑏𝑠𝑝 1 𝑇𝑏𝑠𝑝 ∙

3

1= 3 𝑇𝑏𝑠𝑝 1 𝑇𝑏𝑠𝑝 ∙

1

4=

1

4 𝑇𝑏𝑠𝑝

1

2𝑇𝑠𝑝 ∙

1

2=

1

4 𝑇𝑠𝑝

1

2𝑇𝑠𝑝 ∙

3

1=

3

2 𝑇𝑠𝑝

1

2𝑇𝑠𝑝 ∙

1

4=

1

8 𝑇𝑠𝑝

1 𝑇𝑠𝑝 ∙1

2=

1

2 𝑇𝑠𝑝 1 𝑇𝑠𝑝 ∙

3

1= 3 𝑇𝑠𝑝 1 𝑇𝑠𝑝 ∙

1

4=

1

4 𝑇𝑠𝑝

3

2 𝐶𝑢𝑝𝑠 ∙

1

2=

3

4 𝐶𝑢𝑝

3

2 𝐶𝑢𝑝𝑠 ∙

3

1=

9

2 𝐶𝑢𝑝𝑠

3

2 𝐶𝑢𝑝𝑠 ∙

1

4=

3

8 𝐶𝑢𝑝

It always comes back to fractions!

𝐶𝑎𝑛’𝑡 𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑦 𝑡ℎ𝑖𝑠

5

Section 2.1 – 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

3∙

6

7 6.

4

5∙

20

40 7.

1

3∙

6

11

8. 7

8∙

16

35 9.

11

12∙

12

22 10.

4

7∙

49

56

Multiply the following improper fractions, simplify the answer and write the result as a ratio.

11. 5

3∙

9

4 12.

7

5∙

60

49 13.

9

3∙

22

11

14. 13

8∙

16

24 15.

13

12∙

48

22 16.

15

7∙

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 – 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: 1𝑐𝑚 = 10𝑚𝑚 or 1𝑐𝑚 ∶ 10𝑚𝑚 ↔ 10𝑚𝑚 ∶ 1𝑐𝑚

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!

𝑊ℎ𝑎𝑡 𝑦𝑜𝑢 𝐻𝑎𝑣𝑒 ∗ 𝑅𝑎𝑡𝑖𝑜 = 𝐴𝑛𝑠𝑤𝑒𝑟

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)

1𝑐𝑚 = 10𝑚𝑚

1𝑐𝑚 ∶ 10𝑚𝑚

10𝑚𝑚 ∶ 1𝑐𝑚

1𝑐𝑚

10𝑚𝑚↔

10𝑚𝑚

1𝑐𝑚

1𝑚 = 100𝑐𝑚

1𝑚 ∶ 100𝑐𝑚

100𝑐𝑚 ∶ 1𝑚

1𝑚

100𝑐𝑚↔

100𝑐𝑚

1𝑚

1𝑘𝑚 = 1000𝑚

1𝑘𝑚 ∶ 1000𝑚

1000𝑚 ∶ 1𝑘𝑚

1𝑘𝑚

1000𝑚↔

1000𝑚

1𝑘𝑚

Example:

1𝑐𝑚 = 10𝑚𝑚

1𝑚 = 100𝑐𝑚

1𝑘𝑚 = 1000𝑚

All differ by multiples of 10

BASE 10 SYSTEM

8

Example:

How many centimeters are in 123 meters?

Solution:

123𝑚 ∗100𝑐𝑚

1𝑚

123𝑚 ∗100𝑐𝑚

1𝑚=

123 ∗ 100𝑐𝑚

1= 𝟏𝟐𝟑 𝟎𝟎𝟎𝒄𝒎

Example:

How many 𝑘𝑚 are there in 15 242 𝑐𝑒𝑛𝑡𝑖𝑚𝑒𝑡𝑒𝑟𝑠?

Solution:

Step 1:

15 242𝑐𝑚 ∗1𝑚

100𝑐𝑚=

15242𝑚

100= 152.42𝑚

Step 2:

152.42𝑚 ∗1𝑘𝑚

1000𝑚=

152.42𝑘𝑚

1000= 𝟎. 𝟏𝟓𝟐𝟒𝟐𝒌𝒎

We can do it all in one step, just set up the ratios, continuous multiplication, so the units cancel!

15242𝑐𝑚 ∗1𝑚

100𝑐𝑚∗

1𝑘𝑚

1000𝑚=

15242𝑘𝑚

100 ∗ 1000=

15242𝑘𝑚

100000= 0.15242𝑘𝑚

I use the Ratio of 𝑐𝑚 ∶ 𝑚

I set it up so the meters are on B (since my original is on top)

That way they cancel out

𝑀𝑒𝑡𝑒𝑟𝑠 cancel with 𝑚𝑒𝑡𝑒𝑟𝑠

Just left with 𝐶𝑒𝑛𝑡𝑖𝑚𝑒𝑡𝑒𝑟𝑠

First I get 𝒎𝒆𝒕𝒆𝒓𝒔 using 𝒎: 𝒄𝒎 ratio

This time 𝒄𝒎 is on the bottom

because I want it to cancel out

Now I get 𝒌𝒊𝒍𝒐𝒎𝒆𝒕𝒆𝒓𝒔 using 𝒌𝒎: 𝒎 ratio

This time 𝒎 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 𝑚𝑖𝑙𝑒 = 1760 𝑦𝑎𝑟𝑑𝑠

1𝑚𝑖 ∶ 1760𝑦𝑑𝑠

1760𝑦𝑑𝑠 ∶ 1 𝑚𝑖

1𝑚𝑖

1760𝑦𝑑𝑠↔

1760𝑦𝑑𝑠

1𝑚𝑖

1 𝑚𝑖𝑙𝑒 = 5280 𝑓𝑡

1𝑚𝑖 ∶ 5280𝑓𝑡

5280𝑓𝑡 ∶ 1 𝑚𝑖

1𝑚𝑖

5280𝑓𝑡↔

5280𝑓𝑡

1𝑚𝑖

1 𝑦𝑎𝑟𝑑𝑠 = 3 𝑓𝑒𝑒𝑡

1𝑦𝑑 ∶ 3𝑓𝑡

3𝑓𝑡 ∶ 1𝑦𝑑

1𝑦𝑑

3𝑓𝑡↔

3𝑓𝑡

1𝑦𝑑

1 𝑓𝑜𝑜𝑡 = 12 𝑖𝑛𝑐ℎ𝑒𝑠

1𝑓𝑡 ∶ 12𝑖𝑛

12𝑖𝑛 ∶ 1𝑓𝑡

1𝑓𝑡

12𝑖𝑛↔

12𝑖𝑛

1𝑓𝑡

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 𝑓𝑒𝑒𝑡 are in 64 𝑖𝑛𝑐ℎ𝑒𝑠?

Solution:

64𝑖𝑛 ∗1𝑓𝑡

12𝑖𝑛=

64𝑓𝑡

12= 𝟓. 𝟑𝒇𝒕

Inches cancel

10

Example:

How many inches are there in 3 𝑚𝑖𝑙𝑒𝑠?

Solution:

Multi Step Set-Up

3𝑚𝑖 ∗1760𝑦𝑑𝑠

1𝑚𝑖= 5280𝑦𝑑𝑠

5280𝑦𝑑𝑠 ∗3𝑓𝑡

1𝑦𝑑𝑠= 15840𝑓𝑡

15840𝑓𝑡 ∗12𝑖𝑛

1𝑓𝑡= 𝟏𝟗𝟎 𝟎𝟖𝟎𝒊𝒏

One Step Set-Up

3𝑚𝑖𝑙𝑒 ∗1760𝑦𝑑

1𝑚𝑖∗

3𝑓𝑡

1𝑦𝑑∗

12𝑖𝑛

1𝑓𝑡= 𝟏𝟗𝟎 𝟎𝟖𝟎𝒊𝒏

Example:

How many 𝑓𝑒𝑒𝑡 in 4.5 𝑚𝑖𝑙𝑒𝑠?

Solution:

4.5𝑚𝑖 ∗1760𝑦𝑑𝑠

1𝑚𝑖= 7920𝑦𝑑𝑠

7920𝑦𝑑𝑠 ∗3𝑓𝑡

1𝑦𝑑= 𝟐𝟑𝟕𝟔𝟎𝒇𝒕

One Step

4.5𝑚𝑖 ∗1760𝑦𝑑𝑠

1𝑚𝑖∗

3𝑓𝑡

1𝑦𝑑= 𝟐𝟑𝟕𝟔𝟎𝒇𝒕

𝐶𝑎𝑛𝑐𝑒𝑙 𝑚𝑖𝑙𝑒𝑠

𝐶𝑎𝑛𝑐𝑒𝑙 𝑦𝑑𝑠

𝐶𝑎𝑛𝑐𝑒𝑙 𝑓𝑒𝑒𝑡

Multi-Step 𝐶𝑎𝑛𝑐𝑒𝑙 𝑚𝑖𝑙𝑒𝑠

𝐶𝑎𝑛𝑐𝑒𝑙 𝑦𝑑𝑠

11

Metric to Imperial ↔ 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

(You’ll see an example)

Here are the conversions from system to system

Equation Ratio Fraction (Read Top per Bottom)

1 𝑚𝑖 ≅ 1.609𝑘𝑚

1𝑚𝑖 ∶ 1.609𝑘𝑚

1.609𝑘𝑚 ∶ 1 𝑚𝑖

1𝑚𝑖

1.609𝑘𝑚↔

1.609𝑘𝑚

1𝑚𝑖

1 𝑓𝑡 ≅ 0.305 𝑚

1𝑓𝑡 ∶ 0.305 𝑚 0.305 𝑚 ∶ 1𝑓𝑡

1𝑓𝑡

0.305 𝑚↔

0.305 𝑚

1𝑓𝑡

1 𝑖𝑛 ≅ 2.54𝑐𝑚

1 𝑖𝑛 ∶ 2.54𝑐𝑚

2.54𝑐𝑚 ∶ 1 𝑖𝑛

1𝑖𝑛

2.54𝑐𝑚↔

2.54𝑐𝑚

1𝑖𝑛

Example:

How many kilometers are in 730ft?

Solution:

Since there is NO DIRECT CONVERSION from 𝑘𝑚 to 𝑓𝑒𝑒𝑡, stay in Imperial first

Switch from 𝒇𝒆𝒆𝒕 𝒕𝒐 𝒎𝒊𝒍𝒆𝒔

Then we can switch from 𝒎𝒊𝒍𝒆𝒔 𝒕𝒐 𝒌𝒎 (a DIRECT CONVERSION)

Multi-Step

730𝑓𝑡 ∗1𝑚𝑖𝑙𝑒

5280𝑓𝑡= 0.138𝑚𝑖𝑙𝑒𝑠

0.138𝑚𝑖𝑙𝑒𝑠 ∗1.609𝑘𝑚

1𝑚𝑖𝑙𝑒= 𝟎. 𝟐𝟐𝒌𝒎

One Step

730𝑓𝑡 ∗1𝑚𝑖𝑙𝑒

5280𝑓𝑡∗

1.609𝑘𝑚

1𝑚𝑖𝑙𝑒=

730 ∗ 1.609𝑘𝑚

5280=

1174.57𝑘𝑚

5280= 𝟎. 𝟐𝟐𝒌𝒎

𝐶𝑎𝑛𝑐𝑒𝑙 𝑓𝑒𝑒𝑡 𝐶𝑎𝑛𝑐𝑒𝑙 𝑚𝑖𝑙𝑒𝑠

12

Example:

How many 𝑐𝑒𝑛𝑡𝑖𝑚𝑒𝑡𝑒𝑟𝑠 are there in 42𝑦𝑑𝑠?

Solution:

Multi-Step

We have a direct conversion from centimeters to inches, so let’s go from yards to inches first

42𝑦𝑑𝑠 ∗3𝑓𝑡

1𝑦𝑑= 126𝑓𝑡

126𝑓𝑡 ∗12𝑖𝑛

1𝑓𝑡= 1512𝑖𝑛

1512𝑖𝑛 ∗2.54𝑐𝑚

1𝑖𝑛= 3840.48𝑐𝑚

One-Step

We have a direct conversion from centimeters to inches, so let’s go from yards to inches first

42𝑦𝑑𝑠 ∗3𝑓𝑡

1𝑦𝑑∗

12𝑖𝑛

1𝑓𝑡∗

2.54𝑐𝑚

1𝑖𝑛= 𝟑𝟖𝟒𝟎. 𝟒𝟖𝒄𝒎

Example:

How many 𝑓𝑒𝑒𝑡 are there in 4𝑘𝑚

Solution:

Multi-Step

We have a direct conversion from meters to feet, so let’s go from kilometers to meters first

4𝑘𝑚 ∗1000𝑚

1𝑘𝑚= 4000𝑚

4000𝑚 ∗1𝑓𝑡

0.305𝑚=

4000

0.305𝑓𝑡

4000

0.305𝑓𝑡 = 𝟏𝟑 𝟏𝟏𝟒. 𝟕𝟓𝒇𝒕

One-Step

We have a direct conversion from meters to feet, so let’s go from kilometers to meters first

4000𝑘𝑚 ∗1000𝑚

1𝑘𝑚∗

1𝑓𝑡

0.305𝑚=

4000

0.305𝑓𝑡 = 𝟏𝟑 𝟏𝟏𝟒. 𝟕𝟓𝒇𝒕

All Conversions get set-up the same way. Make sure the Units Cancel and then

just Multiply Across and Divide the Final Fraction.

𝐶𝑎𝑛𝑐𝑒𝑙 𝑦𝑎𝑟𝑑𝑠 𝐶𝑎𝑛𝑐𝑒𝑙 𝑓𝑒𝑒𝑡 𝐶𝑎𝑛𝑐𝑒𝑙 𝑖𝑛𝑐ℎ𝑒𝑠

𝐶𝑎𝑛𝑐𝑒𝑙 𝑘𝑖𝑙𝑜𝑚𝑒𝑡𝑒𝑟𝑠 𝐶𝑎𝑛𝑐𝑒𝑙 𝑚𝑒𝑡𝑒𝑟𝑠 𝐷𝑖𝑣𝑖𝑑𝑒 𝑡𝑜 𝑔𝑒𝑡 𝑡ℎ𝑒 𝐴𝑛𝑠𝑤𝑒𝑟

13

Section 2.2 – 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 𝑘𝑚

2. 6.2 𝑚𝑖𝑙𝑒𝑠

3. 984 𝑦𝑎𝑟𝑑𝑠

4. 784.56 𝑓𝑡

5. 0.003 𝑦𝑎𝑟𝑑𝑠

14

Convert the following measurements to feet.

6. 12 690 𝑚𝑖𝑙𝑒𝑠

7. 0.567 𝑘𝑚

8. 1 234 567 𝑚𝑚

9. 3.4 𝑐𝑚

Convert the following measurement to miles.

10. 43 567 𝑖𝑛

11. 3562 𝑐𝑚

12. 0.392 𝑚

15

Convert the following measurements to meters.

13. 9 𝑚𝑖𝑙𝑒𝑠

14. 15 555 𝑖𝑛

15. 38.76 𝑦𝑑𝑠

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 – 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 𝑡 = 1000𝑘𝑔

1𝑘𝑔 = 1000𝑔

1𝑔 = 1000𝑚𝑔

1𝑡 ∶ 1000𝑘𝑔

1𝑘𝑔 ∶ 1000𝑔

1𝑔 ∶ 1000𝑚𝑔

1𝑡

1000𝑘𝑔↔

1000𝑘𝑔

1𝑡

1𝑘𝑔

1000𝑔↔

1000𝑔

1𝑘𝑔

1𝑔

1000𝑚𝑔↔

1000𝑚𝑔

1𝑔

Imperial

1 𝑇 = 2000𝑙𝑏

1𝑙𝑏 = 16𝑜𝑧

1𝑇 ∶ 2000𝑙𝑏

1𝑙𝑏 ∶ 16𝑜𝑧

1𝑇

2000𝑙𝑏↔

2000𝑙𝑏

1𝑇

1𝑙𝑏

16𝑜𝑧↔

16𝑜𝑧

1𝑙𝑏

1 𝑖𝑛 ≅ 2.54𝑐𝑚

1 𝑖𝑛 ∶ 2.54𝑐𝑚

2.54𝑐𝑚 ∶ 1 𝑖𝑛

1𝑖𝑛

2.54𝑐𝑚↔

2.54𝑐𝑚

1𝑖𝑛

Metric to Imperial

1𝑔 = 0.04𝑜𝑧

1𝑘𝑔 = 2.21𝑙𝑏

1𝑡 = 1.1𝑇

1𝑔 ∶ 0.04𝑜𝑧 1𝑘𝑔 ∶ 2.21𝑙𝑏

1𝑡 ∶ 1.1𝑇

1𝑔

0.04𝑜𝑧↔

0.04𝑜𝑧

1𝑔

1𝑘𝑔

2.21𝑙𝑏↔

2.21𝑙𝑏

1𝑘𝑔

1𝑡

1.1𝑇↔

1.1𝑇

1𝑡

Make sure the Units Cancel and then just Multiply Across and Divide the Final Fraction.

17

Metric

Example:

How many 𝑔𝑟𝑎𝑚𝑠 are in 12𝑘𝑔? How many 𝑔𝑟𝑎𝑚𝑠 in 2342𝑚𝑔? How many 𝑘𝑖𝑙𝑜𝑔𝑟𝑎𝑚𝑠 in 42 758𝑔?

Solution:

12𝑘𝑔 ∗1000𝑔

1𝑘𝑔= 𝟏𝟐 𝟎𝟎𝟎𝒈

2342𝑚𝑔 ∗1𝑔

1000𝑚𝑔=

2342

1000𝑔

𝟐. 𝟑𝟒𝟐𝒈

42 758𝑔 ∗1𝑘𝑔

1000𝑔=

42 758

1000𝑘𝑔

𝟒𝟐. 𝟕𝟔𝒌𝒈

Imperial

Example:

How many 𝑜𝑢𝑛𝑐𝑒𝑠 in 4𝑙𝑏𝑠? How many 𝑝𝑜𝑢𝑛𝑑𝑠 in 3𝑇? How many 𝑜𝑢𝑛𝑐𝑒𝑠 in 12.4𝑇?

Solution:

4𝑙𝑏𝑠 ∗16𝑜𝑧

1𝑙𝑏= 𝟔𝟒𝒐𝒛

3𝑇 ∗2000𝑙𝑏𝑠

1𝑇= 𝟔𝟎𝟎𝟎𝒍𝒃𝒔

12.4𝑇 ∗2000𝑙𝑏𝑠

1𝑇∗

16𝑜𝑧

1𝑙𝑏=

12.4 ∗ 2000 ∗ 16 = 𝟑𝟗𝟔 𝟖𝟎𝟎𝒐𝒛

Metric ↔ Imperial

Example:

How many 𝑔𝑟𝑎𝑚𝑠 in 17𝑜𝑢𝑛𝑐𝑒𝑠? How many 𝑝𝑜𝑢𝑛𝑑𝑠 in 42𝑘𝑔? How many 𝑔𝑟𝑎𝑚𝑠 in 1.4𝑇

Solution:

17𝑜𝑧 ∗28.35𝑔

1𝑜𝑧= 𝟒𝟖𝟏. 𝟗𝟓𝒈

42𝑘𝑔 ∗1𝑙𝑏

0.45𝑘𝑔=

42

0.45𝑙𝑏 =

𝟗𝟑. 𝟑𝒍𝒃𝒔

1.4𝑇 ∗2000𝑙𝑏𝑠

1𝑇∗

16𝑜𝑧

1𝑙𝑏=

1.4 ∗ 2000 ∗ 16 = 44 800𝑜𝑧

44 800𝑜𝑧 ∗28.35𝑔

1𝑜𝑧= 𝟏 𝟐𝟕𝟎 𝟎𝟎𝟎𝒈

18

Time

Time conversions work the same, but we need to remember: 60𝑠𝑒𝑐 60𝑚𝑖𝑛𝑠, 𝑛𝑜𝑡 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)

60𝑠𝑒𝑐 = 1𝑚𝑖𝑛

60𝑠𝑒𝑐 ∶ 1𝑚𝑖𝑛

1𝑚𝑖𝑛 ∶ 60𝑠𝑒𝑐

60𝑠𝑒𝑐

1𝑚𝑖𝑛↔

1𝑚𝑖𝑛

60𝑠𝑒𝑐

60𝑚𝑖𝑛 = 1ℎ𝑟

60𝑚𝑖𝑛 ∶ 1ℎ𝑟

1ℎ𝑟 ∶ 60𝑚𝑖

60𝑚𝑖𝑛

1ℎ𝑟↔

1ℎ𝑟

60𝑚𝑖𝑛

24ℎ𝑟 = 1𝑑𝑎𝑦

24ℎ𝑟 ∶ 1𝑑𝑎𝑦

1𝑑𝑎𝑦 ∶ 24ℎ𝑟

1𝑑𝑎𝑦

24ℎ𝑟↔

24ℎ𝑟

1𝑑𝑎𝑦

7𝑑𝑎𝑦 = 1𝑤𝑒𝑒𝑘

7𝑑𝑎𝑦 ∶ 1𝑤𝑒𝑒𝑘

1𝑤𝑒𝑒𝑘 ∶ 7𝑑𝑎𝑦

7𝑑𝑎𝑦

1𝑤𝑒𝑒𝑘↔

1𝑤𝑒𝑒𝑘

7𝑑𝑎𝑦

52𝑤𝑒𝑒𝑘 = 1𝑦𝑒𝑎𝑟

52𝑤𝑒𝑒𝑘 ∶ 1𝑦𝑒𝑎𝑟

1𝑦𝑒𝑎𝑟 ∶ 52𝑤𝑒𝑒𝑘

52𝑤𝑒𝑒𝑘

1𝑦𝑒𝑎𝑟↔

1𝑦𝑒𝑎𝑟

52𝑤𝑒𝑒𝑘

365𝑑𝑎𝑦𝑠 = 1𝑦𝑒𝑎𝑟

365𝑑𝑎𝑦𝑠 ∶ 1𝑦𝑒𝑎𝑟

1𝑦𝑒𝑎𝑟 ∶ 365𝑑𝑎𝑦𝑠

365𝑑𝑎𝑦𝑠

1𝑦𝑒𝑎𝑟↔

1𝑦𝑒𝑎𝑟

365𝑑𝑎𝑦𝑠

Example: How 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 in a 𝑑𝑎𝑦?

Solution:

Example:

How many 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 in a 𝑤𝑒𝑒𝑘? How 𝑤𝑒𝑒𝑘𝑠 in 40 320 𝑚𝑖𝑛𝑢𝑡𝑒𝑠?

1𝑤𝑒𝑒𝑘 ∗7𝐷𝑎𝑦

1𝑊𝑒𝑒𝑘∗

24ℎ𝑟

1𝐷𝑎𝑦∗

60𝑚𝑖𝑛𝑠

1ℎ𝑟=

60𝑠𝑒𝑐

1𝑚𝑖𝑛

= 𝟔𝟎𝟒 𝟖𝟎𝟎𝒔𝒆𝒄

40 320𝑚𝑖𝑛𝑠 ∗1ℎ𝑟

60𝑚𝑖𝑛∗

1𝑑𝑎𝑦

24ℎ𝑟∗

1𝑤𝑒𝑒𝑘

7𝑑𝑎𝑦=

40 320

60 ∗ 24 ∗ 7

=40 320

10 080𝑤𝑒𝑒𝑘 = 𝟒 𝒘𝒆𝒆𝒌𝒔

1𝑑𝑎𝑦 ∗24ℎ𝑟

1𝑑𝑎𝑦∗

60𝑚𝑖𝑛𝑠

1ℎ𝑟= 𝟏𝟒𝟒𝟎𝒎𝒊𝒏𝒔

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

𝐹 =9

5𝐶 + 32

𝐶 =5

9(𝐹 − 32)

Example:

What is 32℃ in Fahrenheit What is 101℉ in Celsius

Solution:

𝐹 =9

5(32) + 32

𝑭 = 𝟖𝟗. 𝟔℉

𝐶 =5

9(𝐹 − 32)

𝐶 =5

9(101 − 32)

𝐶 =5

9(69)

𝑪 = 𝟑𝟖. 𝟑℃

There is a point where Fahrenheit and Celsius values are equal: −𝟒𝟎℃ = −𝟒𝟎℉

Sub in the 32℃

Sub in the 101℉

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 𝒎𝒆𝒕𝒆𝒓𝒔/𝒔𝒆𝒄𝒐𝒏𝒅 is a car travelling at: 𝟕𝟎𝒌𝒎/𝒉𝒓

Solution:

70𝑘𝑚

1ℎ𝑟∗

1000𝑚

1𝑘𝑚∗

1ℎ𝑟

60𝑚𝑖𝑛𝑠∗

1𝑚𝑖𝑛

60𝑠𝑒𝑐=

70 ∗ 1000𝑚

60 ∗ 60𝑠𝑒𝑐=

70 000𝑚

3600𝑠𝑒𝑐= 𝟏𝟗. 𝟒

𝒎

𝒔

Example:

The speed of light is 299 792 458 𝑚𝑒𝑡𝑒𝑟𝑠/𝑠𝑒𝑐𝑜𝑛𝑑

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 458𝑚

1𝑠𝑒𝑐∗

1𝑘𝑚

1000𝑚∗

60𝑠𝑒𝑐

1𝑚𝑖𝑛∗

60𝑚𝑖𝑛

1ℎ𝑟∗

24ℎ𝑟

1𝑑𝑎𝑦∗

365𝑑𝑎𝑦

1𝑦𝑟= 𝟗. 𝟒𝟓 ∗ 𝟏𝟎𝟏𝟐𝒌𝒎/𝒚𝒓

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 – Practice Problems

Perform the following MASS conversions.

1. Convert 2.3𝑇 to 𝑂𝑢𝑛𝑐𝑒𝑠 2. Convert 23.5𝑙𝑏𝑠 to 𝑚𝑖𝑙𝑙𝑖𝑔𝑟𝑎𝑚𝑠 3. Convert 13.4𝑘𝑔 to 𝑝𝑜𝑢𝑛𝑑𝑠 4. Convert 13 465𝑜𝑧 to 𝑡𝑜𝑛𝑛𝑒𝑠 (𝑀𝑒𝑡𝑟𝑖𝑐) 5. Convert 3.4𝑇 to 𝑚𝑖𝑙𝑙𝑖𝑔𝑟𝑎𝑚𝑠

22

Perform the following TIME conversions.

6. How many 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 are in 3 𝑑𝑎𝑦𝑠? 7. How many 𝑤𝑒𝑒𝑘𝑠 are in 3 𝑎𝑛𝑑 𝑎 ℎ𝑎𝑙𝑓 𝑦𝑒𝑎𝑟𝑠? 8. How many 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 in the 𝑚𝑜𝑛𝑡ℎ𝑠 𝑜𝑓 𝐽𝑢𝑙𝑦 𝑎𝑛𝑑 𝐴𝑢𝑔𝑢𝑠𝑡? 9. How many 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 are in the first 6 𝑚𝑜𝑛𝑡ℎ𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑦𝑒𝑎𝑟?

Perform the following TEMPERATURE conversions

10. How hot is 112℉ in ℃?

11. What is 7℃ in ℉?

12. Prove where Celsius and Fahrenheit are the same.

23

Perform the following conversions of MULTIPLE UNITS.

13. If I can run at 8𝑘𝑚/ℎ𝑟 how fast am I going in 𝑚/𝑠? 14. You watch an ant move 8𝑐𝑚 in 3𝑠𝑒𝑐𝑜𝑛𝑑𝑠, how fast is it travelling in 𝑘𝑚/ℎ𝑟? 15. How long, 𝑖𝑛 𝑚𝑖𝑛𝑢𝑡𝑒𝑠, does it take light to travel 12 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 𝑘𝑚? 16. If you are strong enough to push an object, with constant acceleration at 2 𝑚𝑒𝑡𝑒𝑟𝑠/𝑠𝑒𝑐,

how far can you push it in 2 𝑤𝑒𝑒𝑘𝑠?

24

Extra Work Space

25

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. 𝐴𝑛𝑠𝑤𝑒𝑟𝑠 𝑉𝑎𝑟𝑦

18. 𝐴𝑛𝑠𝑤𝑒𝑟𝑠 𝑉𝑎𝑟𝑦

19. 𝐴𝑛𝑠𝑤𝑒𝑟𝑠 𝑉𝑎𝑟𝑦

1. 324 500 000𝑐𝑚

2. 997 793.3𝑐𝑚

3. 89 977.0𝑐𝑚

4. 23 913.4𝑐𝑚

5. 0.274𝑐𝑚

6. 67 003 200𝑓𝑡

7. 1859.0𝑓𝑡

8. 4047.8𝑓𝑡

9. 0.11𝑓𝑡

10. 0.69𝑚𝑖𝑙𝑒

11. 0.02𝑚𝑖𝑙𝑒

12. 0.0002𝑚𝑖𝑙𝑒

13. 14 493.6𝑚

14. 395.4𝑚

15. 35.5𝑚

16. 𝐴𝑛𝑠𝑤𝑒𝑟 𝑉𝑎𝑟𝑦

1. 73 600𝑜𝑧

2. 9 400 000𝑚𝑔

3. 29.61𝑙𝑏𝑠

4. 0.34𝑡

5. 2 720 000 000𝑚𝑔

6. 259 200𝑠𝑒𝑐𝑜𝑛𝑑𝑠

7. 182𝑤𝑒𝑒𝑘𝑠

8. 89 280𝑚𝑖𝑛𝑠

9. 15 638 400𝑠𝑒𝑐𝑠

10. 44.4℃

11. 44.6℉

12. See written Answer

13. 2.2 𝑚𝑠⁄

14. 0.095 𝑘𝑚ℎ𝑟⁄

15. 0.67𝑚𝑖𝑛𝑠

16. 2 419 200𝑚