1

Dimensional Analysis

Method

Follow along on your Dimensional

Analysis Worksheet and do the

example problem with the

presentation

2

Which is bigger?

• 120

• 5

• 21

Pennies

Quarters

Nickels

Answer = 5

The units are more important than the numbers!

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What is Dimensional Analysis?

• Dimension – means Unit

• Analysis – means Problem Solving

Math using units!

Use the units to find the solution to a

problem!

4

What is Dimensional Analysis?

Dimensional analysis is

based on using “Conversion

Factors” to convert one type

of unit into another.

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What is a conversion factor?

• A conversion factor is an equality written as

a ratio.

• Every equality gives you two different

conversion factors.

cents 10 dime 1

1dime

cents 10r

cents 10

dime 1o

cents 10

dime 1

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What is a conversion factor?

• As a ratio of two equivalent terms, all

conversion factors equal 1.

Since 1 dime = 10 cents,

cents 10

dime 1the ratio

= 1

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How are conversion factors used?

When used correctly in a math problem, they allow us to convert from one type of unit to another!

cents 40 dime 1

cents 104dimes

Notice the unit, dime, cancels and we are left with cents!

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Setup

1. Write down given

2. Set up workspace

3. Write down units of answer

How many dimes are in 30 cents?

30 cents = dimes

This is the

single

step

example!

Step 2 Step 3

Step 1

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Solution

4. Do I want to keep it? If you don’t, write the unit on the bottom of the next line.

5. What can I change it into? Write that unit on top and then include the numbers.

6. Repeat until the answer to #4 is yes!

30 cents = dimes cents

dime 1

10 Step 4

Step 5

Step 6 – cents cancel and we

are left with dimes so “yes”

go to step 7

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Calculate

7. Everything on top is multiplied and everything on

the bottom is divided.

8. Plug the numbers in the calculator and let it do all

the work!

30 cents = dimes cents

dime 1

10

3 dimes )10(

)1)(30(Step 7

Step 8

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More info – Step 4

• Step 4 – Do I want to keep it?

– The it refers only to the unit. Don’t recopy the

number!

– The units are part of the calculation, so to

cancel a unit it must appear on the top and

bottom of a divisor.

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More info - Step 5

• Step 5 -What can I change it into?

– This really means “what do I know it equals”.

In the example, we wrote dime on top because

we know that 10 cents = 1 dime. We also know

100 cents = 1 dollar, but while we could write

dollar on top, it wouldn’t help us solve the

problem.

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More info – Step 5 continued

– In this step, you are finishing what you started

in step 4. You are actually completing what is

termed a “conversion factor” (a ratio of two

equivalent values). All conversion factors then

equal 1.

• 60 seconds = 1 minute can be written:

onds

uteor

ute

onds

sec60

min1

min1

sec60Both equal 1

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Homework

Use this technique to complete the

Dimensional Analysis worksheet!

Each problem is graded on the work –

NOT the answer. You are learning a

technique – show it if you want any

points!

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Multi-Step

Problems

16

= 17.8 cm

km cm

m

100

1

m

km

1000

1 .000178

A pencil is 17.8 cm long. What is its length in km?

Two step metric conversions

(prefix unit to a prefix unit)

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Other Multi-Step Problems

= 2.5 wks s

wks

days

1

7

days

hours

1

24

hours

min

1

60 1512000

min

s

1

60

1500000 s

How many seconds are in 2.5 weeks?

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Complex

Dimensional

Analysis

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1. Square and Cubic unit conversions

= 3 cm3

m3 cm

m

100

1

cm

m

100

1

cm

m

100

1 .000003

Or

3 10-6 m3

How many m3 are in 3 cm3?

cm•cm•cm

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2. Complex Units (the hidden equality)

What does 60 mi/hr (this is a complex unit) really mean?

60 miles = 1 hr

1hr

60miles

60miles

1hror

It is a bridge to convert time and distance units!

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2. Complex Units

a. Using as a conversion factor

The density of gold is 19.3 g/mL. What is the mass if the

volume is 244.8 L?

= 244.8 L

g L

mL

1

1000

mL

g

1

19.3 4.72 106

19.3 g = 1 mL

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2. Complex Units b. Using as a given

The density of gold is 19.3 g/mL, what is it in cg/kL?

=

19.3 g = 1 mL

cg

kL

19.3 g

1 mL g

cg 100

1

mL

L

1000

1

L

kL

1000

1

1.93 109