Dimensional Analysis

Math Made Manageable

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What is Dimensional Analysis?

An organized methodology for examining the relationship between measurements (and convert between them) in order to express the same (or related) value differently.

Okay, so what is a value? A measurement expressed in terms of a number and a

unit.For example: just saying “55” is meaningless.But saying “55 inches” provides us with information.

Objective & Consequences Our original value and our answer are equal. We are only changing the way we express

these values. This means that we must multiply by “1” at

each step in the process (remember this!). Conversion Factors: reflect some equality.

12 inches = 1 foot 2.54 cm = 1 inch

Rules (and steps) of the Game

1. Start with what you know.a. A single unit (if possible).

2. At each step: Fill in units first.a. Canceled Unit: Diagonally Across.b. New Unit (from a conversion factor): into other spot.

3. Then fill in numbers using the conversion factor.

a. Top value must equal bottom value.b. Cancel diagonal units.c. Check remaining units: Is it what we’re looking for? If not Go back to rule 2 and start another step.

4. Multiply left to right (on both the top and bottom).5. Divide the top answer by the bottom answer.

A Simple Problem Convert 2.25 feet into

centimeters (cm). Rule 1: Start with what you know.

Here we have only a single unit. Lets place it on the top (i.e. in the

numerator) of our first step.

2.25 ft

A Simple Problem Convert 2.25 feet into centimeters (cm).Rule 2: Fill in Units first.

Place the unit needing cancelled diagonally across.i.e. on the bottom (i.e. pt 2: in the denominator).

2.25 ft ft

A Simple Problem Convert 2.25 feet into centimeters (cm).Rule 2: Fill in Units first.

New unit comes from a conversion factor.Remember 12 in = 1 foot.

2.25 ft in ft

A Simple ProblemConvert 2.25 feet into centimeters (cm).Rule 3: Fill in numbers.

Numbers also come from the conversion factor.Remember 12 in = 1 foot.Cancel and check remaining units

2.25 ft 12 in 1 ft Inch is not centimeter, so we

know we have at least one more step.

Note that ‘12’ is with‘inch’ both here…

And here!

A Simple ProblemConvert 2.25 feet into

centimeters (cm).Repeat rules 2 & 3:

Fill in units.Fill in numbers.Cancel and check remaining units.

2.25 ft 12 in 1 ft

A Simple ProblemConvert 2.25 feet into centimeters (cm).Repeat rules 2 & 3:

Fill in units.Fill in numbers.Cancel and check remaining units.

2.25 ft 12 in cm 1 ft in

A Simple ProblemConvert 2.25 feet into centimeters (cm).Repeat rules 2 & 3:

Fill in units.Fill in numbers.Cancel and check remaining units.

2.25 ft 12 in 2.54 cm 1 ft 1 in

cm is the unit we are looking for.

So there are no more steps.

A Simple ProblemConvert 2.25 feet into centimeters (cm).Rules 4 & 5:

Multiply left to right.• On both top and bottom.

Divide top answer by bottom answer.

2.25 ft 12 in 2.54 cm 68.58 cm 1 ft 1 in 1

Or just 68.6 cm (remember sig figs!)

A Complex ProblemConvert 40.0 miles per hour into meters per sec.Rule 1: Start with what you know.

Here we have a compound starting unit.• We avoid this if we can, but in this case it’s not possible

Per means divide; hour goes on bottom.

40 mi 1 hr

A Complex Problem Convert 40.0 miles per hour into meters per sec. Rules 2 & 3: Some helpful conversion factors.

1 mile = 1.609 km (or 1 km = 0.6214 miles) 1 km = 1000 m (or 1 m = 0.001 km) 1 hr = 60 min and 1 min = 60 sec

40 mi 1 hr

A Complex Problem Convert 40.0 miles per hour into meters per sec. Rules 2 & 3: Some helpful conversion factors.

1 mile = 1.609 km (or 1 km = 0.6214 miles) 1 km = 1000 m (or 1 m = 0.001 km) 1 hr = 60 min and 1 min = 60 sec

40 mi 1.609 km 1 hr 1 mi

Remember to cancel and check remaining units afterEach step.

A Complex Problem Convert 40.0 miles per hour into meters per sec. Rules 2 & 3: Some helpful conversion factors.

1 mile = 1.609 km (or 1 km = 0.6214 miles) 1 km = 1000 m (or 1 m = 0.001 km) 1 hr = 60 min and 1 min = 60 sec

40 mi 1.609 km 1000 m 1 hr 1 mi 1 km

Now lets fill in the rest.

But note: hr is on the Bottom, here.

A Complex Problem Convert 40.0 miles per hour into meters per sec.

40 mi 1.609 km 1000 m 1 hr 1 min 1 hr 1 mi 1 km 60 min 60 sec

Rules 2 & 3: Some helpful conversion factors. 1 mile = 1.609 km (or 1 km = 0.6214 miles) 1 km = 1000 m (or 1 m = 0.001 km) 1 hr = 60 min and 1 min = 60 secAnd note: hr is on

the top, here.

A Complex Problem Convert 40.0 miles per hour into meters per sec.

Rules 4 & 5: Remember. Multiply across the top and bottom first.

40 mi 1.609 km 1000 m 1 hr 1 min 1 hr 1 mi 1 km 60 min 60 sec

divide the top answer by the bottom answer.

64360 m3600 sec

Or 17.9 meters / sec

THEN

A Complex Problem Convert 40.0 miles per hour into meters per sec. Rules 4 & 5: Remember.

We could have used different conversion factors.

40 mi 1 km 1 m 1 hr 1 min 1 hr 0.6214 mi 0.001 km 60 min 60 sec

40 m2.237 sec

Still 17.9 meters / sec

A Dosage Calculation Problem

A patient requires 95 milligrams of Amoxicillin. The amoxicillin syrup contains 63.5 mg in each 5 ml dose. Your measuring spoons are delineated in fluid ounces. How many fluid ounces will you give.

The trick to word problems is learning to parse your information and organize

your data.

A Dosage Calculation Problem

A patient requires 95 milligrams of Amoxicillin. The amoxicillin syrup contains 63.5 mg in each 5 ml dose. Your measuring spoons are delineated in fluid ounces. How many fluid ounces will you give. What conversion factors do we have?

From the problem: 5 ml syrup = 63.5 mg amoxicillin.

Others you should know: 1 fluid oz = 29.6 ml

A Dosage Calculation Problem

Convert 95 mg Amoxicillin to ? Fluid oz of syrup.

Rule 1: Start with the single unit that you know.

Given: 5 ml syrup = 63.5 mg amoxicillin. Others you should know: 1 fluid oz = 29.6

ml

95 mg Amox

Rules 2 & 3: Fill in units and numbers with conversion factors.

5 ml syrup 63.5 mg Amox

1 fl. oz.29.6 ml syrup

Cancel and Check!

A Dosage Calculation Problem

Convert 95 mg Amoxicillin to ? Fluid oz of syrup.

95 mg Amox

5 ml syrup 63.5 mg Amox

1 fl. oz.29.6 ml syrup

Rules 4 & 5: Remember. Multiply across the top and bottom first. divide the top answer by the bottom answer.

475 fl. oz. 1880

THEN

Answer: 0.25 fluid ounce

A Dosage Calculation ProblemA doctor prescribes 85 mg of medicine for each 10 kg of patient mass. Your patient weighs 175 pounds and the medicine comes in tablets containing 225 mg each. How many tablets will you give. What conversion factors do we have?

From the problem: 10 kg patient mass = 85 mg medicine.1 tablet = 225 mg medicine.

Others you should know: 1 kg = 2.20 lbs

A Dosage Calculation ProblemHow many tablets for a 175 lb patient?

Rule 1: Start with the single unit that you know.

175 lb patient

Rules 2 & 3: Fill in units and numbers with conversion factors.

1 kg 2.20 lb

85 mg med10 kg patient

Cancel and Check!

Conversion Factors: 1 kg = 2.20 lbs10 kg patient mass = 85 mg medicine.1 tablet = 225 mg medicine.

1 tablet225 mg med

A Dosage Calculation ProblemHow many tablets for a 175 lb patient?

175 lb patient 1 kg 2.20 lb

85 mg med10 kg patient

1 tablet225 mg med

Rules 4 & 5: Remember. Multiply across the top and bottom first. divide the top answer by the bottom answer.

14875 tablets 4950

THEN

Answer: 3 tablets