Chapter 1:Unit 14. Solving Problems with single or multi-step Conversion Factors

Problem Solving with Conversion Factors

This process is called dimensional analysis. A general problem solving method in which the different units are associated with numbers and final result is calculate by converting one unit to another unit. calc. is set up from unit cancellation to target unit.

Step#1: Identify the units of known or given quantity and the units of target quantity
Step#2: Figure out the conversion factors required from given unit to target unit.
Step#3: Multiply the given quantity by one or more conversion factors in such a manner that the original units are canceled leaving only the desired unit.

Watch this out to solve drug dosage related problems that require conversion factors.

Example Problem:
(A)     A dose of 250 mg of acetaminophen is prescribed for a 20-kg child. How many ml of Children’s Tynenol (100. mg of acetaminophen per 2.5 ml) are needed?

Step#1: Identify given and target quantity and numbers:

Given: 250mg             target: ml
Step#2: Write the conversion factors:

Provided in the problem:                    2.5 ml
                                                                100 mg

Step#3. Multiply the given quantity by the conversion factor.

250 mg ×  2.5 ml      =  6.3ml (lowest 2 sig fig)
100 mg

Check: the answer is reasonable. Since the required dose is larger than standard dose, volume would be larger than standard.

Nurses Conversion units for drugs
Drug Calculations 

Drug calculations vary depending on whether you are dealing with liquid or solid medications, or if the dose is to be given over a period of time. In this section I will go over each of these situations in turn.

It is very important that you know how drug dosages are worked out, because it is good practice to always check calculations before giving medication, no matter who worked out the original amount. It is far better to point out a mistake on paper than overdose a patient.

a)  Tablets

Working out dosage from tablets is simple.

Formula for dosage:

Total dosage required  = Number of tablets required
Dosage per tablet

Note-If your answer involves small fractions of tablets, it would be more sensible to try to find tablets of a different strength rather than try to make 1/3  of a tablet for example.
Examples

1. A patient needs 500mg of X.  X comes in 125mg tablets. How many tablets per day does he need to take?

Total dosage required is 500mg,
Dosage per tablet is 125mg
So our calculation is 5 0 0 =
4 He needs 4 tablets a day

b)  Liquid Medicines
Liquid medicines are a little trickier to deal with as they will contain a certain dose within a certain amount of liquid, such as 250mg in 50ml, for example.
To work out the dosage, we use the formula:
What you want   × What it’s in
What you’ve got

Examples
2)We need a dose of 500mg of Y. Y is available in a solution of 250mg per 50ml.

In this case,

What we want = 500
What we’ve got = 250 What it’s
in = 50

So our calculation is 5 0 0 × 50 =100
250

We need 100ml of solution.
3) We need a dose of 250mg of Z is available in a solution of 400mg per 200ml. In this case,

What we want = 250
What we’ve got = 400
What it’s in = 200

So our calculation is 2 5 0 × 200 = 125
400

We need 125ml of solution.
Medicine over Time
1) Tablets/liquids
This differs from the normal calculations in that we have to split our answer for the total dosage into 2 or more smaller doses.

Look at Example 1 again. If the patient needed the 500mg dose to last the day, and tablets were taken four times a day, then our total of 4 tablets would have to be split over 4 doses.

Total amount of liquid/tablets for day = Amount to be given per dose
Number of doses per day

We would perform the calculation: 4÷4=1 So
he would need 1 tablet 4 times a day.
2)  Drugs delivered via infusion
For calculations involving infusion, we need the following information:

• The total dosage required
• The period of time over which medication is to be given
• How much medication there is in the solution

Example
4)A patient is receiving 500mg of medicine X over a 20 hour X is delivered in a solution of 10mg per 50ml.
What rate should the infusion be set to?
Here our total dosage required is 500mg Period of time is 20 hours

There are 10mg of X per 50ml of solution

Firstly we need to know the total volume of solution that the patient is to receive. Using the formula for liquid dosage we have:

500 ×50=2500 So the patient needs to receive 2500mls.
10
We now divide the amount to be given by the time to be taken:              2 5 0 0

The patient needs 2500mls to be given at a rate of 125mls per hour

d)  Drugs labelled as a percentage
Some drugs may be labelled in different ways to those used earlier.
V/V and W/V
Some drugs may have V/V or W/V on the label.
V/V means that the percentage on the bottle corresponds to volume of drug per volume of solution
i.e 15% V/V means for every 100ml of solution, 15ml is the drug.
W/V means that the percentage on the bottle corresponds to the weight of drug per volume of solution.  Normally this is of the form ‘number of grams per number of millilitres’. So in this case 15% W/V means that for every 100ml of solution there are 15 grams of the drug.

If we are converting between solution strengths, such as diluting a 20% solution to make it a 10% solution, we do not need to know whether the solution is V/V or W/V.
Examples
5. We need to make up 1 litre of a 5% solution of We have stock solution of 10%. How much of the stock solution do we need? How much water do we need?

We can adapt the formula for liquid medicines here:

What we want           ×  What we want it to be in
What we’ve got

We want a 5% solution. This is same as 5/100 or 1/20.
We’ve got a 10% solution. This is the same as 10/100 or 10 .
We want our finished solution to have a volume of 1000ml. Our
formula becomes: ( 1/20)   * 1000
                                  (1/10)

(using the rule for dividing fractions)

= 1/2×1000 = 500 .

We need 500mls of the A solution.

Which means we need 1000 – 500 = 500mls of water.

(Alternatively you can use the fact that a 5% solution is half the strength of a 10% solution to see that you need 500ml of solution and 500ml of water)
6.)You have a 20% V/V solution of drug The patient requires 30ml of the drug. How much of the solution is required?

20% V/V means that for every 100ml of solution we have 20ml of drug F. Using our formula:

What you want   × What it’s in
What you’ve got

This becomes 30 ×100 = 150
We need 150mls of solution.
7)Drug G comes in a W/V solution of 5%. The patient requires 15 grams of How many mls of solution are needed?

5% W/V means that for every 100mls of solution, there are 5 grams of G. Using the formula gives us
15 ×100 = 300

300mls of solution are required.

Other Dimensional analysis problem example:
B)A person is driving a car with a speed of 229.8 km/h. What is the speed in
1)Miles per hour
2)Feet per second

Given 1 km= 0.6214 mile 1000m= 1km,  1 m= 3.28 ft, 60 sec=1min and 60 min= 1 hr
1)Step#1: Identify given and target quantity and numbers:
Given: 229.8km/hr                  target: mi/hr
Step#2: Write the conversion factors:

Provided in the problem:                    0.6214 mi
                                                            1 km
Step#3. Multiply the given quantity by the conversion factor.

229.8 km ×  0.6214 mi      =  143.4mi/hr (lowest 4 sig fig used in problem)
1 km
Step#1: Identify given and target quantity and numbers:

Given: 229.8km/hr                  target: ft/s

Step#2: Write the conversion factors:

Provided in the problem:                    3.28 ft & 0.6214 mi
                                                                   1 m          1 km

1 hr,              1 min

60 min           60 sec

Step#3. Multiply the given quantity by the conversion factor.

229.8 km ×  1000m  × 3.28 ft   × 1 hr       × 1 min
     1hr                       1 km         1 m        60 min      60 sec

=  209 ft/s ( lowest 3 sig fig used in calc).

For more information watch this out:

For more practice visit:

Questions:

1. If one teaspoon contains 5.0 ml, how many teaspoons of Children Tylenol (100. mg of acetaminophen per 2.5 ml) are needed for a child with a dose of 240 mg?

2. A patient is prescribed 0.150mg of a drug that is available in 25 μg tablets.How many tablets are needed?

3. how many milliliters of Children’s Mortin (100 mg of ibuprofen per 5 ml) are needed to give a child a dose of 180 mg?

4. A chemist synthesized 0.510 kg of aspirin in the lab. If the normal doses of aspirin is two 325 mg tablets, how many doses did she prepare?

5.  A patient requires 3.0 pt of blood during surgery. How many liters does this correspond to?

given  1qt= 2 pt, 1 L=1.06 qt

6. A patient is prescribed 2.0 g of medication to be taken four times a day. If the medicine is available in 500 mg tablets, how many tablets are needed in a 24. hr period?

Ans: 1. 9 ml

2.6 tablets

3. 1.2 teaspoon

4. 785 doses

5. 1.4 L

6. 16 tablets

• For more practice: Go to the following website for online tutorial on Unit conversion:
  http://joneslhs.weebly.com/Unit Conversion Tutorial                
• Go to the following website: http://joneslhs.weebly.com
• Click on the Learn button on the left. Read the tutorial first.  When you think that you understand the idea, go back to the Main Menu and click on One Step Conversions.
• One Step Conversions
  • For problems 1, 2, and 3 write down what the completed problem looks like. Cancel the units that cancel.  Circle the unit that doesn’t cancel.  Write down the answer to the problem.
  • 

For problems 4-9, you can just write down the answer once you have solved it.

4. Calculated Answer:
5. Calculated Answer:
6. Calculated Answer:
7. Calculated Answer:
8. Calculated Answer:
9. Calculated Answer:

For problem 10, solve it on paper here.  Then type in the calculated answer to see if you are correct.

10. Solved problem and answer:

WHEN YOU ARE DONE, SHOW YOUR WORK TO YOUR INSTRUCTOR BEFORE MOVING ON TO A MORE CHALLENGING LEVEL.

Multi-Step Conversions                      Name: ___________________________________

• For problems 1, 2, and 3 write down what the completed problem looks like. Cancel the units that cancel.  Circle the unit that is the one left at the end.  Write down the answer to the problem.

For problems 4-10, you can just write down the answer once you have solved it.

4. Calculated Answer:
5. Calculated Answer:
6. Calculated Answer:
7. Calculated Answer:
8. Calculated Answer:
9. Calculated Answer:

For problem 10, solve it on paper here.  Then type in the calculated answer to see if you are correct.

10. Solved problem and answer:

WHEN YOU ARE DONE, SHOW YOUR WORK TO YOUR INSTRUCTOR BEFORE MOVING ON TO A MORE CHALLENGING LEVEL.

Name: ___________________________________

Double Unit Conversions

• Read the directions on the first problem to see how to get started. Work through the challenging problems recording your answer for each one.   Don’t forget units!

1. Calculated Answer:
2. Calculated Answer:
3. Calculated Answer:
4. Calculated Answer:
5. Calculated Answer:
6. For problem 6, solve it on paper here. Then type in the calculated answer to see if you are correct.

Cubed and Squared Conversions

• Read the directions on the first problem to see how to get started. Work through the challenging problems recording your answer for each one. Don’t forget units!

1. Calculated Answer:
2. Calculated Answer:
3. Calculated Answer:
4. For problem 4, solve it on paper here. Then type in the calculated answer to see if you are correct.

The following classroom resources has been taken from AACT.

Cupcake Conversions, from Bench to Bakery

Background

All consumer products start as a small batch to formulate ideal qualities but are scaled up to mass production for consumer purchase. In this process troubleshooting is essential to maintain quality and consistency of product.

You will take on the role of a successful baker who has an award winning recipe for cupcakes which is going to be scaled up to commercial baking. To achieve this end goal you will look at conversions from English units to Metric units and then standardize all units to grams regardless of ingredients. Finally you will look at issues on a microscale of production and see how they would affect macroscale baking of the product.

Objectives

• Convert measurements from English to the International Units of Measurement (Metric).
• Scale up measurements for a standard recipe to a larger quantity.
• Trouble shoot problems encountered with scaling up a product from bench to bakery (small to larger quantities).

Activity 1

1. Read the recipe for a vanilla cupcake below:

Vanilla CupcakesThe following recipe yields 20-25 cupcakes.Ingredients:·         2 cups of flour·         ½ teaspoon of salt·         2 teaspoons of baking powder·         ½ cup of unsalted butter, softened·         ¾ cup of sugar·         2 eggs·         1 cup of whole milk·         1 teaspoon vanilla extract (optional)Directions for baking:·         Preheat oven to 375oF; line muffin cups with papers.·         Beat and mix butter and sugar until it becomes a light and fluffy homogenous mixture. Beat in eggs one at a time.·         Mix baking powder, salt and flour.·         Add the flour mixture alternating with milk; beat well.·         Stir in the vanilla.·         Divide evenly among pans and bake for 18 minutes.·         Let cool in pans.

2. You live in a global society and you realize that this recipe should be out there for the rest of the world. Unfortunately, the rest of the world (except for the United States, Liberia and Myanmar) doesn’t use the English Standard of measurement. They all use the International System of Measurement, or the Metric scale. Please convert each of the measurements below from English Standard to Metric.a)How many grams are in 2 cups of wheat flour if 1 cup is 120.00 grams? Remember to use dimensional analysis to solve this problem.

Example:

Starting unit x Conversion = Answer

2.00 Cups of Flour                  x          120.00 grams =         240. g

1.00 Cup of Flour

Note that the units are cancelled because anything divided by itself is the value of 1.00.

b) How many grams are in ½ teaspoon of salt if 5.00 grams of salt are in 1.00 teaspoon?

c) How many grams are in 2 teaspoons of baking powder if 4.60 grams of baking powder are in 1.00 teaspoon?d)

d) How many grams are in ½ cup of unsalted butter if there are 227.00 grams of butter in 1.00 cup?

e) How many grams of sugar are in ¾ cup of sugar if there are 200.00 grams of sugar in 1.00 cup?

f) What is the mass of 2 eggs if the mass of an average egg is 2.00 ounces? There are 28.50 grams per ounce.  You must first convert from the number of eggs to ounces and then the number of ounces to grams.

g) How many grams of milk are in 1 cup of milk if there are 473.176 mL of milk in 2.00 cups and the density of milk is 1.027 grams/mL? You must first convert the cups of milk to milliliters and then convert milliliters to grams using the density.  Remember use dimensional analysis.

h) How many grams of vanilla extract are in 1 teaspoon vanilla extract if 1.00 teaspoon is 0.15 fluid ounces and 1.00 fluid ounce is 28.35 grams? You must first convert the teaspoon to fluid ounces and then the fluid ounces to grams.  Remember to use dimensional analysis.

1. Convert the baking temperature of 375 ^oF to Celsius.
   a) Why Celsius? Watch the video to understand the difference between the two scales.
   b) Use the following formula for the conversion:
   T_(°C) = (T_(F) – 32) x 5/9j) Successful bakeries don’t just make one batch of anything. In order to be competitive this recipe must be scaled up. How much of each ingredient would be required to make 200 cupcakes? Fill in the table below with your scale up information.

Ingredients	Amount Calculated (g) for Single Batch or 24 Cupcakes	Amount Calculated (g) for 8.33 Batches or 200 Cupcakes
Flour
Salt
Baking Powder
Unsalted Butter
Sugar
Eggs
Milk
Vanilla Extract

Activity 2

Scaling up is not as simple as taking the basic ingredients and then multiplying by a factor to get the total quantity. So many variables can affect the outcome.

1. Using the graphic organizer below, brainstorm at least three issues that could arise when baking cupcakes (i.e. what could go wrong) at the micro-level (home/test kitchen).

Heating	Ingredients	Mixing	Pan Selection

2. Using the graphic organizer below, brainstorm at least three issues that could arise when baking cupcakes (i.e. what could go wrong) at the macro-level (commercial bakery).

Heating	Ingredients	Mixing	Pan Selection

3. Share your thoughts with at least one other person in class. Fill in ideas that you didn’t have in your graphic organizer.
4. Be prepared to share your ideas during the teacher lead discussion.
5. Answer the following questions:
   a) After listening to other students and watching the slide show summarize your findings in a concise set of directives to the bakery manager to:
   1. Ensure that the final product (the 200 cupcakes) maintains its quality and consistency.
   2. Provide the final recipe, in grams, to the bakery manager.b) Putting yourself in the role of the bakery manager explain why you will still need to do testing to ensure quality control.

Answer Key: Cupcake Conversions, from Bench to Bakery

*Note that not all answers are calculated using significant digits.

Activity 1

3. Read the recipe for a vanilla cupcake below:

Vanilla CupcakesThe following recipe yields 20-25 cupcakes.Ingredients:·         2 cups of flour·         ½ teaspoon of salt·         2 teaspoons of baking powder·         ½ cup of unsalted butter, softened·         ¾ cup of sugar·         2 eggs·         1 cup of whole milk·         1 teaspoon vanilla extract (optional)Directions for baking:·         Preheat oven to 375oF; line muffin cups with papers.·         Beat and mix butter and sugar until it becomes a light and fluffy homogenous mixture. Beat in eggs one at a time.·         Mix baking powder, salt and flour.·         Add the flour mixture alternating with milk; beat well.·         Stir in the vanilla.·         Divide evenly among pans and bake for 18 minutes.·         Let cool in pans.

4. You live in a global society and you realize that this recipe should be out there for the rest of the world. Unfortunately, the rest of the world (except for the United States, Liberia and Myanmar) doesn’t use the English Standard of measurement. They all use the International System of Measurement, or the Metric scale. Please convert each of the measurements below from English Standard to Metric.k) How many grams are in 2 cups of wheat flour if 1 cup is 120.00 grams? Remember to use dimensional analysis to solve this problem.

Example:

Starting unit x Conversion = Answer

2.00 Cups of Flour               x      120.00 grams     =      240. g

1.00 Cup of Flour

Note that the units are cancelled because anything divided by itself is the value of 1.00.
l) How many grams are in ½ teaspoon of salt if 5.00 grams of salt are in 1.00 teaspoon?

0.5 teaspoons of salt    x      5.00 grams         =                      2.50 g

1.00 teaspoons of salt

 m) How many grams are in 2 teaspoons of baking powder if 4.60 grams of baking powder are in 1.00 teaspoon?

2.00 teaspoons of baking powder     x      4.60 grams         =      9.20 g

1.00 teaspoons of baking powder

n) How many grams are in ½ cup of unsalted butter if there are 227.00 grams of butter in 1.00 cup?

0.50 Cups of butter      x      227.00 grams     =      113.5 g

1.00 Cup of butter

o) How many grams of sugar are in ¾ cup of sugar if there are 200.00 grams of sugar in 1.00 cup?

.75 Cups of sugar        x      200.00 grams     =      150.00 g

1.00 Cup of sugar

p)  What is the mass of 2 eggs if the mass of an average egg is 2.00 ounces? There are 28.50 grams per ounce. You must first convert from the number of eggs to ounces and then the number of ounces to grams.

2 eggs       x      2.00 ounce x       28.50 grams       =      114 g

1 egg                1 ounce

q) How many grams of milk are in 1 cup of milk if there are 473.176 mL of milk in 2.00 cups and the density of milk is 1.027 grams/mL? You must first convert the cups of milk to milliliters and then convert milliliters to grams using the density.  Remember use dimensional analysis.

1 cup of milk               x      473.176 mL        x       1.027 grams       =        242.98 g

2 cups of milk     1 mL

r) How many grams of vanilla extract are in 1 teaspoon vanilla extract if 1.00 teaspoon is 0.15 fluid ounces and 1.00 fluid ounce is 28.35 grams? You must first convert the teaspoon to fluid ounces and then the fluid ounces to grams.  Remember to use dimensional analysis.

1 tsp vanilla        x      0.15 fl. ounce      x       28.35 grams       =        4.25 g

1 tsp                         1 fl. ounce

s)  Convert the baking temperature of 375 ^oF to Celsius.
a)Why Celsius? Watch the Temperature Guys video to understand the difference between the two scales.
b) Use the following formula for the conversion:
T_(°C) = (T_(F) – 32) x 5/9

190.5 °C
t)  Successful bakeries don’t just make one batch of anything. In order to be competitive this recipe must be scaled up. How much of each ingredient would be required to make 200 cupcakes? Fill in the table below with your scale up information.

Ingredients	Amount Calculated (g) for Single Batch or 24 Cupcakes	Amount Calculated (g) for 8.33 Batches or 200 Cupcakes
Flour	240	2000
Salt	2.50	20.83
Baking Powder	9.20	76.67
Unsalted Butter	113.5	945.83
Sugar	150.00	1250
Eggs	114	950
Milk	242.98	2024.83
Vanilla Extract	4.25	35.42

Activity 2

Scaling up is not as simple as taking the basic ingredients and then multiplying by a factor to get the total quantity. So many variables can affect the outcome.

6. Using the graphic organizer below, brainstorm at least three issues that could arise when baking cupcakes (i.e. what could go wrong) at the micro-level (home/test kitchen). Answers will vary, refer to PPT.

Heating	Ingredients	Mixing	Pan Selection

7. Using the graphic organizer below, brainstorm at least three issues that could arise when baking cupcakes (i.e. what could go wrong) at the macro-level (commercial bakery). Answers will vary, refer to PPT.

Heating	Ingredients	Mixing	Pan Selection

8. Share your thoughts with at least one other person in class. Fill in ideas that you didn’t have in your graphic organizer.
9. Be prepared to share your ideas during the teacher lead discussion.
10. Answer the following questions:
    c) After listening to other students and watching the slide show summarize your findings in a concise set of directives to the bakery manager to:
    1. Ensure that the final product (the 200 cupcakes) maintains its quality and consistency.
    2. Provide the final recipe, in grams, to the bakery manager.

Answers will vary, some possibilities include:

• Production baker needs to calibrate ovens.
• Production baker might have different equipment so it will be different at larger scales so he/she will still need to trouble shoot batches of cupcakes.
• He/she should change variables, collect data, and send the results back to the original baker so that trouble shooting can occur back at the micro level. This is what happens in industry with product development.d) Putting yourself in the role of the bakery manager explain why you will still need to do testing to ensure quality control.