1. Proportional Relationships
Chapter 14

2. Intro to ratios and proportions
Ratio is a relationship between two quantities
Proportions are statements that two ratios are equal
Ratio = 30 mi/1 gal or 1 gal/30 mi
Proportion 1 cake/2 oz choc = 2 cake/4 oz
Proportional statements are equations due to the = sign

3. Problem solving A Proportions
How many oz of chocolate are needed to make 5 cakes if 1 cake uses 2 oz?
2 oz choc = X
1 cake cakes
To solve for X cross multiply and divide
Or 2 oz choc x 5 cakes = 1 cake x X
Or X = (2 oz x 5 cakes)/ 1 cake
X = 10oz

4. Problem solving 2 Dimensional Analysis
1 cake = 2 oz of choc
So 1 cake/2 oz of choc or 2 oz of choc/cake
We want to determine chocolate in 5 cakes
We want to eliminate cakes
2 oz of choc cake = 10 oz of choc
1 cake

5. Keeping track of units
In proportions units of the denominator and numerator must be the same
The dimensional analysis, the denominator and numerator will be different
In both cases, unneeded units will cancel and needed units are left.
If this is not the case, the problem is wrong

6. Basics of Dimensional Analysis
Makes use of the cancellation properties of fractions
Both units and numbers may be cancelled
Can be used with anything that has an equivalence statement.
The conversion factors are derived from equivalence statements.

7. Cancellation of Fractions
Numbers may be cancelled in the numerator and denominator if they are the same or multiples of one another
Ex: x 4 = 4 3
Ex: 3 x 4(1)= 3 x 1 = 1.5
8(2)

8. Cancellation of Units
Cancellation of units allows conversion of one unit to another
12 in x 3 ft 36 in
1 ft
120 g x 4 cups = 480 g
1 cup

9. Equivalence Statements
Set one quantity equal to another.
Ex: 12 in = 1 ft
3 ft = 1 yd
10 mm = 1 cm
These statements will always have an equal sign in them.

10. Conversions Factors Convert one unit to another
Are formed from the equivalence statement by dividing each side by the other
There are two conversion factors for each equivalence statement.
Use the conversion factor that will give you the desired unit and cancel the other units.

11. Deriving Conversion Factors
Equivalence statement: 12 in = 1 ft
Conversion factors: 12 in/1 ft or 1 ft/12 in
Equivalence statement: 1 cup = 120 g
Conversion factors: 1cup/120g or g/1cup

12. Percents Percents are familiar ratios
The % sign symbolizes a fraction and the word percent means “of every hundred”
I like to say a % is parts per hundred
% is the number of units studied X 100
Total units

13. Rules for percent % to decimal Decimal to % % to fraction
Move decimal two places to the left
Decimal to %
Move the decimal two places to the right
% to fraction
Write % as fraction with denominator of 100
Fraction to %
Divide numerator by denominator and multiply by 100 and add % sign

14. Continued Calculating the % of a number What is 25% of 200?
Convert the percent to a decimal
Multiply the decimal times the number of interest
What is 25% of 200?
25% = 0.25
.25 x 200 = 50

15. Percents to Laboratory Solutions
How would you prepare 300 ml of a 25% ethanol solution?
Ethanol content = .25 x 300 ml = 75 ml
Water content = total – ethanol
Water content = 300 ml – 75 ml = 225 ml
To prepare the solution, mix 225 ml of water and 75 ml of ethanol

16. Density
Density, d, is the ratio between the mass and volume of a material
Density = mass/volume
May express in various units
Density changes with temperature due to expansion of the material
Benzene d20o = g/ml
Density is g/ml at 20oC

17. Continued Densities are often compared to water
The density of water is 1.00 g/ml
Immiscible liquids of different densities will form layers
Density = mass/volume
Mass = density x volume
Volume = mass/density

18. Concentration and Dilution
Concentration is the amount of a particular substance in a stated volume of a solution or mixture
Solute is the substance that is dissolved or dissociated
Solvent is the substance that dissolves the solute
Amount is how much of a substance is present

19. Practice
How do you make a 400 ml solution that has a concentration of 10 g of KBr in 100 ml of total solution?
Amount of solute:
10 g of KBr g of sol. = 40 g
100 g of solution
Dissolve KBr in 100 g H2O and fill contain.

20. Dilutions
Dilution is when one substance is added to another to reduce the concentration of the first
The original solution is called the stock solution
Many ways to define solutions
Need to be sure you are talking about the total of the final mixture or amount of diluent (substance that dilutes)

21. Dilution Terminology
1 part of food coloring combined with 9 parts of water means the food coloring is 1 part in 10 ml of total volume or 1/10 food coloring
An undiluted substance is called 1/1
When talking about dilutions, the symbol(:) means parts.
1 ml of food color + 9 ml of water (1:9)
This is not a 1:10 dilution

22. Examples A 1:2 dilution means there are three total parts of volume
½ dilution means that there are two total parts of volume
½ = 1:1
1:2 = 1/3
1:3:5 means 1 part, 3 parts b, and 5 parts c

23. Dilutions and proportional relationships
Dilution and proportion are related
1 ml/10 ml = 10 ml/100 = 1/10
100 ml of enzyme are added to 900 ml of buffer. What is the dilution?
30 ml of protein are placed in 400 ml of hepron. What is the dilution?

24. Calculations for Preparing one dilution
How do you prepare 10 ml of a 1/10 dilution of food coloring?
What does 1/10 mean?
How much diluent is needed?
How much food coloring is needed?

25. Other calculations 100 ml of a 1/5 dilution of food coloring?
= x =
Add 20 ml of food color and 80 of water

26. Dilution and Concentration
Concentration of diluted tubes is important
RULE: The concentration of a diluted solution is determined by multiplying the concentration of the original solution times the dilution expressed as a fraction.

27. Examples
A solution containing 5 mg of enzyme/ ml of solution is diluted 1/10. What is the concentration of the diluted solution?
5 mg = .5 mg/ml 10

28. Dilution Series
Dilution series is a group of solutions that have the same components but at different concentrations.
Accomplished by making each diluted solution independently of the others starting with a stock solution.
Serial dilutions are similar to a dilution series but the dilution factor is consistent from step to step

29. Independent dilution
Make a 1/10, 1/50 and 1/100 dilution of food coloring from an original bottle.
1/10 dilute 1 ml to 10 total ml of volume
1/50 dilution
10 ml = 0.2 ml of fc in 10 ml of vol 50
1/100 dilution
10 ml = 0.1 ml in 10 ml of vol
100

30. Non independent dilutions
The concentration of a diluted solution in the final tube is determined by multiplying the concentration of the original solution times the dilution in the first tube, times the dilution in the second tube and so on.
A 1 ml sample of enzyme was diluted 1/10 by the producer. The end user diluted this mixture 1/5. If the original material was 10 mg/ml, what is the concentration of the final dilution

31. Continued
Describe the dilution of a solution containing 15 g of HCl/liter to .15mg/l. We need 2 ml of the final solution.
Are there more than one way to do this dilution?

32. Serial Dilution
A serial dilution is a series of dilutions that all have the same dilution factor. (all are 1/10 or ½ dilutions)
Could the previous problem be done as a serial dilution?