What is a ratio?

• The ratio of male students to female students at a school is 2:3.

• The ratio of juice concentrate to water is 1:3.

• Josie rode her skateboard 5 miles per hour.

What is the difference between a ratio and a fraction?

• Can a ratio always be interpreted as a fraction?

Some ratios or rates can’t be written as fractions

• Josie rode her skateboard 5 miles per hour.

• There is no “whole”, and so a fraction does not really make sense.

What is a proportion?

Proportions

• A comparison of equal fractions

• A comparison of equal rates

• A comparison of equal ratios

Ratios and Rates

• If a : b = c : d, then a/b = c/d.• If a/b = c/d, then a : b = c : d.

• Example:• 35 boys : 50 girls = 7 boys : 10 girls• 5 miles per gallon = 15 miles using 3

gallons

Exploration 6.3

• #1 Do a and b on your own. Then, discuss with a partner.

Additive vs Multiplicative relationships

• This year Briana is making $30,000. Next year she will be making $32,000.

• How much more will she be making next year?

• What is her increase in salary?

• How does her salary next year compare with her salary this year?

We can add fractions, but not ratios

• On the first test, I scored 85 out of 100 points.

• On the second test, I scored 90 out of 100 points.

• Do I add 85/100 + 90/100 as

• 175/200 or 175/100?

Exploration 5.18• When will a fraction be equivalent to a

repeating decimal and when will it be equivalent to a terminating decimal?

• Why does a fraction have to have a repeating or terminating decimal representation?

• #5

What is the meaning of?

“proportional to”

To determine proportional situations…

• Start easy:• I can buy 3 candy bars for $2.00.• So, at this rate, 6 candy bars should cost…• 9 candy bars should cost…• 30 candy bars should cost…• 1 candy bar should cost… this is called a unit

rate.

To determine proportional situations

• Cooking: If a recipe makes a certain amount, how would you adjust the ingredients to get twice the amount?

• Maps (or anything with scaled lengths) If 1 inch represents 20 miles, how many inches represent 30 miles?

• Similar triangles.

To solve a proportion…

• If a/b = c/d, then ad = bc. This can be shown by using equivalent fractions.

• Let a/b = c/d. Then the LCD is bd.

• Write equivalent fractions:a/b = ad/bd and c/d = cb/db = bc/bd

• So, if a/b = c/d, then ad/bd = bc/bd.

To set up a proportion…

• I was driving behind a slow truck at 25 mph for 90 minutes. How far did I travel?

• Set up equal rates: miles/minute• 25 miles/60 minutes = x miles/90 minutes.• Solve: 25 • 90 = 60 • x; x = 37.5 miles.

Reciprocal Unit Ratios

• Suppose I tell you that can be exchanged for 3 thingies.

• How much is one thingie worth? • 4 doodds/3 thingies means

1 1/3 doodads per thingie.• How much is one doodad worth?• 3 thingies/4 doodads means

3/4 thingie per doodad.

Exploration 6.4

• Part 1: a, b, c, e, f– Solve each of these on your own and then

discuss with your partner/group.

Ratio problems

• Suppose the ratio of men to women in a room is 2:3

• If there are 10 more women than men, how many men are in the room?

• If there are 24 men, how many women are in the room?

• If 12 more men enter the room, how mnay women must enter the room to keep the ration of men to women the same?

Strange looking problems

• I see that 1/4 of the balloons are blue, and there are 6 more red balloons than blue.

• Let x = number of blue balloons, and so x + 6 = number of red balloons.

• Also, the ratio of blue to red balloons is 1 : 3• Proportion: x/(x + 6) = 1/3• Alternate way to think about it. 2x + 6 = 4x

x x + 6

Let’s look again at proportions

• Explain how you know which of the following rates are proportional?

• 6/10 mph• 1/0.6 mph• 2.1/3.5 mph• 31.5/52.5 mph• 240/400 mph• 18.42/30.7 mph• 60/100 mph