21
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 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• 175/100?
What is the meaning of?
“proportional to”
To determine proportional situations…
• 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: units?
Reciprocal Unit Ratios• Suppose I tell you that 4 doodads can be
exchanged for 3 thingies. • How much is one thingie worth? • 4 doodads/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:
Determine whether each situation is a proportional relationship or not. Can it be solved using a proportion?
Can you write an equation that relates the quantities in the situation?
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 many
women must enter the room to keep the ration of men to women the same?
Proportional Reasoning• 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
