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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