1. What is the Title of Lesson 2-6? 2. What is a ratio? 3. What is a proportion? 4. What is the difference between a rate and a unit rate? 5. What are means and extremes?
Objectives: By the end of the next class, students will be able to: Analyze and compare ratios and rates. Solve proportions using equivalent ratios (fractions) and rates. Find the means and extremes of a proportion with 85% or above mastery.
A ratio is a comparison of two numbers by division. The ratio of x to y can be written in 3 different ways. x: y x to y or x y An equation that has 2 equal ratios (fractions) is called a proportion. Example: 2: 5 = 6: 15 2 =
To determine if 2 ratios (fractions) are equivalent, you can write them in simplest form. See Example 1: Another way to determine if 2 ratios are equal is by using cross products. If cross products are =, then the ratios form a proportion. (This means that the product of the means must equal the product of the extremes.) In 2: 5 = 6:15 2 = and 6 are called the means (the middle terms of the proportion). 15 and 2 are called the extremes (the first and last terms of the proportions). Does 5 6 = 15 2 30 = 30
So, 2 = , Are these fractions equivalent? 7 9 = 3 14 Put a ? over the = sign 63 42 So 3 Individual Practice – p , 3
Solve each proportion. If necessary round to the nearest tenth. 4. n = (n) = 9(6) 27n = n = 2
The ratio of 2 measurements having different units of measure is called a rate. For example, 45 miles in 5 hours is called a rate. miles hour A rate that tells how many of one item is being compared to 1 of another item is called a unit rate. In the example above 9 miles per hour. 7. Jennie ran the first 6 miles of a marathon in 58 minutes. If she is able to maintain the same pace, how long will it take her to finish the 26.2 miles? What units of measure are we comparing?
miles minutes How will this proportion be setup? 6 miles = minutes x minutes 6x = 58(26.2) 6x = x = minutes How many hours and minutes would this be? It would take her 4 hours and 13.3 minutes to finish the race.
8. Maps: On a map, North Carolina, Raleigh and Asheville are about 8 inches apart. If the scale is 1 inch = 12 miles, how far apart are the cities? The rate will be inches miles 1inch = 8 inches 12 miles m miles 1m = 96 Cross Products m = 96 miles Simplify The cities are about 96 miles apart.
33. Menu: On Monday, a restaurant made $545 selling 110 hamburgers. If they sold 53 hamburgers on Tuesday, how much did they make? The rate will be cost hamburgers cost 545 = x Hamburgers x = 28, 885 Cross products Divide both sides by 110 x = Simplify
TThe made about $ on Tuesday for selling 53 hamburgers. Proportions Using the Distributive Property. 27. x – 3 = (x – 3) = 30 Cross products 10x – 30 = 30 Distributive Property Add 30 to both sides 10x = 60 Simplify Divide 10 on both sides x = 6 Simplify
Shared Practice: Now try 29, 31 and 35
37. 4v + 7 = 6v (6v + 2) = 10(4v + 7) 90v + 30 = 40v + 70 Try 39
44. Diaries: In a survey, 36% of the students said that they kept an electronic diary. There were 900 students who kept an electronic diary. How many students were in the survey? What units of measure are we using? students keeping an electronic diary total students 36 = x 36x = 90, x = 2,500
There were 2500 total students in the survey.