Lesson 8 2 Solving Rate Problems with Proportions

Lesson 8. 2 Solving Rate Problems with Proportions

Math Message �Solve problems 1 -3 on journal page 282

Math Message � Problem 1: • The left side, • 8 miles 1 hour �Is the original rate • The right side, • 24 miles 3 hours �Is the equivalent rate • 8 miles 24 miles 1 hour 3 hours • Is a proportion � Until now, you have used proportions to only summarize solutions after finding them by some other method. � In this lesson, you will learn how to use proportions to solve rate problems.

Introduction �Hannah can type at the rate of 40 words per minute. How long would it take her to type a 200 -word essay? Words 40 200 Minutes 1 t �This is an open proportion because it is neither true not false.

Solution �If we use the Identify Property of Multiplication, we can solve this problem. 40 words • 5 200 words 1 minute • 5 t minutes (5) �Since t=5, Hannah would need 5 minutes to type the essay.

Practice � Amador ran 200 meters in 45 seconds. At this pace, how long would it take him to run 1, 600 meters? Meters 200 1, 600 Seconds 45 s � Since 200 * 8 = 1, 600 and 45 * 8 = 360, it would take Amador 360 seconds, or 6 minutes to run 1, 600 meters.

Practice �Madra’s grandmother’s recipe for 48 oatmeal cookies calls for 2 eggs. How many cookies can be made with 3 eggs? Cookies 48 c Eggs 2 3 �Since 1 egg will make 24 cookies then 3 eggs can make 72 cookies.

Practice �In March 2000, the Big Dig construction project in Boston was spending about $4 million per day. At this rate, how long would it take to spend $100 million? Dollars 4 million 100 million Days 1 d �Since 4 * 25 is 100 and 1 * 25 is 25, it would take 25 days to spend $100 million.

8. 3 Solving Proportions by Cross Multiplication

Math Message �Complete 286. the problems on journal page

Definitions �Cross products are found by multiplying the numerator of each fraction by the denominator of the other fraction �Cross multiplication is the process of finding cross products.

Equivalent Fractions/Cross Products � 3 5 6 10 � 10 12 12 � 10*3=30 � 10*8=80 � 5*6=30 � 12*5 -60 � Cross products equal; fractions are equivalent � 5 8 Cross products don’t equal; fractions are not equivalent

Using Cross Multiplication to Solve Proportions � Step 1 � Because • Cross multiply. Note that the cross product of 6 and x is written as 6 x. � 5 6 x 18 we want the two fractions in the proportion to be equivalent, we also want the two cross products to be equal. � That is, we want the product 6*x to equal the product 18*5 � 18*5=6*x

Using Cross Multiplication to Solve Proportions � Step 3 • Solve the equation from � Step • Write 15 in place of x in Step 2 the proportion: � 5 � 18*5=6*x � 90=6 x � 90/6=x � 15=x 4 6 15 18

Practice �Complete the even problems on page 288 and 289 in your math journal with a partner

Assignment �Complete math boxes 8. 2 and 8. 3 and page 249 & 250