Solving Unit Rate Problems

Lesson 3 Solving Unit Rate Problems 1 Getting the idea A rate is a comparison, or ratio, of two quantities with different units. For example, a store sells $15 3 T-shirts for $15. The comparison $15 to 3 T-shirts is a rate. This rate can also be written as ​ _______ 3 T-shirts ​. When a rate compares a quantity to one unit of another quantity, the rate is a unit rate. $15 The rate ​ _______ 3 T-shirts ​is not a unit rate because the rate compares the cost to more than one T-shirt. In a unit rate, the second quantity (or denominator) should be 1 unit. A speed limit is a unit rate. For example, the speed limit of 55 miles per hour compares the distance to 1 hour. A speed of 55 miles per hour is equivalent to traveling 55 miles in each 1-hour time span. This 55 miles rate can also be written as ​ ______ 1 hour ​. $15 3 T-shirts 55 miles 1 hour not a unit rate unit rate Every rate can be written as a unit rate. Duplicating any part of this book is prohibited by law. © 2015 Triumph Learning, LLC Example 1 A store sells 3 T-shirts for $15. What is the cost per T-shirt? Strategy Step 1 Use a diagram to find the unit rate. Write the given rate as a fraction. Because the question asks for the cost per T-shirt, the rate compares dollars to T-shirts. $15 ​ _______ 3 T-shirts ​ Step 2 Draw a diagram to represent the rate. There are 15 dollars, so divide the top rectangle into 15 equal parts. There are 3 T-shirts, so divide the bottom rectangle into 3 equal parts. dollars T- shirts Lesson 3: Solving Unit Rate Problems 23 313NASE_Mth_G6_D1_PDF Round.indd 23 20/05/14 2:27 PM Step 3 Use the diagram to find the unit rate. The unit rate is the rate that compares the cost to one T-shirt. dollars T- shirts $5 There are five dollars for one T-shirt. So, the unit rate is ______ ​ 1 T-shirt​, or $5 per T-shirt. Solution The cost per T-shirt is $5. Example 2 Changying bikes 36 miles in 3 hours. How many miles per hour does she travel? Strategy Step 1 Use an equation. Write a rate that compares distance to time. 36 miles Distance _______ ​  time ​5 ​ ______ 3 hours ​ Step 2 Write an equation. The unit rate is the number of miles per 1 hour. The given rate and the unit rate are equivalent. 36 miles ? miles _____ ​ ______ 3 hours ​5 ​ 1 hour ​ Step 3 Use division to find the missing number. Because 3 hours 4 3 5 1 hour, write an equivalent rate by dividing 36 miles by 3: 36 miles 4 3 12 miles ______ ​ __________ 3 hours 4 3 ​5 ​  1 hour ​ Solution 12 miles Changying travels ​ ______ 1 hour ​, or 12 miles per hour. Note that in both Example 1 and Example 2, the unit rate can be found by dividing the numerator by $15 the denominator and simplifying the fraction. The unit rate for _______ ​ 3 T-shirts​is 15 4 3 5 5 dollars. The unit 36 miles rate for ​ ______ 3 hours ​is 36 4 3 5 12 miles per hour. This is because the quotient can be written as a fraction over a denominator of 1. To find a unit rate, write the ratio as a fraction and divide the numerator by the denominator. 24 Domain 1: Ratios and Proportional Relationships 313NASE_Mth_G6_D1_PDF Round.indd 24 20/05/14 2:27 PM Example 3 1 Ramon drives at a rate of 60 miles per hour. He stops for lunch after driving for ​ __ 2 ​hour. How many miles 1 did Ramon drive in __ ​ 2​hour? Strategy Use a diagram to find the distance. Step 1 Draw a diagram to show the given unit rate. 60 miles The given unit rate is ​ ______ 1 hour ​. miles 0 60 0 1 hours Step 2 Label the known and unknown quantities on the diagram. 1 1 Ramon drove for __ ​ 2 ​hour. Show __ ​ 2​on the number line for hours. miles 0 60 hours 0 Step 3 1 1 2 Use the diagram to find the unknown quantity. The tick mark divides 60 into two equal parts. Since 60 4 2 5 30, each part must represent 30. Duplicating any part of this book is prohibited by law. © 2015 Triumph Learning, LLC miles 0 30 60 0 1 2 1 hours Solution 1 In __ ​ 2 ​hour, Ramon traveled 30 miles. Lesson 3: Solving Unit Rate Problems 25 313NASE_Mth_G6_D1_PDF Round.indd 25 20/05/14 2:27 PM Example 4 Eva pays a company $30 every 6 months to host her Web site. (a) How much does Eva pay per month? (b) How much does Eva pay for 9 months? Strategy Step 1 Use equations to find the equivalent rates. Write the given rate. Cost $30 _______ ​ ____ Time ​5 ​ 6 months ​ Step 2 Use division to find the cost per month. Divide 30 by 6 to simplify the fraction and find the unit rate. 30 ​ ___ 6 ​5 30 4 6 5 5 $5 5 This is equal to __ ​ 1 ​, or ​ ______ 1 month .​ The unit rate is $5 per month. Step 3 Write an equation that can be used to answer part b. Part b asks for the cost for 9 months. Use the unit rate to write an equation. $5 ? _______ ​ ______ 1 month ​5 ​ 9 months ​ Step 4 Use multiplication to find the missing number. Because 1 month 3 9 5 9 months, write an equivalent rate by multiplying the numerator and denominator by 9: $5 3 9 $45 _______ ​ _________ 1 month 3 9 ​5 ​ 9 months ​ Solution 26 Eva pays $5 per month and $45 for 9 months. Domain 1: Ratios and Proportional Relationships 313NASE_Mth_G6_D1_PDF Round.indd 26 20/05/14 2:27 PM 2 COACHED EXAMPLE A water pump can pump 250 gallons from a pool in 5 minutes. (a) What is the unit rate of the pump in gallons per minute? (b) How long will it take to pump 450 gallons from the pool? (a) The unit rate compares The given rate is to gallons in . minutes. Write the given rate as a fraction. gallons ___________ ​  ​ minutes Divide the numerator by the denominator to find the unit rate. 4 5 The unit rate is (b) Use the . rate to write an equation to solve the problem. gallons 450 gallons ___________ ​  1 minute ​5 ​ _________ ? minutes ​ Multiply the numerator and denominator by gallons 3 to find the equivalent fraction. 450 gallons       ​5 ​ ___________​ ​ __________________ Duplicating any part of this book is prohibited by law. © 2015 Triumph Learning, LLC 13 It takes minutes minutes minutes to pump 450 gallons from the pool. The unit rate of the pump is gallons per minute. It takes 450 gallons of water from the pool. minutes to pump Lesson 3: Solving Unit Rate Problems 27 313NASE_Mth_G6_D1_PDF Round.indd 27 20/05/14 2:27 PM 3 LESSON PRACTICE 1 Four muffins cost $12. Complete the model to find the cost per muffin. dollars muffins The cost per muffin is 2 . Select a rate that is a unit rate. Circle all that apply. 36 cookies A. ​ ________ 3 batches ​ B. 60 gallons every 4 hours C. $40 per ticket 10 miles D. ​ ______ 10 hours ​ 1 E. ​ __ 2 ​cup per serving F. 15 people for every row 12 roses G. ​ ______ 1 vase ​ 3 Draw a line from the rate to its unit rate. 225 feet • • 18 feet per minute 450 feet • • 30 feet per minute 36 feet • • 45 feet per minute 210 feet • • 50 feet per minute A. ​ _______ 5 minutes ​ B. ​ _______ 9 minutes ​ C. ​ _______ 2 minutes ​ D. ​ _______ 7 minutes ​ 4 A line-painting truck is painting a line on the side of a roadway. The truck paints at a rate of 1 8 miles per hour. How many miles can be painted in __ ​ 4​hour? 1 miles can be painted in ​ __ 4 ​hour. 28 Domain 1: Ratios and Proportional Relationships 313NASE_Mth_G6_D1_PDF Round.indd 28 20/05/14 2:27 PM 5 Twelve boxes of pencils cost $16. Which equation could you use to find the cost per box? boxes 12 boxes A. ​ _______ ​5 ​ __________ ​ $16 $1 $16 $1 __________​ B. ​ _______ 12 boxes ​5 ​  boxes 12 boxes $ C. ​ _______ ​5 ​ ______ 1 box ​ $16 $16 $ ______ D. ​ _______ 12 boxes ​5 ​  1 box ​ 6 Cassie drives 165 miles in 3 hours. Circle the numbers that make the rates below equivalent to the rate 165 miles in 3 hours. 55 Cassie drives 60 miles per hour. 65 Duplicating any part of this book is prohibited by law. © 2015 Triumph Learning, LLC 275 Cassie drives 300 miles in 5 hours. 325 7 2 Troy’s recipe for bagels makes 18 bagels per batch. Troy makes __ ​ 3​batch of bagels. How many bagels does Troy make? Complete the model to solve the problem. bagels 0 18 0 1 batches Troy makes bagels. Lesson 3: Solving Unit Rate Problems 29 313NASE_Mth_G6_D1_PDF Round.indd 29 20/05/14 2:27 PM 8 Are the rates equivalent? Select True or False for each equation. A. 35 feet for every 7 seconds 5 7 feet for every second ○ True ○ False ______ B. ​ _______ 5 cases ​5 ​  1 case ​ ○ True ○ False C. $12 per hour 5 $48 for every 4 hours ○ True ○ False ○ True ○ False 120 cans 1 teacher 25 cans 6 teachers _________ D. ​ ________ 15 students ​5 ​ 90 students ​ 9 Takashi ran 200 yards in 40 seconds. Ana ran 150 yards in 25 seconds. Part A Can you tell who runs faster without rewriting the rates? Explain why or why not. Part B Who ran faster? Show your work. 30 Domain 1: Ratios and Proportional Relationships 313NASE_Mth_G6_D1_PDF Round.indd 30 20/05/14 2:27 PM 10 Use numbers from the box to complete the table so that the rates are equivalent. 10 15 90 Dollars Hours 1 5 18 180 12 45 216 Duplicating any part of this book is prohibited by law. © 2015 Triumph Learning, LLC 360 11 The speed limit is 50 miles per hour. Kyle is driving to a baseball game that starts in 2 hours. Kyle is 130 miles away from the baseball field. If Kyle drives at the speed limit, will he arrive in time? Show your work. 12 A new issue of Padma’s favorite magazine comes out every month. If she buys the magazine at the bookstore, the cost is $5 per month. If she subscribes, the cost is $57 per year. Should Padma buy each issue separately or should she get a subscription? Show your work. Lesson 3: Solving Unit Rate Problems 31 313NASE_Mth_G6_D1_PDF Round.indd 31 20/05/14 2:27 PM