Lesson 4 Problem Solving: Solving Word Problems 3 L. 4.pdf · Problem Solving: Solving Word...

Unit 3 • Lesson 4 255

How much for just one?Oftenwemaywanttoknowthevalueofjustoneofsomething.Thevalue of one of something is called the unit rate.

We use this concept often without actually thinking about unit rate. For instance,inthegrocerystore,itiseasierifwecomparethecostofjustone unit, such as an ounce or a pound.

Ifweseeasignadvertisingfourboxesofcerealfor$10,wemightaskourselveshowmuchitcostsforjustonebox.

Let’s look at a problem involving unit rate.

Example 1

Find the unit rate.

Problem:

Itcosts$6tobuy3cartonsofmilk.Whatisthecostof1carton?

Set up a proportion with a variable.

CostCarton $6

3 = x1

Complete the proportion by finding the value of x.

CostCarton $6

3 = $21

One carton of milk costs $2. The unit rate is $2 per carton.

x is the cost for 1 carton.

Problem Solving: Solving Word Problems Using Unit Rates

Problem Solving: Solving Word Problems Using Unit Rates

Lesson 4

256 Unit 3 • Lesson 4

Lesson 4

Anotherwaywetalkaboutunitrateiswhenweusethetermmiles per hour.Thistermmeansthenumberofmileswetravelinonehour.Milesperhourisaunitrate.It’sthevalueofjustoneofsomethingeveryonehour. Example 2 shows this situation.

Example 2

Find the unit rate.

Problem:

Quentin’sdaddrovehimtoasoccertournament.Theydrove240milesin4hours.Abouthowfardidtheydriveinjust1hour?Whatwasthe rate in miles per hour?

Set up a proportion with a variable.

MilesHour 240

4 = m1

Complete the proportion by finding a value for m.MilesHour 240

4 = 601

We see now that m=60.ThismeansQuentinandhisdaddrove60milesin1hour.Anotherwayofsayingthisis“60milesperhour.”

The unit rate is 60 miles per hour.

Noticethateachproportionstartswiththeunitswritteninwords.

Writing the words out is a good habit to practice with this type of problem.Wewanttoremindourselveswhatthenumbersstandfor.Thatway, when we solve for a variable, we know exactly what that variable represents. In the example about Quentin and his dad, the variable m stands for miles. In the previous example, the variable x stands for the cost in dollars.

Example 3 shows another problem involving unit rate.

m is the miles traveled in 1 hour.

Unit 3 • Lesson 4 257

Lesson 4

Example 3

Find the unit rate.

Problem:

We are at the state fair. We use tickets to pay for the rides. Each ride requires the same number of tickets. We can take 4 rides on 20 tickets. How many tickets does it take for 1 ride?

Set up a proportion with a variable.

TicketsRide 20

4 = t1

Complete the proportion by finding the value for t.

TicketsRide 20

4 = 51

We need 5 tickets to take 1 ride.

t stands for the number of tickets.

We can see that t = 5.

258 Unit 3 • Lesson 4

Lesson 4

How do we solve word problems using unit rates?Sometimes we have to compute the unit rate for something to find the betterdeal.Weusuallyexpectthatitemsmarked“3for$ ” or“5for$ ”arebetterdealsthanbuyingjustoneitem.However, this is not always the case.

Tocomparethesedifferentpricingmethods,wefindtheunitrate.Example 1 shows how we analyze this type of situation to determine the better deal.

Example 1

Find the unit rate.

Problem:

Monicaneedssoup.Thestorehasaspecial—5cansfor$10.Ifshebuysjust1canofsoup,thecostis$2.20.Whichisthebetterdeal?

Wecomparethecostof1canofsoupfor$2.20tothespecialdealof5cansfor$10.Wesetupaproportiontofindtheunitrate.

CostNumber of Cans $10

5 = x1

• Firstweaskourselves,“1·?=5?”Theansweris5.

• Thenweneedtofindthevalueforx in the statement, “x ·5=$10.”

It’s$2.Wecompletetheproportionbyfillinginthevalueforx.

CostNumber of Cans $10

5 = $21

The unit price is $2.

Wenowcomparetheunitrateof$2tothecostfor1can—$2.20.Buyingsoupat5for$10isthebetterdealbecauseit’s$2percan,not$2.20percan.

Grocery stores often use this pricing method. We expect items marked “3for$ ” to be the best deals. However, that is not always the case.

Unit 3 • Lesson 4 259

Lesson 4

Example 2 shows a different situation. Sometimes the special pricing methods are not the best deal.

Example 2

Find the unit rate.

Problem:

Marcusneedstoweartiesforhisnewjob.Atthedepartmentstore,tiesare$19eachor3for$60.

Weneedtocomparethecostofjust1tietothespecialdealbyfindingthe unit rate. We set up the proportion like this:

CostNumber of Ties $60

3 = x1

• Firstweaskourselves,“1·?=3?”Theansweris3.

• Thenweneedtofindthevalueforx in this statement, “x ·3=$60.”

Thecostis$20.Wecompletetheproportionbyfillinginthevalueforx.

CostNumber of Ties $60

3 = $201

Whenwecomparethetwopricingmethods,weseethatthe“specialdeal” is not the better deal.

The unit rate is $20 per tie.

Ifwebuythetiesindividually,theyare$19pertie.

Problem-Solving ActivityTurntoInteractive Text, page 108.

Reinforce UnderstandingUse the mBook Study Guide to review lesson concepts.

260 Unit 3 • Lesson 4

Lesson 4

Activity 1

Simplify the ratios.

1. 714

12 2. 3

12 14 3. 4

20 15

4. 515

13 5. 8

48 16

Activity 2

Set up the unit rate problems as proportions. Tell what the variable represents. Show the units in words. Find the unit rate.

1. Nguyenpaid$40for4CDs.Ifall4costthesameamount,whatwasthecost ofjust1CD?

2. Sheldonpaid$180for6pairsofcontactlenses.Whatwasthecostofjust 1 pair of lenses?

3. Brittcando45sit-upsin3minutes.Howmanysit-upscanshedoinjust 1 minute?

Activity 3

Tell the better deal in each case by finding the unit rate.

1. What’sthebetterdeal,1applefor$.50,or3for$1?

2. What’sthebetterdeal,1pairofjeansfor$60or3for$200?

3. What’sthebetterdeal,1T-shirtfor$19or2for$40?

4. What’sthebetterdeal,1CDfor$15or2for$25?

5. What’sthebetterdeal,3juicesfor$1or$.75for1juiceinthevendingmachine?

Model 612 Answer: 6

12 = 12

Model Itcosts$8for4sandwiches.Howmuchdoesitcostforjust1sandwich?

Answer: CostSandwich $8

4 = x1 X stands for the cost of 1 sandwich.

CostSandwich $8

4 = $21 Thecostofonesandwichis$2.

Homework

CostCD

$404

$101

CostPair

$1806

$301

Sit upsMinutes

453

151

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Unit 3 • Lesson 4 261

Lesson 4

Activity 4 • Distributed Practice

Solve.

1. 45 + 2

5 = x 15 2. 35 − w = 19

3. 4.75 + 2.98 = z 4. a + 385 = 410

5. 12 · 3

5 = b 3

10 6. 139.7 − 48.19 = c

7. 45 ÷ 1

5 = d 8. e ÷ 7 = 50

Homework

Copyright2010byCambiumLearningSoprisWest®.Allrightsreserved.Permissionisgrantedtoreproducethispageforstudentuse.