,. Which of thesee>.p! IflJl IS has Ihe greatestvalue?
Choose lhe Idle' 01the eorred answer.
Problem Solving on Tests
,,
~H_~i
... !
,, ,,
,, ,, ,,
Aapl1lunt DnIw a model toshow how i to find Ihe ao is,'''''. j
,
, !
~41 ;
... _. __.~..~.... -._._._. __._._._._.~... _ ..~._._._-_..
Mult~Cho,..
, ,, , ,,
A2S 83' e4' 0 9 ~
..._._._.._ Ql!P.!'.!.I!!'!l!!&_~
,, , ,
i ,, ,
! ,,
,
, b Suppose you hirethe same number 1. of siogersand dancenand pay$160
J 0.06 < 0.01 1 more lor !he singefs. How many
- .i~iili;;;:---'ã:-=C'~' ="''11 siogersand dancersdid)'Oll tlife?
.. ". I c Yourbudget lor per!OliUelSis 54.400.
! Youwant 10spend. least $4200. aod
; must h _ 8t least2 of eadl lund of
~ perlonner. How many of e&c:h kind
; could you hire?
; ~~
•..., .
, ,
Learn About It
Henry Ford invented away to make automobiles quickly, By 1927, the assembly line completed one Model Tautomobile fNery 24 seconds. How many Model Ts could theIlSSelTlbly line complete in 15 minute$ (900 seconds)?
Estimate Too Large
.1"~ PIKe thefirSt .'" -- .'" T• • ..1"!too Adjust.
0 ligd "'Ihe 0 Try3. 0
" 0 37 A 12
quotient.Try 4. 3 ,,1900 "jiOO
• "jiOO -121 -12'
24fgoiij -33
'" '"
- .. " - '" -, ..
"
.. " " " '" '" " .-
".IOO~ 3.~ I • 100IIrge. ,.~
"""* t I I
SoIutlon: Theycould make 37 Model Tsin 15nW1uIe5.
Estimate Too Small
II>cfwiIion . .
... "- ...
, ... /
..1"!too PlIIce thefirSt
o ligd. Try 4.
16)849•
=-"
20 20 "
. . . 100 ...
.T"
o Adjust., Try 5.
161849 -80 •
.,.~"
.,o. o Try 2."
16)849
-"I-33..
"
17 16 2.lOOsmal.
..1"!too Adjust.Try 3.
o 53 AI
16)849
-", - .. .. ,
, "
3.~
Solution: 849 + 16'" 53 A1.
118
Divide. Cheek your answers.
Guided Practice
1. 64)558
4. 420 + 46
2. 64)316
$.782+16
3.27)139 6. 650 + 24
.,,\\(Yourself
• DidI estimateIt.-~f$1
digit 01 Ihequotient?
• Isthe remainderklss than Ihe divisor?
Explain Your Thinking'" What should you do if)'O\lr estimated quotient is too large? too small?
Practice and Problem Solving Divide. Cheek your answers.
7. 45)230 8. 75)626
11. 64)439 12. 181176
HI. 16!t20 HI. 93)362
10. 618 + 32 20. 314 + 63
9. 23)823
13. 73)431
17. 29)203
21. 816 + 27
10. 24)620
14. 67)408
18. 43)808 22. 629 + 17
X Algebra. Functions Copy and complete.
23 Rule' Divide by 12 24. Rule' Divide by25
Input "",,,"'
"
'" ..
",
Input
"""'"'
'" ,
,
"'"
tnput
"""'"'
"" ''0 "
"
"" "
.... "
Solve.
~. A ModelTcould travel at amaximum speed of45 miles per hour. At that rate, how long would it take for a ModelTto travel a distance of 315miles?
27. Ford's 1914assembly line could tum out a Model Tin 93 minutes. Aboot howmany times faster than the normal
728 minlJtes is that?
c 35gallons o 250 gallons A 25 gallons
B 26 gallons Multiple Choice
34. AModel Tcould go 19miles on a gallon ofgas. How much gas would it use togo 475miles?PI-5.~2) 20. 0.624
31. 0.6
33. 0.060
Mixed Review and Test Prep
28. 2.067
30. 1.376
32. 6.041
ropen Response
Name the place value of the digit 6 in each number.p. 1.~5l
E.<tr<lPrilC1lCeSee page 133.setC, Chapter 5 Lesson 4 119
Division With Greater
Numbers
Objective DMd&a1'No>digit ....mb<lr into adividendwilt>uptosI.digits.
Learn About It
In the 18505. stagecoaches delivered letters and packages OUI West Suppose a stagecoach traveled from Missouri to California and back in 42 days. If il traveled 5.416 miles. aboul how many miles did the stagecoach travel each day?
Find 5,416 + 42.
",TI1", Estimate the fi,st digit , / ' Think
"
0 of the quolient. Then 42)5,416
'"
divide the hundreds. -" Multiply. 1
" 40)4,000 . /
12 _ Subtract 54
"
Try 1 hundred. Compare. 12, "
Think
"ur... 0 Bring downDivide the tens.the tens. 42)5,416" 42)5,416" JO
"- 40)1,200
- 4 2l - 4:2J
Try 3 tens.
'" '"
'" __ Est"nate Os - 84 __ Mul\Jply.:2
"
too Iorge.
" Subtract. , 21
"'
Try 2 tens, Compare. 37 .
"
,,111... Bring down the ones.
'" 128 R40 / Think
0 Dividethe ones. 42)5,416 42}5,416 ,
~I ~I '- 40)360
Try9 ones. ,,,
'"
-" -" I I
'" ", MUlliply B "
I
-378_ Estimale is _ 336 __ Subtract. 376 336
I
too lor9\!. ----:w Compare.40< 42
Try 8 ones.
Solution: Tile slagecoacillraveled between 126 and 129 miles each day.
120
Other Examples A. Zeros In the Quotient
Find 72.096'" 24.
B. Three-Digit Divisor Find 74.530'" 256.
Think 3.000 25)75.000
"''''
24)72,096
-n
00
- "09
- "
"
-" o
Think 300 250)75,000
""'"
256)74.530 - 512
2333 - 2304 290 - 256
"
Guided Practice
DIvide. Cheek.
I. 14)5.634 2. 42)5.425
.. ~\t. YourSelf
• WhemsllouldIp1aoo thelirst d9l?
• IS lhe 8Stimate<lllig~
toolargeor too smal?
3. 29.622'" 12 4. 91.233'" 731
Explain Your Thinking'" E.>cplain how to use multiplication and addition to check }'(lur answer in E.>cercise 4.
Practice and Problem Solving Divide. Cheek.
s. 17)5.185 6. 48)2.400 7. 7317.408
8. 35)11.144 II. 59)35.424 10. 91J27,636
II. 9.427 ... 31 12. 9.454 ... 47 13. 7.664'" 58 14. 493.438 ... 16 15. 682.675'" 25 1s. 75,223'" 729
X Algebra. Equations II q is the quotient and r is the remainder, write and solve a division problem lor eaeh equation.
17. 20q + r" 3.221 20. 52q+r=89.162
18. 35q + r- 7.805 21.11q+r=1.090
111. 29q + r" 16.258
22. 15q + r = 3.333
Chapter 5 Lesson 5 121
Solve.
23. June wallIS to place a bordef along the top of her bedroom. Sile needs 576 inches. How many packages will she need to buy if each package contains 10 feet of border?
25. What is the ma~imum number of digits there could be in the quotient of a frve- digit divideJld divided by a three-digit divisor? Give e~amples to show your reaSOJliJlg.
Choose a Computation Method
24. Tile variables a. b. and c in tile division problem shown below represent
3 diflereJlt digits. The division has been started for you. Complete the division.
,
abJa,ccb - , b
,
Totalleoglh 2.8t2 miles
St•••cuch a•• I••
M...tal Malh • Esllmatlon • Paper and Pencil. Calculalor
Data Thelable shows Information aboullhe Bullerfield Overland Siage Company. Use Ihe lable lor Problems 26-26. Then uplaln which method you choae.
25. II each 01 the stations on the route was the same distance apart. about how far is the distance between stations?
21. If the fare is based on the length of a trip. about how much would it cost to travel 700 miles?
28. When the Pony Express began delivering mail in 1860. its riders look an average of 12 days 10 cover 1.866 miles. On average. who covered more miles in a day. Pony Express riders or Siagecoach drivers? How did you decide?
Numberotsralions
'"
~lbR8
12)500
. ,"
." '0 00
ãn
"
122
29. What's Wrong? Stagecoach horses might be changed every 12 miles. To find the number 01 changes made in 500 miles. Jared divided 500 by 12.
Jared's work is shown althe left.
Whal did he do wrong?
30. Create and Solve Do some research about transportation in the 1BOOs. Write a problem based on }'Qur researcl1. Trade problems witl1 a classmate and solve.
How to Play 2 Players
What You'll Need •Iou' se1Sotnumber cards (Learning Too 6)
• divisioo lrames (Leeming Too 30)
\ 2 3
\\l, 5 6 1 8 9
~
Arrange your cards in a division frame so the
quotient will bethe least possible one you can make.
Take turns dealing the cards, repeating Sleps 1 to 3. The first player 1o gel a total ot 10 ~n1S wins,
Shullie the cards and gi\18 fi\18 cards toeach player.
Divide to find your quotient and compare wrth the other playe(s quotient The smaller quotient gets I ~n1.
e o o
Quotient Quest
Check your understanding for Lessons 1-5.
Divide, Check your answer. ~1, 2.'."'" 51
I. 600.;. 60 ,. 15)107
2. 25.000 + 500
e. 34)884
3. 420,000 .;. 7.000
7. 4.290 + 56
••
8. 36.247 + 29
Work backward to solve each problem. ~31 8. Maggie rides her horse 3 more miles
than Tasha. Emmaline rides 5 mOfe miles than Tasha. Catherine rides 2 fewer miles than Emmaline. Emmaline rides 6 miles. How many miles does each girl ride?
10. At one shop. there are 2 fewer jackets with beads than jackets with fringe.
There are 3 fewer plain jackets than jackets with beads. How many jackets with Iringe are there if there are 15 plain jackets?
I Extra Pr1Ictice al edup\ace.comJmap I Chapter 5 Lesson 5 123
• _ _ l/'aUs....-n_
Algebra
Order of Operations
Obje<:tive Usethe order01 ope,aliona 10$implifye>cpre$$ions.
Vocabulary
.~.operations simplitV
Learn About It
Simplify. 8 +(4 )( 24) + 32
To simplify an e~pression when there are more than two operations in the e~pressiOIl. you must use a set of rules called the order of operations.
The order of operations tells you in which order to perform the operatiolls wtlell simplifying.
Some people use this sentence as a memory device to help them remember the order of operations: e.lease I:~cuse my dear Aunt Sally.
Order of Operations
1. Simplify the terms w~hinearentheHS.
2. Simplify the terms w~hvponents.
3. Multiplyand glvlde fromleft 10right.
4. Add aooJubtractfrom left toright.
Find 8 -+- (4 )( 24) + 32.
Simplify within the parentheses.
(4 )( 24) 24
" "
Divide 96by 32.
,,'" -"3
o
Add 6.
3
+8
"
Solution: 8 -+- (4 X 24) -+- 32 - 11
Jdnet
Using Ihe order of operations 10 simplify an expressiOfl ensures that the expression has the same \flIlue.
Here are the ways two students
simplified 14 - (5 -+- 2) X 2. Delit::l
14-(5+2)X2
= 14-1X2
= 14 - 14
-0 14-(5+2)X2
~ 14-1X2
=1x2
=14 Solution: Delia used the correct Ol'der
of operations. After simplifying the terms within the parentheses.
she multiplied 7 and 2 before subtracting.
Problem: W'ho used the order of operations correctly?
124
Another Example
Parentlleses and Exponents (12 + 4)' x (4 x 5) - (8 - 4) + 5'
• 3 2
X 20 - 4 + :1~'~~~~~S"TIPI~Y....,ttl... parentheses.
_ 9 x 20 - 4 + 25 S,~I~y exponents_
.180 - 4 +25 MuKIllly,
_ 176 + 25 _ 201 Acld and S<.btra<;t trom left to nghl
Guided Practice
Simplify.
1.5+(8-6)
3.25+(2+20)-40
2. (14+2)x5
4. (12 + 13) x (8 + 4)
Mll YOUrself
• Whichoperat"", should Istart_?
• HaYII IsimpIiIiedthe
""pressioncompletely?
Explain Your Thinking'" How would you solve Exercise 2 if YQu did notllave parentheses to show which operations go together?
Practice and PrOblem Solving
Simplify.
5. (12+6)-3
8. (27 - 2) - (20 .;. 5)' 11.5+(124-2)+32 Write:>, <:. or '" for each • •
6.9+(18+9)X6 7. (8X2)+42
II. (21-3).;.(4+5)+(3X5) 10. (8'+6)+(10-3) 12. 1295 - (49 - 42) x 14 13. (65'" 5) + 42
14.11 +(32-7).(11 +32)-7 15. (36 + J2) x 8 . (24 x 2) - 42
Mental Math Use mental malh 10 simplify.
18. (7+2)-9+8
20. (5x5)x2x(2x2)
15.26 - (18 + 6 ) . (24 + 3) + 22
17. 1.822 - (153 + 22). (1.822 - 153) + 22
111. (4 +2) +(3 +3) +(7 - 1) 21. (4+4)X4+(2X2)
X Algebra. expressions Evaluale Ihe expression. given J{ '" 2 and y '" 6.
22. x2+(y_x) 25. 2{x+Y) - xã
23. (y2-4)+x
26. (xã yf
24.3.1'+ r
27. (4x+3)-y
Cllapter 5 Lesson 6 125
Solve.
28. A steamboat has 112 passef1gers. At one StOp.
2 groups of 8 passellgers get off the boat to sightsee aoo24 more passengers go shopping.
How many passengCfS are left on the boat?
29. Use parentheses to write two different expressions with equal values.
30. What's Wrong? When Alexander simplified the expression S2+8 x 3 - 4. he said the result was95. Did he follow the order of operations?
If he did not. which rule did he fail 10 follow.
and what should the result have been?
31. You Decide Copythis expressioo on paper and decide where you want the parentheses to go.
Simplify.
4x5+6-8+2+22
Tell a parmer the value of your expression. Have your paltner find where the parentheses belong.
32. Use parentheses to make Ihe sentence true.
5x8+4+2-3O
33. Use the numbers 3. 4. 6, aoo 8 to make each sentence true.
a. • x~-~+.-20
b. <--~x ~x~-120
- - - - - -
Mixed Review and Test Prep Open Response
Add or subtract.p.2.1.-=->3)
34.34,229 + 6.183
35. 72.405 + 8.924 + 16.308 36.2.419-1.728
126
37. Ellplain the steps you would take to simplify this expression. Then simplify. Show your work.
P\.5.I.-=->Bl
(4 + 3) x 8 - (18 + 2) + 32
STEP 3 Divide33 by8, Wfite the last remainder in the quotient.
STEP 2 Divide27tens by 8.Write the remainder in the dividend.
Number Sense
Other Ways to Divide
There is more than one way to solve most problems. Two Other ways to find a quotient are short division and repeated subtraction,
Way 0 Short Diyision Find 7,473 +8.
STEP 1 Divide 74 hundreds by 8.Write the remainder in the dividend,
(
Think
74+8-9R2 )
,
Write: 8)74'73
Think 27+S-3R3
"
Write: 8)74'7'3
934 R1 Write: 8)74'7'3
Way e Repeated SUbtraction Find 224 + 56.
STEP 1 Start with 224. Subtract 56 repeatedly.
STEP 2 Coont howmany limes you subtracted 56.
".
->6 ,>6 ->6
'"
->6
>6 ->6
o
You subtracted 56four times. so there ara4 groups of56 in 224.
56+56+56+56 _ 224 4x56_224 224+56-4
Divide. Show your work. Use short division or repeated SUbtraction.
1. 5.622 + 9 2. 7)23.401 3. 28}224 4. 5.103 + 567
!s. 10.251 + 3 e. 8)25.082 7. 27!l62 a. 2.808 + 351
II. When is it easier to use shOfl division?
When is it easier to use repeated subtraction?
E.xplain.
Chapter 5 Lesson 6 127
What is the quesUon?
How many groups of hikers were ltIere?
What do you know?
• There were 103 hikers. • A maximum of 8 people were in a group.
Problem-Solving Application
Interpret Remainders r--~~~~~---.
Obtectiva SoIwproblems irlYoMf>gremainders.
Wilen you solve a problem with a remainder, yQu need
to decide how to Interpret the remainder.
Problem One summer 103 hikers signed up to hike part of Lewis and Clar1c.'s route. If a maximum of 8 people
could be in a group. how many groups were there?
Oivide 103 by 6. De<:ide how you will interpret Ihe remainder.
• Will you increase lha quotient? Increastlthe quotient when you mustindude lhe remainder.
• Will you drop tile remainder? Ofop the remainder when you do not need to include ~.
• Will your remainder beIhe answer? Use tho remainder as the aMWElr when you want to know how many are left over.
• Filld the number 01 full groups of hikers.
• Decide how to use the remainder.
The 7 remaining people made their own group.
Add 1 more group 10 your answer.
Solution; There were 13 groups on the hike.
12 R7 8)103
12+1-13
Look back at the problem.
Does Ihe answer make sense?
128
Guided Practice
UM lheAskYourself quesUons 10 help you solveI!!lOI;:h problem.