Download Chapter 3 - Crestwood Local Schools

Rounding and Estimating
PRE-ALGEBRA LESSON 3-1
Round each number to the underlined place value.
a. 89.63
90
b. 579,122
600,000
3-1
c. 0.83476
0.8348
Rounding and Estimating
PRE-ALGEBRA LESSON 3-1
(For help, go to Skills Handbook page 727.)
Use the number 27.3865. Write the value of the given digit.
1. 2
2. 3
3. 8
4. 6
Check Skills You’ll Need
3-1
Rounding and Estimating
PRE-ALGEBRA LESSON 3-1
Solutions
1. 27.3865; 2 tens
2. 27.3865; 3 tenths
3. 27.3865; 8 hundredths
4. 27.3865; 6 thousandths
3-1
Rounding and Estimating
PRE-ALGEBRA LESSON 3-1
a. Round 8.7398 to the nearest tenth.
tenths place
8.7398
less than 5
Round down to 7.
8.7
3-1
Rounding and Estimating
PRE-ALGEBRA LESSON 3-1
(continued)
b. Round 8.7398 to the nearest integer.
nearest integer is ones place
8.7398
5 or greater
Round up to 9.
9
Quick Check
3-1
Rounding and Estimating
PRE-ALGEBRA LESSON 3-1
Estimate to find whether each answer is reasonable.
a. Calculation
$115.67
$ 83.21
+$ 59.98
$258.86
Estimate
$120
$ 80
+$ 60
$260
The answer is close to the estimate. It is reasonable.
3-1
Rounding and Estimating
PRE-ALGEBRA LESSON 3-1
(continued)
b. Calculation
$176.48
–$ 39.34
$147.14
Estimate
$180
–$ 40
$140
The answer is not close to the estimate. It is not reasonable.
Quick Check
3-1
Rounding and Estimating
PRE-ALGEBRA LESSON 3-1
You are buying some fruit. The bananas cost $1.32,
the apples cost $2.19, and the avocados cost $1.63. Use
front-end estimation to estimate the total cost of the fruit.
Add the
front-end digits.
+
.30
Estimate by
rounding.
.20
.60
1.10 = 5.10
The total cost is about $5.10.
Quick Check
3-1
Rounding and Estimating
PRE-ALGEBRA LESSON 3-1
Estimate the total electricity charge:
March: $81.75; April: $79.56; May: $80.89.
3 months
The values cluster around $80.
80 • 3 = 240
The total electricity charge is about $240.00.
Quick Check
3-1
Rounding and Estimating
PRE-ALGEBRA LESSON 3-1
Round to the underlined place.
1. 6.557
2. 3.0448
6.6
3.04
Estimate.
3. $4.95 + $.89
about $6.00
4. 4.589 + 5.098 + 5.179
about 15
3-1
Estimating Decimal Products and Quotients
PRE-ALGEBRA LESSON 3-2
Round to the nearest whole number and to the nearest tenth.
a. 4.07
4, 4.1
b. 0.708
1, 0.7
3-2
c. 364.825
365, 364.8
Estimating Decimal Products and Quotients
PRE-ALGEBRA LESSON 3-2
(For help, go to Lesson 3-1.)
Round to the nearest one.
1. 145.89
2. 199.27
3. 101.06
4. 28.45
Check Skills You’ll Need
3-2
Estimating Decimal Products and Quotients
PRE-ALGEBRA LESSON 3-2
Solutions
nearest integer
1. 145.89
5 or greater; round up to 6
146
nearest integer
2. 199.27
less than 5; do not change
199
nearest integer
3. 101.06
less than 5; do not change
101
nearest integer
4. 28.45
less than 5; do not change
28
3-2
Estimating Decimal Products and Quotients
PRE-ALGEBRA LESSON 3-2
Estimate 6.43 • 4.7.
6.43
6
4.7
6 • 5 = 30
6.43 • 4.7
5
Round to the nearest integer.
Multiply.
30
Quick Check
3-2
Estimating Decimal Products and Quotients
PRE-ALGEBRA LESSON 3-2
Joshua bought 3 yd of fabric to make a flag. The fabric
cost $5.35/yd. The clerk said his total was $14.95 before tax.
Did the clerk make a mistake? Explain.
5.35
5
5 • 3 = 15
Round to the nearest dollar.
Multiply 5 times 3, the number of
yards of fabric.
The sales clerk made a mistake. Since 5.35 > 5, the actual cost
should be more than the estimate. The clerk should have charged
Joshua more than $15.00 before tax.
Quick Check
3-2
Estimating Decimal Products and Quotients
PRE-ALGEBRA LESSON 3-2
The cost to ship one yearbook is $3.12. The total cost
for a shipment was $62.40. Estimate how many books were in
the shipment.
3.12
3
Round the divisor.
62.40
60
Round the dividend to a multiple of 3
that is close to 62.40.
60 ÷ 3 = 20
Divide.
The shipment is made up of about 20 books.
Quick Check
3-2
Estimating Decimal Products and Quotients
PRE-ALGEBRA LESSON 3-2
Is 3.29 a reasonable quotient for 31.423 ÷ 5.94?
5.94
31.423
6
Round the divisor.
30
Round the dividend to a multiple of 6
that is close to 31.423.
30 ÷ 6 = 5
Divide.
Since 3.29 is not close to 5, it is not reasonable.
Quick Check
3-2
Estimating Decimal Products and Quotients
PRE-ALGEBRA LESSON 3-2
Estimate each product or quotient.
2. 983.24 • 2.41
1. $4.78 ÷ 0.891
about $5
about 2,000
3. –5.89 ÷ (–2.7)
4. 24.69 • 0.7
about 2
about 25
5. 20.498 • 4.908
about 100
3-2
Mean, Median, and Mode
PRE-ALGEBRA LESSON 3-3
Order from greatest to least: 0.677, 0.855, 0.760, 0.078, 0.541
0.855, 0.760, 0.677, 0.541, 0.078
3-3
Mean, Median, and Mode
PRE-ALGEBRA LESSON 3-3
(For help, go Skills Handbook page 723.)
Write the numbers from least to greatest.
1. 8, 6, 4, 9, 3, 5, 6
2. 72, 68, 69, 71, 72
3. 112, 101, 98, 120, 101
4. 3.74, 3, 3.7, 3.3, 37
Check Skills You’ll Need
3-3
Mean, Median, and Mode
PRE-ALGEBRA LESSON 3-3
Solutions
1. 3, 4, 5, 6, 6, 8, 9
2. 68, 69, 71, 72, 72
3. 98, 101, 101, 112, 120
4. 3, 3.3, 3.7, 3.74, 37
3-3
Mean, Median, and Mode
PRE-ALGEBRA LESSON 3-3
Six elementary students are participating in a one-week
Readathon to raise money for a good cause. Use the graph. Find
the (a) mean, (b) median, and (c) mode of the data if you leave
out Latana’s pages.
a. Mean:
sum of data values
number of data values
=
40 + 45 + 48 + 50 + 50
5
=
233
5
= 46.6
The mean is 46.6.
3-3
Mean, Median, and Mode
PRE-ALGEBRA LESSON 3-3
(continued)
b. Median: 40 45 48 50 50
Write the data in order.
The median is the middle number, or 48.
c. Mode: Find the data value that
occurs most often.
The mode is 50.
Quick Check
3-3
Mean, Median, and Mode
PRE-ALGEBRA LESSON 3-3
How many modes, if any, does each have? Name them.
a. $1.10 $1.25 $2.00 $2.10 $2.20 $3.50
No values are the same, so there is no mode.
b. 1 3 4 6 7 7 8 9 10 12 12 13
Both 7 and 12 appear more than the other data values.
Since they appear the same number of times, there are two modes.
c. tomato, tomato, grape, orange, cherry, cherry, melon, cherry, grape
Cherry appears most often.
There is one mode.
Quick Check
3-3
Mean, Median, and Mode
PRE-ALGEBRA LESSON 3-3
Use the data: 7%, 4%, 10%, 33%, 11%, 12%.
a. Which data value is an outlier?
The data value 33% is an outlier.
It is an outlier because it is 21% away from the closest data value.
b. How does the outlier affect the mean?
77
6
12.8
Find the mean with the outlier.
44
6
7.3
Find the mean without the outlier.
12.8 – 7.3 = 5.5
The outlier raises the mean by about 5.5 points.
3-3
Quick Check
Mean, Median, and Mode
PRE-ALGEBRA LESSON 3-3
Which measure of central tendency best describes
each situation? Explain.
a. the monthly amount of rain for a year
Mean;
since the average monthly amount of rain for a year is not likely to
have an outlier, mean is the appropriate measure.
When the data have no outliers, use the mean.
b. most popular color of shirt
Mode;
since the data are not numerical, the mode is the appropriate measure.
When determining the most frequently chosen item, or when the data
are not numerical, use the mode.
3-3
Mean, Median, and Mode
PRE-ALGEBRA LESSON 3-3
(continued)
c. times school buses arrive at school
Median;
since one bus may have to travel much farther than other buses, the
median is the appropriate measure.
When an outlier may significantly influence the mean, use the median.
Quick Check
3-3
Mean, Median, and Mode
PRE-ALGEBRA LESSON 3-3
Which measure of central tendency best describes each situation?
1. numbers of legs on the animals in a zoo
mode
2. favorite digits (from 0 to 9) of the students in a class
mode
3. numbers of days-per-student that students are absent from school
median
4. test scores
mean
3-3
Using Formulas
PRE-ALGEBRA LESSON 3-4
Evaluate each expression for r = 4, s = 6, t = 2.
a. 4r
16
b. 18 ÷ s + t
c. (r + s) • 2
20
5
3-4
Using Formulas
PRE-ALGEBRA LESSON 3-4
(For help, go to Lesson 1-3.)
Evaluate each expression for x = 3 and y = 4.
1. 2x + 2y
2. 2x + y
3. 2(x + y)
4.
x+y
2
Check Skills You’ll Need
3-4
Using Formulas
PRE-ALGEBRA LESSON 3-4
Solutions
1. 2x + 2y = 2(3) + 2(4)
=6+8
= 14
2. 2x + y = 2(3) + (4)
=6+4
= 10
3. 2(x + y) = 2(3 + 4)
= 2(7)
= 14
4.
x+y
3+4
=
2
2
7
= 2
= 3.5
3-4
Using Formulas
PRE-ALGEBRA LESSON 3-4
Suppose you ride your bike 18 miles in 3 hours.
Use the formula d = r t to find your average speed.
d = rt
Write the formula.
18 = (r )(3)
Substitute 18 for d and 3 for t.
18
3r
=
3
3
Divide each side by 3.
6=r
Simplify.
Your average speed is 6 mi/h.
Quick Check
3-4
Using Formulas
PRE-ALGEBRA LESSON 3-4
Use the formula F = n + 37, where n is the number
4
of chirps a cricket makes in one minute, and F is the
temperature in degrees Fahrenheit. Estimate the temperature
when a cricket chirps 76 times in a minute.
n
F = 4 + 37
76
Write the formula.
F = 4 + 37
Replace n with 76.
F = 19 + 37
Divide.
F = 56
Add.
The temperature is about 56°F.
Quick Check
3-4
Using Formulas
PRE-ALGEBRA LESSON 3-4
Find the perimeter of a rectangular tabletop with a
length of 14.5 in. and width of 8.5 in. Use the formula for the
perimeter of a rectangle, P = 2 + 2w.
P = 2 + 2w
Write the formula.
P = 2(14.5) + 2(8.5)
Replace
P = 29 + 17
Multiply.
P = 46
Add.
with 14.5 and w with 8.5.
The perimeter of the tabletop is 46 in.
Quick Check
3-4
Using Formulas
PRE-ALGEBRA LESSON 3-4
Use the formula P = 4s to find the perimeter of a square with side s.
1. s = 5.6 m
2. s = 9.3 in.
22.4 m
37.2 in.
3. Find the side of a square with a perimeter of 164 yd.
41 yd
3-4
Solving Equations by Adding or Subtracting Decimals
PRE-ALGEBRA LESSON 3-5
Separate the digits 1 through 9 into three groups that have
the same sum.
1, 5, 9; 2, 6, 7; 3, 4, 8
3-5
Solving Equations by Adding or Subtracting Decimals
PRE-ALGEBRA LESSON 3-5
(For help, go to Skills Handbook page 730.)
Simplify.
1. 2.8 + 7.06
2. 0.65 + 1.8
3. 4.52 – 2.48
4. 3.7 – 0.62
Check Skills You’ll Need
3-5
Solving Equations by Adding or Subtracting Decimals
PRE-ALGEBRA LESSON 3-5
Solutions
1.
2.80
+ 7.06
9.86
2.
0.65
+ 1.80
2.45
3.
4.52
– 2.48
2.04
4.
3.70
– 0.62
3.08
3-5
Solving Equations by Adding or Subtracting Decimals
PRE-ALGEBRA LESSON 3-5
Solve 6.8 + p = –9.7.
6.8 + p = –9.7
6.8 – 6.8 + p = –9.7 – 6.8
p = –16.5
Check
Subtract 6.8 from each side.
Simplify.
6.8 + p = –9.7
6.8 + (–16.5)
–9.7
Replace p with –16.5.
–9.7 = –9.7
Quick Check
3-5
Solving Equations by Adding or Subtracting Decimals
PRE-ALGEBRA LESSON 3-5
Ping has a board that is 14.5 ft long. She saws off a
piece that is 8.75 ft long. Use the diagram below to find the
length of the piece that is left.
x + 8.75 = 14.5
x + 8.75 – 8.75 = 14.5 – 8.75
Subtract 8.75 from each side.
x = 5.75
Simplify.
The length of the piece that is left is 5.75 ft.
3-5
Quick Check
Solving Equations by Adding or Subtracting Decimals
PRE-ALGEBRA LESSON 3-5
Solve –23.34 = q – 16.99.
–23.34 = q – 16.99
–23.34 + 16.99 = q – 16.99 + 16.99
–6.35 = q
Add 16.99 to each side.
Simplify.
Quick Check
3-5
Solving Equations by Adding or Subtracting Decimals
PRE-ALGEBRA LESSON 3-5
Quick Check
Alejandro wrote a check for $49.98. His new account
balance is $169.45. What was his previous balance?
Words previous balance minus check is new balance
Let p = previous balance.
Equation
p
–
$49.98 =
$169.45
p – 49.98 = 169.45
p – 49.98 + 49.98 = 169.45 + 49.98
p = 219.43
Add 49.98 to each side.
Simplify.
Alejandro had $219.43 in his account before he wrote the check.
3-5
Solving Equations by Adding or Subtracting Decimals
PRE-ALGEBRA LESSON 3-5
Solve each equation.
2. –1.01 = c – 9
1. a + 10 = 7.9
– 2.1
7.99
3. s – (–2.6) = 1.6
4. 3.02 + d = 2.91
–1
– 0.11
5. –23.7 = 13.3 + g
– 37
3-5
Solving Equations by Multiplying or Dividing Decimals
PRE-ALGEBRA LESSON 3-6
Lucinda buys 12 gal of gasoline at $1.95 per gallon. How
much does the gas cost?
$23.40
3-6
Solving Equations by Multiplying or Dividing Decimals
PRE-ALGEBRA LESSON 3-6
(For help, go to Skills Handbook page 765.)
Find each product.
1. 2.6(4.5)
2. 3.2(0.15)
3. 11.03(0.6)
4. 8.003(0.6)
Check Skills You’ll Need
3-6
Solving Equations by Multiplying or Dividing Decimals
PRE-ALGEBRA LESSON 3-6
Solutions
1.
2.6
 4.5
2.
130
+ 1040
11.70 = 11.7
3.
0.15
 3.2
30
+ 450
0.480 = 0.48
11.03
4. 8.003
6.618
4.8018
 0.6
 0.6
3-6
Solving Equations by Multiplying or Dividing Decimals
PRE-ALGEBRA LESSON 3-6
Solve –6.4 = 0.8b.
–6.4 = 0.8b
–6.4
0.8b
=
0.8
0.8
–8 = b
Check
Divide each side by 0.8.
Simplify.
–6.4 = 0.8b
–6.4
0.8(–8)
Replace b with –8.
–6.4 = –6.4
Quick Check
3-6
Solving Equations by Multiplying or Dividing Decimals
PRE-ALGEBRA LESSON 3-6
Every day the school cafeteria uses about 85.8
gallons of milk. About how many days will it take for the
cafeteria to use the 250 gallons in the refrigerator?
Words
daily milk
consumption
times
number
of days
is
250 gallons
=
250
Let x = number of days.
Equation
85.8
•
x
3-6
Solving Equations by Multiplying or Dividing Decimals
PRE-ALGEBRA LESSON 3-6
(continued)
85.8x = 250
85.8x
250
=
85.8
85.8
Divide each side by 85.8.
x = 2.914 . . . Simplify.
x
3
Round to the nearest whole number.
The school will take about 3 days to use 250 gallons of milk.
Quick Check
3-6
Solving Equations by Multiplying or Dividing Decimals
PRE-ALGEBRA LESSON 3-6
Solve –37.5 =
–37.5 =
–37.5(–1.2) =
c
.
–1.2
c
–1.2
c
(–1.2)
–1.2
45 = c
Multiply each side by –1.2.
Simplify.
Quick Check
3-6
Solving Equations by Multiplying or Dividing Decimals
PRE-ALGEBRA LESSON 3-6
A little league player was at bat 15 times and had
a batting average of 0.133 (rounded to the nearest thousandth). The
batting average formula is a = h , where a is the batting average, h is
n
the number of hits, and n is the number of times at bat. Use the
formula to find the number of hits she made.
h
a= n
h
0.133 = 15
Replace a with 0.133 and n with 15.
3-6
Solving Equations by Multiplying or Dividing Decimals
PRE-ALGEBRA LESSON 3-6
(continued)
h
0.133(15) = 15 (15)
1.995 = h
2
h
Multiply each side by 15.
Simplify.
Since h (hits) represents an integer,
round to the nearest integer.
The little league player made 2 hits.
Quick Check
3-6
Solving Equations by Multiplying or Dividing Decimals
PRE-ALGEBRA LESSON 3-6
Solve each equation.
p
1. 9b = –30.6
2. –10.8 = –2.5
–3.4
27
3. 2.45 = –0.7k
4.
–3.5
t
= 240
3.7
888
5. y ÷ (–0.3) = 146.7
– 44.01
3-6
Using the Metric System
PRE-ALGEBRA LESSON 3-7
Use mental math to find each product or quotient.
a. 8.75 × 100
875
b. 9.6 × 100
960
c. 4.6 ÷ 10
0.46
d. 0.94 × 100
94
e. 4.06 ÷ 100
0.0406
3-7
Using the Metric System
PRE-ALGEBRA LESSON 3-7
(For help, go to Skills Handbook page 734.)
Find each product or quotient.
1. 5  100
2. 14.06 ÷ 1,000
3. 0.294  10
4. 0.9 ÷ 100
Check Skills You’ll Need
3-7
Using the Metric System
PRE-ALGEBRA LESSON 3-7
Solutions
1. 5 • 100
Move the decimal point
2 places to the right. 500.
2. 14.06 ÷ 1000
Move the decimal point
3 places to the left. 0.01406
5 • 100 = 500
14.06 ÷ 1000 = 0.01406
3. 0.294 • 10
Move the decimal point
1 place to the right. 02.94
4. 0.9 ÷ 100
Move the decimal point
2 places to the left. 0.009
0.294 • 10 = 2.94
0.9 ÷ 100 = 0.009
3-7
Using the Metric System
PRE-ALGEBRA LESSON 3-7
Choose an appropriate metric unit. Explain your choice.
a. the width of a textbook
Centimeter; the width of a textbook is about two
hands, or ten thumb widths, wide.
b. the mass of a pair of glasses
Gram; glasses have about the same mass as
many paperclips, but less than this textbook.
c. the capacity of a thimble
Milliliter; a thimble will hold only a small amount of water.
Quick Check
3-7
Using the Metric System
PRE-ALGEBRA LESSON 3-7
Choose a reasonable estimate. Explain your choice.
a. capacity of a drinking glass: 500 L or 500 mL
500 mL; a drinking glass holds less than a quart
of milk.
b. length of a hair clip: 5 m or 5 cm
5 cm; the length of a hair clip would be about 5
widths of a thumbnail.
c. mass of a pair of hiking boots: 1 kg or 1 g
1 kg; the mass is about one half the mass of your
math book.
Quick Check
3-7
Using the Metric System
PRE-ALGEBRA LESSON 3-7
Complete each statement.
a. 7,603 mL =
L
7,603 ÷ 1,000 = 7.603
To convert from milliliters
to liters, divide by 1,000.
7,603 mL = 7.603 L
b. 4.57 m =
cm
4.57  100 = 457 cm
To convert meters to
centimeters, multiply by 100.
4.57 m = 457 cm
Quick Check
3-7
Using the Metric System
PRE-ALGEBRA LESSON 3-7
A blue whale caught in 1931 was about 2,900 cm
long. What was its length in meters?
Words
length in
centimeters
÷
centimeters
per meter
=
length in
meters
Equation
2,900
÷
100
=
29
The whale was about 29 m long.
Quick Check
3-7
Using the Metric System
PRE-ALGEBRA LESSON 3-7
Write the metric unit that makes each statement true.
1. 23 kg = 23,000
g
2. 970 cm = 9.7
m
Complete each statement.
3.
g = 42 mg
0.042
4.
km = 5,000 m
5
3-7
Problem Solving Strategy: Act It Out
PRE-ALGEBRA LESSON 3-8
Describe the pattern. Then give the next three terms.
3, 6, 12, 24, 48, . . .
Starting with 3, each number is two times the number before it: 96, 192, 384.
3-8
Problem Solving Strategy: Act It Out
PRE-ALGEBRA LESSON 3-8
(For help, go to Lesson 1-7.)
Write a rule for each number pattern. Find the next three numbers in
the pattern.
1. 0, 6, 12, 18,…
2. –18, –9, 0, 9,…
3. 0, 2, 1, 3, 2, 4, 3,…
4. 7, 6, 8, 7, 9, 8, 10,…
Check Skills You’ll Need
3-8
Problem Solving Strategy: Act It Out
PRE-ALGEBRA LESSON 3-8
Solutions
1. 0, 6, 12, 18,…
Start with 0 and add 6 repeatedly.
0, 6, 12, 18, 24, 30, 36
+6 +6 +6 +6 +6 +6
2. –18, –9, 0, 9,…
Start with –18 and add 9 repeatedly.
–18, –9, 0,
9,
18, 27, 36
+9 +9 +9 +9 +9 +9
3-8
Problem Solving Strategy: Act It Out
PRE-ALGEBRA LESSON 3-8
Solutions (continued)
3. 0, 2, 1, 3, 2, 4, 3,…
Start with 0. Alternately add 2 and subtract 1.
0, 2, 1, 3, 2, 4, 3, 5, 4, 6
+2 –1 +2 –1 +2 –1 +2 –1 +2
4. 7, 6, 8, 7, 9, 8, 10,...
Start with 7. Alternately subtract 1 and add 2.
7, 6, 8, 7, 9, 8, 10, 9, 11, 10
–1 +2 –1 +2 –1 +2 –1 +2 –1
3-8
Problem Solving Strategy: Act It Out
PRE-ALGEBRA LESSON 3-8
Marta gives her sister one penny on the first day of October,
two pennies on the second day, and four pennies on the third day. She
continues to double the number of pennies each day. On what date will
Marta give her sister $10.24 in pennies?
Days after
the first
Number of
pennies
Amount
0
1
2
3
4
5
1
2
2•2= 4
4•2= 8
8 • 2 = 16
16 • 2 = 32
$0.01
$0.02
$0.04
$0.08
$0.16
$0.32
3-8
Problem Solving Strategy: Act It Out
PRE-ALGEBRA LESSON 3-8
(continued)
You can tell from the pattern in the chart that you just need to
count the number of 2’s multiplied until you reach 1,024, which
is $10.24 in pennies.
2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 = 1024
10 twos = 10 days after the first penny is given
Marta will give her sister $10.24 in pennies on October 11.
Quick Check
3-8
Problem Solving Strategy: Act It Out
PRE-ALGEBRA LESSON 3-8
Solve using any strategy.
1. On Monday, Jon reads page 45 of his book and continues to
read until he finishes page 89. How many page does he read
on Monday?
89 – 45 + 1, or 45 pages
2. Marion put one penny in a bank. Each day after that she put in
the bank double the number of pennies from the previous day
until the bank was full. If the bank was full on the 8th day, when
was the bank only half-full?
7th day
3-8