Transcript 5. - Crestwood Local Schools

Comparing and Ordering Rational Numbers
PRE-ALGEBRA LESSON 5-1
Write the prime factorization of 364.
22 x 7 x 13
5-1
Comparing and Ordering Rational Numbers
PRE-ALGEBRA LESSON 5-1
(For help, go to Lesson 4-3.)
Write the prime factorization of each number.
1. 20
2. 125
3. 45
4. 186
5. 621
6. 1,575
Check Skills You’ll Need
5-1
Comparing and Ordering Rational Numbers
PRE-ALGEBRA LESSON 5-1
Solutions
1. 20
2 • 10
2•2•5
20 = 22 • 5
2. 125
5 • 25
5•5•5
125 = 53
3. 45
3 • 15
3•3•5
45 = 32 • 5
4. 186
2 • 93
2 • 3 • 31
186 = 2 • 3 • 31
5. 621
3 • 207
3 • 3 • 69
3 • 3 • 3 • 23
621 = 33 • 23
6. 1,575
3 • 525
3 • 3 • 175
3 • 3 • 5 • 35
3•3•5•5•7
1,575 = 32 • 52 • 7
5-1
Comparing and Ordering Rational Numbers
PRE-ALGEBRA LESSON 5-1
Today, the school’s baseball and soccer teams had
games. The baseball team plays every 7 days. The soccer team
plays every 3 days. When will the teams have games on the
same day again?
7, 14, 21, 28, 35, 42, . . .
List the multiples of 7.
3, 6, 9, 12, 15, 18, 21, . . .
List the multiples of 3.
The LCM is 21. In 21 days both teams will have games again.
Quick Check
5-1
Comparing and Ordering Rational Numbers
PRE-ALGEBRA LESSON 5-1
Find the LCM of 16 and 36.
16 = 24
36 = 22 • 32
LCM = 24 • 32
= 144
Write the prime factorizations.
Use the greatest power of each factor.
Multiply.
The LCM of 16 and 36 is 144.
Quick Check
5-1
Comparing and Ordering Rational Numbers
PRE-ALGEBRA LESSON 5-1
Find the LCM of 5a4 and 15a.
5a4 = 5 • a4
15a = 3 • 5 • a
LCM = 3 • 5 • a4
= 15a4
Write the prime factorizations.
Use the greatest power of each factor.
Multiply.
The LCM of 5a4 and 15a is 15a4.
Quick Check
5-1
Comparing and Ordering Rational Numbers
PRE-ALGEBRA LESSON 5-1
Graph and compare the fractions in each pair.
a. 7 , 3
8
8
3
8
7
8
3
3
7
is on the left, so < .
8
8
8
1
1
b. – , –
3
6
–1 –1
3
6
1
1
– 1 is on the right, so – 6 > – 3 .
6
Quick Check
5-1
Comparing and Ordering Rational Numbers
PRE-ALGEBRA LESSON 5-1
The softball team won 6 of its games and the hockey
7
team won 7 of its games. Which team won the greater fraction
9
of its games?
Step 1
Step 2
Find the LCM of 7 and 9.
7 = 7 and 9 = 32
LCM = 7 • 32 = 63
Write equivalent fractions with a denominator of 63.
6 • 9 54
=
7 • 9 63
7 • 7 49
=
9 • 7 63
Step 3
Compare the fractions.
54
49
6
7
>
, so
>
63
63
7
9
The softball team won the greater fraction of its games.
5-1
Quick Check
Comparing and Ordering Rational Numbers
PRE-ALGEBRA LESSON 5-1
Order 3 , 1 , and 2 from least to greatest.
7
4
3
3
3 • 12
36
=
=
7
7 • 12
84
1
1 • 21
21
=
=
4
4 • 21
84
The LCM of 7, 4, and 3 is 84.
Use 84 as the common denominator.
2
2 • 28
56
=
=
3
3 • 28
84
21
< 36 < 56 , so 1 < 3 < 2 .
84
84
84
4
7
3
Quick Check
5-1
Comparing and Ordering Rational Numbers
PRE-ALGEBRA LESSON 5-1
Find the LCM of each pair of numbers.
1. 8, 6
2. 12, 16
24
48
3. Compare and order 3 , – 8 , and – 3 from least to greatest.
16
10
– 8 < – 3 < 3
10
16
16
5-1
16
Fractions and Decimals
PRE-ALGEBRA LESSON 5-2
Without actually dividing, which expression is the largest?
6.1 ÷ 2.8
12.6 ÷ 1.1
9.305 ÷ 2.99
12.6 ÷ 1.1
5-2
Fractions and Decimals
PRE-ALGEBRA LESSON 5-2
(For help, go to Skills Handbook page 728.)
Write the decimals in order from least to greatest.
1. 2.41, 0.241, 24.1, 12.4
2. 1.030, 13.03, 1.300, 1.003
3. 0.1, 0.01, –0.1, –0.01
Check Skills You’ll Need
5-2
Fractions and Decimals
PRE-ALGEBRA LESSON 5-2
Solutions
1. 0.241, 2.41, 12.4, 24.1
2. 1.003, 1.030, 1.300, 13.03
3. –0.1, –0.01, 0.01, 0.1
5-2
Fractions and Decimals
PRE-ALGEBRA LESSON 5-2
1
The fuel tank of Scott’s new lawn mower holds 2 gal of
gasoline. Scott poured 0.4 gal into the tank. Did Scott fill the tank?
1
= 1 ÷ 2 = 0.5
2
1
Since 2 = 0.5 and 0.5 > 0.4, Scott did not fill the tank.
Quick Check
5-2
Fractions and Decimals
PRE-ALGEBRA LESSON 5-2
Write each fraction as a decimal. State the block of
digits that repeats.
a.
5
6
5 ÷ 6 = 0.83333 …
= 0.83
Divide.
Place a bar over the digit that repeats.
5
= 0.83; the digit that repeats is 3.
6
7
b. 11
7 ÷ 11 = 0.636363 …
= 0.63
Divide.
Place a bar over the block of digits that repeats.
7
= 0.63; the block of digits that repeats is 63.
11
5-2
Quick Check
Fractions and Decimals
PRE-ALGEBRA LESSON 5-2
Write the numbers in order, from least to greatest.
–0.8, 3 , – 5 , 0.125
12
4
3 ÷ 12 = 0.25
–5 ÷ 4 = –1.25
Change the fractions to decimals.
–1.25 < –0.8 < 0.125 < 0.25
Compare the decimals.
5
3
From least to greatest, the numbers are – 4 , –0.8, 0.125, and 12 .
Quick Check
5-2
Fractions and Decimals
PRE-ALGEBRA LESSON 5-2
Write 1.72 as a mixed number in simplest form.
1.72 = 1 72
100
72 ÷ 4
= 1 100 ÷ 4
1.72 = 1
18
25
Keep the whole number 1.
Write seventy-two hundredths as a fraction.
Divide the numerator and denominator
of the fraction by the GCF, 4.
Simplify.
Quick Check
5-2
Fractions and Decimals
PRE-ALGEBRA LESSON 5-2
Write 0.18 as a fraction in simplest form.
n = 0.18
Let the variable n equal the decimal.
100n = 18.18
Because 2 digits repeat, multiply each side by
102, or 100.
100n = 18.18
– n = 0.18
99n = 18
The Subtraction Property of Equality lets you
subtract the same value from each side of the
equation. So, subtract to eliminate 0.18.
99n
18
=
99
99
n = 18 ÷ 9
99 ÷ 9
2
= 11
Divide each side by 99.
Divide the numerator and denominator by the
GCF, 9.
Simplify.
2
As a fraction in simplest form, 0.18 = 11 .
5-2
Quick Check
Fractions and Decimals
PRE-ALGEBRA LESSON 5-2
1. Order 6 , – 1 , 0.625, –0.35 from least to greatest.
5
3
–0.35, – 1 , 0625, 6
3
5
2. Write 17 as a decimal.
20
0.85
Write each decimal as a fraction or mixed number in simplest form.
3. 6.04
4. 0.21
1
7
33
6 25
5-2
Adding and Subtracting Fractions
PRE-ALGEBRA LESSON 5-3
Five consecutive multiples of 6 have a sum of 270. What are they?
42, 48, 54, 60, 66
5-3
Adding and Subtracting Fractions
PRE-ALGEBRA LESSON 5-3
(For help, go to Lesson 5-1.)
Find the LCM of each group of numbers or expressions.
1. 4, 8
2. 9, 18
3. 2n, 10
4. 3, 6, 9
5. 8, 5, 4
6. 10, n
Check Skills You’ll Need
5-3
Adding and Subtracting Fractions
PRE-ALGEBRA LESSON 5-3
Solutions
1.
4 = 22
8 = 23
LCM = 23
=8
2.
9 = 32
18 = 2 • 32
LCM = 2 • 32
= 18
3.
2n = 2 • n
5=5
LCM = 2 • 5 • n
= 10n
4.
3=3
6=2•3
9 = 32
LCM = 2 • 32
= 18
5.
8 = 23
5=5
4 = 22
LCM = 23 • 5
= 40
6.
10 = 2 • 5
n =n
LCM = 2 • 5 • n
= 10n
5-3
Adding and Subtracting Fractions
PRE-ALGEBRA LESSON 5-3
Find each sum or difference. Simplify if possible.
a. 4 + 2
9
9
4 2
4+2
+ =
9 9
9
6
9
2
=
3
=
12
Add the numerators.
Simplify.
Simplify.
5
b. b – b
12 – 5
12
5
–
=
b
b
b
7
= b
Subtract the numerators.
Simplify.
Quick Check
5-3
Adding and Subtracting Fractions
PRE-ALGEBRA LESSON 5-3
Simplify each difference.
a.
1 3
–
6 4
1 3
– = 1•4–3•6
6 4
6•4
4 – 18
=
24
= –14
24
= –7
24
Rewrite using a common denominator.
Use the Order of Operations to simplify.
Simplify.
Simplify.
2
5
b. y – 16
2
– 5 = 2 • 16 – 5 • y
y
16
y • 16
= 32 – 5y
16y
Rewrite using a common denominator.
Simplify.
5-3
Quick Check
Adding and Subtracting Fractions
PRE-ALGEBRA LESSON 5-3
Suppose one day you rode a bicycle for 3 1 hours, and
2
1
jogged for 1 hours. How many hours did you exercise?
4
1
1
7
5
3 +1 = +
2
4
2
4
Write mixed numbers as improper fractions.
7•4+5•2
Rewrite using a common denominator.
2•4
28 + 10
=
Use the Order of Operations to simplify.
8
38
=
8
Write as a mixed number.
= 46
8
= 43
Simplify.
4
3
You exercised for 4 hours.
Quick Check
4
=
5-3
Adding and Subtracting Fractions
PRE-ALGEBRA LESSON 5-3
Find each sum or difference.
2
10
1. 24 + 24
3
1
2. 5 4 – 2 8
1
2
5
38
3. – 3 – 1
5
3
4. 4 + 3
7
2c
– 14
8c + 21
14c
15
5-3
Multiplying and Dividing Fractions
PRE-ALGEBRA LESSON 5-4
Write each number as an improper fraction.
a. 4 1
5
21
5
b. 3 7
8
31
8
5-4
Multiplying and Dividing Fractions
PRE-ALGEBRA LESSON 5-4
(For help, go to Skills Handbook page 738.)
Write each mixed number as an improper fraction.
1. 2
1
3
2. 3
3
10
3. 1
4
9
4. 4
4
5
5. 7
7
8
6. 5
1
7
Check Skills You’ll Need
5-4
Multiplying and Dividing Fractions
PRE-ALGEBRA LESSON 5-4
Solutions
1
1
1. 2 3 = 2 + 3
6
3
1
10
6+1
3
7
23 = 3
4
=
3
33
5-4
9
4
= 9+ 9
10
30 + 3
10
3 10 = 10
4
3. 1 9 = 1 + 9
3
= 30 +
1
= 3+ 3
=
3
2. 3 10 = 3 + 10
=
4
9+4
9
13
19 = 9
Multiplying and Dividing Fractions
PRE-ALGEBRA LESSON 5-4
Solutions (continued)
4
4
4. 4 5 = 4 + 5
4
7
7
5. 7 8 = 7 + 8
1
1
6. 5 7 = 5 + 7
=
4
20
+ 5
5
=
7
56
+ 8
8
=
1
35
+ 7
7
=
20 + 4
5
=
56 + 7
8
=
35 + 1
7
24
45 = 5
7
63
78 = 8
5-4
1
36
57 = 7
Multiplying and Dividing Fractions
PRE-ALGEBRA LESSON 5-4
Find 2 • 5 .
3
2
5
2•5
•
=
3
7
3•7
=
10
21
7
Multiply the numerators.
Multiply the denominators.
Simplify.
Quick Check
5-4
Multiplying and Dividing Fractions
PRE-ALGEBRA LESSON 5-4
a. Find 3 • 2 .
4
3
3 2
3 12
• =
•
4 3 24 13
1
=
1
2
Divide the common factors.
Multiply.
5 3w
• .
w 17
5 • 3w = 5 • 3w 1
w 17
17
1w
b. Find
=
15
17
Divide the common factors.
Multiply.
Quick Check
5-4
Multiplying and Dividing Fractions
PRE-ALGEBRA LESSON 5-4
Keesha’s desktop is a rectangle 3 1 ft long and 1 1 ft
2
2
wide. What is the area of her desktop?
A =3
1
1
•1
2
2
Area of a rectangle = length • width.
1
1
7
3
and 1 as improper fractions, and .
2
2
2
2
=7• 3
Write 3
21
4
Multiply.
2
=
2
1
= 54
Write as a mixed number.
The area of Keesha’s desk is 5
1 2
ft .
4
Quick Check
5-4
Multiplying and Dividing Fractions
PRE-ALGEBRA LESSON 5-4
3
7
a. Find ÷
.
5 20
3
7
3
10
÷
=
•
5 10
5
7
Multiply by the reciprocal of the divisor.
2
3
10
=
•
7
15
Divide the common factors.
= 6
Multiply.
7
5-4
Multiplying and Dividing Fractions
PRE-ALGEBRA LESSON 5-4
(continued)
27
9
b. Find
÷ 4q .
8q
9
4q
27
27
÷
=
•
4q
9
8q
8q
3
1
Multiply by the reciprocal of the divisor.
1
27
=
• 4q
91
2 8q 1
Divide the common factors.
= 3
Multiply.
2
= 11
2
Write as a mixed number.
Quick Check
5-4
Multiplying and Dividing Fractions
PRE-ALGEBRA LESSON 5-4
Find 4 1 ÷ (–3 3 ).
2
4
8
1
3
9
27
÷ (–3 ) =
÷ (– )
2
8
2
8
9
8
• (– )
2
27
=
Change to improper fractions.
Multiply by –
8
27
, the reciprocal of – .
27
8
4
1
9
8
=
• –
273
12
4
3
= – , or –1
1
3
Divide the common factors.
Simplify.
Quick Check
5-4
Multiplying and Dividing Fractions
PRE-ALGEBRA LESSON 5-4
Simplify each expression.
1. –
3
7
•
8
12
2.
– 7
21
1
,
or
1
20
20
32
12g
3z
5z
÷
4
7
5
3. 25 •
2g
4. 5 3 ÷ 1 11
8
16
6
1
,
or
1
5
5
10
1
,
or
3
3
3
5-4
Using Customary Units of Measurement
PRE-ALGEBRA LESSON 5-5
In 2002, a disc jockey broadcast her show for 27 1 h each week.
2
How many hours did she broadcast that year?
1,430 h
5-5
Using Customary Units of Measurement
PRE-ALGEBRA LESSON 5-5
(For help, go to Lesson 5-4.)
Find each product.
1. 4 •
3. 1
1
12
1
8
•
4
1
2. 3
1
16
•
4
1
4. 4
1
1
•
2
3
Check Skills You’ll Need
5-5
Using Customary Units of Measurement
PRE-ALGEBRA LESSON 5-5
Solutions
1. 4 • 1 = 4 • 1
12
1 12
2. 3 1 • 16 = 13 • 16
4
1
4
1
4
= 12
=
1
= 3
1
8
208
4
= 52
5
8
3. 1 4 • 1 = 4 • 1
1
1
9
1
4. 4 2 • 3 = 2 • 3
9
40
= 4
= 6
= 10
= 2
3
1
=12
5-5
Using Customary Units of Measurement
PRE-ALGEBRA LESSON 5-5
Choose an appropriate unit of measure. Explain your
choice.
a. weight of a hummingbird
Measure its weight in ounces because a hummingbird is very light.
b. length of a soccer field
Measure its length in yards because it is too long to measure in feet
or inches and too short to measure in miles.
Quick Check
5-5
Using Customary Units of Measurement
PRE-ALGEBRA LESSON 5-5
Use dimensional analysis to convert 68 fluid ounces to cups.
68 fl oz
1
68 fl oz =
17
•
1c
8 fl oz
Use a conversion factor that changes
fluid ounces to cups.
=
68 fl oz • 1 c
2 8 fl oz
Divide the common factors and units.
=
17
c
2
Simplify.
1
2
= 8 c
There are 8
Write as a mixed number.
1
c in 68 fl oz.
2
Quick Check
5-5
Using Customary Units of Measurement
PRE-ALGEBRA LESSON 5-5
Fred’s fruit stand sells homemade lemonade in 6 1 -pint
2
bottles for $1.99. Jill’s fruit stand stand sells homemade
lemonade in 3 1 -qt containers for the same price. At which stand
2
do you get more lemonade for your money?
3
1
7
qt =
qt •
2
2
7 qt
= 2 •
1
= 7 pt
1
2 pt
1 qt
Use a conversion factor that
changes quarts to pints
2 pt
1 qt
Divide the common factors and units.
Multiply.
Since 7 pints > 6 1 pints, you get more lemonade for your money
2
Quick Check
at Jill’s stand.
5-5
Using Customary Units of Measurement
PRE-ALGEBRA LESSON 5-5
Choose an appropriate unit of measure. Explain your choice.
1. weight of a dog
Pounds; dogs are too heavy to measure in ounces and too
light to measure in tons.
2. length of a football field
Yards; it is too long to measure in feet or inches and too short
to measure in miles.
3. Use dimensional analysis to convert 1 1 mi to ft.
2
7,920 ft
5-5
Problem Solving Strategy: Work Backward
PRE-ALGEBRA LESSON 5-6
Use mental math to decide whether or not each pair of fractions is
equal: 6 and 36 , 4 and 16 , 6 and 18
7
42 5
20 15
20
yes, yes, no
5-6
Problem Solving Strategy: Work Backward
PRE-ALGEBRA LESSON 5-6
(For help, go to Lesson 5-2.)
Order from least to greatest.
1.
1 1 1 1 1
2 , 3 , 4 , 5 , 10
3
3
3
3
3. 7 , 5 , 13 , 3
2.
7 5 13 3
3, 3, 3 , 3
3
3
3
3
4. – 7 , – 5 , – 13 , – 5
Check Skills You’ll Need
5-6
Problem Solving Strategy: Work Backward
PRE-ALGEBRA LESSON 5-6
Solutions
1.
1 1 1 1 1
2 , 3 , 4 , 5 , 10
2.
7 5 13 3
3, 3, 3 , 3
0.50, 0.3, 0.25, 0.20, 0.10
2.3, 1.6, 4.3, 1
1 1 1 1
1
,
, , ,
10 5 4 3 2
3 5 7 13
3 , 3 , 3 , 3
3. 3 , 3 , 3 , 3
7 5 13 3
4. – 3 , – 3 , – 3 , – 3
7
5
13
5
0.43, 0.6, 0.23, 1
–0.43, –0.6, –0.23, 1
3 7
3 3
,
,
5 3
5 , 3
–3, –3 ,–3 , – 3
3
5
7
13
5-6
Problem Solving Strategy: Work Backward
PRE-ALGEBRA LESSON 5-6
Your flight leaves the airport at 10:00 A.M. You must
arrive 2 hours early to check your luggage. The drive to the
airport takes about 90 minutes. A stop for breakfast takes about
30 minutes. It will take about 15 minutes to park and get to the
terminal. At what time should you leave home?
Move the hands of the clock to find the time you should leave home.
Write the starting time for each event.
5-6
Problem Solving Strategy: Work Backward
PRE-ALGEBRA LESSON 5-6
(continued)
You should leave
home at 5:45 A.M.
Quick Check
5-6
Problem Solving Strategy: Work Backward
PRE-ALGEBRA LESSON 5-6
Solve by working backward.
1. Jason has a total of 212 sheep. Last spring, 52 lambs were born, but
9 died. How many sheep did he have before the lambs were born?
169 sheep
2. The bus picks up Tom every morning at 7:20 A.M. Tom walks his dog
every morning before school for 36 minutes. It takes him 25 minutes
to get ready for school and 7 minutes to walk to the bus stop. At
what time does Tom need to wake up in order to get to the bus stop
on time?
6:12 A.M.
5-6
Solving Equations by Adding or Subtracting Fractions
PRE-ALGEBRA LESSON 5-7
Find the sum or difference. If possible, write each answer in lowest
terms.
a. 4 + 2 = ?
5
b. 7 5 – 5 2 = ?
6
5
11
21
5
2
5-7
6
Solving Equations by Adding or Subtracting Fractions
PRE-ALGEBRA LESSON 5-7
(For help, go to Lesson 5-3.)
Find each sum or difference.
1. 1
3
7
–2
4
8
2. 3
5
7
+2
8
12
3. 5
3
1
–3
4
8
4. –4
1
2
–3
6
9
Check Skills You’ll Need
5-7
Solving Equations by Adding or Subtracting Fractions
PRE-ALGEBRA LESSON 5-7
Solutions
1. 1
3
7
–2
4
8
2. 3
7
23
7 • 8 – 4 • 23
–
=
8
8
4•8
=
29
55
29 • 12 + 8 • 55
+
=
8
12
8 • 12
56 – 92
32
=
–36
348 + 440
96
788
= 32
9
5
7
+4
8
12
= 96
1
20
= – 8 = –1 8
5
= 8 96 = 8 24
5-7
Solving Equations by Adding or Subtracting Fractions
PRE-ALGEBRA LESSON 5-7
Solutions (continued)
3. 5
3
1
–3
4
8
4. –4
23
25
23 • 8 – 4 • 25
–
=
4
8
4•8
=
25
29
– 6 + 9 =
184 – 100
32
84
= 32
=2
1
2
–3
6
9
20
5
=2
32
8
–25 • 9 – 6 • 29
6•9
=
–225 – 174
54
=
–399
54
21
7
= –7 54 = –7 18
5-7
Solving Equations by Adding or Subtracting Fractions
PRE-ALGEBRA LESSON 5-7
One school recycles about 1 of its waste paper. The
3
student council set a goal of recycling 3 of the school’s waste
4
paper by the end of the year. By how much does the school need
to increase its paper recycling to reach the goal?
Words
fraction school
recycles
Let
Equation
n
1
3
plus
the
increase
is
student
goal
= the increase.
+
5-7
n
=
3
4
Solving Equations by Adding or Subtracting Fractions
PRE-ALGEBRA LESSON 5-7
(continued)
1
3
+ n =
3
4
1
1
3
1
–
+ n =
–
3
3
4
3
Subtract
1
from each side.
3
n =
3•3–1•4
3•4
Use 3 • 4 as the common denominator.
n =
9 – 4
12
Use the Order of Operations.
n =
5
12
Simplify.
5
To meet the student council goal, the school needs to recycle
more
12
of its waste paper.
5-7
Solving Equations by Adding or Subtracting Fractions
PRE-ALGEBRA LESSON 5-7
(continued)
Check
Is the answer reasonable? The present fraction of paper
waste that is recycled plus the increase must equal the
goal. Since 1 + 5 = 4 + 5
3
12
12
12
= 9 = 3 , the answer
12
4
is reasonable.
Quick Check
5-7
Solving Equations by Adding or Subtracting Fractions
PRE-ALGEBRA LESSON 5-7
Solve x – 2 = 1 .
3
2
3
=
1
9
2
2
+
3
3
=
1
2
+
9
3
Add
x =
1•3+9•2
9•3
Use 9 • 3 as the common denominator.
x =
3 + 18
27
Use the Order of Operations.
x –
x –
9
21 7
x =
27 9
x =
7
9
2
to each side.
3
Divide the common factors.
Simplify.
5-7
Quick Check
Solving Equations by Adding or Subtracting Fractions
PRE-ALGEBRA LESSON 5-7
Solve q – 6 1 = –1 3 .
2
5
1
3
= 1
2
5
1
1
3
1
q – 6 + 6 = 1 + 6
2
2
5
2
q – 6
q = –
q =
8
5
+
13
2
–8 • 2 + 5 • 13
5•2
–16 + 65
10
49
q =
10
9
q = 4
10
q =
Add 6
1
to each side.
2
Write mixed numbers as
improper fractions.
Use 5 • 2 as the common
denominator.
Use the Order of Operations.
Simplify.
Write as a mixed number.
5-7
Quick Check
Solving Equations by Adding or Subtracting Fractions
PRE-ALGEBRA LESSON 5-7
Solve the following equations.
1.
3
13
+s=
7
21
4
21
4. –
5
19
+g=
11
22
7
1 22
2. k +
2
7
=
5
9
3. u –
17
45
5
6
1
5. x – 2 3 = –4
14
–1 15
5-7
4
15
1
3
=
12
4
Solving Equations by Multiplying Fractions
PRE-ALGEBRA LESSON 5-8
Because of a mishap, 3 of the oranges in a crate were ruined. If
10
the oranges originally weighed 50 lb, how many pounds of oranges
were ruined?
35 lb
5-8
Solving Equations by Multiplying Fractions
PRE-ALGEBRA LESSON 5-8
(For help, go to Lesson 5-4.)
Find each product.
1. 4 • 5
7
8
1
2.
5
3. –1 • –
4
1
9
•
10
1
5
–3
2
4. 4 •
2
3
Check Skills You’ll Need
5-8
Solving Equations by Multiplying Fractions
PRE-ALGEBRA LESSON 5-8
Solutions
1. 4 • 5
7
8
2.
9
•
10
5
–3
9 • –5
9
5
•
–
=
10 • 3
10
3
4•5
4
5
=
•
7•8
7
8
–45
=
20
56
= 30
=
5
14
= 6
–9
1
= –1 2
5-8
Solving Equations by Multiplying Fractions
PRE-ALGEBRA LESSON 5-8
Solutions (continued)
1
5
1
3. –1 4 • – 1
5
–5
–4 • 1 =
2
4. 4 2 • 3
–5 • –5
4•1
9•2
9
2
=
•
2•3
2
3
25
=
= 4
1
18
6
=3
=64
5-8
Solving Equations by Multiplying Fractions
PRE-ALGEBRA LESSON 5-8
Solve 7y =
7y =
1
3
1
1
1
• (7y) =
•
7
3
7
y =
1
.
3
1
21
Multiply each side by
1
, the reciprocal of 7.
7
Simplify.
Quick Check
5-8
Solving Equations by Multiplying Fractions
PRE-ALGEBRA LESSON 5-8
Solve 2 w = 13 .
5
15
2
13
w =
5
15
5
2
•
2
w =
5
13
5
•
15
2
Multiply each side by
5
2
, the reciprocal of .
2
5
1
5
13
w =
•
2
15 3
w =
13
6
w = 2
1
6
Divide the common factors.
Simplify.
Write as a mixed number.
Quick Check
5-8
Solving Equations by Multiplying Fractions
PRE-ALGEBRA LESSON 5-8
Solve – 20 c = 4 .
27
–
20
4
c =
27
9
– 27 – 20 c = – 27 4
20
9
27
20 9
3
3
5
reciprocal of – 20 .
27
20
1
27 • 4
c =
205 • 9 1
= –
Multiply each side by – 27 , the
Divide common factors.
Simplify.
Quick Check
5-8
Solving Equations by Multiplying Fractions
PRE-ALGEBRA LESSON 5-8
How many 2 1 -t trucks can you place on a rail car that
2
has a carrying capacity of 15 t?
Words
weight of
each truck
Let
Equation
n
2
times
the number
of trucks
is
carrying
capacity
= the number of trucks.
1
2
•
5-8
n
=
15
Solving Equations by Multiplying Fractions
PRE-ALGEBRA LESSON 5-8
(continued)
2 1 • n = 15
2
5 n = 15
2
2
5
Write 2 1 as 5 .
2
• 5 n = 2 • 15
2
5
2 • 15
n =
51 • 1
= 6
2
Multiply each side by 2 , the reciprocal of 5 .
5
2
3
Divide common factors.
Simplify.
You can place 6 trucks on the rail car.
5-8
Quick Check
Solving Equations by Multiplying Fractions
PRE-ALGEBRA LESSON 5-8
Solve each equation.
1.
5
7
=
b
16
8
5
14
25
5
2. –
g= 8
32
4
3. 15 = –3
1
–5
–4 3
5-8
6
m
13
Powers of Products and Quotients
PRE-ALGEBRA LESSON 5-9
112 x 32 x 22 is the prime factorization for ?.
4,356
5-9
Powers of Products and Quotients
PRE-ALGEBRA LESSON 5-9
(For help, go to Lesson 4-7.)
Simplify each expression.
1. (22)3
2. (32)2
3. (15)4
4. (x3)6
5. (b2)5
6. (a7)4
Check Skills You’ll Need
5-9
Powers of Products and Quotients
PRE-ALGEBRA LESSON 5-9
Solutions
1. (22)3
2. (32)2
3. (15)4
= 22 • 3
= 32 • 2
= 15 • 4
= 26
= 34
= 120
= 64
= 81
=1
4. (x3)6
5. (b2)5
6. (a7)4
= x3 • 6
= b2 • 5
= a7 • 4
= x18
= b10
= a28
5-9
Powers of Products and Quotients
PRE-ALGEBRA LESSON 5-9
Simplify (3z5)4.
(3z5)4 = 34 • (z5)4
Raise each factor to the fourth power.
= 34 • z5 • 4
Use the Rule for Raising a Power to a Power.
= 34 • z20
Multiply exponents.
= 81z20
Simplify.
Quick Check
5-9
Powers of Products and Quotients
PRE-ALGEBRA LESSON 5-9
a. Simplify (–3a)4.
(–3a)4 = (–3)4(a)4
= 81a4
b. Simplify –(3a)4.
–(3a)4 = (–1)(3a)4
= (–1)(3)4(a)4
= –81a4
Quick Check
5-9
Powers of Products and Quotients
PRE-ALGEBRA LESSON 5-9
Find the area of a square with side length x .
4
A = s2
=
x
4
s = length of a side
2
x2
= 2
4
=
x2
16
2
The area of the square is x square units.
16
Quick Check
5-9
Powers of Products and Quotients
PRE-ALGEBRA LESSON 5-9
Simplify each expression.
1. (12x2yz)2
2. (–4ab3)3
3. –(–2g2h)8
144x4y2z2
–64a3b9
–256g16h8
4.
u3
6
u6
36
2
5.
3q8r 6
s7
5
243q 40 r 30
s35
5-9