comparing and ordering rational numbers
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Comparing and Ordering Rational Numbers. PRE-ALGEBRA LESSON 5-1. Write the prime factorization of 364. 2 2 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. - PowerPoint PPT PresentationTRANSCRIPT
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5-1
Comparing and Ordering Rational NumbersComparing and Ordering Rational NumbersPRE-ALGEBRA LESSON 5-1PRE-ALGEBRA LESSON 5-1
Write the prime factorization of 364.
22 x 7 x 13
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Comparing and Ordering Rational NumbersComparing and Ordering Rational NumbersPRE-ALGEBRA LESSON 5-1PRE-ALGEBRA LESSON 5-1
Write the prime factorization of each number.
1. 20 2. 125 3. 45
4. 186 5. 621 6. 1,575
(For help, go to Lesson 4-3.)
Check Skills You’ll Need
5-1
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PRE-ALGEBRA LESSON 5-1PRE-ALGEBRA LESSON 5-1
Solutions
1. 20 2. 125 3. 452 • 10 5 • 25 3 • 152 • 2 • 5 5 • 5 • 5 3 • 3 • 5 20 = 22 • 5 125 = 53 45 = 32 • 5
4. 186 5. 621 6. 1,5752 • 93 3 • 207 3 • 5252 • 3 • 31 3 • 3 • 69 3 • 3 • 175186 = 2 • 3 • 31 3 • 3 • 3 • 23 3 • 3 • 5 • 35
621 = 33 • 23 3 • 3 • 5 • 5 • 71,575 = 32 • 52 • 7
Comparing and Ordering Rational NumbersComparing and Ordering Rational Numbers
5-1
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PRE-ALGEBRA LESSON 5-1PRE-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.
Comparing and Ordering Rational NumbersComparing and Ordering Rational Numbers
Quick Check
5-1
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PRE-ALGEBRA LESSON 5-1PRE-ALGEBRA LESSON 5-1
Find the LCM of 16 and 36.
= 144 Multiply.
16 = 24
36 = 22 • 32Write the prime factorizations.
The LCM of 16 and 36 is 144.
LCM = 24 • 32 Use the greatest power of each factor.
Comparing and Ordering Rational NumbersComparing and Ordering Rational Numbers
Quick Check
5-1
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PRE-ALGEBRA LESSON 5-1PRE-ALGEBRA LESSON 5-1
Find the LCM of 5a4 and 15a.
5a4 = 5 • a4
15a = 3 • 5 • a Write the prime factorizations.
= 15a4 Multiply.
The LCM of 5a4 and 15a is 15a4.
LCM = 3 • 5 • a4 Use the greatest power of each factor.
Comparing and Ordering Rational NumbersComparing and Ordering Rational Numbers
Quick Check
5-1
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PRE-ALGEBRA LESSON 5-1PRE-ALGEBRA LESSON 5-1
Graph and compare the fractions in each pair.
is on the left, so < .38
38
78
b. – 13
, – 16
is on the right, so > .– 16
– 16
– 13
a. 78
38
,
38
78
– 13
– 16
Comparing and Ordering Rational NumbersComparing and Ordering Rational Numbers
Quick Check
5-1
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PRE-ALGEBRA LESSON 5-1PRE-ALGEBRA LESSON 5-1
The softball team won of its games and the hockey
team won of its games. Which team won the greater fraction
of its games?
677
9
Step 1 Find the LCM of 7 and 9.7 = 7 and 9 = 32
LCM = 7 • 32 = 63Step 2 Write equivalent fractions with a denominator of 63.
6 • 97 • 97 • 79 • 7
=54634963
=
Step 3 Compare the fractions.5463
4963
67
79
> >, so
The softball team won the greater fraction of its games.
Comparing and Ordering Rational NumbersComparing and Ordering Rational Numbers
Quick Check
5-1
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PRE-ALGEBRA LESSON 5-1PRE-ALGEBRA LESSON 5-1
Order , , and from least to greatest.37
14
23
37
14
23
3 • 127 • 12
1 • 214 • 21
2 • 283 • 28
3684
2184
5684
=
=
=
=
=
=
The LCM of 7, 4, and 3 is 84.Use 84 as the common denominator.
2184
3684
5684
14
37
23
< < , so < < .
Comparing and Ordering Rational NumbersComparing and Ordering Rational Numbers
Quick Check
5-1
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PRE-ALGEBRA LESSON 5-1PRE-ALGEBRA LESSON 5-1
Find the LCM of each pair of numbers.
1. 8, 6 2. 12, 16
3. Compare and order , – , and – from least to greatest. 316
810
316
24 48
810
– –< <316
316
Comparing and Ordering Rational NumbersComparing and Ordering Rational Numbers
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5-2
PRE-ALGEBRA LESSON 5-2PRE-ALGEBRA LESSON 5-2
Fractions and DecimalsFractions and Decimals
Without actually dividing, which expression is the largest?
6.1 ÷2.8 12.6 ÷ 1.1 9.305 ÷ 2.99
12.6 ÷ 1.1
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Fractions and DecimalsFractions and DecimalsPRE-ALGEBRA LESSON 5-2PRE-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
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Fractions and DecimalsFractions and DecimalsPRE-ALGEBRA LESSON 5-2PRE-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
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Fractions and DecimalsFractions and DecimalsPRE-ALGEBRA LESSON 5-2PRE-ALGEBRA LESSON 5-2
The fuel tank of Scott’s new lawn mower holds gal of
gasoline. Scott poured 0.4 gal into the tank. Did Scott fill the tank?
12
= 1 ÷ 2 = 0.5
Since = 0.5 and 0.5 > 0.4, Scott did not fill the tank.12
12
Quick Check
5-2
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Fractions and DecimalsFractions and DecimalsPRE-ALGEBRA LESSON 5-2PRE-ALGEBRA LESSON 5-2
Write each fraction as a decimal. State the block of
digits that repeats.
a.
b.
56
5 ÷ 6 = 0.83333 … Divide.
Place a bar over the digit that repeats.= 0.8356
= 0.83; the digit that repeats is 3.
711
7 ÷ 11 = 0.636363 … Divide.
= 0.63 Place a bar over the block of digits that repeats.
= 0.63; the block of digits that repeats is 63.7
11 Quick Check
5-2
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Fractions and DecimalsFractions and DecimalsPRE-ALGEBRA LESSON 5-2PRE-ALGEBRA LESSON 5-2
Write the numbers in order, from least to greatest.
–0.8, , , 0.125312
54
–
–1.25 < –0.8 < 0.125 < 0.25 Compare the decimals.
3 ÷ 12 = 0.25Change the fractions to decimals.
–5 ÷ 4 = –1.25
From least to greatest, the numbers are , –0.8, 0.125, and .54
– 312
Quick Check
5-2
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Fractions and DecimalsFractions and DecimalsPRE-ALGEBRA LESSON 5-2PRE-ALGEBRA LESSON 5-2
Write 1.72 as a mixed number in simplest form.
Keep the whole number 1.Write seventy-two hundredths as a fraction.
1.72 = 1 72100
Divide the numerator and denominator of the fraction by the GCF, 4.
72 ÷ 4100 ÷ 4= 1
Simplify.1825
1.72 = 1
Quick Check
5-2
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Fractions and DecimalsFractions and DecimalsPRE-ALGEBRA LESSON 5-2PRE-ALGEBRA LESSON 5-2
Write 0.18 as a fraction in simplest form.
n Let the variable n equal the decimal.= 0.18
As a fraction in simplest form, 0.18 = .2
11
100n = 18.18 Because 2 digits repeat, multiply each side by 102, or 100.
1899
99n99
= Divide each side by 99.
n 18 ÷ 999 ÷ 9
= Divide the numerator and denominator by theGCF, 9.
211= Simplify.
100n = 18.18n = 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.
Quick Check
5-2
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Fractions and DecimalsFractions and DecimalsPRE-ALGEBRA LESSON 5-2PRE-ALGEBRA LESSON 5-2
1. Order , – , 0.625, –0.35 from least to greatest.
2. Write as a decimal.
Write each decimal as a fraction or mixed number in simplest form.
3. 6.04 4. 0.21
65
1720
13
65
13
–0.35, – , 0625,
0.85
61
257
33
5-2
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5-3
PRE-ALGEBRA LESSON 5-3PRE-ALGEBRA LESSON 5-3
Adding and Subtracting FractionsAdding and Subtracting Fractions
Five consecutive multiples of 6 have a sum of 270. What are they?
42, 48, 54, 60, 66
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Adding and Subtracting FractionsAdding and Subtracting FractionsPRE-ALGEBRA LESSON 5-3PRE-ALGEBRA LESSON 5-3
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
(For help, go to Lesson 5-1.)
Check Skills You’ll Need
5-3
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Adding and Subtracting FractionsAdding and Subtracting FractionsPRE-ALGEBRA LESSON 5-3PRE-ALGEBRA LESSON 5-3
Solutions
1. 4 = 22 2. 9 = 32 3. 2n = 2 • n
8 = 23 18 = 2 • 32 5 = 5
LCM = 23 LCM = 2 • 32 LCM = 2 • 5 • n
= 8 = 18 = 10n
4. 3 = 3 5. 8 = 23 6. 10 = 2 • 5
6 = 2 • 3 5 = 5 n = n
9 = 32 4 = 22 LCM = 2 • 5 • n
LCM = 2 • 32 LCM = 23 • 5 = 10n
= 18 = 40
5-3
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Adding and Subtracting FractionsAdding and Subtracting FractionsPRE-ALGEBRA LESSON 5-3PRE-ALGEBRA LESSON 5-3
Find each sum or difference. Simplify if possible.
a.
b.
49
29+
49
29+ =
4 + 29 Add the numerators.
69
= Simplify.
23
= Simplify.
5b
12b
–
– =5b
12b
12 – 5b Subtract the numerators.
7b= Simplify. Quick Check
5-3
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Adding and Subtracting FractionsAdding and Subtracting FractionsPRE-ALGEBRA LESSON 5-3PRE-ALGEBRA LESSON 5-3
Simplify each difference.
a.
b.
16
34
–
2y
– 516
4 – 1824
= Use the Order of Operations to simplify.
–1424
= Simplify.
= –724
Simplify.
2y
– =516
2 • 16 – 5 • yy • 16
Rewrite using a common denominator.
= 32 – 5y16y
Simplify.
16
34
– = 1 • 4 – 3 • 66 • 4
Rewrite using a common denominator.
Quick Check
5-3
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Adding and Subtracting FractionsAdding and Subtracting FractionsPRE-ALGEBRA LESSON 5-3PRE-ALGEBRA LESSON 5-3
Suppose one day you rode a bicycle for 3 hours, and
jogged for 1 hours. How many hours did you exercise?14
12
You exercised for 4 hours.34
12
3 + = +1 14
72
54
Write mixed numbers as improper fractions.
= 7 • 4 + 5 • 22 • 4
Rewrite using a common denominator.
= 28 + 108
Use the Order of Operations to simplify.
=
= 4 68
388
Write as a mixed number.
= 34
4 Simplify.
Quick Check
5-3
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Adding and Subtracting FractionsAdding and Subtracting FractionsPRE-ALGEBRA LESSON 5-3PRE-ALGEBRA LESSON 5-3
Find each sum or difference.
1. 2. 5 – 2
3. – – 4.
34
224
1024+
18
35
13
32c
47 +
12
583
1415
– 8c + 2114c
5-3
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5-4
PRE-ALGEBRA LESSON 5-4PRE-ALGEBRA LESSON 5-4
Multiplying and Dividing FractionsMultiplying and Dividing Fractions
Write each number as an improper fraction.
a. 4 b. 315
78
215
318
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Multiplying and Dividing FractionsMultiplying and Dividing FractionsPRE-ALGEBRA LESSON 5-4PRE-ALGEBRA LESSON 5-4
Write each mixed number as an improper fraction.
1. 2 2. 3 3. 1
4. 4 5. 7 6. 5
(For help, go to Skills Handbook page 738.)
13
310
49
45
17
78
Check Skills You’ll Need
5-4
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Multiplying and Dividing FractionsMultiplying and Dividing FractionsPRE-ALGEBRA LESSON 5-4PRE-ALGEBRA LESSON 5-4
Solutions
1. 2. 3.
2 = 2 +13
13
= +63
13
=6 + 1
3
2 =13
73
=30 + 3
10
3 = 3 +3
103
10
= +3010
310
3 =3
103310
1 = 1 +49
49
= +99
49
=9 + 4
9
1 =49
139
5-4
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Multiplying and Dividing FractionsMultiplying and Dividing FractionsPRE-ALGEBRA LESSON 5-4PRE-ALGEBRA LESSON 5-4
Solutions (continued)
4. 5. 6.
4 = 4 +45
45
= +45
=20 + 4
5
4 =45
205
245
7 = 7 +78
78
= +78
=56 + 7
8
7 =78
568
638
5 = 5 +17
17
= +17
=35 + 1
7
5 =17
357
367
5-4
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Multiplying and Dividing FractionsMultiplying and Dividing FractionsPRE-ALGEBRA LESSON 5-4PRE-ALGEBRA LESSON 5-4
Find • .23
57
23
57
• = 2 • 5 Multiply the numerators.
= 1021 Simplify.
Multiply the denominators.3 • 7
Quick Check
5-4
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Multiplying and Dividing FractionsMultiplying and Dividing FractionsPRE-ALGEBRA LESSON 5-4PRE-ALGEBRA LESSON 5-4
34
23
a. Find • .
23
34
34
• = 23
23
•1 1
12Divide the common factors.
= 12
Multiply.
b. 5w
3w17
Find • .
5w
3w17
• = •5w
3w171
1Divide the common factors.
1517
= Multiply.
Quick Check
5-4
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Multiplying and Dividing FractionsMultiplying and Dividing FractionsPRE-ALGEBRA LESSON 5-4PRE-ALGEBRA LESSON 5-4
Keesha’s desktop is a rectangle 3 ft long and 1 ft
wide. What is the area of her desktop?
12
12
A Area of a rectangle = length • width.= 3 • 1 12
12
= 72
32
• Write 3 and 1 as improper fractions, and .12
12
72
32
Multiply.= 214
Write as a mixed number.14= 5
The area of Keesha’s desk is 5 ft2.14
Quick Check
5-4
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Multiplying and Dividing FractionsMultiplying and Dividing FractionsPRE-ALGEBRA LESSON 5-4PRE-ALGEBRA LESSON 5-4
a. Find ÷ .35
720
35
35
710÷ = •
107 Multiply by the reciprocal of the divisor.
= • 351
107
2
Divide the common factors.
= 67
Multiply.
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Multiplying and Dividing FractionsMultiplying and Dividing FractionsPRE-ALGEBRA LESSON 5-4PRE-ALGEBRA LESSON 5-4
(continued)
b. Find ÷ .9
4q
= 32
Multiply.
278q
94q÷ = •
4q9 Multiply by the reciprocal of the divisor.
278q
278q
= • 1
4q9
1
Divide the common factors.278q
3 1
2 1
= 1 12
Write as a mixed number.
Quick Check
5-4
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Multiplying and Dividing FractionsMultiplying and Dividing FractionsPRE-ALGEBRA LESSON 5-4PRE-ALGEBRA LESSON 5-4
Find 4 ÷ (–3 ).12
38
4 ÷ (–3 ) = ÷ (– ) Change to improper fractions.278
12
38
92
= • (– ) Multiply by – , the reciprocal of – . 827
278
827
92
1
1
= • – Divide the common factors. 827
92
4
3
= – , or –1 Simplify.43
13
Quick Check
5-4
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Multiplying and Dividing FractionsMultiplying and Dividing FractionsPRE-ALGEBRA LESSON 5-4PRE-ALGEBRA LESSON 5-4
Simplify each expression.
1. 2.
3. 4.12g25
52g
•
3z4
5z7÷
712•
38
–
1116÷ 15
38
– 732 , or 1
2120
120
13, or 3
103
65 , or 1
15
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5-5
PRE-ALGEBRA LESSON 5-5PRE-ALGEBRA LESSON 5-5
Using Customary Units of MeasurementUsing Customary Units of Measurement
12
In 2002, a disc jockey broadcast her show for 27 h each week.
How many hours did she broadcast that year?
1,430 h
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Using Customary Units of MeasurementUsing Customary Units of MeasurementPRE-ALGEBRA LESSON 5-5PRE-ALGEBRA LESSON 5-5
Find each product.
1. 4 • 2. 3 •
3. 1 • 4. 4 •
(For help, go to Lesson 5-4.)
14
112
81
14
161
12
13
Check Skills You’ll Need
5-5
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Using Customary Units of MeasurementUsing Customary Units of MeasurementPRE-ALGEBRA LESSON 5-5PRE-ALGEBRA LESSON 5-5
Solutions
1. 2.
3. 4.
4 • = •41
112
112
=4
12
13=
3 • = •161
161
14
134
=208
4
= 52
=404
= 10
1 • = •54
14
81
81 4 • = •
12
13
92
13
=96
=32
= 112
5-5
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Using Customary Units of MeasurementUsing Customary Units of MeasurementPRE-ALGEBRA LESSON 5-5PRE-ALGEBRA LESSON 5-5
Choose an appropriate unit of measure. Explain your
choice.
a. weight of a hummingbird
b. length of a soccer field
Measure its weight in ounces because a hummingbird is very light.
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
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Using Customary Units of MeasurementUsing Customary Units of MeasurementPRE-ALGEBRA LESSON 5-5PRE-ALGEBRA LESSON 5-5
Use dimensional analysis to convert 68 fluid ounces to cups.
68 fl oz = •Use a conversion factor that changes fluid ounces to cups.
68 fl oz 1
1 c8 fl oz
17
= Divide the common factors and units.68 fl oz • 1 c 8 fl oz2
17 2 = c Simplify.
= 8 c Write as a mixed number.12
There are 8 c in 68 fl oz.12 Quick Check
5-5
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Using Customary Units of MeasurementUsing Customary Units of MeasurementPRE-ALGEBRA LESSON 5-5PRE-ALGEBRA LESSON 5-5
Fred’s fruit stand sells homemade lemonade in 6 -pint
bottles for $1.99. Jill’s fruit stand stand sells homemade
lemonade in 3 -qt containers for the same price. At which stand
do you get more lemonade for your money?
12
12
Since 7 pints > 6 pints, you get more lemonade for your money
at Jill’s stand.
12
3 qt = qt • Use a conversion factor that changes quarts to pints
12
72
2 pt1 qt
= • Divide the common factors and units.1 2 pt
1 qt7 qt 2 1
= 7 pt Multiply.
Quick Check
5-5
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Using Customary Units of MeasurementUsing Customary Units of MeasurementPRE-ALGEBRA LESSON 5-5PRE-ALGEBRA LESSON 5-5
Choose an appropriate unit of measure. Explain your choice.
1. weight of a dog
2. length of a football field
3. Use dimensional analysis to convert 1 mi to ft.
Pounds; dogs are too heavy to measure in ounces and too light to measure in tons.
Yards; it is too long to measure in feet or inches and too short to measure in miles.
7,920 ft
12
5-5
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5-6
PRE-ALGEBRA LESSON 5-6PRE-ALGEBRA LESSON 5-6
Problem Solving Strategy: Work Backward Problem Solving Strategy: Work Backward
Use mental math to decide whether or not each pair of fractions is
yes, yes, no
equal: and , and , and3642
67
45
1620
615
1820
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Problem Solving Strategy: Work Backward Problem Solving Strategy: Work Backward PRE-ALGEBRA LESSON 5-6PRE-ALGEBRA LESSON 5-6
Order from least to greatest.
1. 2.
3. 4.
12
(For help, go to Lesson 5-2.)
110,
13
14
15, , ,
73
133,
53
33, ,
37
313,
35
33, , – 3
7 , – 35
313, – , – 3
5
Check Skills You’ll Need
5-6
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3. 4.
Problem Solving Strategy: Work Backward Problem Solving Strategy: Work Backward PRE-ALGEBRA LESSON 5-6PRE-ALGEBRA LESSON 5-6
Solutions
1. 2.12
110,
13
14
15, , ,
73
133,
53
33, ,
37
313,
35
33, , – 3
7 , – 35
313, – , – 3
5
0.50, 0.3, 0.25, 0.20, 0.10
12
110 ,
15
14
13, , ,
2.3, 1.6, 4.3, 1
33
133,
53
73, ,
0.43, 0.6, 0.23, 1
73
35,
35
33, ,
–0.43, –0.6, –0.23, 1
– 33 , – 3
53
13, – , –37
5-6
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Problem Solving Strategy: Work Backward Problem Solving Strategy: Work Backward PRE-ALGEBRA LESSON 5-6PRE-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
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Problem Solving Strategy: Work Backward Problem Solving Strategy: Work Backward PRE-ALGEBRA LESSON 5-6PRE-ALGEBRA LESSON 5-6
(continued)
You should leave home at 5:45 A.M.
Quick Check
5-6
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Problem Solving Strategy: Work Backward Problem Solving Strategy: Work Backward PRE-ALGEBRA LESSON 5-6PRE-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?
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?
169 sheep
6:12 A.M.
5-6
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5-7
PRE-ALGEBRA LESSON 5-7PRE-ALGEBRA LESSON 5-7
Solving Equations by Adding or Subtracting FractionsSolving Equations by Adding or Subtracting Fractions
Find the sum or difference. If possible, write each answer in lowest
terms.
a. + = ? b. 7 – 5 = ? 45
25
56
26
1 15
2 12
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Solving Equations by Adding or Subtracting FractionsSolving Equations by Adding or Subtracting FractionsPRE-ALGEBRA LESSON 5-7PRE-ALGEBRA LESSON 5-7
Find each sum or difference.
1. 1 – 2 2. 3 + 2
3. 5 – 3 4. –4 – 3
(For help, go to Lesson 5-3.)
34
712
78
58
34
18
16
29
Check Skills You’ll Need
5-7
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Solving Equations by Adding or Subtracting FractionsSolving Equations by Adding or Subtracting FractionsPRE-ALGEBRA LESSON 5-7PRE-ALGEBRA LESSON 5-7
Solutions
1. 1 – 2 2. 3 + 434
78
58
= – = –198
18
238
78 – =
7 • 8 – 4 • 234 • 8
56 – 9232=
–3632=
712
348 + 44096=
78896=
5512+ =
29 • 12 + 8 • 558 • 12
298
= 8 = 82096
524
5-7
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Solving Equations by Adding or Subtracting FractionsSolving Equations by Adding or Subtracting FractionsPRE-ALGEBRA LESSON 5-7PRE-ALGEBRA LESSON 5-7
Solutions (continued)
3. 5 – 3 4. –4 – 334
18
16
–225 – 17454=
–39954=
299+ =
–25 • 9 – 6 • 296 • 9
256
= –7 = –72154
718
184 – 10032=
=
258– =
23 • 8 – 4 • 254 • 8
234
= 2 = 22032
8432
58
29
–
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Solving Equations by Adding or Subtracting FractionsSolving Equations by Adding or Subtracting FractionsPRE-ALGEBRA LESSON 5-7PRE-ALGEBRA LESSON 5-7
One school recycles about of its waste paper. The
student council set a goal of recycling of the school’s waste
paper by the end of the year. By how much does the school need
to increase its paper recycling to reach the goal?
13
34
Words plus is
Let n = the increase.
Equation + n =
fraction school recycles
theincrease
student goal
34
13
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Solving Equations by Adding or Subtracting FractionsSolving Equations by Adding or Subtracting FractionsPRE-ALGEBRA LESSON 5-7PRE-ALGEBRA LESSON 5-7
(continued)
+ n =13
34
13
– + n = – Subtract from each side.13
34
13
13
n = Use 3 • 4 as the common denominator.3 • 3 – 1 • 4 3 • 4
n = Use the Order of Operations.9 – 4 12
n = Simplify. 512
To meet the student council goal, the school needs to recycle more of its waste paper.
512
5-7
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Solving Equations by Adding or Subtracting FractionsSolving Equations by Adding or Subtracting FractionsPRE-ALGEBRA LESSON 5-7PRE-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 + = + = = , the answer
is reasonable.
912
13
34
512
412
512
Quick Check
5-7
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Solving Equations by Adding or Subtracting FractionsSolving Equations by Adding or Subtracting FractionsPRE-ALGEBRA LESSON 5-7PRE-ALGEBRA LESSON 5-7
Solve x – = .23
19
x – + = + Add to each side.23
19
23
23
23
x – = 23
19
x = Use 9 • 3 as the common denominator. 1 • 3 + 9 • 2 9 • 3
x = Use the Order of Operations.3 + 18 27
x = Divide the common factors.2127
7
9
x = Simplify.79
Quick Check
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Solving Equations by Adding or Subtracting FractionsSolving Equations by Adding or Subtracting FractionsPRE-ALGEBRA LESSON 5-7PRE-ALGEBRA LESSON 5-7
Solve q – 6 = –1 .12
35
q – 6 = 1 12
35
q = Use the Order of Operations.–16 + 65 10
q = Simplify.4910
q =
–8 • 2 + 5 • 13 5 • 2
Use 5 • 2 as the common denominator.
q = – + Write mixed numbers as improper fractions.
85
13 2
q = 4 Write as a mixed number. 910
Quick Check
5-7
q – 6 + 6 = 1 + 6 Add 6 to each side.12
35
12
12
12
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Solving Equations by Adding or Subtracting FractionsSolving Equations by Adding or Subtracting FractionsPRE-ALGEBRA LESSON 5-7PRE-ALGEBRA LESSON 5-7
Solve the following equations.
1. + s = 2. k + = 3. u – =
4. – + g = 5. x – 2 = –4
37
1321
25
79
112
34
511
1922
13
415
421
1745
56
7221
1415–1
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5-8
PRE-ALGEBRA LESSON 5-8PRE-ALGEBRA LESSON 5-8
Solving Equations by Multiplying FractionsSolving Equations by Multiplying Fractions
310Because of a mishap, of the oranges in a crate were ruined. If
the oranges originally weighed 50 lb, how many pounds of oranges
were ruined?
35 lb
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PRE-ALGEBRA LESSON 5-8PRE-ALGEBRA LESSON 5-8
(For help, go to Lesson 5-4.)
Find each product.
1. 2.
3. 4.
•47
910
58 •
53–
–1 •51–
14 •
234
12
Check Skills You’ll Need
5-8
Solving Equations by Multiplying FractionsSolving Equations by Multiplying Fractions
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PRE-ALGEBRA LESSON 5-8PRE-ALGEBRA LESSON 5-8
Solutions
1. 2.•47
58
910 •
53–
•47
58 =
4 • 57 • 8
= 2056
= 514
910 •
53– =
9 • –510 • 3
=–4530
= –112
–96=
5-8
Solving Equations by Multiplying FractionsSolving Equations by Multiplying Fractions
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PRE-ALGEBRA LESSON 5-8PRE-ALGEBRA LESSON 5-8
Solutions (continued)
3. 4.–1 •51–
14 •
234
12
=254
•54 =
–5 • –54 • 1–
–51
= 614
•92
23 =
9 • 22 • 3
= 186
= 3
5-8
Solving Equations by Multiplying FractionsSolving Equations by Multiplying Fractions
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PRE-ALGEBRA LESSON 5-8PRE-ALGEBRA LESSON 5-8
Solve 7y = .13
• (7y) = • Multiply each side by , the reciprocal of 7.13
17
17
17
7y = 13
y = Simplify. 121
Quick Check
5-8
Solving Equations by Multiplying FractionsSolving Equations by Multiplying Fractions
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PRE-ALGEBRA LESSON 5-8PRE-ALGEBRA LESSON 5-8
52
w = • Divide the common factors.1315
1
3
w = 2 Write as a mixed number.16
Solve w = .25
1315
• w = • Multiply each side by , the reciprocal of .25
52
52
52
25
1315
w = 25
1315
w = Simplify.13 6
Quick Check
5-8
Solving Equations by Multiplying FractionsSolving Equations by Multiplying Fractions
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PRE-ALGEBRA LESSON 5-8PRE-ALGEBRA LESSON 5-8
Solve – c = .49
2027
– c = 49
2027
c = Divide common factors.27 • 420 • 9
13
15
= – Simplify.35
– – c = – Multiply each side by – , the
reciprocal of – .
49
2720
2027
2720
2720
2027
Quick Check
5-8
Solving Equations by Multiplying FractionsSolving Equations by Multiplying Fractions
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PRE-ALGEBRA LESSON 5-8PRE-ALGEBRA LESSON 5-8
How many 2 -t trucks can you place on a rail car that
has a carrying capacity of 15 t?
12
Words times is
Let n = the number of trucks.
Equation 2 • n = 15
weight ofeach truck
the number of trucks
carrying capacity
12
5-8
Solving Equations by Multiplying FractionsSolving Equations by Multiplying Fractions
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PRE-ALGEBRA LESSON 5-8PRE-ALGEBRA LESSON 5-8
(continued)
12 2 • n = 15
n = Divide common factors.2 • 155 • 1
3
1
n = 15 Write 2 as .52
52
12
• n = • 15 Multiply each side by , the reciprocal of .52
52
25
25
25
= 6 Simplify.
You can place 6 trucks on the rail car. Quick Check
5-8
Solving Equations by Multiplying FractionsSolving Equations by Multiplying Fractions
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– g =
Solve each equation.
1. 2. 3. 15 = –3 m= b
PRE-ALGEBRA LESSON 5-8PRE-ALGEBRA LESSON 5-8
78
516
2532
58
613
514
45–
13–4
5-8
Solving Equations by Multiplying FractionsSolving Equations by Multiplying Fractions
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5-9
PRE-ALGEBRA LESSON 5-9PRE-ALGEBRA LESSON 5-9
Powers of Products and QuotientsPowers of Products and Quotients
112 x 32 x 22 is the prime factorization for ?.
4,356
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Powers of Products and QuotientsPowers of Products and QuotientsPRE-ALGEBRA LESSON 5-9PRE-ALGEBRA LESSON 5-9
Simplify each expression.
1. (22)3 2. (32)2 3. (15)4
4. (x3)6 5. (b2)5 6. (a7)4
(For help, go to Lesson 4-7.)
Check Skills You’ll Need
5-9
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Powers of Products and QuotientsPowers of Products and QuotientsPRE-ALGEBRA LESSON 5-9PRE-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
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Powers of Products and QuotientsPowers of Products and QuotientsPRE-ALGEBRA LESSON 5-9PRE-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
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Powers of Products and QuotientsPowers of Products and QuotientsPRE-ALGEBRA LESSON 5-9PRE-ALGEBRA LESSON 5-9
a. Simplify (–3a)4.
b. Simplify –(3a)4.
(–3a)4 = (–3)4(a)4
= 81a4
–(3a)4 = (–1)(3a)4
= (–1)(3)4(a)4
= –81a4Quick Check
5-9
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Powers of Products and QuotientsPowers of Products and QuotientsPRE-ALGEBRA LESSON 5-9PRE-ALGEBRA LESSON 5-9
Find the area of a square with side length .x4
A = s2 s = length of a side
x2
42=
x2
16=
The area of the square is square units. x2
16
= x4
2
Quick Check
5-9
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Powers of Products and QuotientsPowers of Products and QuotientsPRE-ALGEBRA LESSON 5-9PRE-ALGEBRA LESSON 5-9
Simplify each expression.
1. (12x2yz)2 2. (–4ab3)3 3. –(–2g2h)8
4. 5.u3
62
144x4y2z2 –64a3b9 –256g16h8
u6
36
3q8r s7
56
243q r s35
40 30
5-9