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Chapter 5 Study Guide/ Notes
5-1 Prime Factorization
Vocabulary:
-Prime number—a whole # greater than 1, that has exactly two factors: 1 and itself.
-Composite number—a whole # greater than 1, that has more than two factors
PRIME numbers: 2, 3, 5, 7, 11, 13, 17, 19…
You can find the prime factorization in more than one way.
EX: Factor Tree: Split the number you are factoring into two factors. Continue to split each factor until
there are only prime factors left. Circle the prime numbers. Write the prime factorization using
exponents.
24
12
6 X
X
2
2
3 X 2
24: 2 X 2 X 2 X 3 = 2³ X 3
EX: Upside-Down Division (subsequent division): Divide by prime numbers until only prime numbers
remain. Write the prime factorization using exponents.
2
24
2 12
2 6
3
24: 2 X 2 X 2 X 3 = 2³ X 3
Practice:
Determine whether each number is prime or composite.
1) 27
2) 31
Composite
Prime
3) 60
4) 56
Composite
Composite
Find the prime factorization of each number or expression. Use any method you prefer.
5)
36
6)
2² X 3²
7)
128
2⁴ X 3
8)
2⁷
9)
34a²b³
2 X 17 X a X a X b X b X b
48
15mn
3X5XmXn
10)
20p²
2² X 5 X p X p
5-2 Greatest Common Factor
You can find the Greatest Common Factor (GCF) a few different ways also.
Ex: Listing Factors: List all of the factors of each number. Circle the common factors. Find the largest
of those numbers. That is the greatest common factor.
Find the GCF of 20 and 24.
20: 1, 2, 4, 5, 10, 20
24: 1, 2, 3, 4, 6, 8, 12, 24
GCF: 4
Ex: Upside-Down Division: Write the set of numbers with an upside-down division bar. Divide the
numbers by a prime factor. Repeat until there are no more common factors. Multiply the numbers on
the left to find the greatest common factor.
2 20
24
2 10
12
5
6
GCF: 2X2= 4
Practice: Find the greatest common factor (GCF) of each set of numbers.
1) 40, 50
10
2) 32, 48
16
3) 45, 75
15
4) 36, 60, 84
12
5-3) Simplifying Fractions
You can write fractions in simplest form using a few different methods
Ex: Dividing by common factors.
8
÷ 2
4
÷ 2
2
20
÷ 2
10
÷ 2
5
Ex: Upside-Down Division.
8
2
20
8 / 20
2
2 4 / 10
5
2 / 5
Practice: Write the fraction in simplest form. You may use either method.
1) 9
12
2) 48
64
3/4
3/4
3) 15
25
3/5
To find equivalent fractions, the easiest way is to MULTIPLY the numerator and denominator by the
same number.
Ex: 1 X 2
2
4 X2
8
Practice: Find 3 equivalent fractions of the fraction given.
4) 1
5) 3
3
5
2/6, 3/9, 4/12, etc.
6/10, 9/15, 12/20
5-4 Fractions and Decimals:
Fractions  Decimals: divide the numerator (top #) by the denominator (bottom #).
Ex: 1
4
. 25
4 )1.00
-8
20
If you notice that the decimal is beginning to repeat, use bar notation and put a line over the first part
of the sequence that repeats.
Ex: If the decimal looks like: .333333…, write it like this  .3
If the decimal looks like: .1232323…, write it like this .123
If you are given a mixed number, write the “large number” in front of the decimal, then, divide the
fractional part as before.
Decimals Fractions: To help, read the decimal aloud. Then, write as a fraction and simplify.
Ex: 0.48 = “forty-eight hundredths” = 48 = 24 = 12
100 50
25
Practice: Write each fraction as a decimal.
1) 2
5
0.4
2) 3 ⅝
3) 3
20
3. 625
0.15
Write each repeating decimal using bar notation.
4) 4.20320320…
4.203
5) 0.444444…
0.4
6) 0.2345454545…
0.2345
Write each decimal as a fraction.
7) 0.75
8) 0.34
9) 0.2
3/4
17/50
1/5
10) 5.8
5 4/5
5-5 Fractions and Percents:
FractionsPercents: Write the fraction as an equivalent fraction with 100 as the denominator. Then
write the numerator with a % symbol.
Ex: 3 x 5
=
20 x 5
15
= 15%
100
Practice: Write each fraction as a percent.
1) 17
20
2) 3
5
85%
3) 2
25
60%
8%
Percents Fractions: Remove the % symbol. Then, write the number with a denominator of 100 and
simplify.
Ex: 48% = 48
÷ 2 = 24
÷ 2 = 12
100 ÷ 2 = 50
÷ 2 = 25
Practice: Write each percent as a decimal. (*Should say write as a FRACTION. Here are
answers for both decimal and percent:)
4) 40%
5) 6%
6) 24%
0.4 = 2/5
0.06 = 3/50
0.24 = 6/25
5-6 Percents and Decimals:
PercentsDecimals: Remove the % symbol. Then move the decimal 2 places to the LEFT.
Ex: 16%  16 1 6 .  0.16
Practice: Write each percent as a decimal.
1) 76%
2) 8.5%
0.76
0.085
3) 92 ½ %
0.925
DecimalsPercents: Move the decimal 2 places to the RIGHT. Then attach the % symbol.
Ex: 0.332 0 . 3 3 2  33.2  33.2%
Practice: Write each decimal as a percent.
4) 0.67
67%
5) 0.418
41.8%
6) 0.2
20%
5-7 Least Common Multiple
Vocabulary:
-multiple- the product of a number and any whole number
-least common multiple (LCM)-Given the multiples of two numbers, this is the smallest number they
have in common
You can find the least common multiple (LCM) in a few different ways.
Ex: Listing Multiples: List the multiples of each number. Find the smallest multiple they have in
common.
6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66…
10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100…
LCM of 6 and 10 is 60.
**Ex: Upside-Down Division: divide both numbers by a prime factor. Multiply all numbers on the
outside (making an “L” shape).
2
6
10
3
5
2 X 3 X 5 = 60
With 3 numbers: divide by a prime factor that goes into at least 2 of the 3 numbers given. Bring the
number down, if the factor does not go into it. Multiply all numbers on the outside (making an “L”
shape)
Ex:
2
5
4
10
15
2
5
15
2
1
3
Practice: Find the least common multiple (LCM) of each set of numbers.
1) 12, 16
2) 30, 45
48
90
4) 8, 12, 16
3) 45, 63
189
5) 12, 16, 36
48
144
5-8: Comparing and Ordering Rational Numbers
Comparing two fractions: Fractions must have a common denominator to compare them. 1) Find the
LCM of the two denominators. 2) Rewrite the fractions with the common denominator. 3) Determine
how many times you need to multiply the original denominator of the fraction on the LEFT to get to
the new denominator. Multiply the numerator by that number (12 X 3 = 36, so we multiply 7 X 3 to get
21). 4) Determine how many times you need to multiply the original denominator of the fraction on
the RIGHT to get to the new denominator. Multiply the numerator by that number. 5) COMPARE the
numerators.
Ex: Compare using <, >, or =.
7
8
12
18
2
12
18
3 6
9
2
21
36
>
16
36
3
LCM= 36
Comparing a mixed set of decimals, fractions, and percents: To compare, 1) write each number in the
same form (writing each as a decimal requires the least work usually). 2) Write the numbers in order
from least to greatest. 3) Be sure to put the numbers back into the ORIGINAL FORM when you give
your final answer.
Ex: Order the numbers from least to greatest.
0.74, 70%, 3
4
0.74 0.70 0.75  0.70, 0.74, 0.75  70%, 0.74, ¾
Practice: Compare using < , >, or =.
1) 7
10
2
3
>
2) 16
20
40
50
=
3) 7
5
14
11
>
Order the numbers from LEAST to GREATEST. (Make sure to put numbers back into their original forms
when giving your answer.)
4) 0.23, 19%, 1/5
5) 8/10, 81%, 0.805
19%, 1/5, 0.23
8/10, 0.805, 81%