Download Study Session Test 1 Chapter 3

Study Session Test 1 Chapter 3
February 3, 2016
Chapter 3 Section 1: The Least Common Multiple and
Greatest Common Factor
Natural number factors of a number divide that number evenly (there is no remainder).
Note:
• 1 is a factor of every number
• 2 is a factor of a number if the the last digit of the number is either 2, 4, 6, 8, or 0.
Ex: 436 ends in 6, therefore 2 is a factor of 436.
• 3 is a factor of a number if the sum of the digits is divisible by 3.
Ex: The sum of the digits of 489 is 4 + 8 + 9 = 21. 21 is divisible by 3, therefore 3 is
a factor of 489.
• 4 is a factor of a number if the last two digits of the number is divisible by 4.
Ex: 556 ends in 56. 56 is divisible by 4, therefore 4 is a factor of 556.
• 5 is a factor of a number if the the last digit of the number is either 5 or 0.
Ex: 520 ends in a 0 therefore 5 is a factor of 520.
• 6 is a factor of a number is 2 and 3 are factors of the number.
• 8 is a factor of a number if the last 3 digits of the number is divisible by 8.
Ex: The last three digits of 77184 is 184 which is divisible by 8. Therefore 8 is a factor
of 77184.
• 9 is a factor of a number if the sum of the digits of the number is divisible by 9.
Ex: The sum of the digits of 6993 is 6 + 9 + 9 + 3 = 27. 27 is divisible by 9, therefore
9 is a factor of 6993.
• 10 is a factor of a number if the number ends in 0.
• A number is a factor of itself.
1
A prime number is a natural number greater than 1 that has exactly two natural number
factors, 1 and the number itself.
A number that is not prime is a composite number.
The prime factorization of a number is the expression of the number as a product of its
prime factors.
The least common multiple (LCM) is the smalles common multiple of two or more numbers.
The greatest common multiple (GCF) is the largest common factor of two or more
numbers.
Chapter 3 Section 2: Introduction to Fractions
A fraction can represent the number of equal parts of a whole.
A proper fraction is a fraction that is less than 1.
A mixed number is a number greater than 1 with a whole-number part and a fractional
part.
An improper fraction is a fraction greater than or equal to 1. The numerator of an improper fraction is greater than or equal to the denominator.
Writing an improper fraction as a mixed number of whole number.
2
Writing a mixed number as an improper fraction.
Chapter 3 Section 3 Writing Equivalent Fractions
Writing the simplest form of a fraction means writing it so that the numerator and denominator have no common factors other than 1.
Equal fractions with different denominators are called equivalent fractions
To find the order relation between two fractions with the same denominator, compare
the numerators. The fraction that has the smaller numerator is the smaller fraction.
Chapter 3 Section 4 Multiplication and Division of Fractions
To multiply two (or more) fractions, multiply the numerators together and multiply the
denominators together:
a·c
a c
· =
b d
b·d
To divide two fractions, we use “Keep Change Flip”
a c
a d
÷ = ·
b d
b c
Chapter 3 Section 5 Addition and Subtraction of Fractions:
Adding or Subtraction Fractions with the Same Denominator:
a c
a+c
+ =
b b
b
3
a−c
a c
− =
b b
b
Adding or Subtracting Fractions with Different Denominators:
a c
ad + bc
+ =
b d
bd
a c
ad − bc
− =
b d
bd
Chapter 3 Section 6 Operations on Positive and Negative Fractions
Note:
The product of two numbers with the same sign is positive.
The product of two numbers with different signs is negative.
The quotient of two numbers with the same sign is positive.
The quotient of two numbers with different signs is negative.
Chapter 3 Section 7 The Order of Operations Agreement and Complex Fractions
Order of Operations Agreement:
Step 1. Perform operations inside grouping symbols such as parenthesis and fraction bars.
Step 2. Simplify exponential expressions.
Step 3. Do multiplication and division as they occur from left to right.
Step 4. Do addition and subtraction as they occur from left to right.
Examples with Detailed Solutions
1. Find all the factors of 56
2. Find the prime factorization of 56
3. Find the least common multiple (LCM) of 10 and 12.
4. Find the greatest common factor (GCF) of 4, 10, and 24.
5. Write the improper fraction
17
as a mixed or whole number.
4
4
6. Write the mixed number 6
1
as an improper fraction.
2
7. Find the order relation between
3
2
and .
5
8
8. Multiply the given numbers.
5 7
·
14 8
9. Divide the given numbers.
1
7
7 ÷2
8
16
10. Add the given numbers.
3
7
+
12 16
11. Subtract the given numbers.
3
5
−
14 42
12. Multiply the given numbers.
1
3
−2 · −
3
7
13. Add the given numbers.
3 2
− +
4 5
14. Simplify.
3
·
10
3
2
2
+
3
3
15. Simplify.
7
12
1 5
2 −
3 6
4−3
Solution to # 1:
1 and 56 are factors of 56. Since 56 ends in 6, 2 is also a factor of 56. Note: 56 = 1 · 56,
56 = 2 · 28, 56 = 4 · 14, and 56 = 7 · 8. Hence, the factors of 56 are: 1, 2, 4, 7, 8, 14, 28, 56.
Solution to # 2:
56 = 7 · 8
=2·2·2·7
Therefore, the prime factorization of 56 is 2 · 2 · 2 · 7 or 23 · 7
5
Solution to # 3:
2
3
10 =
2
12 = 2 · 2 3
5
5
To find the LCM multiply the largest product of each prime number. Hence, the LCM
of 10 and 12 is 2 · 2 · 3 · 5 = 4 · 15 = 60.
Solution to # 4:
2
3 5
4=
2·2
10 =
2
5
24 = 2 · 2 · 2 3
To find the GCF multiply the smallest product of each prime number in each column
that does have a blank space. Hence, the GCF of 4, 10, and 24 is 2.
Solution to # 5:
4
4 17
16
1
17
1
Hence,
=4 .
4
4
Solution to # 6:
1
6·2+1
6 =
2
2
12 + 1
=
2
13
=
2
Solution to # 7:
To determine the order relation between the two fractions, we must write each fraction as
an equivalent fraction with the same denominator. The least common denominator of 5 and
8 is 2 · 2 · 2 · 5 = 40.
2
2 8
= ·
5
5 8
2·8
=
5·8
16
=
40
6
3
3 5
= ·
8
8 5
3·5
=
8·5
15
=
40
When two fractions have the same denominator, the larger fraction will have the larger
numerator. Hence,
3
2
>
5
8
Solution to # 8:
5 7
5·7
· =
14 8
14 · 8
5·7
=
2·2·2·2·7
5
=
16
Solution to # 9:
Please not that in order to multiply or divide two mixed numbers, you must first change
each mixed number to an improper fraction.
1
7 · 8 + 7 2 · 16 + 1
7
÷
7 ÷2 =
8
16
8
16
63 33
=
÷
8
16
63 16
·
=
8 33
63 · 16
=
8 · 33
3·3·7·8·2
=
8 · 3 · 11
42
=
11
Solution to # 10:
To add two or more fractions, we must first ensure that each fraction has the same denominator. To find the least common denominator (LCD) we find the least common multiple of
12 and 16:
2
3
12 =
2·2
3
16 = 2 · 2 · 2 · 2
7
Hence the LCD is 2 · 2 · 2 · 2 · 3 = 48.
7
3
7 4
3 3
+
=
· +
·
12 16
12 4 16 3
3·3
7·4
+
=
12 · 4 16 · 3
9
28
+
=
48 48
28 + 9
=
48
37
=
48
Solution to # 11:
To subtract two fractions we must first ensure that each fraction has the same denominator.
To find the least common denominator (LCD), we must find the least common multiple
(LCM) of 14 and 42:
2 3 7
14 = 2
7
42 = 2 3 7
Hence, the LCD is 2 · 3 · 7 = 42.
5
3 3
5 1
3
−
=
· −
·
14 42
14 3 42 1
3·3
5·1
=
−
14 · 3 42 · 1
9
5
=
−
42 42
9−5
=
42
4
=
42
2
=
21
8
Solution to # 12:
3
1 3
1
=2 ·
−2 · −
3
7
3 7
2·3+1 3
=
·
3
7
7 3
= ·
3 7
7·3
=
3·7
21
=
21
=1
7
3
Note that and are reciprocals of each other. The product of a number and its reciprocal
3
7
is 1.
Solution to # 13:
3 2
3 5 2 4
− + =− · + ·
4 5
4 5 5 4
3·5 2·4
=−
+
4·5 5·4
8
15
=− +
20 20
−15 + 8
=
20
−7
=
20
9
Solution to # 14:
3
·
10
3
2
2
3
2 2 2
2
+ =
·
· ·
+
3
3
10
3 3 3
3
3·2·2·2
2
=
+
10 · 3 · 3 · 3 3
2
24
+
=
270 3
24
2 90
=
+ ·
270 3 90
24
2 · 90
=
+
270 3 · 90
180
24
+
=
270 270
24 + 180
=
270
204
=
270
3 · 2 · 2 · 17
=
3·3·3·2·5
34
=
45
10
Solution to # 15:
4 3 · 12 + 7
7
−
12 = 1
12
2·3+1 5
1 5
2 −
−
3 6
3
6
4 43
−
= 1 12
7 5
−
3 6
4 12 43
·
−
= 1 12 12
7 2 5
· −
3 2 6
48 43
−
= 12 12
14 5
−
6
6
48 − 43
12
=
14 − 5
6
5
= 12
9
6
5 6
=
·
12 9
5·6
=
12 · 9
5
=
18
4−3
11
Problems For You To Try
Below are problems for you to try. For these problems, you are only given the
solution, you must fill in the steps yourself.
1. Find all the factors of 48
2. Find the prime factorization of 36
3. Find the least common multiple of 3, 5, and 10.
4. Find the greatest common factor (GCF) of 45 and 80
5. Write the improper fraction
6. Write the mixed number 5
11
as a mixed or whole number.
9
5
as an improper fraction.
17
7. Find the order relation between
17
1
and
.
18
12
8. Multiply the given numbers.
1 5
3 ·7
2 7
9. Divide the given numbers.
2
5 ÷9
5
10. Add the given numbers.
7
2 1
+ +
3 5 12
11. Subtract the given numbers.
11
7
−
15 20
12. Multiply the given numbers.
1
7 · (−6)
2
13. Add the given numbers.
1
7
5
+
+ −
8
2
12
14. Simplify
5 1
−
6 2
2
−
1
12
12
15. Simplify
1
2
4 −1
3
6
5
1
6 −3
8
4
Solutions:
1. 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
2. 2 · 2 · 3 · 3 or 22 · 32
3. 30
4. 5
5. 1
2
9
6.
90
17
7.
17
1
>
18
12
8. 27
9.
3
5
10.
29
20
11.
23
60
12. -45
13.
17
24
14.
1
36
15.
28
27
13