Download Chapter 5 Notes - Sacred Heart School

Chapter 5 Notes
Number Theory and Fractions
Prime and Composite numbers
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A prime number is a number greater than 1 that has exactly two factors, itself and 1.
A composite number is a number greater than 1 that has more than two factors.
The numbers 1 and 0 are neither prime or composite
Examples
Prime: 3, 5, 11, 19
Composite: 12 (factors 1,2,3,4,6,12) and 25 (factors 1,5,25)
Prime Factorization
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Every composite number can be written as the product of prime factors and this is called prime
factorization. Use a factor tree to find the prime factors.
To make a factor tree
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Start with the composite number
Choose any 2 factors of that number
Continue factoring until all the branches show prime numbers.
Arrange the prime factors in order from least to greatest.
Example
36
↙↘
9
4
↙↘ ↙↘
3
3 2
22 ×32
2
How to find equivalent fractions
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Multiply or divide the numerator (top number in a fraction) and the denominator (bottom
number of the fraction) by the same number
Examples
5 = 15
6 18
The denominator 6 is multiplied by 3 to get the new denominator 18. To find
the equivalent numerator multiply the numerator 5 by 3 to get 15.
8 =1
32 4
The numerator 8 is divided by 8 to get the new numerator 1. To find the
equivalent denominator divide the denominator 32 by 8 to get 4.
To find the Greatest Common Factor (GCF)
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List all the factors of each number
Find the common or shared factors
Choose the greatest common factor
Example
8: 1, 2, 4, 8
12: 1, 2, 3, 4, 6, 12
20: 1, 2, 4, 5, 10, 20
Common factors: 1, 2, 4
Greatest common factor: 4
To rename a fraction in simplest form:
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Find the GCF of the numerator and denominator
Divide the numerator and denominator by their GCF
If you divided and they are still not in simplest form, divide again.
Example
20 →factors of 20: 1, 2, 4, 5, 10, 20
32 →factors of 32: 1, 2, 4, 8, 16, 32
GCF of 20 and 32 is 4.
20 ÷ 4 = 5
32 ÷ 4 8
Mixed Numbers and Improper Fractions
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A mixed number has a whole number and a fraction
An improper fraction has a numerator equal to or greater than its denominator
To rename a mixed number as an improper fraction:
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Multiply the whole number by the denominator
Add the product to the numerator
Write the sum over the denominator
Example
3 ½ = (2 x 3) + 1 = 7
2
2
To rename an improper fraction as a whole number or as a mixed number
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Divide the numerator by the denominator
If there is a remainder, write it over the denominator and express in simplest form
Example
38 = 38÷4 = 9 R2 = 9 2/4 = 9 ½
4
To find the least common multiple (LCM)
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Write out the multiples of both numbers starting with the larger one
Go back and forth writing multiples until you find the lowest number, except 0, that is common
to both
Example
Multiples of 12: 12, 24, 36, 48, 60
Multiples of 30: 30, 60
The LCM od 12 and 30 is 60.
To compare fractions
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When fractions have a common denominator, compare their numerators to find the larger one.
When the fractions have unlike denominators, find the two fractions least common
denominator (least common multiple of denominators), rename them using this least common
denominator, and compare the two new fractions.
Example
5 > 4
6 5
The LCM of 5 and 6 is 30. 5/6 renamed with 30 as the denominator is 25/30. 4/5
renamed with 30 as the denominator is 24/30. This makes 5/6 greater than 4/5.
To order fractions
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Finds the least common denominator (LCD) of all the fractions
Rename each fraction using the LCD
Compare the numerators and write the fractions in order
If you are ordering mixed numbers, compare the whole numbers first
Example
2 = 16
3
24
3 = 18
4 24
5 = 15
8
24
From least to greatest 5/8, 2/3, ¾
From greatest to least ¾, 2/3, 5/8
To rename a fraction as a decimal
 Divide the numerator by the denominator.
 Place the decimal point after the numerator and in the quotient
 Divide. Add zeros as needed.
Example
0.75
4 3.00
To rename a mixed number as a decimal
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Separate the mixed number into a whole number part and a fraction part
Rename the fraction part as a decimal
Add the whole number part and the decimal
Example
0.0625
16 1.0000
9 + 0.0625 = 9.0625
To rename a decimal as a fraction
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Read the given decimal
The place value names the denominator
The amount names the numerator.
Write your fraction and put in simplest form
Example
0.35 is thirty-five hundredths = 35 = 7
100 20