Math 5 Section 1. Fractions, Decimals and Percents Prime Factorization 70 = 5 7 Prime numbers have exactly two factors, one and the number itself. Factorization can be done with a tree or using the box method. Be sure to type your answer as a product of primes. 540 = Least Common Multiple 1 and 0 LCM = 60 LCM s are used to find Least Common Denominators. You can use factor trees or the box method to find the LCM. 1 and 18 LCM = Equivalent Fractions Building Fractions During the process of making common denominators, you will need to build fractions. When you multiply the top and bottom of a fraction by the same number, you are multiplying by one, which is why the fractions are equivalent. 5 = = 7= 3 1 14 8 3 Least Common Denominators 1 3 5 7 1 5 and and and 8 6 8 7 9 Simplifying Fractions Simplifying or reducing fractions is done by dividing the top and bottom by the same number (GCF). 14 4 36 1 7 108
Converting Fractions to Decimals Divide. You will either get a remainder of zero, or you will see a repeating pattern, or you will round your answer. 5 8 9 9 7 Converting Decimals to Fractions 0.75 0. 4 0. 107 Converting Decimals to Percents Multipy by 100 or move the decimal two places right. 0.1 0. 349 0. 7 Converting Percents to Decimals Divide by 100 or move the decimal two places left. 37 % 8.5% 137 %
Math 5 Section 1.3 The Number System and the Real Number Line Natural Numbers 1,, 3, These are the numbers used for counting. Whole Numbers 0, 1,, 3, This is the set of natural numbers with 0 included. Integers,-, -1, 0, 1,, positive integers(natural numbers) plus zero plus negative integers(opposites of the positive integers) Rational Numbers terminating or repeating decimals can be re-written as a fraction(integer/integer 0) 3 3.4 8 Irrational Numbers non-terminating and non-repeating decimals can not be re-written as a fraction π 10 1 10 6 0.3 Real Numbers The set of all numbers corresponding to points on the number line. Rational numbers plus irrational numbers. Inequality Symbols < is less than > is greater than Absolute Value The absolute value of a number is its distance from zero on a number line. Examples: 1. Write the set of integers between -3 and 1, not including -3 or 1. 17. Graph on a number line. 4 3. Use either < or > to write a true sentence. 17 0. 4. Write an inequality with the same meaning as 3< 5. 5. Evaluate each expression. 7 13 0
Math 5 Sections 1.4 & 1.5 Operations with Integers, Fractions & Decimals Adding and Subtracting Real Numbers Rules for Adding Real Numbers Adding two positive numbers add, the answer is positive Adding two negative numbers add absolute values, the answer is negative Adding a positive and a negative number subtract absolute values, answer is sign of larger number Rules for Subtracting Real Numbers Add the opposite keep first number the same, change subtraction to addition, change second number to the opposite number, and then follow adding rules Examples: Evaluate each expression. You must correctly change each subtraction question into an addition question before evaluating. 1. 9+ 3. ( 3) 3. 4.13 6. 4. 5 ( 11) 5. 3. 5.941 6. 1+ ( 1) 7. 1 5 + 6 8 8. 9 ( 7) + 5 9. 8+ 4 11+ 6 ( 1) 10. 3 1 4 8
Multiplying and Dividing Real Numbers Rules for Multiplying Real Numbers Multiplying two positive numbers multiply, the answer is positive Multiplying two negative numbers multiply, the answer is positive Multiplying a positive and a negative number multiply, answer is negative Any number times zero equals zero Rules for Dividing Real Numbers Follow same sign rules as multiplication Division with zero Zero divided by any number (except zero) is equal to zero 0 7= 0 0 = 0 1 Any number (except zero) divided by zero is undefined 6.3 0= undefined 5 0 = undefined Examples: Evaluate each expression. 1. 7 6. 3 3 5 3. 4.1( 9.5) 4. 1 9 9 3 10 5. Evaluate 5x when x= 6. 100 ( 0) 7. 9 0 8. 5 3 9. 4.41 6.3 4 4
Math 5 Section 1.6 Properties of Real Numbers expression 5+ a 7 + 3y 4z 5 equivalent expressions Two expressions that have the same value for all allowable replacements 5+ = 4+ 3 5x= 7x x Properties of Real Numbers: commutative property a+ b= b+ a a b= b a 4+ 3= 3+ 4 5= 5 associative property a+ ( b+ c) = ( a+ b) + c a ( b c) = ( a b) c + (3+ 4) = (+ 3) + 4 (6 ) 5= 6 ( 5) distributive property ab ( + c) = a b+ a c 4(7+ ) = 4 7+ 4 3( x 9) = 3 x 3 9 = 3x 7 identity properties a+ 0= a 7+ 0= 7 Inverse properties a+ ( a) = 0 7 + ( 7) = 0 a 1= a 81 = 8 1 a = 1 a 3 = 1 3
Examples: 1. Convert each measurement. 13 feet = inches 130 feet = yards. Write an equivalent expression. commutative property associative property y+ 8= (8 x) y 3. Multiply. 4( x+ 3) 5 ( 16) 8 y 4. Factor. 7x+ 8 3 4y 9a+ 7b+ 81 5. Combine like terms. 1a+ 3a 9y 17y 8x 5x+ 6+ 3y y 4
Math 5 Sections 1.7 & 1.8 - Simplifying Expressions Order of operations: 1. Do all calculations within grouping symbols before operations outside. (P). Evaluate all exponential expressions. (E) 3. Do all multiplications and divisions in order from left to right. (MD) 4. Do all additions and subtractions in order from left to right. (AS) PEMDAS Examples: 1. +. 1m ( 4m 6) 3 (7 4) 10 5 3. 4{ 8 ( x 3) + 9 4( 3x ) + 6 } 4. 19 5( 3) + 3 6. [ 7( 4 8) + 16] [ 5 ( 7 8) ] 5. 5 + 7 7. ( 3x+ y) (5x 4y)