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Chapter 3.notebook January 20, 2016 What is a rational number? Write 4 as a 5 decimal. You must be specific about three things! 0.8 State two rational numbers between 5.4 and 5.5 Express A as a mixed fraction. A 5 5.41 6 5 8/10 5.42 Write two equivalent fractions! 8 9 16/18 Oct 313:53 PM What is a rational number? Write 4 as a 5 decimal. State two rational numbers between 5.4 and 5.5 You must be specific about three things! Express A as a mixed fraction. A 5 6 Write two equivalent fractions! 8 9 Oct 313:53 PM 1 Chapter 3.notebook January 20, 2016 Rational Numbers any # that can be written as fraction any number including decimal that repeats → any # that ends Sep 1010:59 AM Express each fraction as a decimal, then sort as a repeating or terminating decimal. 5 9 6 27 27 33 8 5 20 10 18 12 Oct 315:25 PM 2 Chapter 3.notebook The numerator is LARGER than the denominator. 7 3 January 20, 2016 Improper vs. Mixed Fractions This is a Improper Fraction Mixed Fraction Integer + Fraction 2 31 Oct 315:27 PM Arrange the numbers from least to greatest. Change the numbers to decimals! 0.375, 0.555..., 2.5, 1.25, 0.7, 2.666... Least... 2.5 1.25 0.375 0.555... 0.7 2.666... ...Greatest Oct 2710:29 PM 3 Chapter 3.notebook January 20, 2016 Which rational number is larger?? (Decimals may be used on this side.) (NO Decimals please!!.) 13 16 12 15 2 3 0.8125 0.8 3 4 8 12 9 12 16 24 18 24 Oct 315:43 PM 1) Identify wether the number is rational or nonrational 2 3 1.234567..... 1.66 2.25 2)Express each fraction as a decimal a) 4 5 b) 9 c) 3 6 11 3)Express decimal as a fraction mixed fraction a) 0.1 b) 4.75 c) 3.222 Feb 27:51 PM 4 Chapter 3.notebook January 20, 2016 Sept. 14 3.2 Adding Rational Numbers Write each mixed number as an improper fraction: 1) 3 3 5 2) -5 5 6 Copy this DOWN and Hand this IN 3) Put the fractions in order from least to greatest , 12 , 45 , 11 15 2 32 ,1 20 Oct 299:43 AM Adding Fractions When adding fractions you need a COMMON DENOMINATOR: 5 6 + 1) 8 8 = 2) 8 + 4 7 7 = Oct 2910:22 AM 5 Chapter 3.notebook January 20, 2016 Determine the sum of each of the following 1) ( (= 3 + 3 7 7 2) a) 2.7 + 1.8 b) 3.7 + 4.5 c) 2.7 + (8.7) Nov 19:01 PM What happens if the denominators are different? Find a Common Denominator by determining the LCM. L owest C ommon M ultiple Oct 303:39 PM 6 Chapter 3.notebook January 20, 2016 Multiples Find a common denominator: 4 + 8 5 3 + = 5 3 5 x 1 = 5 5 x 2 = 10 5 x 3 = 15 5 x 4 = 20 LCM 3 x 1 = 3 3 x 2 = 6 3 x 3 = 9 3 x 4 = 12 3 x 5 = 15 Oct 303:43 PM What about mixed numbers? 1 3 3 5 2 + 2 Step 1: Write each mixed number as an improper fraction. + Step 2: Find a common denominator, and then add numerators. + Oct 304:02 PM 7 Chapter 3.notebook January 20, 2016 Practice! 1) 1 2 2 3 1 4 (-1 ) +(-2 ) 5 + (-3 ) 7 8 2) Oct 304:28 PM 1) Identify whether the number is rational or nonrational i) 3 4 3.286754.... 0.33 1.66 1.85 2)Express each fraction as a decimal(Show all work) a) 1 5 b) 2 15 c) 4 13 3)Express decimal as a mixed fraction in simplest form. (Show all work) a) 0.4 c) 1.125 b) 3.2 4) Determine each sum. (Show all work) a) 6 5 + 56 )) 2 5 b) 8 3 c) + 5 4 3 27 + 2 14 Oct 182:06 PM 8 Chapter 3.notebook January 20, 2016 4) On December 18th, the temperature in Miramichi was 21.6oC. By noon the next day, the temperature increased by 3.7 oC. a) What was the temperature at noon on December 19 th? b) On December 17th, the temperature was 2.1oC less than (colder than) that of December 18th. What was the temperature on the 17th? Sep 1210:18 PM When subtracting Rational Numbers you must have a ... Ex) This look similar to adding Rational Numbers = Same Denominators Oct 187:52 PM 9 Chapter 3.notebook January 20, 2016 You try ... (Remember to write all solution in simplest form) 1) 2) 3) Oct 188:48 PM When denominators are different you have to find a "common denominator" LCM How By determining the LCM Lowest Common Multiple (of the denominators) Oct 188:27 PM 10 Chapter 3.notebook January 20, 2016 You try... 2) 1 2 5 5 1) 2 + 2 3) 3 Oct 189:04 PM STEP 1) Write each mixed number as an inproper fraction STEP 2) Find common denominators and then subtract like before STEP 3) Reduce all fractions Oct 188:56 PM 11 Chapter 3.notebook January 20, 2016 Your Turn 2) 1) Oct 189:23 PM Multiplying Rational Numbers What rules do we use to multiply integers? Indicate if the answer will be negative or positive. How do you know? (4) x 3 = negative (3) x (6) = positive 2 x 8 = positive Nov 16:01 PM 12 Chapter 3.notebook January 20, 2016 Copy down When multiplying integers, we use the following rules: (a negative #) x (a positive #) = (a negative #) a negative # x a negative # = a positive # a positive # x a positive # = a positive # So, when the signs are opposite, the product is negative Copy down and when the signs are the same, the product is positive! Nov 18:10 PM Now, let's take a look at Fractions . What rules do we use to multiply fractions? Evaluate the following expression. How did you get your answer? = 6 x 8 5 x 7 = 48 35 When multiplying fractions, we use this rule: Multiply the numerator by the numerator then Multiply the denominator by the denominator ** Then, of course, REDUCE!! (if possible) Nov 16:01 PM 13 Chapter 3.notebook January 20, 2016 Try these out! Use what you know about multiplying integers & fractions to evaluate the following expressions. Don't forget to ALWAYS reduce if possible! (1.5) x (1.8) Nov 16:12 PM Jan 911:09 AM 14 Chapter 3.notebook January 20, 2016 Multiplying Rational Numbers in Fraction Form Determine the product: Look for common factors in the numerators and denominators. 11 and 44 have a common factor 11. 7 and 21 have a common factor 7. Divide numerator and denominator by their common factors. The signs are the same, so the product is positive! First, we simplify: 1 3 1 4 = Our rule for multiplying fractions is: numerator by numerator denominator by denominator So, our new expression, looks like this: 1 x 3 = 1 x 4 3 4 = Nov 16:27 PM Multiplying Rational Numbers in mixed number Form Determine the product. The signs are different, so the product is negative! Write the mixed numbers as improper fractions: = 8 _ 7 ( ) 3 ( ) 4 8 _ 7 ( ) 3 ( ) 4 2 = 1 = (2)(7) (3)(1) = 14 3 = 4 23 Nov 16:31 PM 15 Chapter 3.notebook January 20, 2016 9 • Every nonzero fraction has a reciprocal. • Fractions with a denominator of "0" are undefined. • To find the reciprocal of a fraction, you simply flip the fraction !! 10 16 Chapter 3.notebook January 20, 2016 Express each division question as a multiplication question !!!! 11 Dividend Quotient Divisor 12 17 Chapter 3.notebook January 20, 2016 Express division as multipication by multiplying the dividend by the reciprocal of the divisor !! = 13 math #1 14 18 Chapter 3.notebook January 20, 2016 #2 15 #3 16 19 Chapter 3.notebook January 20, 2016 Jan 911:12 AM Dividend Missing (__) ÷ 4 = 3 Think: Division is the inverse of Multiplication. What # goes in the blank? Any division statement can be written as an equivalent multiplication. OR (__) = 3 x 4 To Solve for Missing Dividend take Divisor X Quotient Now with Rational #s You Try A) 5 3 = ( ) ( ) 11 7 B) ____ ÷ 12.6 = 4.2 Nov 145:40 PM 20 Chapter 3.notebook January 20, 2016 Divisor Missing Decimals Think: Quotient is negative thus the BLANK must be what sign? ________ 15 ÷ (__) = 5 What # goes in the blank? To solve for missing Divisor take Dividend ÷ Quotient OR 15 = 5 x (__) Erase to see 15 = 5 x (__) 5 5 or 15 x 1 = (__) 5 Multiply by the reciprocal reciprocal of 5 is 1 5 3 = (__) You Try 1) 2.5 ÷ ___ = 5 2) 1.16 ÷ ___ = 0.2 What about fractions??? Nov 145:40 PM Divisor Missing &Fractions 18 6 ( ) ( ) = 49 7 ( ) ( ) ÷ 18 49 = 6 7 The Quotient is Positive Thus the divsor is _______ Divsor = Dividend ÷Quotient Use the strategy of multilpying by the reciprocal 6 49 ( ) = ( ) 7 x 18 Simplify 1 6 49 ( ) = ( ) 7 x 18 7 1 3 7 ( ) =( ) 3 Nov 147:01 PM 21 Chapter 3.notebook January 20, 2016 Your Turn 3 15 ( ) = ( ) 26 2 A) b) 12 5 ( ) = ( ) 21 8 Nov 147:54 PM Remember from operations "BEDMAS".....order of In the order that they appear Recall Evaluate the following 1) (5) 3[18 ÷ (3)]2 Oct 194:10 PM 22 Chapter 3.notebook January 20, 2016 Evaluate the following: It is no difference with decimals....follow BEDMAS With decimals you may need to round your final answers 1) (1.3) + 0.8 ÷ (0.2) x 5 2) (3.6) 1.7 ÷ [0.6 (0.8)] 2 = (1.3) + (4) x 5 = (3.6) 1.7 ÷ [1.4]2 = (3.6) 1.7 ÷ 1.96 = (1.3) + 20 = 21.3 (3.6) 0.867346938 =(3.6) 0.867346938 this number does not terminate =4.467346939 Oct 199:36 PM 1) 3.3 [(5) +1]3 2) (2.7) x (3.4 8.5) 6.1 ÷ 7 Oct 199:54 PM 23 Chapter 3.notebook = = = January 20, 2016 erase (0.8) + 4 x 1.5 (0.8) + 6 6.8 Nov 213:47 PM Remember fractions are just numbers erase to see solutions 1) Step 1) BRACKETS find common denominator then add the #s in brackets Common Denominator = 6 = Reduce sum if possible = Step 2) Multiply next = Step 3) Divide next = multiply by the reciprocal of the divisor. = x reduce at this point to work with smaller fractions = Step 4) Subtract.....find common denominator = 12 47 = Oct 1910:02 PM 24 Chapter 3.notebook January 20, 2016 Erase to reveal answer 1) Remember to switch mixed to improper fractions Make common denominators inside brackets Complete Brackets Multiply Divide Oct 1910:38 PM 4. Please erase to reveal answer AS BEDM 4 12 = [ 112 [ = = ( 14 ( x 12 = 1 4 = 1 = ( 8 ( 1 = 32 ( ( 4 = 31 4 Nov 214:06 PM 25 Chapter 3.notebook January 20, 2016 Page 144 & 145 If the question deals with fractions you must work with fractions (no calculator) As soon as you see a decimal you can use a calculator #2 (without calculator) # 3(c,d) #5 (a,c) #7(a,b,c) (without calculator) #10(b,c) (without calculator) #14 (b, d) #18(ac) #19(b,d) #21 (without calculator) #23a,c,d,g 26