decimals

decimals - Fifth Grade Fifth Math Course I Math Fractions...

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Unformatted text preview: Fifth Grade Fifth Math Course I Math Fractions and Decimals 1 Fractions What are fractions? They are not whole numbers They since they represent a part or a portion of a whole. portion What part of the figure below is What shaded? shaded? One out of the three equal One sections is shaded. So we write 1/3 (one-third) of the whole figure is shaded figure 2 Fractions Every fraction has three parts. Every The top number is the numerator, the bottom number is numerator the the denominator, and the middle denominator and line is the fraction line. fraction 5 NUMERATOR (tells how many parts you have) NUMERATOR 7 DENOMINATOR (tells the total number of parts) DENOMINATOR Fraction line 3 Fractions What part of the What figure is shaded? shaded? 4 Proper Fractions Proper fractions have a value Proper that is less than 1, the numerator is less than the denominator denominator 3 5 7 11 101 301 9 19 5 Improper Fractions Fractions that have a Fractions value that is equal to 1 or great than 1. or Four out of four Four sections are shaded. 4/4 (four-fourths) is shaded. shaded. This represents the This whole object. So 4 =1 4 8 29 =1 6 =1 8 29 Improper Fractions A 4 4 of figure A is shaded and 3 4 of Figure B is shaded. So a total of 7 is shaded. 4 7 So, = 1 3 4 4 We have one We whole object and 3 of 4 another that is shaded shaded 7 B Fractions What part of the figure below is What shaded? shaded? What part of the figure is not What shaded? shaded? 8 Fractions Which of the following fractions Which are proper fractions? are Which of the following fractions Which are improper fractions? are 12 , 7 , 7 8 3, 4 9 , 15 , 10 6 16 3 9 Fractions If, on a true/false test, you get 17 correct If, answers out of 20 questions, a) what fraction did you get correct? b) what fraction did you get wrong? fraction If, in a class of 30 students, 17 are If, females, a) what fraction are females? b) what fraction are males? what If 37 out of 100 golf balls found in a lake on If a golf course are orange, what fraction of the golf balls are not orange? the 10 Equivalent Fractions 3 Of the block is shaded Of 4 6 Of the block is shaded Of 8 The shaded sections represent the same amount of The 3 6 each block. So those two fractions, 4 and 8 , represent the same amount of shaded area. They have the same value. have 6 3 and 6 are equivalent. 3 = 4 8 4 8 3 6 You can change 4 to 8 by multiplying its numerator and denominator by 2. You can change 6 to 3 by 8 4 dividing its numerator and denominator by 1 Both 2. 1 procedures yield fractions that have the same value. procedures Equivalent Fractions By multiplying the numerator and By denominator of a fraction by the same number, you raise the fraction to higher terms, getting an equivalent fraction. fraction. x2 3 4 = x2 x 10 x3 6 8 1 3 = 3 9 5 8 x3 = x 10 12 50 80 Equivalent Fractions 3 8 Change to a fraction with a Change numerator of 12. numerator 3 Change 2 to a fraction with a Change denominator of 18. denominator 5 = 35 11 = 33 6 ? 13 ? 4 7 = 16 1= 6 ? 54 2 9 =? 3= 8 24 13? ? 45 Reducing Fractions To reduce a fraction to lower To terms, you must find a number terms you other than 1 that divides evenly into both the numerator and denominator of the fraction. You must continue to divide both the numerator and denominator until no whole number except 1 divides evenly into each. divides 14 Reducing Fractions Reduce ÷2 30 = 42 30 42 15 21 ÷2 2 divides evenly into both 30 and 42 30 42 to lowest terms ÷3 15 = 21 5 7 ÷3 3 divides evenly into both 15 and 21 has been reduced to it’s lowest terms 5 (prime numbers) lowest 7 15 Reducing Fractions Reduce 12 24 ÷3 12 = 24 ÷3 to lowest terms ÷2 4 8 = ÷2 ÷2 2 4 = 1 2 ÷2 16 Greatest Common Factor When reducing fractions, you When must find the greatest common factor factor Ex: GCF of 12 and 30 12 1 x 12 2 x 6 1, 2, 3, 4, 6 and 12 are factors of 12 3x4 30 1 x 30 2 x 15 1, 2, 3, 5, 6, 10, 15, and 30 are factors 1, 3 x 10 of 30 of 5x6 17 Greatest Common Factor Find the Greatest Common Factor 6 and 10 5 and 8 12 and 15 10 and 20 To reduce a fraction to its simplest To form, divide the numerator and denominator by the Greatest Common Factor Common 12 1, 2, 3, 4, 6, 12 12 ÷ 3 = 4 15 1, 3, 5, 15 15 ÷ 3 18 5 Reducing Fractions Reduce to lowest terms 140 = 9= 910 12 100 150 36 90 = 18 30 = = 12 36 = 19 Mixed Numbers Improper fractions have a value that is greater than or equal to 1. Every improper fraction can be expressed as either a fraction whole number or as the sum of a whole number and a proper fraction (mixed number mixed number) number The fraction line not only separates the The numerator and denominator, it also implies a division – the numerator divided by the denominator denominator Some improper fractions give whole Some number results number 4 = 4 4 =1 15 = 3 15 = 5 4 3 20 Mixed Numbers 7 4 Some improper fractions give Some mixed number results mixed =4 7 19 = 5 19 5 3 = 1 R3 = 1 4 4 = 3 R4 = 3 5 Since we are dividing Since by 4, the remainder is 3 3 out of 4, or 4 . So the mixed number is 3 1 and 4 Since we are dividing Since by 5, the remainder is 4 out of 5, or 4 . So 5 the mixed number is 4 3 and 5 21 Mixed Numbers Simplify 18 3 = 39 7= 29 = 7 22 Mixed Numbers & Improper Mixed Fractions Fractions You can change a whole You number to an improper fraction by putting the whole number over a denominator of 1 15 = 15 1 100 = 100 1 7= 7 1 82 1 82 = 23 Converting mixed numbers Converting into improper fractions into Consider how we changed Consider improper fractions into mixed numbers numbers 7 4 =4 7 3 = 1 R3 = 1 4 If you have a mixed number, If you can change it to an improper fraction improper + 13 x4 = 4x1+3 4 = 7 4 24 Converting mixed numbers to Converting improper fractions improper Change the following mixed numbers to Change improper fractions improper 2 6 7 = 1 140 2 = 3= 54 1= 10 6 2= 122 3 25 Multiplying Fractions The easiest operation with The fractions is multiplication, since to multiply fractions you just have to multiply their numerators and multiply their denominators to get the numerator and the denominator of the answer. of 1x3 2 5 = 1x3 = 3 2x5 10 26 Multiplying Fractions 6 x 42 = 6 x 42 = 252 15 63 15 x 63 945 Now we must reduce the answer ÷3 ÷3 ÷7 252 945 = ÷3 84 315 = ÷3 28 105 = 4 15 ÷7 The reducing process in that example was quite The involved since the numerator and denominator of the fraction were large numbers. However, if we reduced the fractions before we multiplied, the solution would be a little easier. solution 27 Multiplying Fractions 6 x 42 15 63 6 15 2 5 and 6 x 42 15 63 = Reduces to Therefore 42 63 reduces to 2x 5 2 3 = 3 x 14 = 3 x 14 = 42 70 25 70 x 25 1750 Reduce the answer answer ÷2 42 1750 = ÷2 ÷7 21 875 = ÷7 2 3 3 125 28 4 15 Cross Canceling in Fractions 3 70 14 25 The original fractions and are not The reducible. However, the product 42 can 750 be reduced to lower terms. What if1we could reduce the fractions before we before multiplied? We can reduce the numerator of the 2nd fraction with the denominator of of the 1st fraction. the 1 3 x 14 = 3 x 1 70 25 5 x 25 5 = 3 125 This process is called cross canceling and This can ONLY be used when multiplying fractions. You must divide the same number evenly into any numerator and any denominator. denominator. 29 Cross Canceling Ex: 2 x 18 x 15 = ? 10 26 2 18 9 15 x x 1 10 26 Change 2 to the fraction 2 and reduce 1 18 to 9 10 5 1 2 x9 1 5 1 x 15 26 5 3 13 19 x x 3 = 27 11 13 13 27 = 2 1 13 13 Cross cancel the 2 and the 26; cross Cancel the 5 and the 15. Multiply the remaining fractions and Change the improper fraction to a mixed number 30 Multiplying Fractions 3 4 12 15 x x 20 7 = 8 25 x 75 100 = 31 Dividing Fractions To divide fractions, invert (find find the reciprocal of) the divisor (2nd the fraction) and change the division into multiplication. into 3÷2 5 1 = 3x 5 1 2 32 Dividing Fractions 2÷5 3 7 = 10 ÷ 7 = 15 27 35 ÷ 28 = 6 10 ÷ 45 = 3 33 Multiplying and Dividing Mixed Multiplying Numbers Numbers To multiply and divide mixed To numbers, first change them into improper fractions and then multiply. multiply. Ex: 11 x 5 2 = ? 9 11 x 47 = 517 57 4 1 9= 9 9 33 Ex: Ex: 4 5 15 x 16 4 3 1 4 1 x 51 =? (Cross cancel) 3 5 x 4 = 20 = 20 =1 34 1 1 Multiplying and Dividing Mixed Multiplying Numbers Numbers Solve the problems: 72 3 x 10 1 = ? 2 8 x 25 =? 16 10 2 ÷ 6 2 15 5 =? 95 6 =? ÷ 12 35 Fractional Parts of Numbers Fractions are often used to find a part of a number Fractions or quantity. You might encounter phrases such as 4 “ 5 of those surveyed”, “ 1 off the retail price”, or 3 “ 3 of the class.” In statements like these, you are 4 finding a part of a total amount. The word “of” in this usage indicates multiplying the fraction and the number. number. 4 of 4725 means of 5 4 5 1 of 3 of $630 means 3 of 4 of 36 means 1 3 3 4 x 4725 x 630 x 36 36 Fractional Parts of Numbers Solve these problems: 2 3 5 16 of 951 = ? of 9 = ? Three fifths of those surveyed preferred “Colgate” Three toothpaste. If 100,000 people were surveyed, how many preferred “Colgate” toothpaste? many 1 A store is advertising a discount of 4 off all retail store prices. What would be the cost of a $304 item after such a discount? such 37 Adding and Subtracting Like Adding Fractions Fractions “Like” fractions are fractions with Like” the same denominator the 2 1 3 Ex: 5 + 5 = 5 Ex: Adding and subtracting fractions Adding is straightforward. Leave the denominator the same and add or subtract the numerators. or 38 Adding and Subtracting Like Adding Fractions Fractions Solve these problems 2+3 7 7 +1 7 =? 5-2=? 6 6 5+2 8 8 +3 8 =? 63 - 11 + 8 = ? 25 25 25 39 Finding the Least Common Finding Denominator Denominator Before you can add or subtract Before “unlike” fractions (different denominators), you must first find the Least Common Denominator (LCD) Least The Least Common Denominator is The Least the smallest whole number that is evenly divisible by the denominators of other fractions of 40 Finding the Least Common Finding Denominator Denominator To find the LCD, find the To multiples of the numbers in the denominator and choose the lowest common number lowest Ex: Ex: 3+5 8 6 Multiples =? of: 6: 6, 12, 18, 24, 30, 36, etc 6: 24 8: 8, 16, 24, 32, 40, 48, etc 8: 24 41 Adding and Subtracting Unlike Adding Fractions Fractions 3+1 4 5 4, 8, 12, 16, 20, 24, 28, … =? 5, 10, 15, 20, 25, 30, 35, … The LCD is 20, since 20 is the The smallest number that is divisible by both 4 and 5. We have to change 3 and 1 into 20ths 4 5 before we can add them. before 3=? 4 20 x5 1= ? 5 20 x4 15 20 4 20 add 19 20 42 Adding and Subtracting Unlike Adding Fractions Fractions 2-1=? 3 2 1+2 2 5 +9=? 10 3+5 4 6 +3 2 - 2 3 =? 43 Adding and Subtracting Unlike Adding Fractions Fractions Find the Least Common Find Denominator. Denominator. Change each fraction to a Change fraction with the LCD as its denominator. denominator. Add or subtract those like Add fractions. fractions. Reduce your answer and Reduce change any improper fraction into a mixed number. into 44 Adding and Subtracting Mixed Adding Numbers Numbers 32 7 + 18 = ? 21 Change the mixed numbers to Change improper fractions improper The LCD is 21. Change each The fraction to a denominator of 32. fraction Add the like fractions, simplify Add and reduce. and 3 2 = 23 = 7 7 69 21 + 1 8 = 29 = 21 21 29 21 add 98 = 4 14 42 21 21 = 3 45 Adding and Subtracting Mixed Adding Numbers Numbers 21 12 +32 3 - 34 8 5 =? =? 46 ten t hs hun tho dredth usa s ndt Ten -th hs Hu ousa ndr n ed- dt hs t mil lion housa ths ndt hs Hu n Ten dred Bill -billio billion ion ns s s Hu n Ten dred Mil -milli millio lion ons ns Hu s n Ten dred Th -thou thous ous sa an and nds ds s Hu n Ten dred s On s es Reading and Writing Decimals 678, 310, 456, 127 . 3 4 1 0 1 9 (and) Any digit to the right of the ones place has Any a value that is less than one – a fraction value For example, in 0.576, For 0.576 5 The 5 means 5 tenths 10 The 7 means 7 hundredths 7 100 The 6 means 6 thousandths 6 47 1000 Reading and Writing Decimals Numbers to the right of the decimal point Numbers are read like whole numbers and are give a value according to the position of its last digit. digit. tenths hundredths thousandths 7 10 0.7 is read seven tenths 0.2 3 is read twenty-three hundredths 23 100 0.1 7 7 is read one hundred seventy-seven thousandths 1 48 77 1000 Reading and Writing Decimals If we have numbers to the right and left of If the decimal point, we really have a mixed number. number. Ex: 5.3 is read “five and three tenths” Ex: 16.08 is read sixteen and eight Ex: hundredths hundredths 0.17 – write in words 426.9 – write in words Seven tenths = ? Forty-five and six thousandths = ? Twelve hundredths = ? 49 Adding Decimals Place the numbers in a column so that their decimal Place points are line up. points Add the digits that have the same place value. Place the decimal point in the answer below the Place other decimal points. other Ex: To add 5.3 + .076 + 12.21 5.3 5.3 .076 .076 +12.21 17.586 17.586 If any of the numbers you are trying to add are If whole numbers without a decimal point, you can simply place a decimal point at the end of the whole number number Ex: 26 = 26. 5 = 5. 5. 50 Adding Decimals 52 + 3.01 + 0.035 + 1.58 = ? 307.52 + 136 + .65 + 28 + 1.17 + 0.2 = ? 0.01 + 0.002 + 0.0003 + 0.00004 = ? 51 Subtracting Decimals Line up the decimal points before subtraction 27.973 – 2.241 = ? 27.973 - 2.241 2.241 25.732 25.732 Subtraction is easier when both numbers have the Subtraction same number of decimal digits. We sometimes have to give an alternate representation for a decimal number. decimal Ex: 0.7 can be written in different ways 0.70 0.70 0.700 0.700 0.7000 0.7000 Decimal points added after the decimal point do not Decimal change the value of the decimal number change 52 Subtracting Decimals .5762 – .34 = ? .5762 .5762 - .3400 .3400 .2362 .2362 56.4 – 3.27 = ? 189 – 13.567 = ? 53 Multiplying Decimals To understand how to multiply To decimal numbers, we must look at the fractions which the decimal numbers represent. numbers Ex: 0.3 x 0.71 = ? 3 and 0.71 = 71 0.3 = 100 10 3 So, 0.3 x 0.71 = 10 x So, 71 100 213 = = 0.213 0.213 1000 The number of decimal digits in the The product is the total of the number of decimal digits in the numbers being 54 multiplied. multiplied. Multiplying Decimals 19.86 x 0.089 = ? 1 9.86 9.86 x 0.089 17874 17874 15888 1.76754 has 2 decimal digits has 3 decimal digits The answer has a total of The 5 decimal digits decimal To place the decimal in the answer, start To from the right and move 5 places to the left left 43.6 x 2.7 = ? 6.075 x 3.14 = ? 55 Dividing Decimals If the divisor is a whole number, the decimal point is If placed in the answer above its position in the dividend and the division is done ignoring the decimal point decimal 52.8 ÷ 4 = ? 52.8 13.2 13.2 4 52.8 Place the decimal point above the decimal point in the dividend. If the divisor is not a whole number, you can make it If a whole number by moving its decimal point to the right. You must also move the decimal point the same number of places to the right in the dividend. same 2.38 ÷ .7 = ? 2.38 .7 3.4 3.4 .7 2 . 3 8 1. 1. 2. 3. 4. Move the decimal point to change the Move divisor into a whole number divisor Move the point the same in the dividend Place the decimal point in the answer Place above the moved decimal point above Do the division ignoring the decimal Do 56 points points Dividing Decimals 32.36 ÷ .04 = ? 32.36 .04 82.2 ÷ 6 = ? 82.2 57 Converting Fractions and Converting Decimals Decimals To convert a fraction to a To decimal, remember that the fraction line indicates division – divide the numerator by the denominator. denominator. 3 =3÷4 4 .75 .75 4 3.00 3.00 58 Converting Decimals to Converting Fractions Fractions To convert a decimal number to a To fraction, use the place value of the digits to the right of the decimal point – tenths, hundredths, thousandths, ten-thousandths, etc ten-thousandths, 9 Ex: 0.9 reads nine tenths 10 Ex: 0.35 reads thirty-five Ex: hundredths 35 hundredths 100 2 Ex: 6.2 reads 6 and two tenths 6 Ex: 10 59 Converting Decimals and Converting Fractions Fractions Write the fraction as a decimal 3 5 7 10 5 8 Write as reduced fractions or Write mixed numbers mixed 0.32 5.375 76.8 60 ...
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This note was uploaded on 11/11/2011 for the course MATH 110 taught by Professor Staff during the Winter '08 term at BYU.

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