Pre-Algebra Notes Unit Five: Rational Numbers and Equations

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1 Pre-Algebra Notes Unit Five: Rational Numbers and Equations Rational Numbers Rational numbers are numbers that can be written as a quotient of two integers. Since decimals are special fractions, all the rules we learn for fractions will work for decimals. The only difference is the denominators for decimals are powers of 0; i.e., 0, 0, 0, 0, etc... Students normally think of powers of 0 in standard form: 0, 00, 000, 0,000, etc. In a decimal, the numerator is the number to the right of the decimal point. The denominator is not written, but is implied by the number of digits to the right of the decimal point. The number of digits to the right of the decimal point is the same as the number of zeros in the power of 0: 0, 00, 000, 0,000 Therefore, one place is tenths, two places are hundredths, and three places are thousandths. Examples: ) 0.56 places ) 0.5 places ). place The correct way to say a decimal numeral is to: ) Forget the decimal point (if it is less than one). ) Say the number. ) Then say its denominator and add the suffix ths. Examples: ) 0.5 Fifty-three hundredths ) 0.70 Seven hundred two thousandths ) 0. Two tenths ) 5.6 Five and sixty-three hundredths When there are numbers on both sides of the decimal point, the decimal point is read as and. You say the number on the left side of the decimal point, and then the decimal point is read as and. You then say the number on the right side with its denominator. McDougal Littell, Chapter 5 Pre-Algebra, Unit 05: Rational Numbers and Equations Page of

2 Examples: ) Write 5.0 in word form. Fifteen and two hundred three thousandths ) Write 7.08 in word form. Seven and four hundred eighty-three ten-thousandths ) Write 7.5 in word form. Two hundred forty-seven and forty-five hundredths Converting Fractions to Decimals: Terminating and Repeating Decimals Syllabus Objective: (.7) The student will write equivalent representations of fractions, decimals, and percents. A rational number, a number that can be written in the form of a (quotient of two integers), will b either be a terminating or repeating decimal. One way to convert fractions to decimals is by making equivalent fractions. Convert to a decimal. Since a decimal is a fraction whose denominator is a power of 0, look for a power of 0 that will divide into evenly. 5 0 Since the denominator is 0, we need only one digit to the right of the decimal point, and the answer is 0.5. Convert to a decimal. Again, since a decimal is a fraction whose denominator is a power of 0, we look for powers of 0 that the denominator will divide into evenly. won t go into 0, but will go into 00 evenly Since the denominator is 00, we need two digits to the right of the decimal point, and the answer is McDougal Littell, Chapter 5 Pre-Algebra, Unit 05: Rational Numbers and Equations Page of

3 There are denominators that will never divide into any power of 0 evenly. Since that happens, we look for an alternative way of converting fractions to decimals. Could you recognize numbers that are not factors of powers of ten? Using your Rules of Divisibility, factors of powers of ten can only have prime factors of or 5. That would mean, whose prime factors are and, would not be a factor of a power of ten. That means that will never divide into a power of 0 evenly. For example, a fraction such as 5 will not terminate it will be a repeating decimal. Not all fractions can be written with a power of 0 as the denominator. We need to look at another way to convert a fraction to a decimal: divide the numerator by the denominator. Convert 8 to a decimal. This could be done by equivalent fractions since the only prime factor of is However, it could also be done by division Doing this division problem, we get 0.75 as the equivalent decimal. Convert 5 to a decimal. This could not be done by equivalent fractions since one of the factors of is. We can still convert it to a decimal by division Six is repeating, so we can write it as 06.. The vinculum is written over the digit or digits that repeat. Convert to a decimal. This would be done by division or 0. 6 McDougal Littell, Chapter 5 Pre-Algebra, Unit 05: Rational Numbers and Equations Page of

4 Converting Decimals to Fractions Syllabus Objective: (.7) The student will write equivalent representations of fractions, decimals, and percents. To convert a decimal to a fraction: ) Determine the denominator by counting the number of digits to the right of the decimal point. ) The numerator is the number to the right of the decimal point. ) Simplify, if possible. Examples: ) Convert 0.5 to a fraction = 5 ) Convert 0.6 to a fraction But what if we have a repeating decimal? ) Convert 8. to a mixed number and improper fraction or 5 5 While the decimals 0. and 0. look alike at first glance, they are different. They do not have the same value. We know 0. is three tenths,. How can we say or write 0. as a fraction? 0 As we often do in math, we take something we don t recognize and make it look like a problem we have done before. To do this, we eliminate the repeating part the vinculum (line over the ). Convert 0. to a fraction McDougal Littell, Chapter 5 Pre-Algebra, Unit 05: Rational Numbers and Equations Page of

5 Let s let x = 0... Notice, and this is important, that only one number is repeating. If I multiply both sides of the equation above by 0 (one zero), then subtract the two equations, the repeating part disappears. 0x. x 0. 9x x x is the equivalent fraction for 0. Convert 0.5 to a fraction. The difficulty with this problem is the decimal is repeating. So we eliminate the repeating part by letting x Note, three digits are repeating. By multiplying both sides of the equation by 000 (three zeros), the repeating parts line up. When we subtract, the repeating part disappears. 000x x x x or 999 Convert 0. to a fraction. Note, one digit is repeating, but one is not. By multiplying both sides of the equation by 0, the repeating parts line up. When we subtract, the repeating part disappears. McDougal Littell, Chapter 5 Pre-Algebra, Unit 05: Rational Numbers and Equations Page 5 of

6 0x. x. 9x. 9x x or which simplifies to Ready for a short cut? Let s look at some patterns for repeating decimals. 0. or or or? 9? or or or?? It is easy to generate the missing decimals when you see the pattern! Let s continue to look at a few more repeating decimals, converting back into fractional form. Because we are concentrating on the pattern, we will choose NOT to simplify fractions where applicable. This would be a step to add later ? 07. 9? 08.? ? ? 056.? ? ? 0. 56? The numerator of the fraction is the same numeral as the numeral under the vinculum. We can also quickly determine the denominator: it is 9 ths for one place under the vinculum, 99 ths for two places under the vinculum, 999 ths for three places under the vinculum, and so on. McDougal Littell, Chapter 5 Pre-Algebra, Unit 05: Rational Numbers and Equations Page 6 of

7 But what if the decimal is of a form where not all the numerals are under the vinculum? Let s look at a few The numerator is generated by subtracting the number not under the vinculum from the entire number (including the digits under the vinculum). We still determine the number of nines in the denominator by looking at the number of digits under the vinculum. The number of digits not under the vinculum gives us the number of zeroes Note that again we chose not to simplify fractions where applicable as we want to concentrate on the pattern Comparing and Ordering Rational Numbers Syllabus Objectives: (.6) The student will compare and order rational numbers. (.7) The student will write equivalent representations of fractions, decimals, and percents. (.8) The student will explain the relationship among equivalent representations of rational numbers. We will now have fractions, mixed numbers and decimals in ordering problems. Sometimes you can simply think of (or draw) a number line and place the numbers on the line. Numbers increase as you go from left to right on the number line, so this is particularly helpful when you are asked to go from least to greatest. If placement is not obvious (for instance, when values are very close together), it may be advantageous to write all the number in the same form (decimal or fractional equivalents), and then compare. Order the numbers 5 5, 0.,.,,, from least to greatest. Let s first rewrite all improper fractions as mixed numbers. 5 5 ; ; McDougal Littell, Chapter 5 Pre-Algebra, Unit 05: Rational Numbers and Equations Page 7 of

8 Now let s place the values on the number line From least to greatest, the order would be 5,,, 0.,,.. Sometimes writing the numbers in the same form will assist you in ordering. Order 7, 05.,, 5,. from least to greatest. 8 Find the decimal equivalents, then compare Lining up the decimals, the order from least to greatest is: Using the original forms: 7 05.,,,,. 8 Or find the fractional equivalents, then compare Having found a common denominator, the order from least to greatest is: ,,,, Using the original forms: 7 05.,,,,. 8 McDougal Littell, Chapter 5 Pre-Algebra, Unit 05: Rational Numbers and Equations Page 8 of

9 Adding and Subtracting Fractions with Like Denominators Let s add to. Will it be 8? Why not? If we did, the fraction would indicate that we 8 have two equal size pieces and that 8 of these equal size pieces made one whole unit. That s not true. Let s draw a picture to represent this: Notice the pieces are the same size. That will allow us to add the pieces together. Each rectangle has equally sized pieces. Mathematically, we say that is the common denominator. Now let s count the number of shaded pieces. Adding the numerators, a total of equally sized pieces are shaded and pieces make one unit. We can now show: or Find the sum of. 9 9 Since the fractions have the same denominator, we write the sum over McDougal Littell, Chapter 5 Pre-Algebra, Unit 05: Rational Numbers and Equations Page 9 of

10 Find the difference of and. 5 5 Since the fractions have the same denominator, we write the difference over Writing these problems with variables does not change the strategy. Simplify the variable expression. 5 x x 5 x x 5 x 7 x x Adding and Subtracting Fractions with Unlike Denominators Let s first review the ways to find a common denominator. We find the least common denominator by determining the least common multiple. Strategy : Multiply the numbers. This is a quick, easy method to use when the numbers are relatively prime (have no factors in common). Find the LCM of and 5. Since and 5 are relatively prime, LCM would be 5 or 0. Strategy : List the multiples. Write multiples of each number until there is a common multiple. Find the LCM of and 6.,, 6, 8, 60, 6,, 8, 6, 8 is the smallest multiple of both numbers; therefore, 8 is the LCM. Strategy : Prime factorization. Write the prime factorization of both numbers. The LCM must contain all the factors of both numbers. Write all prime factors, using the highest exponent. Find the LCM of 60 and and 7 The LCM is 5 60 McDougal Littell, Chapter 5 Pre-Algebra, Unit 05: Rational Numbers and Equations Page 0 of

11 This strategy can also be shown by using a Venn diagram. Find the LCM of 6 and 5. Draw a Venn diagram, placing common factors in the intersection. The LCM is the product of all the factors in the diagram. Factors of 6 Factors of 5 5 Multiply all factors in diagram for the LCM: As the numbers in the denominator become larger, this strategy can become cumbersome. That is when the value of the following strategy becomes evident. Strategy : Simplifying/Reducing Method. Write the two numbers as a single fraction; then reduce and find the cross products. The product is the LCM. Find the LCM of 8 and. 8 ;cross products are 8 or 7. The LCM is 7. When adding or subtracting fractions, LCM is referred to as the Least Common Denominator (LCD). We have several ways to find a common denominator. Methods of Finding a Common Denominator. Multiply the denominators.. List multiples of each denominator, use a common multiple.. Find the prime factorization of the denominators, and find the Least Common Multiple.. Use the Simplifying/Reducing Method. Use this method when. the denominators are prime numbers or relatively prime.. the denominators are small numbers.. the denominators are small numbers; some will advise to never or seldom use this method.. the denominators are composite numbers/ large numbers. McDougal Littell, Chapter 5 Pre-Algebra, Unit 05: Rational Numbers and Equations Page of

12 Let s add to. Will it be 7? Why not? If we did, the fraction would indicate that we 7 have two equal size pieces and that 7 of these equal size pieces made one whole unit. That s just not true. Let s draw a picture to represent this: Notice the pieces are not the same size. Making the same cuts in each rectangle will result in equally sized pieces. That will allow us to add the pieces together. Each rectangle now has equally sized pieces. Mathematically, we say that is the common denominator. Now let s count the number of shaded pieces. 7 From the drawing we can see that is the same as and has the same value as. Adding the numerators, a total of 7 equally sized pieces are shaded and pieces make one unit. If we did a number of these problems, we would be able to find a way of adding and subtracting fractions without drawing the picture. Algorithm for Adding/Subtracting Fractions. Find a common denominator.. Make equivalent fractions.. Add/Subtract the numerators.. Simplify (reduce), if possible. McDougal Littell, Chapter 5 Pre-Algebra, Unit 05: Rational Numbers and Equations Page of

13 Using the algorithm, let s try one. 5 Multiply the denominators to find the least common denominator, 5 5. Now make equivalent fractions and add the numerators These problems can also be written horizontally Let s try a few. Using the algorithm, first find the common denominator, and then make equal fractions. Once you complete that, you add the numerators and place that result over the common denominator and simplify, if possible. Remember, the reason you are finding a common denominator is so you have equally sized pieces. To find a common denominator, use one of the strategies shown. Since the denominators are relatively prime, use the multiply the denominators method McDougal Littell, Chapter 5 Pre-Algebra, Unit 05: Rational Numbers and Equations Page of

14 5 8 To find the common denominator, use the Simplifying/Reducing Method, 8 LCD ; Writing these problems with variables does not change the strategy. d d Simplify the expression. 5 The LCD is 5. Making equivalent fractions, we have: d 5d 5 d 6d 5 5 d 5 It is customary to write these problems in a horizontal format like this d d 5d 6d d = 5 If the denominators are larger composite numbers, using the reducing method to find the common denominator may make the work easier. Simplify the expression. 5c 7c 8 McDougal Littell, Chapter 5 Pre-Algebra, Unit 05: Rational Numbers and Equations Page of

15 Using the Simplifying/Reducing method: 8, 8 7, so the LCD is 7. 5c 0c 8 7 7c c 7 c 7 or 5c 7c 0c c c = 7 Another nice feature of using the Simplifying/Reducing Method is that you do not need to compute what 8 7 or 7 because we can see the number in the cross products. That is, we can identify 8 times is 7, so we multiply 5c by to obtain the new numerator ( 5c 0c). Likewise, since times is 7, we determine the other numerator as 7c c. Evaluate the expression. x 7x x x 8x 5 0 7x x 0 0 x x 5 0 5x 7x or 0 6 or x 7x x x x x 5x 7x or McDougal Littell, Chapter 5 Pre-Algebra, Unit 05: Rational Numbers and Equations Page 5 of

16 Multiplying Fractions and Mixed Numbers Multiplying fractions is pretty straight-forward. So, we ll just write the algorithm for it, give an example and move on. Algorithm for Multiplying Fractions and Mixed Numbers. Make sure you have proper or improper fractions.. Cancel, if possible.. Multiply numerators.. Multiply denominators. 5. Simplify (reduce), if possible. 5 Since is not a fraction, we convert it to can be written as 7 5 Now what I m about to say is important and will make your life a lot easier. We know how to reduce fractions, so what we want to do now is cancel with fractions. That s nothing more than reducing using the commutative and associative properties. Using the commutative property, we can rewrite this as Using the associative property, we can rewrite this as Simplify. Then multiply and simplify, as a mixed number Rather than going through all those steps, we could take a shortcut and cancel. McDougal Littell, Chapter 5 Pre-Algebra, Unit 05: Rational Numbers and Equations Page 6 of

17 Now rather than going through all those steps, using the commutative and associative properties, we could have taken a shortcut and cancelled. 7 5 To cancel, we would look for common factors in the numerator and the denominator and divide them out. In our problem, there is a common factor of. By dividing out a, the problem looks like this: Let s look at another one. 7 or Rewrite as improper fractions Cancel 8 and 9 by common factor of Cancel 0 and 5 by common factor of Multiply numerators, multiply denominators, simplify. When variables are added to these problems, the strategy remains the same. Simplify the expression. n n 7 n n n 7 Cancel the and by common factor of. Multiply the numerators and denominators. McDougal Littell, Chapter 5 Pre-Algebra, Unit 05: Rational Numbers and Equations Page 7 of

18 Dividing Fractions and Mixed Numbers Before we learn how to divide fractions, let s revisit the concept of division using whole numbers. When I ask, how many s are there in 8, I can write that mathematically three ways To find out how many s there are in 8, we will use the subtraction model: 8 6 Mathematically, we say 8 =. 0 Now, how many times did we subtract? Count them: there are subtractions. So there are twos in eight. The good news is, division has been defined as repeated subtraction That won t change because we are using a different number set. In other words, to divide fractions, I could also do repeated subtraction. Another way to look at this problem is using your experiences with money. How many quarters are there in $.50? Using repeated subtraction we have: 0 How many times did we subtract? Six. Therefore, 6. But this took a lot of time and space. McDougal Littell, Chapter 5 Pre-Algebra, Unit 05: Rational Numbers and Equations Page 8 of

19 A visual representation of division of fractions would look like the following. 8 We have. Representing that would be Since the question we need to answer is how many ' 8 s are there in, we need to cut this entire diagram into eighths. Then count each of the shaded one-eighths. As you can see there are four. So We have 5. Representing that would be 6 Since the question we need to answer is how many ' s are there in 5, we need to use the cuts 6 for thirds only. Then count each of the one-thirds. As you can see that are. So 5. 6 McDougal Littell, Chapter 5 Pre-Algebra, Unit 05: Rational Numbers and Equations Page 9 of

20 Be careful to choose division examples that are easy to represent in visual form. Well, because some enjoy playing with numbers, they found a quick way of dividing fractions. They did this by looking at fractions that were to be divided and they noticed a pattern. And here is what they noticed. Algorithm for Dividing Fractions and Mixed Numbers. Make sure you have proper or improper fractions.. Invert the divisor ( nd number).. Cancel, if possible.. Multiply numerators. 5. Multiply denominators. 6. Simplify (reduce), if possible. The very simple reason we tip the divisor upside-down, then multiply for division of fractions is because it works. And it works faster than if we did repeated subtractions, not to mention it takes less time and less space. 5 5 (Invert the divisor.) Multiply numerators and denominators, and simplify. 9 0 Make sure you have proper or improper fractions Invert the divisor Cancel 0 and by, and cancel 9 and by Multiply numerators and denominators. Simplify. McDougal Littell, Chapter 5 Pre-Algebra, Unit 05: Rational Numbers and Equations Page 0 of

21 Computing with Fractions and Signed Numbers Syllabus Objective: (.9) The student will perform operations with positive and negative fractions and mixed numbers. The rules for adding, subtracting, multiplying and dividing fractions with signed numbers are the same as before, the only difference is you integrate the rules for integers. 7 Invert divisor Multiply numerators and denominators, and simplify Solving Equations and Inequalities Containing Fractions and Decimals Syllabus Objective: (.0) The student will solve equations and inequalities involving fractions and mixed numbers. (.8) The student will solve multi-step inequalities. First Strategy for Solving: You solve equations and inequalities containing fractions and decimals the same as you do with whole numbers; the strategy does not change. To solve linear equations or inequalities, put the variable terms on one side of the equal sign, and put the constant (number) terms on the other side. To do this, use opposite (inverse) operations. Solve: x. 5 6 x x 5 Undo adding one-third by subtracting onethird from both sides of the equation; make equivalent fractions with a common denominator of 5. McDougal Littell, Chapter 5 Pre-Algebra, Unit 05: Rational Numbers and Equations Page of

22 Solve: x 5 x 5 5 x x or Undo dividing by 5 by multiplying both sides by 5. Cancel. Multiply numerators and denominators, and simplify. Solve: 0 x x 0 x x x 7 x 7 x 6 We could have saved a little time by recognizing that multiplying by and then dividing by could have been done in one step by multiplying by the reciprocal. x 0 x x x 6 McDougal Littell, Chapter 5 Pre-Algebra, Unit 05: Rational Numbers and Equations Page of

23 Solve x x x x x Second Strategy for Solving: Another way to solve an equation or inequality with fractions is to clear the fractions by multiplying both sides of the equation or inequality by the LCD of the fractions. The resulting equation/inequality is equivalent to the original. You can also clear decimals by determining the greatest number of decimal places and multiplying both sides of the equation/inequality by that power of 0. Solve x x x x 656 x Original equation. Multiply each side by LCD of 6. Distribute. Simplify. Undo adding 0 by subtracting 0from both sides. x 5 x 5 x Undo multiplying by by dividing by. Simplify. McDougal Littell, Chapter 5 Pre-Algebra, Unit 05: Rational Numbers and Equations Page of

24 Solve x x x x x x. Original inequality. Since greatest number of decimals is, multiply by Distribute and simplify. Undo addition by subtracting 9000 from each side. Simplify. 0 or 000. Undo multiplication by dividing both sides by 875. Reverse the inequality. Simplify. McDougal Littell, Chapter 5 Pre-Algebra, Unit 05: Rational Numbers and Equations Page of

Pre-Algebra Notes Unit Five: Rational Numbers and Equations

Pre-Algebra Notes Unit Five: Rational Numbers and Equations Rational Numbers Rational numbers are numbers that can be written as a quotient of two integers. Since decimals are special fractions, all the