Rational Number is a number that can be written as a quotient of two integers. DECIMALS are special fractions whose denominators are powers of 10.

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1 PA Ch 5 Rational Expressions Rational Number is a number that can be written as a quotient of two integers. DECIMALS are special fractions whose denominators are powers of 0. Since decimals are special fractions, then all the rules we have already learned for fractions should work for decimals. The only difference is the denominators for decimals are powers of 0; i.e., 0,0 2,0,0 4, etc.... Students normally think of powers of 0 in standard form; 0, 00, 000, 0,000. 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 0, 00, 000,.. Therefore, one place is tenths, two places is hundredths, three places is thousandths, and so on. s: ).56 2 places- 56/00 2).52 places ).2 place The correct way to say a decimal numeral is to: ) Forget the decimal point. 2) Say the number. ) Then say its denominator and add the suffix ths.

2 s: ).5 Fifty-three hundredths 2) Seven hundred two thousandths. ).2 - Two tenths 4) 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, the decimal point is read as and, then say the number on the right said with its denominator. Write 5.20 in word form Fifteen and two hundred three thousandths Terminating and Repeating Decimals A rational number written in the form of a/b will either be a terminating or repeating decimal. Convert Fractions to Decimals One way to convert fractions to decimals is by making equivalent fractions. Convert to a decimal. 2

3 Since a decimal is a fraction whose denominator is a power of 0, I look for a power of 0 that 2 will divide into evenly. 2 = 5 0 Since the denominator is 0, I need only one digit to the right of the decimal point, the answer is.5 Convert 4 to a decimal Again, since a decimal is a fraction whose denominator is a power of 0, we look for powers of 0 that that will divide into evenly. 4 won t go into 0, but will go into = 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 2 or 5. That would mean 2, whose prime factors are 2 and would not be a factor of a power of ten. That means that 2 will never divide into a power of 0. The result of that is a fraction such as 5/2 will not terminate it will be a repeating decimal. Because not all fractions can be written with a power of 0 as the denominator, we may want to look at another way to convert a fraction to a decimal. That is to divide the numerator by the denominator. Convert /8 to a decimal. I could do this by equivalent fractions since the only prime factor of 8 is 2. However, we could also do it by division.

4 Doing this problem out, we get.75 How do you know how many places to carry out the division? Your teacher would have to tell you Converting a Decimal to a Fraction To convert a decimal to a fraction you: ) Determine the denominator by counting the number of digits to the right of the decimal point. 2) The numerator is the number to the right of the decimal point. ) Simplify. ) Convert.52 to a fraction..52 = = 25

5 2) Convert.60 to a fraction..6 = ) Convert 8.2 to a fraction. 8.2 = = Convert to fractions Converting a Repeating Decimal to a Fraction While the decimals. and. look alike at first glance, they are different. They do not have the same value. We know. is three tenths, /0. How can we say or write. as a fraction? Like in all the math we do, we take something we don t recognize and make it look like a problem we have done before. To do this, I have to get rid of the repeating part. The vinculum, the line over the. Convert. to a fraction.. =. By letting x =. Notice, and this is important, only one number is repeating. If I multiply both sides of the above equation by 0, then subtract the two equations, the repeating part disappears.

6 0x =. x =. That results in 9x = or x = / Convert.45 to a fraction. The difficulty with this problem is the decimal is repeating. So we get rid of the repeating part by letting x = = Notice, three numbers are repeating. By multiplying both sides of the equation by 000, the repeating parts line up so when I subtract, they disappear. 000x = x = x = 45 or x = 45/999

7 Methods of finding the LCM Method I Make a list. Write multiples of each numbers until there is a common multiple. Find the LCM of 2 and 6. 2, 24, 6, 48, 60, 6, 2, 48, 48 is the smallest multiple of both numbers, therefore 48 is the LCM Method II Method III Prime factorization. Write the prime factorization of both numbers. The LCM has to contain all the factors of both numbers. Write all the prime factors, use the highest exponent. Reduce the fraction. Write the two numbers as a fraction, reduce and cross multiply. The product is the LCM. Find the LCM of 8 and =, 8 4=24, the LCM is 72 4 When adding/subtracting fractions, the LCM is referred to as the Least Common Denominator (LCD). One way of finding a common denominator is to simply multiply the denominators. Find the common denominator for /5 and 7/0. 5 x 0 = 50, a common denominator is 50

8 Adding and Subtracting Fractions With Unlike Denominators Let s add to 4 Would I get 2 7? Why not? If we did, the 2 7 would indicate that we have two equal pieces and that 7 equal pieces made one whole unit. That s not true. Let s draw a picture to represent this: 4 + Notice the pieces are not the same size. Making the same cuts in each cake will result in equally sized pieces. That will allow me to add the pieces together. Each cake now has 2 equally sized pieces. Mathematically, we say that 2 is the common denominator. Now let s count the number of shaded pieces. 4 = 2 + = From the picture we can see that / is the same as 4/2 and 4 has the same value as /2. Adding the numerators, a total of 7 equally sized pieces are shaded and 2 pieces make one unit. If I did a number of these problems, I 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 2. Make equivalent fractions.. Add/Subtract the numerators 4. Bring down the denominator

9 Using the procedure, let s try one Multiply the denominators to find the common denominator, 5 = 5. Now I make equivalent fractions and add the numerators. 5 = = These problems can also be written horizontally. Let s try a few. Using the algorithm, first find the common denominator, then make equal fractions. Once you complete that, you add the numerators and place that result over the common denominator and simplify. Remember, the reason you are finding a common denominator is so you have equally sized pictures. When finding a common denominator, either multiply the denominators or use the reducing method. The reducing method should be use when you have larger composite numbers. Add or subtract the following problems _

10 Writing these problems with variables does not change the strategy. Simplify the expression. d + 2d 5 The CD is 5. Making equivalent fractions, we have d + 2d 5 = 5d 5 + 6d 5 = 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 8 7c 24 Using the reducing method; 8 24 =, the CD is c 8 7c 24 = 20c 72 2c 72 = 4c 72

$Multiplying Fractions Multiplying fractions is pretty straight-forward. So, we ll just write the algorithm for it, give an example and move on. Algorithm for Multiplying Fractions.$

11 Multiplying Fractions 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. Make sure you have proper or improper fractions 2. Cancel, if possible. Multiply numerators 4. Multiply denominators 5. Reduce 2 x 4 5 Since is not a fraction, we convert it to 7, we rewrite it as follows x 4 5 Now what I m about to say is important and will make your life a lot easier. We know how to reduce fractions, what we want to do now is to cancel with fractions. That s nothing more than reducing using the commutative and associative properties. The numerator is 7 x 4, the denominator is 2 x 5. Writing that as a single fraction I have 7 x Multiplying that out, I get. That will need to be simplified. 2 x 5 0 The Commutative Property of Addition allows me to change the order of the numbers. I will rewrite the numerator; 4 x 7. Now, I can rewrite them as separate fractions using 2 x 5 the Associative Property.

12 4 2 x 7 5, I can reduce 4 2 to 4 2 to 2 and rewrite the problem as 2 x 7. The answer is = Now rather than going through all those steps, using the commutative and associative properties, we could have taken a shortcut and cancelled. 7 2 x To do that, we would look for common factors in the numerator and denominator and divide them out. In our problem, there is a common factor of 2. By dividing out a 2, the problem looks like this 7 x 2 5 = 4 5 = Let s look at another one. 5 x 5 9 Rewriting the mixed number as a fraction, we have 8 5 x 5 9. We have a common factor of 5 in the numerator and denominator, we also have a factor of 9 in each. Canceling the 5 s and the 9 s, we have i 5 9 The answer is 2.

13 When variables are added to these problems, the strategy remains the same. Simplify the expression. n 2 4 2n5 7 = 6n7 28 n 2 4 2n5 7 = n7 4 Multiply the following fractions.. 4 x x x x x x 4

14 Dividing Fractions Before we learn how to divide fractions, let s revisit the concept of division using whole numbers. When I ask, how many 2 s are there in 8. I can write that mathematically three ways To find out how many 2 s there are in 8, I will use the subtraction model: Now, how many times did I subtract 2? Count them, there are 4 subtractions. So there are 4 twos in eight. Mathematically, we say 8 2= 4. You want some good news, division has been already 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. 4 8 Using repeated subtraction as we did with whole numbers, we have 4 8 = 5 8, = 4 8, = 8, 8 8 = 2 8, = 8, 8 8 = 0 Notice we subtracted /8 six times. So there 6 (/8) s in ¾.

$If I did enough of these problems, I would notice that if I multiplied the numerator in the first fraction by the denominator in the second fraction and divided that product by the product of the$

15 If I did enough of these problems, I would notice that if I multiplied the numerator in the first fraction by the denominator in the second fraction and divided that product by the product of the first denominator and second numerators, I would get the same answer. Somebody else might describe that by saying flip the divisor then multiply. 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: 2, rewriting that with a common denominator, we have Now, we take another 4, we get. Then from, wow, that s a lot of subtracting. 4 That takes time and space. It turns out I would have to subtract ¼ six times. Ready for a shortcut or would you rather subtract your brains out? 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 Division of Fractions. Make sure you have fractions 2. Invert the divisor (2 nd number). Cancel, 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. Patterns sure do make life a whole lot easier, don t you think? Inverting the divisor

16 = 5 4 = Convert = 7 4 Inverting the divisor = 28 = Another way of describing the procedure for dividing fractions is to rewrite the problem as a multiplication problem using the reciprocal of the divisor. Reciprocals two numbers whose product is one. 2/5 and 5/2 are reciprocals 5 and /5 are reciprocals

17 Solving Equations Containing Fractions Solve equations containing fractions the same way you solve equations with whole numbers. That is, isolate the variable by using the Order of Operations in reverse using the opposite operation. The only difference is using fractional rules to compute. Solve for x. x + /5 = 2/ Subtract /5 from both sides {add ( /5) to both sides}. x = 2 5 x = 2 5 x = 7 5 Solve for x. x 4 = 5 Since we are dividing by 4, we multiply both sides by 4. x 4 = 5 4i x 4 = 4i5 x = 20

18 That same problem could have looked like this: 4 x = 5 Again, multiplying by 4 on both sides, we have 4i 4 x = 4i5 x = 20 Solve for x. x 5 = 8 I could do the problem in two steps, multiplying by 5, then dividing by or I could multiply both sides of the equation by the reciprocal in do the problem in one step. 5 ix 5 = 5 i8 x = 40/ or

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