Chapter 3 Fractional Notation & Mixed Numerals

SS 3.1 Least Common Multiples

A least common multiple (LCM) is exactly what its name implies, a number that is the least multiple that 2 or more numbers have in common. There are 3 methods of finding the least common multiple of a number. I will introduce them in order of preference. Your book also introduces these three methods. We are learning to find a least common multiple so that we can create a common denominator in order to build a higher term for adding fraction (all that I just said here is that we will be creating equivalent fractions).

Method 1 (Prime Factorization Method)Step 1 Factor the numbers using prime factorizationStep 2 Take note of the unique factors in the factorizationsStep 3 Create a LCM by using each unique prime factor the number of times that

it appears the most, for any one number (not the total number of times that it appears!)

Example: Find the LCM of 12 & 15

Example: Find the LCM of 3, 5 and 7

Note: If they are all primes then the LCM is their product!

Example: Find the LCM of 5, 25, 50

Note: If the largest is a multiple of the smaller then the largest is the LCM.

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Method 2 (Listing Multiples and Finding Smallest)Step 1 List out the multiples of each numberStep 2 Circle the smallest one that all have in common, this is the LCM

We will use the same examples from above to find the LCM using the second method. I think that you will see that in most cases, the first method, although it may seem confusing at first, is superior!

Example: Find the LCM of 12 & 15

Example: Find the LCM of 3, 5 and 7

Example: Find the LCM of 5, 25, 50

The final method is a new method to me, and one that would be useful given very large numbers, but one which would probably be a waste of time for smaller numbers. This method is really a combination of the first 2 methods.

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Method 3 (Division by Common Prime)Step 1 Find the smallest prime which all numbers are divisible byStep 2 Divide the numbers by that prime and bring the answers downStep 3 Find a prime that divides at least 2 numbers if possible, and 1 if notStep 4 Divide the numbers by that prime and bring the answers down, if it is not

possible to divide evenly, bring the number itself down.Step 5 Continue the process, until you can divide no moreStep 6 The product of the divisor primes and the numbers remaining (those that

can’t be divided anymore) is the LCM

Example: Find the LCM of 12 & 15

Example: Find the LCM of 5, 25, 50

Example: Find the LCM of 180, 100 & 450

HW p. 143-144 #2-52 even & #53

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SS3.2 Addition and Applications

In addition we have two cases to consider. The first case is the easiest, when the denominators are alike, and the second requires using material from section 3.1 to find a least common denominator (LCD) to add fractions with unlike denominators. The LCD is the LCM of the denominators, so we have already learned how to find the LCD, and in chapter 2, we learned how to build a higher term when we learned about the Fundamental Theorem of Fractions. We will now put all these skills to use to add fractions with unlike denominators!

Adding Fractions with Common DenominatorsStep 1 Add the numeratorsStep 2 Bring along the common denominatorStep 3 Simplify if necessary (reduce or change to mixed number)

Example: 2/5 + 1/5

Example: 3/8 + 1/8

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Let’s just practice building higher terms first. We have already done this in section 2.5, when we were asked to find the missing number. That was when we learned the Fundamental Theorem of Fractions. The only difference now is that we must determine the denominator for ourselves, and that is where finding the LCM comes in.

Steps for Building a Higher TermStep 1 Determine the new denominator by finding LCMStep 2 Set up an equivalent fraction problem using LCM as your new

denominatorStep 3 Find the new numerator by applying the Fundamental Theorem of

Fractions

Example: Build the higher term for each of the following 1/3 & 2/5

Example: Build the higher term for each of the following 2/5, 3/27 & 17/36

Adding Fractions with Unlike DenominatorsStep 1: Find the LCMStep 2: Build equivalent fractions using LCMStep 3: Add the new fractions with common denominators.Step 4: Simplify the fraction if possible

Now let’s practice adding with unlike denominators by putting all the steps together.

Example: 1/3 + 2/5

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Example: 1/4 + 2/3

Example: 3/8 + 5/12 + 8/15

Application problems when adding fractions are no different than addition problems involving whole numbers. We are still looking for the key words of sum, total, more than, greater than, etc., which will indicate to us that the problem is an addition problem.

Example: Brian bought ¾ pounds of cashews, 3/8 pounds of almonds and 3/2 pounds of peanuts. How many pounds of nuts did Brian buy altogether?

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Example: A guitarist’s band is booked for Friday and Saturday nights at a local club. The guitarist is part of a trio on Friday and part of a quintet on Saturday. Thus the guitarist is paid one-third of one-half the weekend’s pay for Friday and one-fifth of one-half the weekend’s pay for Saturday. What fractional part of the band’s pay did the guitarist receive for the weekend’s work? If the band was paid $1200, how much did the guitarist receive?

Notice that some problems may now contain both addition and multiplication!

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HW p. 149-150 #2-52 even & #41SS 3.3 Subtraction, Order and Applications

Subtraction is the same as addition in method. There is no need to go into a detailed explanation, as the steps are the same. Let’s review the steps and then do some practice problems.

Subtracting Fractions with Like/Unlike DenominatorsStep 1: Find the LCMStep 2: Build equivalent fractions using LCM (if like denominators skip to next

step)Step 3: Subtract the new fractions with common denominators (watch order)Step 4: Simplify the fraction if possible

Now let’s practice a few of each type.

Example: 6/8 1/8

Example: 5/6 1/6

Example: ¾ 1/8

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Example: ¾ 1/20

Comparing Fractions Using Inequality Symbols

When denominators are alike there is little to comparing fractions, we simply compare the numerators and the larger numerator is the larger fraction. You can think of this like eating a pie. If you have a calorie-free pie cut into 5 equal pieces and someone tells you that you will either get 1 slice (1/5) or 2 slices (2/5) -- Which would you rather have? Well, obviously 2 slices, because that is more pie!! Think of your fraction problems like this pie.

Example: Use < or > to compare the following fractionsa) ¼ ¾

b) 5/8 7/8

c) 121/131 7/131

If the denominators are not alike, then we must go about the comparison in a different manner. Of course, as the book suggests, we could find the LCD and build the higher terms and then make the comparison based upon the numerators, but there is an easier way. We can simply compare the cross products. The fraction whose cross product is larger, is the larger fraction! Recall the discussion from section 2.5.

If I want to tell which of 2 fractions is larger, I cross multiply

5 1 1 4 4 5

The fraction with the larger cross product is the larger fraction. If the cross product is the same then the fractions are equivalent!

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Example: Use < or > to compare the following fractionsa) ¾ 8/12

b) 5/6 7/9

c) 2/5 5/8

Hmm, let’s compare my method and the book’s method. Which do you like? A simple cross multiplication or finding the LCD, building the higher term and then fiiiiinally comparing the numerators!

Example: Use < or > to compare the following fractions2/12 3/28

Book’s Method

My Method

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Word problems involving fractions and subtraction are no different than word problems involving whole numbers. When we run into a set up where an addend is missing, we see that we have a subtraction problem, and set the equation up appropriately. Remember that the words difference, less than, how much more, etc. indicate subtraction.

Example: Melaine spent ¾ hours listening to rock and jazz music. She spent 1/3 of an hour listening to jazz. How many hours were spent listening to rock? How many minutes is this?

Example: As part of a rehabilitation program, an athlete must swim and walk a total of 9/10 km each day. If one lap in the swimming pool is 3/80 km, how far must the athlete walk after swimming 10 laps, in order to complete the 9/10 km requirement?

HW

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p. 155-156 #2-50 & 56-68 evens & #74-77 allSS 3.4 Mixed Numerals

Mixed numbers (as most people call them) are an addition problem with the addition sign left out. Mixed numbers represent the sum of whole and a fractional part. Let’s take a look at what I am talking about by looking at a pictorial representation of a mixed number and how it relates to an improper fraction and addition.

Write as an addition problem. 2 + 1 = 3 Improper Fraction

+ 2 2 2

How many wholes and how many parts are there in the picture above?There is whole and parts which equals the mixed number

=

As can be seen from the above examples, the mixed number 1 ½ and the improper fraction 3/2 are equivalent, but different ways to write the same number. We need a way to change a mixed number to an improper fraction and vice versa without drawing pictures, however. We will start by changing improper fractions to mixed numbers, since we have already encountered improper fractions.

Improper Fraction Mixed Number Step 1 Divide numerator by denominator and write in remainder form.Step 2 Use remainder as the numerator of the fractional part. Its denominator is the

denominator of the original improper fraction. Write as a mixed number.Step 3 (Note that this step can be avoided, if the improper fraction is reduced to begin with) Reduce

the fractional portionStep 4 Rewrite

Example: Change 3/2 to a mixed number

(Note: Don’t forget the whole number when rewriting!)

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Example: Change 7/2 to a mixed number

Example: Change 14/4 to a mixed number

Example: Change 26/4 to a mixed number

(Note: You can recognize that the last 2 examples are not reduced prior to converting them to mixed numbers and reduce before conversion, but the entire method must be demonstrated!)

From this point on, all improper fractions should be converted to mixed numbers. Any answer with an improper fraction must be converted unless specified!

Mixed Numbers Improper Fractions Step 1 Make sure that the fractional portion of the mixed number is in lowest termsStep 2 Multiply the denominator by the whole number and add the numeratorStep 3 Put the new number from step 2 over the original fractional portion’s denominator

(Note: We will be using this to add, subtract, multiply and divide mixed numbers)

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Example: Change 2 ¾ to an improper fraction

Example: Change 7 ½ to an improper fraction

Example: Change 2 1/8 to an improper fraction

The concept of a mixed number comes in quite handy when dealing with a long division problem where there is a remainder involved. Rather than writing the answer as a whole with a remainder, we can now write the answer as a mixed number. The following describes how this is accomplished.

Long Division Answers as Mixed NumbersStep 1 Divide as normal until a remainder is reachedStep 2 Instead of writing remainder such and such, place the remainder over the divisor

and write next to the whole number portion of the answer.Step 3 Reduce the fractional portion if necessary

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Example: Divide 253251

Example: Divide 18 8

(Note: This problem requires the fractional portion to be reduced, and that is the reason why I provide the steps that I do for converting an improper fraction to a mixed number. I believe in doing this, you will be more prepared for having to reduce the fractional parts of a long division answer where appropriate.)

Even though our author does not have any problems like these in this section, let’s just throw a couple out there to show the importance of converting an improper fraction to a mixed number.

Example: ½ 1/3

Example: 2/3 1/5

HW

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p. 161-162 #2-62 all evens & #64, 70SS 3.5 Addition & Subtraction with Mixed Numerals & Applications

Although your author does not discuss the fact, there are 2 methods for adding mixed numbers. The first method that I will show is the one shown by the author and the one that I want you to use to complete all homework problems. I will also be specifying some problems to be completed using the second method as well. It is very useful to know and be proficient at both methods. At times one method may be superior to the other, so it is important to learn both!

Adding Whole Numbers & FractionsStep 1 Place in columns with whole numbers over whole numbers and fractions over

fractions, if not already positioned this wayStep 2 If there is not a common denominator, find the LCM and build higher term to the

right of the fractionsStep 3 Add fractionsStep 4 Add whole numbersStep 5 If fractional part is improper or needs to be reduced do the appropriate thing.Step 6 Add the whole number and fractional portion together to make a mixed number

Example: 2 9/50 + 3 7/25

Example: 9 4/5 + 11 ½

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Example: 17 2/3 + 103 11/27

Example: 283 + 5 3/10

Example: 5 ¾ + 6 ½

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Subtracting Whole Numbers & FractionsStep 1 Place in columns with whole numbers over whole numbers and fractions over

fractions, if not already positioned this wayStep 2 If there is not a common denominator, find the LCM and build higher term to the

right of the fractionsStep 3 Subtract the fractions WARNING: If the top numerator is not larger than the

bottom numerator you must borrow 1 from your whole number and add it to your current fraction!!

Step 4 Subtract whole numbersStep 5 If fractional part needs to be reduced do so.Step 6 Add the whole number and fractional portion together to make a mixed number

Example: 11 4/5 9 ½

Example: 6 2 7/8

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Example: 15 7/8 5 8/9

Example: 2 1/3 1 1/2

Example: 7 1/8 6 1/3

Now, I will show the second method of adding and subtracting mixed numbers. The nice thing about this method is that there is no carrying or borrowing; it all comes out in the as so to speak. The unfortunate thing is that this method is not very friendly when the whole numbers are large and the denominators are more than a single digit! I will show you an example of what I mean as the last example.

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Adding & Subtracting Mixed Numbers by Converting to Improper FractionsStep 1 Change addends to improper fractions (Be sure to keep the correct order in subtraction)Step 2 Find an LCD if necessaryStep 3 Add or subtractStep 4 Reduce and/or convert to a mixed number where necessary

Example: 6 1/4 5 1/3

(Note: This problem would have required borrowing with the other method!)

Example: 5 1/2 + 3 2/3

(Note: This problem would have required carrying with the other method!)

Example: 1 + 2 1/7

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Example: 3 9/14 2 3/14

(Note: This problem, because the denominators were already the same, would be done with much more easy using the first method!!)

Example: 236 12/25 225 7/8

(Note: The common denominator can be found first, to make operations have less of a chance for error, but I think that you can probably see why the other method involving borrowing is still preferable!)

ApplicationsThe applications in this section are simply fraction word problems involving mixed numbers. You need to remember to look for the words that tell you that you will be adding or subtracting or doing a combination of the two.

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Example: The road to Cancun is 15 7/8 miles long. The road to Kilamanjaro is 7 ¾ miles long. How much longer is the road to Cancun?

Example: A fishmonger sold packages of squid weighing 2 ¾ pounds and 4 2/5 pounds. What was the total weight of the squid?

Example: The recipe calls for 1/8 pound of butter, ¼ pound of sugar, 1 pound of flour and 1/6 pound of chocolate chips. If the mixing bowl weighs 2 ¾ pounds how much will it weigh with all the ingredients in it?

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Example: Joe, a sheet metal worker, worked 31 ½ hours over a 3-day period. If Joe worked 12 ½ on the first day and 11 1/5 on the second, how many did he work during the third?

ExtensionJust so that we get some more practice and extend our skills a little more as well as maintain some skills from chapter 1, let’s do some order of operation problems!

Example: 15 1/2 1 7/8 + (1/2 ¾)

Example: 1 1/3 1/5 1/6 3 1/3

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Example: (2/3)2 + 1 1/2 1/3

Example: 1 / 5 + 1 / 7 1 / 8 1/5 1/3 ½

HWp. 167-170 #2-58 all evens & #65

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Also, do the following using method of converting to improper fractions #8,16,18,22 &26SS 3.6 Multiplication & Division with Mixed Numerals & Applications

The author and I have no differing views in this section!! What I tell you will be the same as in the book.

Multiplying/Dividing Mixed NumbersStep 1 Change factors or dividend & divisor to improper fractionsStep 2 Multiply or Divide as we did with fractions

Multiplication Review Num Num Denom Denom

Division Review Invert DivisorMultiply

Step 3 Convert to Mixed Number and/or Reduce as necessary

Example: 2 1 3/4

Example: 2 1/3 3 8/9

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Example: 1 2/3 1 2/5

Example: 1 1/3 2

Example: 5 3 8/9

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Example: 3/4 1 1/2 1/9

Example: 1/9 (2 1/3 7)

Example: 2 1/2 1/3 1 1/3

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Example: 2 (1 1/3 + 1/2) + 1 ½ (7/8 + 1 ½)

ApplicationsThese word problems are just like those found in the fraction sections. Just remember to look for the key words that tell you to multiply or divide or to do a combination of all operations!

Example: The exercise fanatic jogs 15 ½ miles per day, six days per week. How far does the exercise fanatic jog in a week?

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Example: Stephanie wants to buy a shelf to hold her 30 books. If each is an average of 1 1/8 inches thick, and she buys a shelf that is 27 inches long, will the shelf hold all of her books?

Example: Fahrenheit temperature can be obtained from Celsius temperature by multiplying by 1 4/5 and adding 32. What Fahrenheit temperature corresponds to a Celsius temperature of 20?

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Example: Find the area of the figure

½ in.

6 7/8 in.

6 7/8 in.

HWp. 175-178 #2-52 evens & #57-62 all

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½ in.

SS 3.7 Order of Operations; Estimation

Because we have been including order of operation problems in each section I do not find it necessary to do any more of those problems at this time. I will however, remind you of the term average and discuss estimation.

Average ProblemsStep 1 Add all numbers to be averagedStep 2 Divide sum by the number of addends summedStep 3 Give the answer appropriately

Example: Find the average of 2 1/3 & 3 8/9

Example: Find the average of 1/3 , 2/7 & 8/9

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Estimating Fractional AnswersStep 1 Estimate each number as being close a whole, ½ or zero

A numerator that is small in comparison to the denominator estimates to zero (<1/2)When the denominator is approximately twice the denominator then estimate is ½When the numerator & denominator are nearly the same then the estimate is 1

Step 2 Add/Subtract/Multiply/Divide as asked to do.

The purpose in doing this is so that you can get a reasonable approximation for your answer and thus tell if you are in the ballpark. It may also be useful on standardized tests or multiple-choice tests!

Let’s practice estimating 1st.Example: Estimate

a) 3/13

b) 23/27

c) 15/31

d) 21/50

e) 5 4/5

Now we will practice a few estimates. Note that I will be asking you to estimate to the nearest ½ in all cases.

Example: 2 1/5 3 8/9

Example: 16 1/3 2 3/13 + 25 9/11 4 11/23

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Example: 7 31/61 + 10 11/12 24 3/25

Example: 3/13 2 9/19 4 11/23

HWp. 183-186 #2-40 even & #58-82 even & #83,84,85p. 189-190 all problems (not to be turned in)p. 191-192 all problems except synthesis (not to be turned in)

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