3 COMMON FRACTIONS. 3.1 Common Fractions as Division

Transcription

1 COMMON FRACTIONS Familiarity with fractions is essential for success in any trade. This chapter reviews how to work with fractions.. Common Fractions as Division A common fraction represents division of one number (the numerator) by another number (the denominator). For example, a number represented by the letter a divided by another number represented by the letter b, is written mathematically as: The denominator of a common fraction designates the number of equal-sized parts into which some complete entity one whole is split or divided. The numerator of a fraction designates the number of parts being considered. Examples of common fractions are: a numerator b denominator,,,, and Fractions are sometimes written using a slash (/) rather than a horizontal bar This practice should be avoided in technical work so that fractions are always thought of in terms of division and the order of operations remains clear. Technically, the fraction bar indicates a division operation wherein the numerator (top number) is divided by the denominator (bottom number). Although this division is not actually performed every time a fraction is encountered, thinking of a fraction as a division problem is helpful when converting a common fraction to its decimal or percent form.

2 CHAPTER COMMON FRACTIONS! Say topple it over to help remember the order of division in a fraction. a a b or b a b = The fraction indicates that parts of an entity split into equal parts is under consideration. For example, the first circle in Figure. is split into three equally sized pieces. In the second circle, of the pieces are shaded. We could say that the circle is eating, the shaded region may mean that cut into three equal pieces, were eaten. shaded. If the circle represents an apple pie we have been, or two pieces of a pie that had been FIGURE. Proper and Improper Fractions Fractions can be classified as proper fractions or improper fractions. A fraction with value less than one is called a proper fraction. Its numerator is smaller than its denominator. Mathematically, For a proper fraction, when a is divided by b, the result is a decimal number less than one. a because a b b < < A fraction whose value is greater than one is called an improper fraction. Its numerator must be larger than its denominator. Mathematically, For an improper fraction, when a is divided by b, the result is a decimal number greater than one. a because a b b > > A fraction whose numerator and denominator are equal signifies unity, the number one. For example, 7 or a are cases of unity, because the result would equal 7 a if the actual division were carried out. Note that we do not have to know the value of a since anything divided by itself must equal. TABLE.: Examples of Proper, Improper, and Unity Fractions Proper fractions Improper fractions Unity fractions

3 CONVERTING IMPROPER FRACTIONS AND MIXED NUMBERS 9 Mixed Numbers A mixed number consists of a whole number and a proper fraction. The implied operation between the two parts of a mixed number is addition. For example, means +. Since improper fractions are greater than one, they are often changed to mixed numbers. EXERCISES. Draw shaded circles to illustrate the indicated fraction. a) -- b) c) -- 6 d) e) f) g) -- 6 h). State whether the indicated fraction is a proper fraction, improper fraction, or mixed number. 0 9 a) b) c) d) 7 e) -- f) g) -- h) Converting Improper Fractions and Mixed Numbers Converting an Improper Fraction to a Mixed Number An improper fraction is converted to a mixed number by dividing the numerator by the denominator and retaining the remainder as a proper fraction. For example, the improper fraction is converted to a mixed number as follows: 7 7 = 7 = R,which equals Notice that the remainder is written as the numerator of the fraction part of the final answer. The denominator is unchanged. EXAMPLE.A: Converting an Improper Fraction to a Mixed Number! R stands for remainder. Change to a mixed number. Write as a long division: 6 Divide by : Write answer as whole number and the remainder as a fraction: 6

4 0 CHAPTER COMMON FRACTIONS EXAMPLE.B: Converting an Improper Fraction to a Mixed Number Change to a mixed number: Write as a long division problem and solve: 9 Write answer as whole number and the reminder as a fraction:. Converting a Mixed Number to an Improper Fraction A mixed number is converted to an improper fraction using three steps: STEP : Multiply whole number part by the denominator of the fractional part STEP : Add result of step to numerator of fractional part STEP : Place result of step over the denominator of the fractional part. 9 EXAMPLE.: Converting a Mixed Number to an Improper Fraction + 7 = = This is the reverse of the process for converting improper fractions to mixed numbers. EXERCISE. Convert the indicated improper fraction to a mixed number. 97 a) c) e) g) 60 b) d) f) 0. Convert the indicated mixed number to an improper fraction. 7 9 a) 6 b) c) d) e) 9 f) 9

5 RAISING A COMMON FRACTION TO HIGHER TERMS. Raising a Common Fraction to Higher Terms Any fraction has an infinite number of equivalent forms. Equivalent fractions are equal in value but different in form. Definition To illustrate, consider the following: = = = =... = = FIGURE. Equivalent Fractions A glance at the circles cut into fourths and eighths in Figure. illustrates why the first two fractions are equivalent. However, since a fraction has an infinite number of equivalent forms, a single drawing does not prove the point for all equivalent forms. Even if we sought to prove it in this fashion for a few more forms, drawing the diagrams would be very tedious. Instead, consider that the fourth fraction, 6, is obtained from the first,, by multiplying both the numerator and denominator of by : = 6 Multiplying both numerator and denominator by the same number is the same as multiplying the entire fraction by, the multiplicative identity. Remember that multiplying a number by does not change its value. Multiplication of by in the form of gives the last equivalent form shown: 7 = 00 This process is called raising a fraction to higher terms. It is used to find equivalent forms for a fraction. Equivalent fractions always have denominators which are multiples of the original denominator. Thus, a fraction with a denominator of 7 can only be raised to equivalent fractions with denominator values of,,,, and so on EXAMPLE.A: Raising a Fraction to Higher Terms! Recall that a multiple of a given number is a number that results when the given number is multiplied by any whole number. 6 Raise to an equivalent fraction with a denominator of. Solution: First ask what number must be multiplied by to give. The answer is. Hence, multiply both numerator and denominator by : 6 =

6 CHAPTER COMMON FRACTIONS EXAMPLE.B: Raising a Fraction to Higher Terms? 9 = = ? 9 6 = = Notice that for a mixed number, the process is the same but is performed on just the fractional part of the mixed number:? = = 0 0 EXERCISES. Raise the indicated fraction to an equivalent fraction.??? a) to b) to c) to d) to Raise the indicated fraction to an equivalent fraction.? 6 6 a) to b) to c) to d) to? 6?? 6?.7 The fraction is how many 6ths?. The fraction is how many ths?. Reducing a Common Fraction to Lowest Terms It is usually convenient when working with a fraction to reduce it to lowest terms. A fraction in lowest terms is one whose numerator and denominator cannot be divided by the same whole number. In other words, the numerator and denominator have no common factor. People typically say a number that divides another number evenly is a factor of that number. This is not strictly correct terminology. The phrase in mathematics a divides b is defined to mean that b a gives a whole number quotient, that is, there is no remainder. Hence, the correct definition of factor: Definition A number that divides another number without a remainder is a factor of the other number.

7 REDUCING A COMMON FRACTION TO LOWEST TERMS Fractions can be reduced to lowest terms if the numerator and denominator contain common factors. Common factor pairs divide to give and thereby cancel each other to give a fraction in lowest terms. EXAMPLE.: Reducing a Fraction to Lowest Terms = = = = = = = = 0 0 Notice that in these examples the obvious common factors are canceled first, then other cancellations are made. To reduce a fraction in one step, we need to cancel the greatest common factor (GCF) of numerator and denominator. For example, = = Sometimes the GCF of two numbers is obvious. When the GCF is not obvious, a method called prime factorization can be used. Reducing Fractions Using Prime Factorization Before we can discuss how to reduce fractions using prime factorization, we must learn to recognize prime numbers. Prime numbers are numbers whose only factors are and themselves. The prime numbers from to are:,,,, 7,,, 7, 9,, 9,. Any other number between and has factors other than itself and. For example: = 6 = = 9 = 0 = = = 7 =! The GCF is the greatest common factor of two or more numbers. The number is the only even prime. A number s prime factors are easily determined by performing a series of short divisions on the number, starting with the lowest prime number that divides it and continuing in this manner until no prime factors remain. The following observations are worth noting: Even numbers have a factor of. Numbers whose digits add to a number divisible by are also divisible by. Numbers that end in 0 or are divisible by.

8 . CHAPTER COMMON FRACTIONS EXAMPLE.: Finding Prime Factors of a Number Find the prime factors of 6: 6 = 9 = ( ) ( ) The prime factors of 6 are therefore,,, and. Similarly, the prime factors of are,,,, and because: = = ( ) ( 6) = ( ) () ( ) Notice that divides both 6 and because they are even numbers. Also, divides 6 and because each of them has digits that add to numbers divisible by. In Example. we found the prime factors for two numbers, 6 and. It makes sense now to ask what the GCF of these two numbers is. In most cases, the GCF can be found by inspection. When inspection does not readily reveal the GCF, the GCF can be found methodically using the procedure described next. Procedure for Finding GCF To find the GCF of two or more numbers use the following four step procedure: STEP : Write the prime factorization of each number. STEP : Compare the factors. Note how many times each factor repeats. STEP : For each factor, choose the least number of repeats. STEP : Multiply only the prime factors identified in Step and the result is the GCF for the original pair of numbers EXAMPLE.6A: Finding the GCF Find the GCF of 6 and. Then reduce the fraction. Solution: Prime factors of 6: Prime factors of : In this case, only the actors and appear. Looking at the factors of, we can see that 6 has the fewest so we keep both of them and discard the factors of for. Likewise, looking at the factors of, we can see that has only a single so we keep it and discard the factors of for 6. Finally, we arrive at: = to be the GCF for 6 and 6

9 REDUCING A COMMON FRACTION TO LOWEST TERMS EXAMPLE.6A: Finding the GCF (Continued) To reduce 6/ using the GCF, we rewrite the numerator and denominator as a multiplication using the GCF,. We can then see that the in the numerator cancels the in the denominator and we are left with ¾ as our answer. 6 = = Note that since we reduced the fraction using the GCF, the answer cannot be reduced any further and it is said to be in lowest terms. EXAMPLE.6B: Finding the GCF Find the GCF of 0 and 0. Then reduce the fraction. Solution: Prime factors of 0 are,,,, because: 0 = = () ( 7) = () () ( 9) = () () () ( ) Prime factors of 0 are,,, because: 0 = 7 = () ( ) = () () ( ) Among the factors of, the 0 has the fewest so we keep one ; for factors of, the 0 has the fewest so we keep one ; and for factors of, the 0 does not have any so we do not include it in finding the GCF. Finally, multiplying the factors we kept, we find the GCF to be: GCF = = 6 Rewriting the original fraction with the 6 as a factor in the numerator and denominator, we find the lowest term result: 0 6 = = EXAMPLE.7: Reducing a Fraction Using Prime Factorization Reduce Solution: to lowest terms using prime factorization. Prime factors of are,, Prime factors of are, 7 None of the prime factors of match the prime factors of, so no cancellation is possible and thus is already in lowest terms.

10 6 CHAPTER COMMON FRACTIONS Reducing a Fraction to Lowest Terms Using Prime Factorization Reduce to lowest terms using prime factorization. EXAMPLE.: Solution: Prime factors of are, Prime factors of 60 are,,, The only common factor is and the fewest times it occurs is once so: Reducing Mixed Numbers To reduce a mixed number, reduce only the fractional part. For example: Therefore, = 0 6 EXERCISES.9 List the prime numbers from 0 to Why is the only even prime number?. Does divide 0? 70? = = 60 = = Determine the prime factors of the indicated numbers: a) 7 b) 66 c) 99 d) 6 e) f) g) 9 h) 6. Reduce the indicated fractions to lowest terms: 7 0 a) c) e) g) 0 6 b) d) f) h) 0 6. Addition and Subtraction of Common Fractions The skills learned to convert fractions in one form to their equivalent forms are used in fraction addition and subtraction. 0

11 ADDITION AND SUBTRACTION OF COMMON FRACTIONS 7 Fractions with the Same Denominators Fractions with the same denominator are said to have a common denominator. To add or subtract such fractions, we simply add or subtract the individual numerators and place the sum over the common denominator. If necessary, the result is reduced to lowest terms or converted to mixed number. EXAMPLE.9A: Add: + + Solution: Adding Fractions with Same Denominator Since each of the fractions have a common denominator of, we simply add the numerators and retain the common denominator: EXAMPLE.9B: Add: = = = Adding Fractions with Same Denominator = = =! Do not add or subtract denominators. 6 = = EXAMPLE.0: Subtracting Fractions with Same Denominator Subtract: 7 7 = = = Fractions with Different Denominators To add or subtract fractions whose denominators are not the same, a common denominator must be found and each fraction in the problem must be converted to an equivalent fraction having this denominator. A common denominator of a group of fractions is a single denominator to which all of the fractions can be converted. In a group of fractions, the product of all of the denominators is always a common denominator. However, this common denominator is frequently a very large number that is unwieldy to work with. Often a smaller common denominator can be found. The smallest possible common denominator that can be used is one into which all the original denominators can be divided. This is called the Least Common Denominator, which is abbreviated LCD. The arithmetic of adding and subtracting fractions can be greatly reduced by using the LCD. Often, the LCD can be found by inspection. Otherwise, a procedure similar to the procedure used to find the GCF can be applied as shown in the following examples.

12 CHAPTER COMMON FRACTIONS EXAMPLE.A: Finding the Least Common Denominator Find the LCD (Least Common Denominator) of the following group of fractions: Solution: STEP : Under each denominator place its prime factors: STEP : Determine the greatest frequency of. (underscored), STEP : Determine the greatest frequency of. (double underscored), STEP : Determine the greatest frequency of. (triple underscored) STEP : Find the LCD: The calculated LCD can be proved by dividing it by each of the original denominators to determine if they can divide the LCD = 0 0 = 0 = 0 = Therefore, 0 is the LCD. EXAMPLE.B: Add the fractions:,,,,and 6 Solution: STEP : Write prime factors: Finding the Least Common Denominator STEP : Determine frequencies of primes: occurs at most time occurs at most times occurs at most time 6,,,,,,,, LCD= ( greatest frequency of ) ( greatest frequency of ) ( greatest frequency of ) LCD = ( ) ( ) () = 0 6,,,,,,

13 ADDITION AND SUBTRACTION OF COMMON FRACTIONS 9 Finding the Least Common Denominator (Continued) STEP : Multiply frequent primes by their most frequent occurrences to get LCD: = When the LCD is determined, the next step in the addition and subtraction of fractions with different denominators is to convert the different fractions into equivalent fractions, each with the LCD as its denominator. STEP : Convert to equivalent fractions using the LCD: EXAMPLE.B: 6 0 =, =, =, =, = 6 STEP : Add the equivalent fractions: = = STEP 6: Reduce to lowest terms: Generalized Procedures for Adding and Subtracting Fractions with Different Denominators Now that we know how to find the least common denominator, we can generalize the procedures for adding and subtracting fractions with different denominators into steps: STEP : Find the least common denominator (LCD). STEP : Convert all fractions to equivalent fractions using the LCD. STEP : Add or subtract the converted fractions. STEP : Reduce answer if possible = = Combined Addition and Subtraction of Fractions Whenever both addition and subtraction operations appear in a problem, the operations are performed in the order in which they appear, unless parentheses are present. EXAMPLE.: Combined Addition and Subtraction of Fractions Perform the indicated operations: Solution: The LCD is, as before. Convert and perform operations: = = = =! PEMDAS

14 0 CHAPTER COMMON FRACTIONS EXERCISES. Find the LCD of each of group of fractions. 7 a),, b),,, c) d),, e),, f) 0 9. Perform the indicated addition and subtraction. a) + b) + c) 9 7 d) e) f) g) + h) + 6 i) 7 6 j) + k) l) m) + n) + 9 o) p) + q) ,, 0,,, A tube has an inside diameter of 6 inch and a wall thickness of inch. What is the outside diameter (O.D.)?.7 A bolt is used to fasten two 6 planks. The -inch thickness (nominal) is actually, and the nut for the bolt is inch thick. What is the minimum length of bolt to use if bolts come in length increments of inch?. A special laminated wooden beam is made up of two pieces of planking inch thick, and three pieces of inch nominal lumber inch thick. What 6 is the actual thickness of the beam?.9 A special jet fuel is made in a laboratory in small quantities for testing. If the formula calls for lb of chemical A, lb of chemical B, and lb of chemical C, what is the total weight of the experimental batch of jet fuel?.0 A carpenter is nailing some inch (actual measurement) plywood onto furring strips that are inch thick. If nails come in inch incremental 6 lengths, how long a nail must be used to insure maximum penetration but not go completely through the furring strip?

15 ADDITION AND SUBTRACTION OF COMMON FRACTIONS. A drafters scale (ruler) has the inches divided into halves, quarters, eighths, and sixteenths. How long a line would result if one-half, three-quarters, five-eighths and eleven-sixteenths were added together?. The dimensions in Figure. are given in inches. Find A, B, C, D, E, and F. FIGURE.. The dimensions in Figure. are given in inches. Determine horizontal dimensions, A, B, C, and D. Also determine vertical dimensions, E, F, G, and H. FIGURE.

16 CHAPTER COMMON FRACTIONS. The dimensions in Figure. are given in inches. Determine the overall dimension, A, of the locating pin shown. FIGURE..6 Addition and Subtraction of Mixed Numbers Mixed numbers can be added by either of two methods. In one method, the mixed numbers are converted into improper fractions, which are then added by means of the LCD. In the other method, the fractional parts are added independently from the whole number parts. This method also usually requires an LCD be found, but it is the technique used most often, so we will review it in depth. Consider mixed numbers whose fractional parts have like denominators: 9 + Since mixed numbers themselves have an addition operation implied between the whole number part and the fraction part, the problem could be represented as From this representation it is logical to add whole number to whole number and fraction to fraction. Thus, = + = + = We would not normally write out the addition between the mixed number parts; we only do so here to clarify the process. Notice that the fraction parts add to, which reduces to. Final answers must always be reduced to simplest form. The process is a little more involved for mixed numbers whose fractions have unlike denominators. Consider the problem +. We write this vertically and show all steps toward the solution: = = 0 = = 0 Finally, + =

17 ADDITION AND SUBTRACTION OF MIXED NUMBERS Notice that we had to convert the fraction parts to their equivalent forms with LCD of. The final answer required no reducing. Now we show the other method of mixed number addition, namely, converting the mixed numbers to improper fractions before adding them. Using the same problem, we again show all steps toward the solution. = = = = = = Notice that this method requires some extra steps. This is why it is the least used of the two techniques for mixed number addition. EXAMPLE.: Finally, + = = Mixed Number Addition Perform the indicated addition using prime factorization: Solution : Using prime factorization, we find the LCD is = 6. Thus, 0 7 = 7 = = 6 = Finally, = = + = ! 9 = = The improper fraction in the answer is converted to a whole number and proper fraction. The whole number is then added to the whole number part of the answer. Subtracting mixed numbers can be more complicated than adding them. Consider the problem. As with mixed number addition, the whole number parts are subtracted separately from the fraction parts. However, the subtraction cannot be solved without negative numbers. We are interested in only nonnegative numbers at this point in our review of arithmetic so even though we could resort to using negative numbers to get the answer, it is not necessary to do so. Write the problem in vertical fashion: -- --

18 CHAPTER COMMON FRACTIONS Math Trivia: The minuend is the first number in the subtraction.the subtrahend is the second number. Although is smaller then, the entire subtrahend, is smaller than the minuend, so the problem is certainly manageable in the nonnegative numbers. To solve, we increase by borrowing from. Taking one whole () from and adding it to gives an improper fraction. The rest of the problem is straightforward. = + = = = Finally, + = = In this kind of problem, it is sometimes easier to use the other method of mixed number addition and subtraction, namely, changing the mixed numbers to improper fractions first. = When unlike denominators are involved in the subtraction, the LCD must be found. This adds to the steps, as the next example shows. Subtraction of Mixed Numbers with Unlike Denominators Perform the indicated subtraction: 7 Solution: First the LCD is found ( 7 = ). Then the fractional parts are converted to equivalent fractions with a denominator of : EXAMPLE.: = 0 Finally, = = = = = = = 7 This problem still requires that the fractional portion of the minuend be increased by borrowing from the whole number portion: = 7 + = 7 = = 9 Finally, 7 = = + =

19 ADDITION AND SUBTRACTION OF MIXED NUMBERS EXERCISES. Perform indicated operations: 7 9 a) + b) 6 + c) d) 0 + e) 7 f) g) h) The dimensions of the detail in Figure.6 are given in inch. Calculate the overall length. FIGURE.6.7 The dimensions of the detail in Figure.7 are given in inch.determine the overall length and width.. FIGURE.7

20 6 CHAPTER COMMON FRACTIONS.9 The dimensions of the detail in Figure. are given in inch.determine the overall length and width..0 FIGURE.. Six sheets of steel have the following thickness in inches:,,,, and 6 6 What is the total thickness of the six sheets?. 6. The dimensions of the detail in Figure. are given in inch.determine dimensions A, B, C, and D. FIGURE.9. Eight s, feet long are nailed end to end with each joint having an overlap of inches. What is the total length of the assembly? 6. A rectangular field is 0 feet, inches long by feet, inches wide. How many feet of fencing are needed to enclose the field?

21 MULTIPLICATION AND DIVISION OF COMMON FRACTIONS 7.7 Multiplication and Division of Common Fractions Raising a fraction to its equivalent form with the goal of producing fractions with common denominators in order to add or subtract them is done with fraction multiplication, as we discussed earlier in this chapter. Now we look at fraction multiplication for all purposes. The rule for multiplication of fractions is simple: Multiply all numerators to obtain the numerator of the answer. Multiply all denominators to obtain the denominator in the answer. Reduce answer to lowest terms. EXAMPLE.: Solve: Solution: Fraction Multiplication 6 = = = Fraction Multiplication As seen in Example., the result in fraction multiplication is usually not in lowest terms. However, the process of reducing the result to lowest terms can be simplified if cancellation of common factors is done before multiplication takes place. Cancellation is the dividing of the numerator and denominator of a fraction by a common factor. Note that if all possible cancellation is performed before the multiplication takes place, no reduction will be necessary at the end as the result will already be in lowest terms. EXAMPLE.6: 0 Solve: 6 Solution: Fraction Multiplication using Cancellation Cancellation is shown on the original problem. Write the result of the canceling over the original number and then multiply. We could have canceled thoroughly before multiplying (the two s with the one ) and then the final answer would have been in lowest terms, with no need to reduce further. An optional cancellation is: 0 = = = 6 0 = = 6! Remember, a fraction in lowest terms has no common factors in its numerator and denominator.

22 CHAPTER COMMON FRACTIONS Notice that the cancellation can be done between any number in the numerator and any number in the denominator. The method of dividing fractions is also quite simple, as it follows from the rule for fraction multiplication. The rule of division is: Fraction Division! CAUTION! Do not invert the fraction preceding the division sign.! CAUTION! Cancellation must not be done before inversion. Invert the fraction following the division sign, change the division to a multiplication sign and proceed as in multiplication of fractions EXAMPLE.7A: Solve: 6 Solution: Division of Fractions STEP : Invert second fraction and change to multiplication, = 6 6 STEP : Perform cancellation and reduce, / = = = = 6 6/ EXAMPLE.7B: Solve: 6 Solution: EXERCISES Division of Fractions 6 STEP : Invert, = 6 / / 6/ STEP : Cancel, / / STEP : Multiply and reduce,. Perform the indicated operations: = a) 9 b) 77 c) d)

23 MULTIPLICATION AND DIVISION OF COMMON FRACTIONS 9 7 e) f) g) h) i) 0 9 j) 7 k) 6 l).6 Perform the indicated operations: 7 a) f) b) g) 9 7 c) h) 9 d) i) e) j) Multiply the difference of and by their sum.. A special nonferrous alloy is composed of tin, copper, and zinc. How many pounds of each metal will be used to pour a casting weighing lb?.9 A pipefitting weighs lb. How many fittings are in a box weighing lb if the empty box (tare weight) weighs lb? 9.0 How many holes, spaced on inch centers, can be drilled in the detail 6 shown in Figure.0? 0 FIGURE.0

24 0 CHAPTER COMMON FRACTIONS. Determine the number of discs that can be blanked from a -feet roll of # gage cold rolled steel if each blank is inches in diameter and there is a inch web between each disc. See Figure.. FIGURE.. A millwright must use a rough wooden timber measuring by inches. If a timber that is 7 6 inches thick is required, and equal thickness are cut from each rough side, how thicker the pieces cut off each side? See Figure.. FIGURE.. Masonry walls of cement blocks are made from a standard size block which is inches long, inches high, and inches thick. The mortar 7 joint allowance for each block is i inches. How many blocks will be needed to build a wall that is feet high and 60 feet long?. Multiplication and Division of Mixed Numbers Multiplication of mixed numbers proceeds as shown in the Example., in which the mixed numbers are converted to improper fractions. Multiplication of mixed numbers can also be done directly, however the process is cumbersome and so is not presented here. The division of mixed numbers, on the other hand, must be done by conversion to improper fractions.

25 MULTIPLICATION AND DIVISION OF MIXED NUMBERS EXAMPLE.: Multiplying Mixed Numbers Multiply: by Solution: STEP : Convert to improper fractions: 6 6 STEP : Cancel: = = 0 Dividing Mixed Numbers Divide by 7 Solution: 6 STEP : Convert to improper fractions, 7 STEP : Invert, cancel and multiply, = = STEP : Reduce to lowest terms, = 6 6 EXAMPLE.9: Combined multiplication and division of mixed numbers is accomplished in the same manner as with fractions. EXAMPLE.0: Combined Multiplication and Division of Mixed Numbers 7 Perform the following operations: 6 6 Solution: STEP : Convert to fractions 9 6 STEP : Invert and change division to multiplication 6 9 STEP : Cancel common factors from numerator and denominator

26 CHAPTER COMMON FRACTIONS EXAMPLE.0: Combined Multiplication and Division of Mixed Numbers (Continued) 6 9 STEP : Carry out multiplication = STEP : Change improper fraction to a mixed number: EXERCISES. Perform the indicated operations: 7 a) 6 b) 6 c) d) e) f) g) h) 6 i) j) k) 9 6 l) m) n) 0 o) p) q) An I-beam weighs lb per foot. What does a piece feet, inches weigh?.6 How long a piece of inches hexagon stock is needed to produce 7 pieces, each inch long if each cut requires in waste?.7 A hexagon bar of steel is placed between centers on a lathe. The bar is inches long. If the bar is machined for of its length, how many inches of unmachined hexagon bar remains?. A gasoline storage tank with a capacity of gallons is full. How many gallons will be required to fill the tank?.9 How many 6 inches nipples can be cut from a feet length of galvanized pipe? Allow inch per cut. 6

27 COMPLEX FRACTIONS 9.0 Gasoline is selling for 99 0 cents per gallon. How much will it cost to fill a 0 gallon gas tank if there are already gallon in the tank?. Calculate the weight of brass bars, each feet long if brass weighs 9 lb per foot.. A cement contractor has four piles of reinforcing bars. In the first pile there are thirteen inch bars, feet, 6 inches long, weighting lb per foot. The second pile contains twenty-seven inch bars, feet long. The third pile has nine inch bars, weighing lbs per foot, feet long. The fourth pile has thirty-five inch bars, feet, inches long. How many pounds of steel does the contractor have? 6.9 Complex Fractions Complex fractions are built-up fractions, like these: 9 The fractions within the complex fraction may be either proper or improper. It is also possible that one part of the complex fraction is a whole number. A complex fraction may also be composed of mixed numbers. To solve a complex fraction means to simplify it. As illustrated in the examples, there are several ways to simplify a complex fraction. Each method relies on a rule of fraction reduction, forming equivalent fractions, and fraction division learned in this chapter. EXAMPLE.: Complex Fractions Simplify the complex fraction: Solution: Three methods are shown. Method, Reducing by canceling within the fraction = = = = =

28 CHAPTER COMMON FRACTIONS Complex Fractions (Continued) Method, Rewriting the fraction as a division problem. EXAMPLE.:! Recall that any number divided by itself equals! Recall that the reciprocal of a number is the multiplicative inverse of the number: Simply flip the number over. 9 = = = 9 9 Method, Equivalent fraction method. A complex fraction may also be simplified by rewriting it as an equivalent complex fraction whose denominator part is reduced to. Multiplying the fraction in the numerator and in the denominator by the reciprocal of the denominator does this. 9 Since equals, the value of the complex fraction is not changed. Only its form is different. 6 The denominator now becomes, as = =. 6 Thus, 9 = = = EXERCISES. Simplify: 7 6 a) c) e) 9 g) i) b) d) f) h) j) 6

29 COMPLEX FRACTIONS. Determine the center-to-center distance of the equally spaced holes in Figure... Perform the indicated operations: FIGURE. a) 6 b)

30 6 CHAPTER COMMON FRACTIONS