KFM02Sr01_Fractions
| |
|Mathematics |
| |
|Fractions |
| [pic] |
|Student Text |
|Rev 1 |
|[pic] |©2003 General Physics Corporation, Elkridge, Maryland |
| |All rights reserved. No part of this book may be reproduced in any form or |
| |by any means, without permission in writing from General Physics |
| |Corporation. |
|TABLE OF CONTENTS |
|FIGURES AND TABLES ii |
|OBJECTIVES iii |
|Definition of Fractions 1 |
|Addition and Subtraction of Fractions 2 |
|Multiplication of Fractions 10 |
|Division of Fractions 11 |
|Reducing Fractions 12 |
|Changing the Form of Fractions 12 |
|Calculator Exercises 14 |
|Addition of Fractions + 14 |
|Subtraction of Fractions ( 17 |
|Multiplication of Fractions ( 19 |
|Division of Fractions ( 20 |
|Summary 22 |
|Practice Exercises 23 |
|GLOSSARY 24 |
|EXAMPLE EXERCISE answerS 25 |
|PRACTICE EXERCISE answerS 30 |
|FIGURES AND TABLES |
|Figure 2-1 Example 2/7 + 3/7 2 |
|Figure 2-2 Example 3/4 2 |
|Figure 2-3 Example 6/8 3 |
|Figure 2-4 Example 1/2 × 1/3 10 |
| |
|No Tables |
|OBJECTIVES |
|Upon completion of this chapter, the student will be able to perform the following objectives at a minimum proficiency level of 80%, unless otherwise|
|stated, on an oral or written exam. |
|DEFINE and GIVE EXAMPLES of: |
|proper fractions |
|improper fractions |
|mixed numbers. |
|CONVERT between proper fractions, improper fractions, and mixed numbers. |
|SOLVE mathematical problems involving fractions by: |
|reducing the fraction |
|solving for the Lowest Common Denominator |
|changing the form of the fraction |
|Without a calculator; ADD, SUBTRACT, MULTIPLY, and DIVIDE fractions (proper fractions, improper fractions, and mixed numbers). |
|With an approved calculator; ADD, SUBTRACT, MULTIPLY, and DIVIDE fractions (proper fractions, improper fractions, and mixed numbers). |
Definition of Fractions
Whole numbers are used for counting; that is, describing the number of objects in a group. However, the result of a measurement need not be a whole number, and in fact, rarely is. The number of pages in this book is by definition a whole number, but the weight of the book in pounds is probably not a whole number. We need a method that can describe the magnitude of numbers that lie between the whole numbers. This is achieved through the use of fractions.
Suppose that we have a circle and we divide the circle into 4 equal parts. How can we mathematically describe one of the parts? A fraction is the ratio of 2 whole numbers, and it indicates division.
Each part of the circle is called “one fourth,” and is denoted by the fraction [pic]. A fraction indicates division, and the dividend is called the numerator. The divisor is called the denominator.
|The fraction [pic] means 1 ( 4. |
|[pic] |
Example 2-1
The meaning of a fraction, then, is that some entity is divided into an equal number of parts indicated by the denominator, and then added as many times as indicated by the numerator.
|What is meant by [pic] of the area of a circle? |
|Divide the circle into 16 equal parts. Combine 9 of these parts. The |
|resulting area will be [pic] of the area of the circle. |
Example 2-2
Suppose the circle we had has a total area of 12 square inches. We still divide the circle into 4 equal parts.
|What will be the area of one of the parts? |
|If the total area is divided by 4, the result will be the area of 1 of |
|the parts. |
|Total Area = 12 square inches |
|Number of parts = 4 |
|Area of one part = 12 ( 4 = [pic] |
|= 3 square inches |
Example 2-3
There are three different types of fractions. A proper fraction is one in which the numerator is less than the denominator, and so has a value less than 1. The fractions [pic], [pic], and [pic] are proper fractions; their values are less than 1.
An improper fraction is one in which the numerator is equal to or greater than the denominator. Its value will be equal to or greater than 1. The fractions [pic], [pic], and [pic] are improper fractions; their values are equal to or greater than 1.
A mixed number consists of a whole number and a fraction. The mixed number [pic] or 3 3/4 means the whole number 3 plus the fraction [pic]. This is the type of number that would arise from the measurement of the width of this page; 8 inches plus [pic] inch equals [pic] or 8 ½ inches.
Addition and Subtraction of Fractions
It is often necessary to add and subtract fractions. In adding and subtracting fractions, it is important to remember first what a fraction represents. It means a division of a unit into a number of equal parts as indicated by the denominator, and then adding these parts as many times as indicated by the numerator.
The fraction [pic] means divide by 7 and add 2 times. The fraction [pic] means divide by 7 and add 3 times. If these 2 fractions are added, the result will be the same as dividing by 7 and adding 5 times. The circle below is divided into 7 equal parts. Two of the parts are lightly shaded. Three of the parts are shaded dark. The sum will be 5 of the parts, or [pic] of the area of the circle.
[pic]
Figure 2-1 Example 2/7 + 3/7
Fractions with the same denominator are added or subtracted by adding or subtracting their numerators and placing the result over the common denominator.
|Add [pic]and [pic]. |
|[pic] |
|Subtract [pic] from [pic] |
|[pic] |
Example 2-4
If the fractions to be added or subtracted do not have the same denominator, they cannot be added or subtracted directly as shown above. They must be altered so that they have the same denominator. Consider the fraction[pic]and the circle shown in Figure 2-2. If we divide the circle into 4 equal parts and add 3 of them, the result will be the area of [pic] of the circle.
[pic]
Figure 2-2 Example 3/4
On the other hand, if we divide the circle into 8 equal parts and shade 6 of them, the result will be [pic] of the area of the circle.
[pic]
Figure 2-3 Example 6/8
Notice that this is exactly the same area as was represented by the fraction [pic]. Therefore, the fraction [pic] is equivalent to the fraction [pic]. If we multiply both the numerator and denominator of a fraction by the same number, we do not change the value of the fraction.
Recall that any number multiplied by one is equal to itself.
|5 ( 1 = 5 |
|[pic] |
Example 2-5
Also any number divided by itself is equal to one.
|[pic] |
|[pic] |
Example 2-6
Therefore any number (or fraction) multiplied by 1 (or a number divided by itself) is still equal to the original number.
|[pic] |
|[pic] |
|[pic] |
|[pic] |
Example 2-7
This is the method that must be used when adding or subtracting fractions with different denominators. One of the fractions is altered to an equivalent form so that its denominator is the same as the denominator of the fraction to which it is to be added (or subtracted). The numerators are then added (or subtracted) and placed over the common denominator, as before.
|Add [pic] |
|[pic] |
|[pic] |
Example 2-8
Note that [pic]+ [pic] is NOT equal to [pic].
What we have done in Example 2-8 is to find the lowest common denominator (LCD). The LCD is the smallest number that can be divided by all the denominators in a problem involving several fractions. In the above example, the number 6 is the LCD, since it is the smallest number that can be divided by 6 and 3.
|What is [pic]? |
|The lowest common denominator is 24 |
|[pic] |
|[pic] |
|[pic] |
|[pic] |
|[pic] |
|[pic] |
Example 2-9
The easiest way to solve for the LCD is to factor each denominator to its smallest factors.
Step 1. Determine the smallest factors of each of the denominators to be added or subtracted.
Step 2. Determine the Least Common Denominator (LCD) by determining how many times each factor must be used.
Step 3. Multiply together all of the factors of the Least Common Denominator (LCD).
Step 4. For each denominator to be added or subtracted determine which factor(s) the denominator must be multiplied by to reach the LCD.
Step 5. For each fraction, create a fraction equal to one with each factor required to make the denominator equal to the LCD.
Step 6. Multiply the numerator and denominator of each fraction by the required factor of the LCD.
Step 7. With all fractions now with the same denominator (the LCD) add and subtract the numerators over the least common denominator.
|Add the fractions [pic] |
|Step 1. Determine the smallest factors of each of the denominators to|
|be added or subtracted. |
|4 has factors of 2 × 2 |
|3 has factors of 1 × 3 |
|Step 2. Determine the Least Common Denominator (LCD) by determining |
|how many times each factor must be used. |
|The LCD must have factors of 2 × 2 × 3. |
|Denominator 1 |Number of times |
|Factors of 4 |2 × 2 |
|2 || | |
|Denominator 2 |Number of times |
|Factors of 3 |1 × 3 |
|3 || |
|The LCD must have 2 twos and 1 three. |
|Step 3. Multiply together all of the factors of the Least Common |
|Denominator (LCD). |
|The LCD is equal to 2 × 2 × 3 = 12 |
|(Cont'd in next Column) |
|Step 4. For each denominator to be added or subtracted determine |
|which factor(s) the denominator must be multiplied by to reach the |
|LCD. |
|Least Common Denominator |
|Factors of LCD 12 |2 × 2 × 3 |
|Denominator 1 | |
|Factors of 4 |2 × 2 |
|Missing factors |3 |
|Factors of LCD 12 |2 × 2 × 3 |
|Denominator 2 | |
|Factors of 3 |1 × 3 |
|Missing factors |2 × 2 |
|The first fraction’s denominator must be multiplied by 3 to reach the |
|LCD. |
|The second fraction’s denominator must be multiplied by 2 × 2 to reach|
|the LCD. |
|Step 5. For each fraction, create a fraction equal to one with each |
|factor required to make the denominator equal to the LCD. |
|The first fraction [pic] must be multiplied by [pic]. |
|The second fraction[pic] must be multiplied by [pic] |
|(Cont'd in next Column) |
|Step 6. Multiply the numerator and denominator of each fraction by |
|the required factor of the LCD. |
|The first fraction becomes [pic]. |
|The second fraction becomes [pic] |
|Step 7. With all fractions now with the same denominator (the LCD) |
|add and subtract the numerators over the least common denominator. |
|[pic] |
Example 2-10
|Add the fractions [pic]. |
|Step 1. Determine the smallest factors of each of the denominators to|
|be added or subtracted. |
| |
|Step 2. Determine the Least Common Denominator (LCD) by determining |
|how many times each factor must be used. |
|Denominator 1 | |
|Factors | |
| | |
| | |
| | |
|Denominator 2 | |
|Factors of | |
| | |
| | |
| | |
| | |
| |
|Step 3. Multiply together all of the factors of the Least Common |
|Denominator (LCD). |
|The LCD is equal to |
|(Cont'd in next Column) |
|Step 4. For each denominator to be added or subtracted determine |
|which factor(s) the denominator must be multiplied by to reach the |
|LCD. |
|Least Common Denominator |
|Factors of LCD | |
|Denominator 1 | |
|Factors of | |
|Missing factors | |
|Factors of LCD | |
|Denominator 2 | |
|Factors of | |
|Missing factors | |
| |
| |
| |
|Step 5. For each fraction, create a fraction equal to one with each |
|factor required to make the denominator equal to the LCD. |
|The first fraction [pic] must be multiplied by [pic]. |
|The second fraction [pic] must be multiplied by [pic] |
|(Cont'd in next Column) |
|Step 6. Multiply the numerator and denominator of each fraction by |
|the required factor of the LCD. |
|The first fraction becomes |
|The second fraction becomes |
|Step 7. With all fractions now with the same denominator (the LCD) |
|add and subtract the numerators over the least common denominator. |
| |
Example 2-11
|Calculate the sum of [pic]. |
|Step 1. Determine the smallest factors of each of the denominators to|
|be added or subtracted. |
|Denominator 1 has factors of |
|Denominator 2 has factors of |
|Denominator 3 has factors of |
|Step 2. Determine the Least Common Denominator (LCD) by determining |
|how many times each factor must be used. |
|Denominator 1 |Number of times |
|Factors of | |
| | |
| | |
|Denominator 2 |Number of times |
|Factors of | |
| | |
|Denominator 3 |Number of times |
|Factors of | |
| | |
|The LCD must have |
|Step 3. Multiply together all of the factors of the Least Common |
|Denominator (LCD). |
|(Cont'd in next Column) |
|Step 4. For each denominator to be added or subtracted determine |
|which factor(s) the denominator must be multiplied by to reach the |
|LCD. |
|Least Common Denominator |
|Factors of LCD | |
|Denominator 1 | |
|Factors of | |
|Missing factors | |
|Factors of LCD | |
|Denominator 2 | |
|Factors of | |
|Missing factors | |
|Factors of LCD | |
|Denominator 3 | |
|Factors of | |
|Missing factors | |
|The first fraction’s denominator must be multiplied by |
|The second fraction’s denominator must be multiplied by |
|The third fraction’s denominator must be multiplied by |
|(Cont'd in next Column) |
|Step 5. For each fraction, create a fraction equal to one with each |
|factor required to make the denominator equal to the LCD. |
|The first fraction [pic] must be multiplied by [pic] |
|The second fraction [pic] must be multiplied by [pic] |
|The third fraction [pic] must be multiplied by [pic] |
|Step 6. Multiply the numerator and denominator of each fraction by |
|the required factor of the LCD. |
|The first fraction becomes |
|The second fraction becomes |
|The third fraction becomes |
|Step 7. With all fractions now with the same denominator (the LCD) |
|add and subtract the numerators over the least common denominator. |
Example 2-12
Alternate solution
|Calculate the sum of [pic]. |
|12 = 2 ( 2 ( 3 |
|16 = 2 ( 2 ( 2 ( 2 |
|8 = 2 ( 2 ( 2 |
|To determine the LCD each denominator must contain all factors common to|
|all the other denominators. |
|A simple way to do this is to multiply all the denominators together. |
|12 ( 16 ( 8 = 1,536 |
|However this will not give you the smallest value and would require |
|extra multiplication. |
|Determine the maximum number of times a factor appears in a factor. In |
|this example that means four 2’s and one 3. Thus the LCD is 2 ( 2 ( 2 (|
|2 ( 3 = 48. |
|To convert each fraction to a fraction with the LCD determine which |
|factor(s) is missing from the current denominator |
|12 = 2 ( 2 ( 3 is missing 2(2 |
|16 = 2 ( 2 ( 2 ( 2 is missing 3 |
|8 = 2 ( 2 ( 2 is missing 2(3 |
|(Cont'd in next Column) |
|Multiply each fraction by one in the form of the missing factors divided|
|by the missing factors. |
|12 missing 2 ( 2 or 4 |
|[pic] |
|16 missing 3 |
|[pic] |
|8 missing 2 ( 3 or 6 |
|[pic] |
|Now, with all the denominators the same perform the required |
|mathematical operation. |
|[pic] |
|Convert to a mixed fraction and simplify. |
|[pic] |
|Since 7 has factors of 1 ( 7 and 48 has factors of 2 ( 2 ( 2 ( 2 ( 3 and|
|there are no common factors in this case the fraction is simplified. |
Example 2-13
When fractions are to be added or subtracted, their LCD must be found.
|If we have to remove [pic] inch from a piece of wood [pic] inches wide, |
|what will the resulting width be? This would be determined by |
|subtracting [pic] from [pic]. |
|[pic] |
| |
| |
| |
Example 2-14
Multiplication of Fractions
Multiplying the numerators together to obtain the numerator of the product, and multiplying the denominators together to obtain the denominator of the product multiply fractions.
|Multiply [pic] and [pic]. |
|[pic] |
Example 2-15
|Multiply [pic] and [pic]. |
| |
Example 2-16
Note that the numerators are multiplied directly, as are the denominators. There is no need to find a lowest common denominator, as in the case of addition or subtraction.
The multiplication of [pic] and [pic] can be visualized with regard to the area of the circle. We first divide the area of the circle into two equal parts. This accounts for the fraction [pic]. This area is now divided into three equal parts. This accounts for the fraction [pic]. The final area is [pic] the area of the circle.
[pic]
Figure 2-4 Example 1/2 × 1/3
Division of Fractions
Fractions may be divided by utilizing the rule that the value of a fraction is unchanged if both the numerator and denominator are multiplied by the same number. The operation [pic]divided by [pic] is indicated as:
[pic]
If we multiply numerator and denominator by 3, we obtain:
[pic]
If we now multiply numerator and denominator by 2, we obtain:
[pic]
This is the value of the quotient of 1/2 ( 1/3.
Simply inverting the denominator and multiplying the result by the numerator obtains the same result.
[pic]
The process of inverting means make the numerator the denominator and vice versa. Thus if we invert[pic], we obtain [pic].
|[pic] |
|[pic] |
|[pic] |
Example 2-17
Reducing Fractions
In dealing with fractions, it is normal procedure to express the fraction in such a manner that the numerator and denominator are as small as possible. This is known as reducing a fraction to its lowest terms, and a fraction is said to be reduced to its lowest terms when the numerator and denominator have no common factor other than 1.
Reducing a fraction to its lowest terms is made possible by the fact that the value of the fraction is unchanged if both the numerator and denominator are divided by the same number. Although the fraction[pic] is a perfectly proper fraction, it can be reduced to the fraction [pic] by dividing the numerator and denominator by 2. The fraction is now reduced to its lowest terms since the numerator and denominator have no common factor other than 1.
|Reduce the fraction [pic] to its lowest terms. |
|[pic] |
Example 2-18
Finding the common factors for the purpose of fraction reduction is normally done by a trial-and-error procedure.
To reduce a fraction to its lowest terms factor both the numerator and the denominator into their smallest values.
|Reduce the fraction [pic] to its lowest terms. |
|[pic] |
|Cancel out any factor found in both terms. |
|[pic] |
|That will result in the fraction being reduced to its lowest terms. |
|[pic] |
Example 2-19
Changing the Form of Fractions
In performing operations with fractions, it is sometimes necessary to alter them to an equivalent form for ease of computation. For example, if we wished to multiply the two mixed numbers, 2[pic]and 7[pic], it becomes much easier if we change the mixed numbers to improper fractions first. A mixed number can be changed to an improper fraction by recognizing that a mixed number is just the sum of a whole number and a fraction.
|Change the mixed number 4[pic]to an improper fraction. |
|[pic] |
Example 2-20
Notice in this example that we found the LCD of the whole number and the fraction, and then added them. Multiplying the whole number by the denominator of the fraction and then adding the numerator of the fraction to the result obtains the same result. The total is then placed over the denominator of the fraction.
|Change the mixed number [pic] to an improper fraction. |
| |
Example 2-21
We can now multiply two mixed numbers together.
|Multiply 2[pic]by 7[pic]. |
|[pic] |
|[pic] |
|[pic] |
|Reduced to lowest terms. |
|[pic] is the answer but it should be checked to determine if it is |
|reduced to lowest terms and converted into a mixed number. |
|(Cont'd in next Column) |
|First factor both the numerator and denominator into the smallest terms |
|[pic] |
|In this case the only factor in both terms is a three. Cancel out terms|
|that are in the numerator and denominator. Multiply the remaining |
|factors together in each term. |
|[pic] |
|The fraction is in its lowest terms. Now it must be converted into a |
|mixed number. |
|To convert into a mixed number divide the denominator into the |
|numerator. |
|Change the improper fraction [pic] to a mixed number. |
|133 ( 8 = 16 with a remainder of 5. |
|[pic] |
|So the mixed number becomes [pic]. |
Example 2-22
An improper fraction can be changed to a mixed number by dividing the numerator by the denominator, with any remainder placed over the denominator as the proper fraction.
|Change the improper fraction [pic] to a mixed number. |
| |
| |
Example 2-23
Calculator Exercises
The calculator can be used for solving addition, subtraction, multiplication, and division of fractions. The following are examples of its use.
Addition of Fractions +
|Find the sum of [pic]and [pic]. |
|[pic] |
|Since the denominator is the same, add the numerators and place the sum |
|over the common denominator. |
|[pic] |
|Enter Operation Display |
|2 + 2 |
|3 = 5 |
|The common denominator is 7. |
|Therefore, |
|[pic] |
Example 2-24
|Find the sum of [pic]and [pic]and [pic]. |
|Since the denominators are the same, add the numerators and place the |
|sum over the common denominator. |
|[pic] |
|[pic] |
|Enter Operation Display |
|3 + 3 |
|2 + 5 |
|4 = 9 |
|The common denominator is 5. |
|Therefore, |
| [pic] |
Example 2-25
Solving equations with a fraction on a calculator does not necessarily require using all the rules presented in this chapter. Entering the numbers into the calculator following the proper rules for the calculator will provide you with the correct answer but it will probably be in decimal form. So there are several steps to convert from decimal to fraction form.
|Find the sum of [pic] and [pic]. |
|Enter Operation Display |
|2 ( 2 |
|7 + 0.2857143 |
|3 ( 3 |
|7 = 0.7142857 |
|2 ( 7 + 3 ( 7 = 0.7142857 |
|In a case where the denominator for both fractions is the same, multiply|
|the resultant answer by the denominator to obtain the fractional number.|
|0.7142857 × 7 = 5 |
|Then place the answer (5) over the common denominator (7) to obtain the |
|answer [pic]. |
Example 2-26
|Find the sum of [pic], [pic], and[pic]. |
|Enter Operation Display |
|3 ( 3 |
|5 + .6 |
|2 ( 2 |
|5 + 1 |
|4 ( 4 |
|5 = 1.8 |
|3 ( 5 + 2 ( 5 + 4 ( 5 = 1.8 |
|Since in this case the denominator for all the fractions is 5, multiply |
|the resultant answer by 5 to obtain the fractional number. |
|1.8 × 5 = 9 |
|Then place the answer (9) over the common denominator (5) to obtain the |
|answer [pic]. |
|Unless told not to, convert all improper fractions into mixed numbers. |
|[pic] |
|The sum of [pic], [pic], and [pic] is [pic]. |
Example 2-27
You can solve fractions on the calculator with fractions that have different denominators. However, the answer will result in a decimal number that may not be easily converted into a fraction.
To solve addition of fractions without concern for keeping the answer as a fraction, simply enter all the mathematical operations into the calculator in the order that they occur.
|Find the sum of [pic]. |
|Enter Operation Display |
|1 ( 1 |
|4 + .25 |
|2 ( 2 |
|3 ( 0.9166667 |
|3 ( 3 |
|8 = 0.5416667 |
Example 2-28
Subtraction of Fractions (
|Find the difference between [pic] and[pic]. |
|[pic] |
|Since the denominators are the same, subtract the numerators and place |
|the difference over the common denominator. |
|[pic] |
|Enter Operation Display |
|9 ( 9 |
|7 = 2 |
|For the numerator: |
|The common denominator is 11. |
|Therefore, |
|[pic] |
Example 2-29
|Find the difference between [pic]and[pic]. |
|[pic] |
|To find the lowest common denominator: |
|Enter Operation Display |
|9 ( 9 |
|3 = 3 |
|3 is the common factor of both denominators. Therefore, the lowest |
|common denominator of the two fractions is 9. Hence, in order to |
|subtract these two fractions, the fraction [pic] must have both its |
|numerator and denominator multiplied by 3. |
|To convert the numerator of [pic]: |
|Enter Operation Display |
|2 ( 2 |
|3 = 6 |
|To convert the denominator of [pic]: |
|Enter Operation Display |
|3 ( 3 |
|3 = 9 |
|(Cont'd in next Column) |
|The result is |
|[pic] |
|Therefore, |
|[pic] |
| |
| |
|To solve the numerator of the subtraction: |
|Enter Operation Display |
|14 ( 14 |
|6 ( 8 |
|Place the difference (8) over the common denominator (9). |
|Therefore, |
|[pic] |
Example 2-30
Solving subtraction of fractions using the calculator follows the same basic rules as addition.
|Solve for the difference between [pic] and [pic]. |
|Enter Operation Display |
|14 ( 14 |
|9 ( 1.5555556 |
|2 ( 2 |
|3 = 0.8888889 |
Example 2-31
Multiplication of Fractions (
|Find the product of [pic]and[pic]. |
|[pic] |
|[pic] |
|To solve for the numerator multiply 2 by 5: |
|Enter Operation Display |
|2 ( 2 |
|5 = 10 |
|To solve for the denominator multiply 3 by 7: |
|Enter Operation Display |
|3 ( 3 |
|7 = 21 |
|Place the new numerator over the new denominator. |
|Therefore, [pic] |
Example 2-32
|Find the product of [pic]and[pic]. |
|[pic] |
|[pic] |
|To solve for the numerator multiply 1 by 1: |
|Enter Operation Display |
|1 ( 1 |
|1 = 1 |
|To solve for the denominator multiply 2 by 3: |
|Enter Operation Display |
|2 ( 2 |
|3 = 6 |
|Place the new numerator over the new denominator. |
|Therefore, [pic] |
Example 2-33
You can also solve multiplication of fractions using the calculator without concerns for maintaining fraction form.
|Find the product of [pic] and [pic]. |
|Enter Operation Display |
|2 ( 2 |
|3 ( 0.6666667 |
|5 ( 3.33333… |
|7 = 0.4761905 |
Example 2-34
Or you can multiply the numerators together and then divide through by the denominators.
|Find the product of [pic] and [pic]. |
|Enter Operation Display |
|2 ( 2 |
|5 ( 10 |
|3 ( 3.33333… |
|7 = 0.4761905 |
Example 2-35
Division of Fractions (
|Find the quotient of [pic]divided by[pic]. |
|[pic] |
|[pic] |
|To solve for the numerator multiply the first fraction by the |
|denominator of the second fraction: |
|[pic] |
|The numerator becomes: |
|Enter Operation Display |
|1 ( 1 |
|3 = 3 |
|The denominator becomes: |
|Enter Operation Display |
|2 ( 2 |
|1 = 2 |
|Hence, the numerator becomes: |
|[pic] |
|(Cont'd in next Column) |
|For the denominator: |
|[pic] |
|The numerator becomes: |
|Enter Operation Display |
|1 ( 1 |
|3 = 3 |
|The denominator becomes: |
|Enter Operation Display |
|3 ( 1 |
|1 = 3 |
|Hence, the denominator becomes: |
|[pic] |
|The net result is then: |
|[pic] |
|Therefore, [pic] |
Example 2-36
Dividing fractions by fractions on a calculator typically requires an additional step of grouping the mathematical operations together using parentheses.
|Find the quotient of [pic]divided by [pic]. |
|[pic] |
|Enter Operation Display |
|1 ( 1 |
|2 ( ( 0.5 |
|1 ( 1 |
|3 ) 0.333 |
| = 1.5 |
Example 2-37
Alternately, you can follow the rules of dividing fractions and invert and multiply the divisor (second) fraction.
|Find the quotient of [pic]divided by [pic]. |
|[pic] |
|Multiply the numerators then divide through by the denominators. |
|Enter Operation Display |
|1 ( 1 |
|3 ( 3 |
|2 ( 1.5 |
|1 = 1.5 |
Example 2-38
Summary
Fractions
• Numerator – top number in a fraction
• Denominator – bottom number in a fraction
• Proper fraction – numerator is less than denominator
• Improper fraction – numerator is greater than or equal to denominator
• Mixed number – sum of an integer and a proper fraction
• Fractions, like whole numbers can be:
a. Added
b. Subtracted
c. Multiplied
d. Divided
To add or subtract fraction the denominator must be the same:
Add [pic].
[pic]
[pic]
Subtract [pic]
[pic]
[pic]
[pic]
To multiply fractions, multiply the numerator by the numerator and multiply the denominator by the denominator:
Multiply [pic] and [pic].
[pic]
To divide fractions invert the fraction and multiply using the rules above.
[pic]
After solving problems with fractions, reduce all fractions to lowest terms:
Reduce the fraction [pic] to its lowest terms.
[pic]
Cancel out any factor found in both terms.
[pic]
That will result in the fraction being reduced to its lowest terms.
To solve problems with mixed numbers, multiply the whole number by the denominator and ad to the numerator. Place the sum over the denominator:
Change the mixed number 4[pic]to an improper fraction.
[pic]
Practice Exercises
1. Indicate whether the following numbers are proper fractions, improper fractions, or mixed numbers.
a. [pic] b. [pic] c. [pic]
d. [pic] e. [pic] f. [pic]
g. [pic] h. [pic] i. [pic]
j. [pic] k. [pic] l. [pic]
m. [pic] n. [pic] o. [pic]
2. Change all of the improper fractions in exercise 1 to mixed numbers.
3. Change all of the mixed numbers in exercise 1 to improper fractions.
4. Reduce the following fractions:
a. [pic] b. [pic] c. [pic]
d. [pic] e. [pic] f. [pic]
g. [pic] h. [pic] i. [pic]
j. [pic] k. [pic] l. [pic]
m. [pic] n. [pic] o. [pic]
5. Compute the following:
a. [pic] b. [pic] c. [pic]
d. [pic] e. [pic] f. [pic]
g. [pic] h. [pic] i. [pic]
j. [pic] k. [pic] l. [pic]
6. One branch in a fluid piping system carries [pic] of the total system flow. If the total system flow is 27,000 lbm. per hour, what is the flow in the branch?
7. A chemistry lab has a stock of 175 bottles of chemicals. 60 of these bottles contain sulfuric acid. 55 of them contain hydrochloric acid. What fraction of the bottles contain sulfuric acid? What fraction contains hydrochloric acid?
8. The outside diameter of a pipe is 4[pic] inches; the pipe wall thickness is [pic] inch. What is the inside diameter of the pipe?
|GLOSSARY |
|Denominator |The divisor of a fraction. The bottom number in a fraction. |
| |Denominator - Down |
|Fraction |The ratio of two whole numbers. It indicates division. |
|Improper fraction |A fraction where the numerator is equal to or greater than the denominator. |
|Lowest common denominator (LCD) |The smallest number that can be divided by all the denominators in a problem involving several fractions. |
|Mixed number |A number consisting of a whole number and a fraction. |
|Numerator |The dividend of a fraction. The top number in a fraction. |
|Proper fraction |A fraction where the numerator is less than the denominator, and so has a value less than 1. |
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EXAMPLE EXERCISE answerS
|Add the fractions [pic]. |
|Step 1. Determine the smallest factors of each of the denominators to|
|be added or subtracted. |
|12 has factors of 2 × 2 × 3 |
|21 has factors of 2 × 3 × 7 |
|Step 2. Determine the Least Common Denominator (LCD) by determining |
|how many times each factor must be used. |
|Denominator 1 |Number of times |
|Factors of 12 |2 × 2 × 3 |
|2 || | |
|3 || |
|Denominator 2 |Number of times |
|Factors of 42 |2 × 3 × 7 |
|2 || |
|3 || |
|7 || |
|The LCD must have 2 twos, 1 three, and 1 seven. |
|(Cont'd in next Column) |
|Step 3. Multiply together all of the factors of the Least Common |
|Denominator (LCD). |
|The LCD is equal to 2 × 2 × 3 ×7 = 84 |
|Step 4. For each denominator to be added or subtracted determine |
|which factor(s) the denominator must be multiplied by to reach the |
|LCD. |
|Least Common Denominator |
|Factors of LCD 84 |2 × 2 × 3 × 7 |
|Denominator 1 | |
|Factors of 12 |2 × 2 × 3 |
|Missing factors |7 |
|Factors of LCD 84 |2 × 2 × 3 × 7 |
|Denominator 2 | |
|Factors of 42 |2 × 3 × 7 |
|Missing factors |2 |
|The first fraction’s denominator must be multiplied by 7 to reach the |
|LCD. |
|The second fraction’s denominator must be multiplied by an additional |
|2 to reach the LCD. |
|(Cont'd in next Column) |
|Step 5. For each fraction, create a fraction equal to one with each |
|factor required to make the denominator equal to the LCD. |
|The first fraction [pic] must be multiplied by [pic]. |
|The second fraction[pic] must be multiplied by [pic] |
|Step 6. Multiply the numerator and denominator of each fraction by |
|the required factor of the LCD. |
|The first fraction becomes [pic] |
|The second fraction becomes [pic] |
|Step 7. With all fractions now with the same denominator (the LCD) |
|add and subtract the numerators over the least common denominator. |
|[pic] |
Example 2-11
|Calculate the sum of [pic]. |
|Step 1. Determine the smallest factors of each of the denominators to|
|be added or subtracted. |
|12 has factors of 2 × 2 × 3 |
|16 has factors of 2 × 2 × 2 × 2 |
|8 has factors of 2 × 2 × 2 |
|Step 2. Determine the Least Common Denominator (LCD) by determining |
|how many times each factor must be used. |
|Denominator 1 |Number of times |
|Factors of 12 |2 × 2 × 3 |
|2 || | |
|3 || |
|Denominator 2 |Number of times |
|Factors of 16 |2 × 2 × 2 × 2 |
|2 || | | | |
|Denominator 3 |Number of times |
|Factors of 8 |2 × 2 × 2 |
|2 || | | |
|The LCD must have 4 twos and 1 three. |
|(Cont'd in next Column) |
|Step 3. Multiply together all of the factors of the Least Common |
|Denominator (LCD). |
|The LCD is equal to 2 × 2 × 2 × 3 = 48 |
|Step 4. For each denominator to be added or subtracted determine |
|which factor(s) the denominator must be multiplied by to reach the |
|LCD. |
|Least Common Denominator |
|Factors of LCD 48 |2 × 2 × 2 × 2 × 3 |
|Denominator 1 | |
|Factors of 12 |2 × 2 × 3 |
|Missing factors |2 × 2 |
|Factors of LCD 48 |2 × 2 × 2 × 2 × 3 |
|Denominator 2 | |
|Factors of 16 |2 × 2 × 2 × 2 |
|Missing factors |3 |
|Factors of LCD 48 |2 × 2 × 2 × 2 × 3 |
|Denominator 3 | |
|Factors of 8 |2 × 2 × 2 |
|Missing factors |2 × 3 |
|(Cont'd in next Column) |
|The first fraction’s denominator must be multiplied by 2 × 2 to reach |
|the LCD. |
|The second fraction’s denominator must be multiplied by an additional |
|3 to reach the LCD. |
|The third fraction’s denominator must be multiplied by an additional 2|
|× 3 to reach the LCD. |
|Step 5. For each fraction, create a fraction equal to one with each |
|factor required to make the denominator equal to the LCD. |
|The first fraction [pic] must be multiplied by [pic]. |
|The second fraction[pic] must be multiplied by [pic]. |
|The third fraction[pic] must be multiplied by [pic]. |
|Step 6. Multiply the numerator and denominator of each fraction by |
|the required factor of the LCD. |
|The first fraction becomes [pic]. |
|The second fraction becomes [pic] |
|The third fraction becomes [pic] |
|(Cont'd in next Column) |
|Step 7. With all fractions now with the same denominator (the LCD) |
|add and subtract the numerators over the least common denominator. |
|[pic] |
Example 2-12
|If we have to remove [pic] inch from a piece of wood [pic] inches wide, |
|what will the resulting width be? This would be determined by |
|subtracting [pic] from [pic]. |
|[pic] |
|[pic] |
|[pic] |
|[pic] |
Example 2-14
|Multiply [pic] and [pic]. |
|[pic] |
Example 2-16
|[pic] |
|[pic] |
|[pic] |
Example 2-17
|Reduce the fraction [pic] to its lowest terms. |
|[pic] |
|Cancel out any factor found in both terms. |
|[pic] |
|That will result in the fraction being reduced to its lowest terms. |
|[pic] |
Example 2-19
|Change the mixed number [pic] to an improper fraction. |
|[pic] |
Example 2-21
|Change the improper fraction [pic] to a mixed number. |
|[pic] and a remainder of 4 |
|[pic] |
Example 2-23
PRACTICE EXERCISE answerS
|1. |a. |[pic] |b. |[pic] |c. |[pic] |
| | |proper fraction | |improper fraction | |improper fraction |
| |d. |[pic] |e. |[pic] |f. |[pic] |
| | |proper fraction | |mixed number | |proper fraction |
| |g. |[pic] |h. |[pic] |i. |[pic] |
| | |mixed number | |improper fraction | |mixed number |
| |j. |[pic] |k. |[pic] |l. |[pic] |
| | |improper fraction | |proper fraction | |proper fraction |
| |m. |[pic] |n. |[pic] |o. |[pic] |
| | |mixed number | |proper fraction | |improper fraction |
|2. |b. |[pic] |3. |e. |[pic] |
| |c. |[pic] | |g. |[pic] |
| |h. |[pic] | |i |[pic] |
| |j. |[pic] | |m. |[pic] |
| |o. |[pic] | | | |
|4. |a. |[pic] or | |b. |[pic] or |
| | |[pic] | | |[pic] |
| |c. |[pic] or | |d. |[pic] or |
| | |[pic] | | |[pic] |
| |e. |[pic] or | |f. |[pic] or |
| | |[pic] | | |[pic] |
| |g. |[pic] or | |h. |[pic] or |
| | |[pic] | | |[pic] |
| |i. |[pic] or | |j. |[pic] or |
| | |[pic] | | |[pic] |
| |k. |[pic] or | |l. |[pic] or |
| | |[pic] | | |[pic] |
| |m. |[pic] or | |n. |[pic] or |
| | |[pic] | | |[pic] |
| |o. |[pic] or | | | |
| | |[pic] | | | |
|5. |a. |[pic][pic] |
| |b. |[pic][pic] |
| |c. |[pic][pic] |
| |d. |[pic][pic] |
| |e. |[pic][pic] |
| |f. |[pic] |
| |g. |[pic][pic] |
| |h. |[pic][pic] |
| |i. |[pic] |
| |j. |[pic][pic] |
| |k. |[pic][pic] |
| |l. |[pic][pic] |
6. Flow in the branch = [pic] of the total system flow.
Flow in the branch = [pic]lbm per hour.
Flow in the branch = 4,500 lbm per hour.
7. Fraction of Bottles with Sulfuric Acid [pic]
[pic] or
[pic]
Fraction of Bottles with Hydrochloric Acid [pic]
[pic] or
[pic]
8. Inside Diameter = Outside Diameter – 2 ( (Wall Thickness)
Inside Diameter [pic]
Inside Diameter [pic]
Inside Diameter [pic]
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