Based on Introduction to Fractions and Multiplication and Division of Fractions: Proper Fractions, Impr. Mixed Numbers *

Transcription

1 OpenStax-CNX module: m7 Derived copy of Introduction to Fractions and Multiplication and Division of Fractions: Proper Fractions, Improper Fractions, and Mixed Numbers * Ann Simao Based on Introduction to Fractions and Multiplication and Division of Fractions: Proper Fractions, Impr This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License.0 Wade Ellis Denny Burzynski Abstract This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses proper fractions, improper fractions, and mixed numbers. By the end of the module students should be able to distinguish between proper fractions, improper fractions, and mixed numbers, convert an improper fraction to a mixed number and convert a mixed number to an improper fraction. Section Overview Positive Proper Fractions Positive Improper Fractions Positive Mixed Numbers Relating Positive Improper Fractions and Positive Mixed Numbers Converting an Improper Fraction to a Mixed Number Converting a Mixed Number to an Improper Fraction Now that we know what positive fractions are, we consider three types of positive fractions: proper fractions, improper fractions, and mixed numbers. * Version.: Aug 7, 20 : am

2 OpenStax-CNX module: m7 2 2 Positive Proper Fractions Positive Proper Fraction Fractions in which the whole number in the numerator is strictly less than the whole number in the denominator are called positive proper fractions. On the number line, proper fractions are located in the interval from 0 to. Positive proper fractions are always less than one. The closed circle at 0 indicates that 0 is included, while the open circle at indicates that is not included. Some examples of positive proper fractions are 2,, 20 06, and 27 2 Positive Improper Fractions Positive Improper Fractions Fractions in which the whole number in the numerator is greater than or equal to the whole number in the denominator are called positive improper fractions. On the number line, improper fractions lie to the right of (and including). Positive improper fractions are always greater than or equal to. Some examples of positive improper fractions are 2,, 0, and 6 Note that 2,,, and 0 6. Positive Mixed Numbers Positive Mixed Numbers A number of the form nonzero whole number + proper fraction is called a positive mixed number. For example, 2 is a mixed number. On the number line, mixed numbers are located in the interval to the right of (and including). Mixed numbers are always greater than or equal to. Relating Positive Improper Fractions and Positive Mixed Numbers A relationship between improper fractions and mixed numbers is suggested by two facts. The rst is that improper fractions and mixed numbers are located in the same interval on the number line. The second fact, that mixed numbers are the sum of a natural number and a fraction, can be seen by making the following observations. Divide a whole quantity into equal parts.

3 OpenStax-CNX module: m7 Now, consider the following examples by observing the respective shaded areas. In the shaded region, there are 2 one thirds, or 2. 2 ( ) = 2 There are one thirds, or, or. ( ) = or Thus, = Improper fraction = whole number. There are one thirds, or, or and. ( ) = or and The terms and can be represented as + or Thus, =. Improper fraction = mixed number. There are one thirds, or, or and 2. ( ) = or and 2 The terms and 2 can be represented as + 2 or 2. Thus, = 2. Improper fraction = mixed number. There are 6 one thirds, or 6, or 2. 6 ( ) = 6 = 2 Thus, 6 = 2 Improper fraction = whole number. The following important fact is illustrated in the preceding examples. Mixed Number = Natural Number + Proper Fraction Mixed numbers are the sum of a natural number and a proper fraction. Mixed number = (natural number) + (proper fraction) For example can be expressed as + The fraction 7 can be expressed as + 7. It is important to note that a number such as + 7 does not indicate multiplication. To indicate multiplication, we would need to use a multiplication symbol (such as ) note: 7 means + 7 and not 7, which means times 7 or multiplied by 7.

4 OpenStax-CNX module: m7 Thus, mixed numbers may be represented by improper fractions, and improper fractions may be represented by mixed numbers. 6 Converting Improper Fractions to Mixed Numbers To understand how we might convert an improper fraction to a mixed number, let's consider the fraction,. = + + }{{ } = + + = Thus, =. We can illustrate a procedure for converting an improper fraction to a mixed number using this example. However, the conversion is more easily accomplished by dividing the numerator by the denominator and using the result to write the mixed number. Converting an Improper Fraction to a Mixed Number To convert an improper fraction to a mixed number, divide the numerator by the denominator.. The whole number part of the mixed number is the quotient. 2. The fractional part of the mixed number is the remainder written over the divisor (the denominator of the improper fraction). 6. Sample Set A Convert each improper fraction to its corresponding mixed number. Example Divide by. The improper fraction = 2. Example 2. Divide 6 by. 6

5 OpenStax-CNX module: m7 The improper fraction 6 =. Example. Divide by. The improper fraction = 7 6. Example Divide 0 by. 0 0 = 26 0 = 26 The improper fraction 0 = Practice Set A Convert each improper fraction to its corresponding mixed number. Exercise (Solution on p..) 2 Exercise 2 (Solution on p..) Exercise (Solution on p..)

6 OpenStax-CNX module: m7 6 Exercise (Solution on p..) Exercise (Solution on p..) 7 Exercise 6 (Solution on p..) 6 7 Converting Mixed Numbers to Improper Fractions To understand how to convert a mixed number to an improper fraction, we'll recall mixed number = (natural number) + (proper fraction) and consider the following diagram. Recall that multiplication describes repeated addition. Notice that can be obtained from 2 using multiplication in the following way. Multiply: = Add: + 2 =. Place the over the : The procedure for converting a mixed number to an improper fraction is illustrated in this example. Converting a Mixed Number to an Improper Fraction To convert a mixed number to an improper fraction,. Multiply the denominator of the fractional part of the mixed number by the whole number part. 2. To this product, add the numerator of the fractional part.. Place this result over the denominator of the fractional part. 7. Sample Set B Convert each mixed number to an improper fraction. Example 7. Multiply: = Add: = 7.. Place 7 over : 7.

7 OpenStax-CNX module: m7 7 Thus, 7 = 7. Example Multiply: 6 =. 2. Add: + 2 = 0.. Place 0 over : 0 Thus, 6 2 = Practice Set B Convert each mixed number to its corresponding improper fraction. Exercise 7 (Solution on p..) Exercise (Solution on p..) Exercise (Solution on p..) Exercise 0 (Solution on p..) Exercises For the following problems, identify each expression as a proper fraction, an improper fraction, or a mixed number. Exercise (Solution on p..) 2 Exercise 2 Exercise (Solution on p..) 7 Exercise Exercise (Solution on p..) 6 Exercise 6 Exercise 7 (Solution on p..),00 2 Exercise

8 OpenStax-CNX module: m7 Exercise (Solution on p..) Exercise Exercise 2 (Solution on p..) 0 Exercise 22 2 Exercise 2 (Solution on p..) 0 Exercise 2 Exercise 2 (Solution on p..) 0 For the following problems, convert each of the improper fractions to its corresponding mixed number. Exercise 26 6 Exercise 27 (Solution on p..) Exercise 2 2 Exercise 2 (Solution on p..) Exercise 0 7 Exercise (Solution on p..) 6 7 Exercise 2 2 Exercise (Solution on p..) 6 2 Exercise 6 Exercise (Solution on p..),000 Exercise 6 2 Exercise 7 (Solution on p..) 7 2 Exercise 2 Exercise (Solution on p..) 6 Exercise 0 00

9 OpenStax-CNX module: m7 For the following problems, convert each of the mixed numbers to its corresponding improper fraction. Exercise (Solution on p. 2.) Exercise 2 2 Exercise (Solution on p. 2.) 6 7 Exercise Exercise (Solution on p. 2.) 0 Exercise 6 0 Exercise 7 (Solution on p. 2.) 2 Exercise Exercise (Solution on p. 2.) 2 2 Exercise Exercise (Solution on p. 2.) 20 2 Exercise 2 6 Exercise (Solution on p. 2.) 0 00 Exercise 00,000 Exercise (Solution on p. 2.) 7 Exercise 6 Why does 0 7 not qualify as a mixed number? Hint: See the denition of a mixed number. Exercise 7 (Solution on p. 2.) Why does qualify as a mixed number? note: See the denition of a mixed number.

10 OpenStax-CNX module: m7 0. Exercises For Review Exercise () Round 2,6,000 to the nearest thousand. Exercise () Determine if,26 is divisible by 2 and. Exercise 60 (Solution on p. 2.) () Find the least common multiple of 2 and 6. Exercise 6 () Specify the numerator and denominator of the fraction 2.

11 OpenStax-CNX module: m7 Solutions to Exercises in this Module Solution to Exercise (p. ) 2 Solution to Exercise (p. ) 2 Solution to Exercise (p. ) Solution to Exercise (p. 6) 2 Solution to Exercise (p. 6) Solution to Exercise (p. 6) improper fraction proper fraction mixed number improper fraction mixed number mixed number proper fraction mixed number 2 2 or 6 2

12 OpenStax-CNX module: m because it may be written as 0 n Solution to Exercise (p. 0) 22, where n is any positive whole number.