Copyright © 2010 Pearson Education, Inc. All rights reserved. R.1 – Slide 1.


Prealgebra Review


R.1

Fractions


Objectives

1. Identify prime numbers.

2. Write numbers in prime factored form.

3. Write fractions in lowest terms.

4. Convert between improper fractions and mixed numbers.

5. Multiply and divide fractions.

6. Add and subtract fractions.

7. Solve applied problems that involve fractions.

R.1 Fractions


R.1 Fractions

A fraction is the quotient of two whole numbers. For example: 3

5Numerator

DenominatorFraction bar

If the numerator of a fraction is less than the denominator, we call it a proper fraction. A proper fraction has a value less than 1. If the numerator is greater than or equal to the denominator, the fraction is an improper fraction. An improper fraction that has a value greater than 1 is often written as a mixed number. For example,

.7 1

may be written as 23 3

Improper fraction Mixed number


R.1 Fractions

A whole number is prime if it has exactly two different factors (itself and 1).

Identifying Prime Numbers

A whole number greater than 1 that is not prime is called a composite number.

The first dozen primes are listed here. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37

Some example of composite numbers are 4, 6, 8, 9, 10, 12.

The number 1 is neither prime nor composite.


Example 1 Decide whether each number is prime or composite.

R.1 FractionsIdentifying Prime Numbers

(a) 51

51 is composite, since 51 = 3 · 17.

(b) 97

97 is prime because its only factors are 1 and 97.

(c) 8314

8314 is composite since it has 2 as a factor. 2 · 4157 = 8314


R.1 FractionsWriting Numbers in Prime Factored Form

To factor a number means to write it as the product of two or more numbers. Factoring is the reverse of multiplying two numbers to get the product.

A composite number written using factors that are all prime numbers is in prime factored form.


Example 2 Write each number in prime factored form.

R.1 FractionsWriting Numbers in Prime Factored Form

a) 28

We use a factor tree, as shown below.

28 = 2 · 2 · 7

7

2 2

28

4

b) 36

We use a factor tree, as shown below.

36 = 2 · 2 · 3 · 3

36

4 9

2 2 3 3


R.1 FractionsWriting Fractions in Lowest Terms

A fraction is in lowest terms when the numerator and denominator have no factors in common (other than 1).

Properties of 1Any nonzero number divided by itself is equal to 1;

for example,

Any number multiplied by 1 remains the same; for example, 7 · 1 = 7.

.3

13



Writing a Fraction in Lowest Terms

Step 1 Write the numerator and denominator in prime factored form.

Step 2 Replace each pair of factors common to the numerator and denominator with 1.

Step 3 Multiply the remaining factors in the numerator and in the denominator.

(This procedure is sometimes called “simplifying the fraction.”)


Example 3 Write each fraction in lowest terms.


15a)

21

15 3 3

21 7

51

75

3

7

36b)

60

36 3 3

60 5

121

12 5

3

5



NoteWhen writing fractions in lowest terms, look for the largest common factor in the numerator and the denominator. If none is obvious, factor the numerator and the denominator into prime factors. Any common factor can be used and the fraction can be simplified in stages. For example, .

36 6 6 6 3 3 3

60 10 10 10 51

6

6 5

2

2 51


Example 4

Write as a mixed number.

R.1 FractionsConverting Between Improper Fractions and Mixed Numbers

79 67

633

67

94

79

67

9To convert an improper fraction to a mixed number, divide the numerator by the denominator. Here, divide 67 by 9. Use the quotient and remainder to form the mixed number.


Example 5

Write as an improper fraction.

R.1 FractionsConverting Between Improper Fractions and Mixed Numbers

34

523

5

34

5To convert a mixed number to an improper fraction, multiply the denominator of the fraction by the whole number and add the numerator of the fraction to get the numerator of the improper fraction. To write as an improper fraction, the numerator is

The denominator of the improper fraction is the same as the denominator in the mixed number. The denominator is 5.

34

55 4 3 20 3 23.


R.1 FractionsMultiplying and Dividing Fractions

Multiplying FractionsTo multiply two fractions, multiply the numerators to get the numerator of the product, and multiply the denominators to get the denominator of the product. The product should be written in lowest terms.


Example 6 Find the product, and write it in lowest terms.


5 4

8 15

3

5 4

52 4

1

6

5 4

8 15

1

2 3



Dividing FractionsTo divide two fractions, multiply the first fraction by the reciprocal of the second. The result, called the quotient, should be written in lowest terms.


Multiply by the reciprocal of the second fraction.

Example 7Find the quotient, and write it in lowest terms.


7 21

12 16

7 7 7 16 4

12 12 12 21

7

73 3

4

4

21 16

16 21

4

9


R.1 FractionsAdding and Subtracting Fractions

Adding FractionsTo find the sum of two fractions with the same denominator, add their numerators and keep the same denominator.

If the fractions have different denominators, write them with a common denominator first.


Example 8 Add. Write sum in lowest terms.

R.1 Fractions

3 1

8 8

Adding and Subtracting Fractions

Add numerators; keep the same denominator.

3

8 8

1

8

43

8

1 1

2 Write in lowest terms.


Finding the Least Common Denominator (LCD)Step 1 Factor all denominators to prime factored form.Step 2 The LCD is the product of every (different) factor

that appears in any of the factored denominators. If a factor is repeated, use the greatest number of repeats as factors of the LCD.

Step 3 Write each fraction with the LCD as the denominator, using the second property of 1.



Example 9 Add. Write sum in lowest terms.

R.1 Fractions

3 2

4 5

Adding and Subtracting Fractions

Since 4 and 5 share no common factors, the LCD is 4 · 5 = 20.

5 4

5 4

3 2 3 2 15 8 15 8

4 5 4 5 20 20 20

23 3 or 1

20 20



Subtracting FractionsTo find the difference between two fractions with the same denominator, subtract their numerators and keep the same denominator. If the fractions have different denominators, write them with a common denominator first.


Example 10Subtract. Write differences in lowest terms.


5 2

6 9

Since 3 and 9 share a common factors of 3, the LCD is 2 · 3 · 3 = 18.

3 2

3 2

5 2 5 2 15 4 15 4

6 9 6 9 18 18 18

11

18

Copyright © 2010 Pearson Education, Inc. All rights reserved. R.1 – Slide 1.

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