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Copyright © 2010 Pearson Education, Inc. All rights reserved. R.1 – Slide 4 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

Copyright © 2010 Pearson Education, Inc. All rights reserved. R.1 – Slide 5 R.1 Fractions A fraction is the quotient of two whole numbers. For example: Numerator Denominator Fraction 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, Improper fraction Mixed number

Copyright © 2010 Pearson Education, Inc. All rights reserved. R.1 – Slide 6 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.

Copyright © 2010 Pearson Education, Inc. All rights reserved. R.1 – Slide 7 Example 1 Decide whether each number is prime or composite. R.1 Fractions Identifying 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

Copyright © 2010 Pearson Education, Inc. All rights reserved. R.1 – Slide 8 R.1 Fractions Writing 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.

Copyright © 2010 Pearson Education, Inc. All rights reserved. R.1 – Slide 9 Example 2 Write each number in prime factored form. R.1 Fractions Writing Numbers in Prime Factored Form a) 28 We use a factor tree, as shown below. 28 = 2 · 2 · 7 7 22 28 4 b) 36 We use a factor tree, as shown below. 36 = 2 · 2 · 3 · 3 36 4 9 2233

Copyright © 2010 Pearson Education, Inc. All rights reserved. R.1 – Slide 10 R.1 Fractions Writing 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 1 Any nonzero number divided by itself is equal to 1; for example, Any number multiplied by 1 remains the same; for example, 7 · 1 = 7.

Copyright © 2010 Pearson Education, Inc. All rights reserved. R.1 – Slide 11 R.1 Fractions Writing Fractions in Lowest Terms Writing a Fraction in Lowest Terms Step 1Write the numerator and denominator in prime factored form. Step 2Replace each pair of factors common to the numerator and denominator with 1. Step 3Multiply the remaining factors in the numerator and in the denominator. (This procedure is sometimes called “simplifying the fraction.”)

Copyright © 2010 Pearson Education, Inc. All rights reserved. R.1 – Slide 12 Example 3 Write each fraction in lowest terms. R.1 Fractions Writing Fractions in Lowest Terms

Copyright © 2010 Pearson Education, Inc. All rights reserved. R.1 – Slide 13 R.1 Fractions Writing Fractions in Lowest Terms Note When 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,

Copyright © 2010 Pearson Education, Inc. All rights reserved. R.1 – Slide 14 Example 4 Write as a mixed number. R.1 Fractions Converting Between Improper Fractions and Mixed Numbers To 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.

Copyright © 2010 Pearson Education, Inc. All rights reserved. R.1 – Slide 15 Example 5 Write as an improper fraction. R.1 Fractions Converting Between Improper Fractions and Mixed Numbers To 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.

Copyright © 2010 Pearson Education, Inc. All rights reserved. R.1 – Slide 16 R.1 Fractions Multiplying and Dividing Fractions Multiplying Fractions To 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.

Copyright © 2010 Pearson Education, Inc. All rights reserved. R.1 – Slide 17 Example 6 Find the product, and write it in lowest terms. R.1 Fractions Multiplying and Dividing Fractions

Copyright © 2010 Pearson Education, Inc. All rights reserved. R.1 – Slide 18 R.1 Fractions Multiplying and Dividing Fractions Dividing Fractions To divide two fractions, multiply the first fraction by the reciprocal of the second. The result, called the quotient, should be written in lowest terms.

Copyright © 2010 Pearson Education, Inc. All rights reserved. R.1 – Slide 19 Multiply by the reciprocal of the second fraction. Example 7 Find the quotient, and write it in lowest terms. R.1 Fractions Multiplying and Dividing Fractions

Copyright © 2010 Pearson Education, Inc. All rights reserved. R.1 – Slide 20 R.1 Fractions Adding and Subtracting Fractions Adding Fractions To 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.

Copyright © 2010 Pearson Education, Inc. All rights reserved. R.1 – Slide 22 Finding the Least Common Denominator (LCD) Step 1Factor all denominators to prime factored form. Step 2The 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 3Write each fraction with the LCD as the denominator, using the second property of 1. R.1 Fractions Adding and Subtracting Fractions

Copyright © 2010 Pearson Education, Inc. All rights reserved. R.1 – Slide 23 Example 9 Add. Write sum in lowest terms. R.1 Fractions Adding and Subtracting Fractions Since 4 and 5 share no common factors, the LCD is 4 · 5 = 20.

Copyright © 2010 Pearson Education, Inc. All rights reserved. R.1 – Slide 24 R.1 Fractions Adding and Subtracting Fractions Subtracting Fractions To 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.

Copyright © 2010 Pearson Education, Inc. All rights reserved. R.1 – Slide 25 Example 10 Subtract. Write differences in lowest terms. R.1 Fractions Adding and Subtracting Fractions Since 3 and 9 share a common factors of 3, the LCD is 2 · 3 · 3 = 18.