§ 1.3 Fractions

Numerators and Denominators A quotient of two numbers is called a fraction. The fraction represents the shaded part of the circle. 1 out of 4 pieces is shaded. is read “one-fourth.” numerator denominator

Simplifying Fractions To simplify fractions we can simplify the numerator and the denominator. 2 5 10 · = factors product A fraction is said to be simplified or in lowest terms when the numerator and denominator have no factors in common other than 1.

Prime and Composite Numbers A prime number is a natural number, other than 1, whose only factors are 1 and itself. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 The first 10 prime numbers A natural number, other than 1, that is not a prime number is called a composite number. Every composite number can be written as a product of prime numbers

Product of Primes Example: Write the number 24 as a product of primes. 24 = 4  6 Write 24 as the product of any two whole numbers. If the factors are not prime, they must be factored. 2  2 2  3 24 = 2  2  2  3 When all of the factors are prime, the number has been completely factored.

The Fundamental Principal of Fractions If is a fraction and c is a nonzero real number, then Example: Write the fraction in lowest terms.

Multiplying Fractions To multiply two fractions, multiply numerator times numerator to obtain the numerator of the product. Multiply denominator times denominator to obtain the denominator of the product. Multiplying Fractions

Multiplying Fractions Example: Multiply. Multiply numerators. Multiply denominators. Simplify the product by dividing the numerator and the denominator by any common factors.

Dividing Fractions Two fractions are reciprocals of each other if their product is 1. Dividing Fractions

Dividing Fractions Example: Divide.

Fractions with the Same Denominator To add or subtract fractions with the same denominator, combine numerators and place the sum or difference over the common denominator. Adding and Subtracting Fractions with the Same Denominator

Equivalent Fractions Equivalent fractions are fractions that represent the same quantity. is shaded. is shaded. Equivalent fractions

Equivalent Fractions Example: Write as an equivalent fraction with a denominator of 20. Since 4 · 5 = 20, multiply the fraction by

Fractions without the Same Denominator To add or subtract fractions without the same denominator, first write the fractions as equivalent fractions with a common denominator The least common denominator (LCD) is the smallest number both denominators will divide evenly into. Example: LCD = 24

Fractions without the Same Denominator Example: LCD = 60