Transcript Ch 5
Chapter 5 Fractions and Rational Expressions 5.1 Fractions, mixed numbers and Rational Expressions Name the fraction represented by a shaded region Fraction : A number that describes a part of a whole Numerator = The number written in the top position in a fraction Denominator: The number written in the bottom position in a Fraction Rational Number: A number that can be expressed as a ratio of integers A number that can be expressed in the form ba ,where a and b are integers and b= 0 Graph fractions on a number line Simplify the fractions Rule If the denominator of a fraction is 1, then the fraction can be simplified to the Numerator If the number of fraction is 0, and the denominator is any number other than 0, then the fraction can be simplified to 0 If the denominator is 0 and the numerator is any number other than 0, we say the fraction is undefined A fraction with the same numerator and denominator (other than zero) can be simplified to 1 Write equivalent fractions Equivalent Fractions: To write an equivalent fraction, multiply or divide both the numerator and denominator by the same nonzero number. Use < , > or = to make a true statement Procedure To compare two fractions: 1. Write equivalent fractions that have a common denominator 2. Compare the numerators in the rewritten fractions Write improper fractions as mixed numbers Improper Fraction: A fraction in which the absolute value of the numerator is greater than or equal to the absolute value of the denominator Mixed Number An integer combined with a fraction. Procedure To write an improper fraction as a mixed number: • Divide the denominator into the numerator • Write the results in the following form: remainder Quotient Original denominator Write mixed numbers as improper fractions Procedure 1. Multiply the denominator by the integer 2. Add the resulting product to the numerator to find the numerator of the improper fraction 3. Keep the same denominator 5.2 Simplifying Fractions and Rational Expressions Simplify the fraction to lowest terms Lowest terms: A fraction is in lowest terms when the greatest common factor of its numerator and denominator is 1. Procedure To simplify a fraction to lowest terms, divide the numerator and denominator by their greatest common factor To simplify a fraction to lowest terms using primes 1. Replace the numerator and denominator with their prime factorizations 2. Divide out all the common prime factors 3. Multiply the remaining factors Simplify improper fractions or fractions within mixed numbers Simplify rational expression Rational Expression : A fraction that is a ratio of monomials or polynomials 5.3 Multiplying fractions, mixed numbers, and rational expressions Multiply fractions Multiply and simplify fractions Multiply mixed numbers Multiply rational expressions Simplify fractions raised to a power Rule When a fraction is raised to a power, we evaluate both the numerator and denominator raised to that power. Solve applications involving multiplying fractions Calculate the area of a triangle Calculate the radius and diameter of a circle Calculate the circumference of the circle 5.4 Dividing Fractions, Mixed Numbers, and Rational; expressions Divide fractions Reciprocals : Two numbers whose product is 1 Procedure 1. Change the operation symbol from division to multiplication and change the divisor to its reciprocal. 2. Divide out any numerator factor with any like denominator factor. 3. Multiply the numerator by numerator and denominator by denominator 4. Simplify as needed Complex fraction : An expression that is a fraction with fractions in the numerator and/or denominator Divide mixed numbers Procedure 1. Write the mixed numbers as improper fractions 2. Write the division statement as an equivalent multiplication 3. Divide out by any numerator factor with any like denominator factor. 4. Multiply 5. Simplify as needed 5.5 Least common multiple Find the least common multiple (LCM) by listing LCM = The smallest natural number that is divisible by the given Numbers Procedure To find the LCM by listing , list multiples of the greatest given number until you find a multiple that is divisible by all the other given numbers Find the LCM using prime factorization Procedure 1. Find the prime factorization of each given number. 2. Write a factorization that contains each prime factor the greatest number of times it occurs in the factorizations. Or, if you prefer exponents, the factorization contains each prime factor raised to the greatest exponent that occurs in the factorizations 3. Multiply to get the LCM Find the LCM of a set of monomials Work fractions as equivalent fractions with the least common Denominator (LCD) Adding and Subtracting fractions, Mixed Numbers, and Rational Expressions Add and subtract fraction with the same denominator To add or subtract fractions that have the same denominator 1. Add or subtract the numerators. 2. Keep the same denominator. 3. Simplify Add and subtract rational expressions with the same denominator Add and subtract fractions with different denominators Procedure • Write the fraction as equivalent fractions with a common denominator • Add or subtract the numerators and keep the common denominator • Simplify Add mixed numbers Procedure Method 1: Write as improper fractions, then follow the procedure for adding fractions Method 2: add the integer parts and fraction parts separately Subtract mixed numbers Procedure Method 1: Write as improper fractions, then follows the procedure for adding /subtracting fractions Method 2: Subtract the integer parts and fraction parts separately Add and Subtract signed mixed numbers Solve equations Solve applications 5.8 Solving Equations Use the LCD to eliminate fractions from equations Procedure 1. Simplify both sides of the equations as needed a) Distribute to clear parentheses b) Eliminate fractions by multiplying both sides by the LCD of all the fractions. (Optional) c) Combine like terms 2. Use the addition/subtraction principle of equality so that all variable terms are on one side of the equation and all constants are on the other side. (Clear the variable term that has the lesser coefficient. This will avoid negative coefficients.) then combine like terms. 3. Use the multiplication/division principle of equality to clear any remaining coefficients. 5.8 Solving Equations Use the LCD to eliminate fractions from equations Translate sentences to equations, then solve Solve applications involving one unknown Solve applications involving two unknowns