1. Fractions and Rational Expressions
Chapter 5
Fractions and Rational Expressions

2. 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 ,where a and b are
integers and b= 0

3. 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

4. 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:
Write equivalent fractions that have a common denominator
Compare the numerators in the rewritten fractions

5. Write improper fractions as mixed numbers
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

6. Write mixed numbers as improper fractions
Procedure
Multiply the denominator by the integer
Add the resulting product to the numerator to find the numerator of the improper fraction
Keep the same denominator

7. 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

8. To simplify a fraction to lowest terms using primes
Replace the numerator and denominator with their prime factorizations
Divide out all the common prime factors
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

9. 5.3 Multiplying fractions, mixed numbers, and rational expressions Multiply fractions Multiply and simplify fractions Multiply mixed numbers

10. 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

11. 5.4 Dividing Fractions, Mixed Numbers, and Rational; expressions
Divide fractions
Reciprocals : Two numbers whose product is 1
Procedure
Change the operation symbol from division to multiplication and change the divisor to its reciprocal.
Divide out any numerator factor with any like denominator factor.
Multiply the numerator by numerator and denominator by denominator
Simplify as needed
Complex fraction : An expression that is a fraction with
fractions in the numerator and/or denominator

12. 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

13. 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

14. Find the LCM using prime factorization
Procedure
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)
Write rational expressions with the LCD

15. 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
Add or subtract the numerators.
Keep the same denominator.
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
Add and subtract rational expressions with different denominator

16. 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
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

17. Solving Equations
Use the LCD to eliminate fractions from equations
Procedure
Simplify both sides of the equations as needed
Distribute to clear parentheses
Eliminate fractions by multiplying both sides by the LCD of all the fractions. (Optional)
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.

18. 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