Math 80 Lecture - Chapter 3 Merced College; Don Power

3.1 LCM and GCF

Examples: 36 and 45; 20, 36, and 45

Review of Prime factorization - techniques

Factor tree: All the leaves constitute the factorization

Successive division: start with division by 2, then 3, 5, ... (i.e. by successive primes)

Repeated division by the same prime should be tried before going to the next.

LCM -- How to calculate

Roster technique: List all multiples

Common factor technique (best when there are just two numbers)

LCM is common factor times each of the "uncommon" factors

Prime factorization technique

GCF -- How to calculate

Roster technique: List all factors (start with list of factor pairs)

Successive division by common factors: Product of common factors is GCF

Prime factorization technique

Applications -- #57 vs # 59

Do you want a common number that is...

Less than the given numbers? Use GCF

More than the given numbers? Use LCM

3.2 Introduction to Fractions

Express shaded part as a fraction

Denom shows the number of equal parts in the whole amount, that is, nr pieces for each pie

Num shows how many parts of that size are being considered, that is, nr pieces that are shaded

Proper fractions, improper fractions, mixed numbers

Proper fraction: num < denom

Improper fraction: num ³ denom

Mixed nr: fraction written as the sum of a whole number and a proper fraction

Convert improper fractions to mixed nrs or whole nrs

Relate to division

Convert mixed nr or whole nr to improper fraction

Denom = denom; Num = Denom x whole nr + num

Write equiv fraction with the given denom

Simplest terms: reduce

Divide numerator and denominator by the gcd

or, factor and reduce

[For fractions, it is enough to do a prime factorization of the smaller nr and check to see if those nrs divide the larger nr.]

Compare fractions with < or >

Put over LCD, compare numerators

or, divide (using calculator)

Applications: what is the "fractional part"

3.3 Addition and Subtraction of Signed Fractions

General procedure for + and -:

Write each fraction with a common denominator

This really means multiply each fraction, top and bottom, as necessary to get the LCD

Add or subtract the numerators only

Copy the common denominator

Ex: 2/3 + 4/5 = 10/15 + 12/15 = 22/15

because you multiply the fractions by 5/5 and 3/3 respectively

Fact: the LCD is the LCM of the denominators

Use the techniques from lesson 3.1 to find the LCM

Divide each denom into the LCM to determine what to multiply by.

Shortcut procedure for adding or subtracting: multiply each fraction (top and bottom) by the alternate denominator. In symbols, a/b + c/d = (ad + bc) / bd

Same Ex: 2/3 + 4/5 = (2 x 5 + 4 x 3) / (3 x 5) = 22/15

Problem: If there are common factors, this doesn't result in the lowest common denom

Ex: 4/21 + 2/35 Shortcut would have you multiply 21 x 35 = 735

But: 21 = 7 x 3, 35 = 7 x 5, so LCD is 7 x 3 x 5 = 105

Revision: mult each fraction (top and bottom) by the alternate reduced denom

Note: reduce by dividing each denom by common factor(s)

So 21 and 35 reduce to 3 and 5; mult 4/21 by 5/5 and mult 3/35 by 3/3

Or, factor each denom into common factor and "uncommon" factor

and mult each fraction (top and bottom) by the alternate "uncommon"factor

So 4 / (7 x 3) + 2 / (7 x 5) requires buildup to LCD of 7 x 3 x 5

Prime factor procedure to build up each fraction:

Works for any number of fractions

Requires that all denoms (and the LCD) be factored down to primes

Procedure

Compare each denom with the LCD

Ask, "What is missing from the LCD"

Mult the fraction, top and bottom by the missing factor(s)

Ex: 3/20 − 5/24 + 1/25: Prime factorizations: 20=2²5, 24=2³3, 25=5²;

LCD is 2³3¹5²; 20 needs 2¹3¹5¹, 24 needs 5², 25 needs 2³3¹

Mixed numbers: May involve carrying (for addition) or borrowing (for subtraction)

Addition: Get a common denom for the fracs -- leave whole nr alone

Add whole nrs, add fracs

If fraction >1, convert fraction sum to a mixed number & add whole nr parts

(New num = old num − denom)

Ex:

Subtraction: if fraction part isn't large enough, borrow by:

adding 1 to the fraction (new num = old num+denom)

subtracting 1 from the whole nr

Then subtract the whole numbers and subtract the fractions.

Ex:

Whole number and a fraction: treat it as though you are converting a mixed number to an improper fraction:

−5 + 2/3: denom is 3; num is 3(−5) + 2

3.4 Mult and Div of Signed Fractions

Mult:

Reduce first (to keep numbers smaller)

Any factor in num can reduce any like factor in denom

Then multiply

Num times num

Denom times denom

Division is defined as mult by the reciprocal of the divisor, so

Flip the divisor (2nd fraction, or bottom of a complex fraction)

Change operation to mult

Then do the mult.

Mixed numbers: First, convert to improper fractions

Whole number times fraction:

The whole number is treated as a numerator, so

1. It can reduce with the denominator.

2. It gets multiplied with the numerator

Ex: −4 x 3/14 Divide both the −4 and 14 by 2 to get −2 x 3/7

Then multiply the −2 with the num to get −6/7

Or, write the whole number as a fraction: −4 becomes −4 / 1

3.5 Solving Equations with Fractions

Best approach (usually): Clear fractions (after the parentheses are cleared)

Technique: Multiply every term by the LCD of all the fractions

Reminder: To multiply the LCD (a whole number) times a fraction

First, divide the denom into the LCD [it will always divide evenly]

Then, multiply the result times the numerator

Now the equation no longer has fractions; solve as we did in previous lessons.

Harder approach (usually): Keep the fractions.

Adding/subtracting terms requires adding/subtracting fractions

Last step: Instead of dividing by the coefficient of the variable,

multiply by the reciprocal of the coefficient of the variable.

3.6 Exponents, Complex Fractions, Order of Operations

Objective: Use exponents with fractions.

Exp applies to the sign in the paren as well as both num and denom

Obj: Use the order of operations with fractions

Recap the standard order

For mixed numbers, remember to convert to improper fractions before multiplying, dividing, or applying exponents

Obj: Simplify Complex Fractions: Only technique shown in our text is to add/subt as necessary to get a single term in the num and a single term in the denom. Then divide fracs

Return to:  Merced College; Don Power               Updated 10/05/06 by Don Power