2.1Least Common Multiple Greatest Common Factor

2.1LEAST COMMON MULTIPLE GREATEST COMMON FACTOR

Objective A - Finding the Least Common Multiple

The multiples of a number are the products of that number if you multiply it by 1, 2, 3, 4, 5, 6...

Ex. The multiples of 5 are:Ex. The multiples of 7 are:

When you compare the multiples of two (or more) numbers, if they have any in common, theyare called a common multiple of those numbers.

The Least Common Multiple (referred to as the LCM) is the smallest common multiple of two (or more) numbers.

Looking at the multiples of 5 and 7 above, what are the common multiples listed.

Howto Find the LCM?

A method to find the LCM without listing all the multiples as above:

1)Find the prime factorization of each number.

2)List the prime factorizations of each number in rows in a table with a prime number listed as the heading of each column.

3)Circle the largest product in each column.

4)The LCM is thatlargestproduct - from all circled numbers.

Ex. Find the LCM of 6 and 8.

2 / 3
6 =
8 =

Therefore, the LCM is:

Ex. Find the LCM of 14 and 42.Ex. Find the LCM of 120 and 160.

Objective B - Finding the Greatest Common Factor

Recall that a number that divides a second number evenly is called a factor of the second number.

A number is a common factor of two (or more) numbers if it’s a factor of those numbers.
Factors of 12 are: 1, 2, 3, 4, 6, and 12
Factors of 40 are: 1, 2, 4, 5, 8, 10, 20, and 40
The common factors of 12 and 40 are:

The Greatest Common Factor (referred to as the GCF)is the largest common factor of two or more numbers.
Looking at the factors of 12 and 40 above, what is the GCF?

A method to find the GCF without listing all of the factors as above:

1)Find the prime factorization of each number.

2)List the prime factorizations of each number in rows in a table with a prime number listed as the heading of each column.

3)Circle the smallest product in each column that does not have a blank.

4)The GCF is thatsmallestproduct - from all circled numbers.

Ex. Find the GCF of 18 and 24.

2 / 3
18 =
24 =

Therefore, the GCF is:

Ex. Find the GCF of 15 and 28.Ex. Find the GCF of 24, 36, and 48.

2.2INTRODUCTION TO FRACTIONS

Objective A - Write a Fraction That Represents Part of a Whole

A fraction represents a certain number of equal parts of a whole item.

Divide the circle into 4 equal parts. Shade 3 of them.
This represents of that shape.
The 3 is in the numerator.
The 4 is in the denominator.

A proper fraction is less than one. That means the numerator is less than the denominator. For example, ¾ is a proper fractionbecause it is < 1.

A mixed number is a number that is greater than 1 that has a whole number part and a fractional part. For example, 1¾ is a mixed numberbecause it is > 1.

An improper fraction is greater than or equal to 1. Meaning, the numerator is greater than or equal to the denominator. Note: an improper fraction is not a “bad fraction.” In algebra, fractions that are greater than one are most often left in improper fraction form. For example, 7/3 is an improper fraction because it is ≥ 1.

Objective B - Writing an Improper Fraction as a Mixed Number ora Whole Number

Process to change from an improper fraction to a mixed number:

1)Divide the denominator into the numerator until you get a final remainder.

2)The whole number part of the mixed number is the whole number in the quotient. The fraction part of the mixed number is the remainder over the divisor.

Ex. Write as a mixed number.Ex. Write as a mixed number.

Ex.Write as a mixed or whole number.Ex. as a mixed or whole number.

Representing a Mixed Number as an Improper Fraction

Process to change a mixed number to an improper fraction:

1)Multiply the whole number times the denominator.

2)Add that to the numerator.

3)Write that number “over” the denominator that was there.

Ex. Write as an improper fraction.Ex. Write as an improper fraction.

2.3WRITING EQUIVALENT FRACTIONS

Objective A - Finding an Equivalent Fraction By Rising to Higher Terms

Two fractions can be equal but look different → draw two circles:
Divide one into halves and the other into fourths.
Shadeof the first circle andof the second circle.
Notice the two regions you shaded are the same.

You can take one fraction and make an equivalent fraction by multiplying both the numerator and denominator by the same number.
This is →raising a fraction to higher terms.

Ex. Write a fraction that is equivalent to and has a denominator of 21.

Ex. Write a fraction that is equivalent to and has a denominator of 55.

Note:You can multiply both the numerator and denominator by the same number becauseyou are really just multiplying by the number ___. (Place Answer Here!)

Ex. Write theequivalent fraction ofby using the Multiplication Property of One:

Objective B - Writing a Fraction in Simplest Form (or Simplest Terms)

This is really the reverse process of Objective A.

A fraction in simplest formis when the numerator and denominator have no common factors between them.
Just as you could raise to by multiplying both the numerator and denominator by 2, so you can also simplify to by dividing both the numerator and denominator by 2.

To do this, you divide both the numerator and denominator by a factor that divides evenly into both. The GCF is the best one to use because that will be the only number by which you need to divide.

Sometimes that number will be obvious to you; other times you will need to do some work. When it’s not obvious, the method that always works is this:

1)Factor the numerator into its prime factors.

2)Factor the denominator into its prime factors.

3)Cancel any factors that are the same.

Ex. Write in simplest form. (Also called reducing the fraction.)

Ex. Write in simplest form. Ex. Write in simplest form.

2.4ADDITION OF FRACTIONS AND MIXED NUMBERS

Objective A - Adding Fractions with the Same Denominator

Process:
1. Add the numerators.
2. Write that sum over the same denominator.

Ex. Ex. Ex.

Objective B - Adding Fractions With the Unlike Denominators

Process:
1. Find the LCM of the denominators.
2. Write each fraction as an equivalent fraction with the LCM as the denominator.
3. Add the fractions once they have the same denominator.

Ex.Ex.

Ex. Ex.

Objective C - Adding Whole Numbers, Mixed Numbers, and Fractions

Case 1:

To add a whole number and a fraction, you end up with a mixed number.

Ex. 5 + =

Case 2:

To add a whole number and a mixed number, you add the whole numbers and write the fraction part next to the sum so you have a mixed number.

(Note: It can be helpful to rewrite the problem vertically.)

Ex. 9 + =

Case 3:

To add two mixed numbers, add the fractional parts and then add the whole numbers. Always reduce the fraction part to simplest form.

(Note: It can be helpful to rewrite the problem vertically.)

Ex.

Ex. Ex.

2.5SUBTRACTION OF FRACTIONS AND MIXED NUMBERS

Objective A - Subtracting Fractions with the Same Denominator

Process:
1. Subtract the numerators.
2. Write that sum over the same denominator.

Ex. Ex.

Objective B - Subtracting Fractions With the Unlike Denominators

Process:
1. Find the LCM of the denominators.
2. Write each fraction as an equivalent fraction with the LCM as the denominator.
3. Subtract the fractions once they have the same denominator.

Ex.Ex.

Ex. Ex. What is less than?

Objective C - Subtracting Whole Numbers, Mixed Numbers, and Fractions

Case 1:

Subtracting mixed number without borrowing.

(Note: It can be helpful to write the problem vertically.)

Ex.Ex.

Case 2:

Subtracting a whole number from a mixed number: you to borrow 1 from the whole number. Note: It can be helpful to write the problem vertically.

Ex.

Case 3:

Subtracting two mixed numbers when you do need to borrow: this happens when the numerator of the “top” fraction is smaller than the numerator of the “lower” fraction.

Note: It can be helpful to write the problem vertically.

Ex.

Ex. In this example, you need to find the LCM for the denominators first.

2.6MULTIPLICATION OF FRACTIONS AND MIXED NUMBERS

Objective A - Multiplying Fractions

Procedure:
1. Multiply the numerators to find the new numerator.
2. Multiply the denominators to find the new denominator.
3. You can write the prime factorization of both the numerator and denominator.
4. Cancel all common factors between the numerator and denominator – this will reduce the final fraction to simplest terms.

Ex. Ex.

Objective B - Multiplying Whole Numbers, Mixed Numbers, and Fractions

Procedure:
1. If there is a whole number, write it “over 1” so it looks like a fraction.
2. If there is a mixed number, change it to an improper fraction.
3. Multiply the two fractions like you did in Objective A.

Ex. Ex.

2.7DIVIDING FRACTIONS AND MIXED NUMBERS

Objective A Dividing Fractions

Reciprocal: The reciprocal of a fraction means that the numerator and denominator have been ______.

What is the reciprocal of?What is the reciprocal of 4?

What is the reciprocal of?What is the reciprocal of?

Division is actually easy for fractions – you just change every division problem into a multiplication problem. (“Flip and Multiply”)

Ex. = =

Notice that it is the second fraction (the divisor fraction) that gets “flipped.”

Ex. Ex.

Objective B - Dividing Whole Numbers, Mixed Numbers, and Fractions

Procedure:
1. If there is a whole number, write it “over 1” so it looks like a fraction.
2. If there is a mixed number, change it to an improper fraction.
3. Divide the two fractions like you did in Objective A.

Ex. Ex.

2.8ORDER, EXPONENTS, AND ORDER OF OPERATIONS AGREEMENT

Objective A - Identifying the Order Relation between Two Fractions

Just as you can graph whole numbers on a number line, so you can also graph fractions on a number line. You place a solid dot one the number line where the fraction is positioned.

Ex. Graph.

Instead of dividing each interval into fifths, divide into any set of equal increments.
Divide the following number line into thirds and graph the fractions and.

When you compare fractions on the number line, the smaller fraction is on the ______, and the larger fraction is on the ______.

If two fractions have the same denominator, it is easy to determine which the smaller fraction is: Compare the numerators and the fraction with the smaller numerator is the smaller fraction.

Ex. Place the correct inequality symbol (< or >) between these fractions:

If two fractions have different denominators, you first need to find the LCM and write them with the same denominator. Then you can compare the numerators and the fraction with the smaller numerator is the smaller fraction.

Place the correct inequality symbol ( or ) between these fractions:

Ex.Ex.Ex.

Objective B - Fractions with Exponents

You can write with an exponent – what is it?

So too, it is possible to write a fraction with an exponent – the exponent just means that you are multiplying it by itself that many times.

Write the following with an exponent:

a. = b.

To simplify an exponential expression like one of these, you need to multiply the numbers, cancel any common factors, and simplify the resulting fraction.

Ex. Simplify

Objective C - Using the Order of Operations Agreement with Fractions

Remember PEMDAS from Chapter #1.
Simplify the following:

Ex.Ex.

Ex. Ex.