GCF Those Fractions, LCM

GCF, LCM, and Fractions

Three very important concepts are taught in elementary school arithmetic classes:  Greatest Common Factor (GCF), Least Common Multiple (LCM), and fractions.  However, many students I have worked with have a very poor conceptualization of these ideas by the time they reach Algebra.  Fractions are used continuously throughout Algebra and beyond, and most teachers don't have the time in the curriculum to go back and teach a full lesson on fractions. Students often shut down when they see fractions, but they aren't as scary as they seem.

Fractions are used in many aspects of life, such as chemistry, baking, economics, business, statistics, etc.   Fractions are used all the time, but people often don't think in terms of fractions.  This is where many students get lost.  I wanted to post a quick review of fractions as a reminder for students who have forgotten.  Before we get into fractions, I'm going to review GCF and LCM, because these are necessary for converting fractions to workable forms.

GCF

  The Greatest Common Factor (GCF) is the largest factor that two numbers share.  The easiest method I've used in finding the GCF between two numbers is to draw a factor tree.  For example:

Example 1:  Find the GCF of 20 and 42

The factors of 20 and of 42 can be represented as factor trees, like so:

                                                      20                                                  42
                                                   /       \                                              /      \
                                                4          5                                          6       7
4 and 5 can be multiplied together to get 20, so those are factors of 20.  6 and 7 can be multiplied to get 42, so those are factors of 42.  However, to find the GCF, we have to factor the factors down to prime numbers.

                                                     20                                                   42
                                                   /       \                                              /      \
                                                4          5                                          6       7
                                              /   \                                                   /   \
                                            2     2                                               2    3

The bottom number on each branch is a prime factor of the number we stared with.  Therefore, the prime factorization of 20 is 2*2*5.  The prime factorization of 42 is 2*3*7.  The only factor that both prime factorizations share is 2, therefore the GCF(20,42) = 2.

Practice with 1)  GCF(5, 7)     2) GCF(10, 20)    3) GCF(30, 385) Answers are at the end of the rant.

Check out this GCF video

LCM

The Least Common Multiple (LCM) is the combination of the prime factorizations of two numbers, without repeating shared factors.  I also use factor trees for LCM.  For example:

Find the LCM of 24 and 120.

                                                24                                                        120
                                              /      \                                                     /      \
                                            6       4                                                  2       60
                                         /   \     /    \                                                        /     \
                                       2    3   2     2                                                   2       30
                                                                                                                      /    \
                                                                                                                    2     15
                                                                                                                          /     \
                                                                                                                       3        5

These trees show us the prime factorization of 24 is 2 * 2 * 2 * 3, or (2^3) * 3.  The prime factorization of 120 is 2 * 2 * 2 * 3 * 5 or (2^3) * 3 * 5.  Both factor trees contain (2^3) and 3.  120 also contains 5.  To find the LCM, we multiply all shared factors together and all non-shared factors together.

The shared factors are (2^3) and 3.  The non-shared factor is 5.  (2^3) * 3 * 5 = 120.  Therefore, the LCM(24, 120) = 120.

Practice problems.  1) LCM(30, 63)    2) LCM(8, 24)   3) LCM (6, 15)  Answers are at the end of the rant.

Check out this LCM video

Fractions

When working with fractions, there are four simple rules to keep in mind:

1)  To add or subtract fractions, they must have the same denominator, found by using the LCM.  Then the numerators are added together, and the denominator stays the same.

2) To multiply fractions, multiply the numerators together to form the new numerator, and multiply the denominators together to form the new denominator.

3) To divide fractions, use the reciprocal of the fraction you are dividing by, then multiply as normal.

4) To simplify fractions, find the GCF of the numerator and denominator, then divide both the numerator and denominator by the GCF.

Quick review:  The number on top of a fraction is the numerator and the number on bottom is the denominator.

Fraction Examples:

Adding Fractions

To add the fractions (1/3) and (1/6) we must find a common denominator.  The numerator for both fractions is 1.  The denominator of the first fraction is 3 and the denominator of the second fraction is 6.  We find the LCM(3, 6) to find a common denominator.  LCM(3, 6) = 6.  Therefore, (1/6) is in the form it needs to be in, but (1/3) is not.

3 had to be multiplied by 2 to get the LCM of 6.  Therefore, we multiply the numerator and denominator of (1/3) by 2 to get a common denominator.  1 * 2 = 2 and 2 * 3 = 6.  Therefore, (1/3) = (2/6)

Now we add (2/6) + (1/6) = (3/6).  We add the numerators of 2 and 1 to get 3 as the new numerator.  The denominator of 6 is unaffected when adding or subtracting.

(3/6) can be simplified by finding the GCF of 3 and 6.  GCF(3, 6) = 3.  Therefore, dividing the numerator of 3 by 3, we get 1.  Dividing the denominator of 6 by 3, we get 2.  The simplified fraction is then (1/2).

Adding Fractions practice problems:  1) (1/3) + (1/3)  2) (1/4) + (1/5)  3) (2/7) + (7/8) Answers are at the end of the rant.

Here's a video for adding fractions:

Multiplying Fractions

Example:  Multiply (2/5) by (3/7).  Multiplying is one of the easiest operations to perform on fractions.  Simply multiply the numerators together, then the denominators together.

The numerators are 2 and 3, multiplying them together gets a new numerator of 6.  The denominators are 5 and 7.  Multiplying them together gets 35.  The new fraction is then (6/35).

Can (6/35) be simplified?  Find the GCF(6, 35) = 1.  This means that (6/35) is in its simplest form.

Multiplying Fractions practice problems: 1) (3/5) * (4/5)  2) (2/3) * (3/2)  3) (1/8) * (7/8) Answers are at the end of the rant.

Here's another video, for multiplying videos. He demonstrates the cross-canceling method.

Dividing Fractions

Example:  Divide (3/5) by (4/5).  To perform this division, we first use the reciprocal of (4/5).  This means simply switch the numerator and denominator.  Therefore, the reciprocal of (4/5) is (5/4).

Now we multiply (3/5) by the reciprocal of (4/5), which is (3/5) * (5/4) = (15/20).

Can (15/20) be simplified?  What is the GCF(15, 20)?  The prime factorization of 15 is 3 * 5.  The prime factorization of 20 is 2 * 2 * 5.  The only shared factor is 5, therefore GCF(15, 20) = 5.

Now we divide both the numerator and denominator by the GCF.  15 divided by 5 is 3.  20 divided by 5 is 4.  Therefore, (15/20) in simplest form is (3/4).

Dividing Fractions practice problems:  1) (1/2) / (4/5)  2) (27/56) / (27/56)  3) (3/4) / (5/6)

Here's a video for dividing fractions:




I hope this has been a helpful review.  If anyone would like further clarification, send me a comment and I'll respond when I can.  Here is a helpful website for teaching fractions visually:  http://www.teachingfractions.co.uk/

GCF practice problem answers:
Answers:

1) GCF(5,7) = 1 because both 5 and 7 are prime numbers, the only factor that they both share is 1.  Any number can be factored as 1 * itself.  For example 1 * 5 = 5 and 1 * 7 = 7.  1 is a factor of all numbers except 0.

2) GCF(10, 20) = 10.  The prime factorization of 10 is 2 * 5.  The prime factorization of 20 is 2 * 2 * 5.  Since both prime factorizations contain a 2 and a 5, the GCF is 2*5 = 10.

3) GCF(30, 385) = 5.  The prime factorization of 30 is 2 * 3 * 5.  The prime factorization of 385 is 5 * 7 * 11.  The only number both prime factorizations share is 5.


LCM practice problem answers:

1)  LCM(30, 63) = 1890.  This is because the prime factorization of 30 is 2 * 3 * 5.  The prime factorization of 63 is 7 * 9.  These two prime factorizations do not share any factors, so the LCM must include all factors from both prime factorizations.  Therefore LCM(30, 63) = 2 * 3 * 5 * 7 * 9 = 1890.

2) LCM(8, 24) = 24.  The prime factorization of 8 is 2 * 2 * 2 or (2^3).  The prime factorization of 24 is 2 * 2 * 2 * 3 or (2^3) * 3.  Both prime factorizations contain (2^3).  24 contains a factor of 3, but 8 does not.  Therefore, the LCM(8, 24) = (2^3) * 3 = 24.

3) LCM(6, 15) = 30.  The prime factorization of 6 = 2 * 3.  The prime factorization of 15 = 3 * 5.  Both prime factorizations contain 3.  The first contains 2, the second contains 5.  Multiplying the factors that are shared by the factors that are not shared, we get LCM(6, 15) = 3 * 2 * 5 = 30.


Adding Fractions practice problem answers:
1) To add (1/3) + (1/3) check that the denominators are the same.  The denominator of both fractions is 3.  Therefore, simply add the numerators 1 + 1 = 2.  Therefore (1/3) + (1/3) = (2/3)

2) To add (1/4) + (1/5) we need a common denominator.  The LCM(4,5) = 20.  To make (1/4) have a denominator of 20, multiply the numerator and denominator by 5.  (1/4) = (5/20).  To make (1/5) have a denominator of 20, multiply the numerator and denominator by 4.  (1/5) = (4/20).  Now add (5/20) + (4/20) = (9/20).  Check that (9/20) is in simplest form, which it is because GCF(9, 20) = 1.


3) To add (2/7) and (7/8), find LCM (7, 8) = 56.  Therefore (2/7) = (16/56) (by multiplying 2 by 8 and 7 by 8).  (7/8) = (49/56) (by multiplying the 7 by 7 and 8 by 7).  Now add (16/56) + (49/56) to get (65/56).  Is (65/56) in simplest form?  GCF(65, 56) = 1.  (65/56) is an improper fraction, but is acceptable most of the time in Algebra.

Multiplying Fractions Practice Problem answers:
1) To multiply (3/5) by (4/5), multiply the numerators, 3 and 4, together to get 12.  Multiply the denominators 5 and 5 together to get 25.  The answer is then (12/25).  Is (12/25) in simplest form?  GCF(12,25) = 1, so yes it is.

2) To multiply (2/3) by (3/2), multiply the numerators, 2 and 3, to get 6.  Multiply the denominators 3 and 2, to get 6 again.  Therefore, the answer is (6/6).  A fraction is another way of signifying division, and anything divided by itself is 1.  Therefore, (6/6) in simplest form is just 1.
*note, (2/3) and (3/2) are reciprocals of each other, because their numerators and denominators are the same numbers, only reversed.  Whenever you multiply reciprocals, the answer is always 1.

3) To multiply (1/8) * (7/8), multiply the numerators 1 and 7 to get 7, and the denominators 8 and 8 to get 64.  Therefore, the answer is (7/64).  Since 7 is prime and is not a factor of 64, this is in simplest form.

Dividing Fractions practice problem answers:
1) Dividing (1/2) by (4/5) is the same as multiplying (1/2) by the reciprocal of (4/5), which is (5/4).  Multiplying (1/2) by (5/4), multiply the numerators 1 and 5, and multiply the denominators 2 and 4.  The answer is then (5/8), which is in simplest form.

2) Before trying to calculate large numbers, look for similarities in the two fractions.  To divide (27/56) by (27/56), notice that they are both the same number.  Any number divided by itself is 1.  Therefore, (27/56) / (27/56) = 1.

3) (3/4) / (5/6) is the same as (3/4) * (6/5), since (6/5) is the reciprocal of (5/6).  Multiply the numerators 3 and 6 to get 18.  Multiply the denominators 4 and 5 to get 20.  Therefore, (3/4) / (5/6) = (3/4) * (6/5) = (18/20).  The GCF(18, 20) = 2, so dividing 18 by 2 and 20 by 2 gives the simplest form of (18/20) to be (9/10), which is the final answer.

This is the end of my rant.