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Fraction How-TOs

with a minimal amount of background theory.

Let us go over that again.  

In the diagram, we see a single third can be divided into four parts of equal size and value.  Each of those parts is a twelfth and equals a fourth of a third.   Four quarters of a divisible object is the object. So  one third is four times a quarter of itself.  That is

Whence two thirds would be twice as much:  

Thus different fractions 

Here the same part of the length, namely 2 thirds may be described in two different ways

Thus different fractionals may describe the same part or fraction of a whole. When they do, the fractional are said to be equivalent. That is, they have the same value. 

Note: Same value and same meaning are slightly different.  The fraction 2 thirds and 8 twelfths have the same value for many purposes. Taking two thirds of a cake (literally dividing it into thirds and taking two of those thirds) and taking 8 twelfths of a cake, the same cake, may physically represent different operations. 

Extension: In the above diagram, we see a single length 1 can be divided into three parts of equal size and value.  Each of those parts is a third.   Three thirds has the same size as one.  Thus 

Thus 1 has the same value as the fraction 3 thirds.  Thus a single fractional may be also be a numeral, that is a different way to express a whole number, here the number one. 

Note Again: Same value and same meaning are slightly different.  The fraction 3 thirds and the number one  have the same value for many purposes. But three thirds of a apple (literally dividing it into thirds and taking all three of those thirds) and taking  a whole apple are different.  A whole apple with its skin intact will last longer than three thirds, each with the interior of the apple exposed to the air. 

Simplest Form of a Fraction or Fractional

A fraction is said to be expressed in simplest form when it is given or represented by a fractional in which the numerators and denominators (tops and bottom) have no common divisors - or are relatively prime. The aim of "simplification" for a fraction or fractional is to find an fractional with the same value or meaning in simplest form. 

Example of simplification

From cancellation of the common factor 2, we see the fractionals 

3    and   6
4             8

are equivalent. They have the same value.  The equal sign is use to indicate two numbers or fractions have the same value above and below.

The common factor cancellation property, algebraic description: 

is a way to recognize equivalent fractions. To apply it, we look for a common factor N (like the number 2 above) to cancel.

The left and right hand side in foregoing equation are said to be equivalent fractions.

Replacing the left hand side by the right hand side in a calculation is called a simplification, a reduction, a cancellation or a lowering of terms.  On the other hand, replacing the right hand side by the left hand side is called raising terms.  Raising of terms is useful in the addition and multiplication of fractions. 

Note:  Since N× M = M× N, the denominator can be written in two forms.  Likewise the numerator N× B = B× N can be written in two forms.  So may describe three more ways to simplify:

Some Video Examples of raising terms and of simplification (lowering terms) follow.

1. [Play Video] 2-3 minutes A few examples of Simplifying Fractions - lowering terms by canceling common factors until there are no more common factors, so that the numerator and denominator are relatively prime, that is there prime decompositions have no primes in common.
2. [Play Video] 3-4 minutes. Equivalent fractions - Lowering and raising terms (the values of numerators and denominators) to obtain equivalent fractions. Simplification involves lowering terms - canceling common factors or divisors on top and bottom. Addition & subtraction of fractions may involve raising terms to obtain a common denominators. See below.

For more text examples, visit www.purplemath.com and www.mathsisfun.com

Raising Terms - Theory:

Numerical View:   Observe a  fifth of a quarter is a twentieth.   Dividing each quarter of a divisible object into five parts of equal value presumably gives 4 times 5 parts of the object of equal value, that is 4 × 5 = 20 parts of equal value.   Now a quarter is given by five fifths of itself or five times a twentieth.

That is 

So  one quarter is  five twentieths

and umpteen (say 3) quarters will be umpteen 5 twentieths

Generalization - Optional Reading

Understanding the following is or could be a step in developing your algebraic thinking skills. 

So  one N-th is  M times an (M × N) th

and umpteen (say P) quarters will be umpteen P times one N-th or P times M times an (M × N) th

Rules for addition and subtraction

Here the denominator M can be any common multiple of B and D. To apply these formula efficiently, remember the smallest common multiple usually gives less work in the simplification of the right hand sides.

Example of Addition:

In this example, M = 24 = the least common multiple of the the two denominators 8 and D = 12 while A = 5 and C = 7. So M/A = 24/8 = 3 and M/D = 24/12 = 2.

[Play Video] 3 minutes  Another example of how to add fractions with and without the least common denominators with an explanation that not using the LCD (least common denominator)  leads to ratios that can be simplified. So use of LCDs is advised.

How to Choose M:

Method 1: List the first B multiples of D, and list the first D multiples of B.  The number B x D = D x B is the last number in each list. Let M  < B x D be the smallest number in both lists. That number will be the smallest common multiple of B and D.

Subexample: Let B = 8 and D = 12 as above.

1	2	3	4	5	6	7	8	9	10	11	12
8	16	24	32	40	48	56	64	72	80	88	96
12	24	36	48	60	72	84	96

The number 24 is the smallest in both lists. So 24 = the least common multiple of 8 and 12.  Shortcut: Calculate the first D multiples of the smallest number B until the first multiple of D appears.

Method 2: From the prime factorizations of B and D form a product of primes where each prime in the product appears to the greatest power that occurs in the prime decomposition of B and D.

Subexample:  B = 8 = 2³ and D = 12 = 3×2².  Then M = 3×2³3 = 8 × 4 = 24 as before.

Method 1  works best with pairs of numbers < 15. Each list is then  < 15 numbers long. Method 2 works best if you know how to obtain the prime factorization of a number.

Real Player Videos

1. [Play Video] 5 minutes. How to add fractions using common denominators. Here the common dominators is the lowest or least common denominator (LCD) and its given by the least common multiple (LCM) of the denominators in the fractions added together.  Here the listing multiples method is used to compute the LCM. The alternative of not using the LCD for the fractions is explored to show what happens when the LCD is not used.
   
2. [Play Video] 3 minutes - Another example of the listing multiples method to find the LCM and thus the LCD for the sum of two fractions.
   
3. [Play Video] 4 minutes - Factorization method to obtain  a common denominator, here the LCM and thus the LCD for the sum of two fractions. See if you can recognize the GCD of the denominators here. It is not mentioned here. In this example,  the LCD is given by a product that does not have to be evaluated explicity due to cancellation of common terms after addition of fractions.
   
4. [Play Video] 5 minutes - How to use Prime Factorization or Decomposition for LCM and LCD for a pair of denominators, an example.

[Play Video] 3 minutes - Comparison of Fractions Size or Magnitude, and (?) more examples of the use of common denominators in addition and subtraction.

To compare two fractions with unlike denominators

express both over a common denominator M. Then

A B

=

A(MB) M

and

C D

=

C(MD) M

Then compare the numerators.

The case M = BD gives

The first fraction A/B is then

1. less than the second fraction C/D if AD < BD. 
2. greater than the second fraction C/D if AD > BD, and
3. has the same value as the second fraction C/D if AD = BD

In case III, we may also say the two fractions are equivalent.

In general, when

A B

=

A(MB) M

and

C D

=

C(MD) M

The first fraction A/B is then

1. less than the second fraction C/D
       if A(MB) < C(MD) 
2. greater than the second fraction C/D
       if A(MB) > C(MD), and
3. has the same value as the second fraction C/D
       if  A(MB) = C(MD)

In case III, we may also say the two fractions are equivalent.

Rules and Efficient Methods for Multiplication

The slogan multiply the tops and multiply the bottoms represents the direct statements and use of the product calculation rule (here algebraically described)

But before you calculate the products A×C aand  B×D of the numerators and denominators, factor A, B, C and D to cancel any common divisors. 

NOTE:  The above rules give the first arithmetic step in a two step process where the second step is simplification. . But for less work, we may combined the first step with the secoond through the cancellation of common divisors (you may like to call them factors) in the tops and bottoms.  

For example

In this example, we did not calculate 12×15 nor 25×16. Instead we factored further the factors 12, 15, 25 and 16 in these products in order to apply cancellation rules for simplifying fractions. 

To apply the formula efficiently

factor A, C, B and D, so that you can start fraction simplification process (reducing terms) before you do multiplication of the numerators and denominators

1. [Play Video] 2-3 minutes. Multiplying Fractions with  cancellation of  common factors done first (recommended) or not. If not,  more simplification needs to be done.
   
2. [Play Video] 3-4 minutes. Equivalent fractions - Lowering and raising terms (the values of numerators and denominators) to obtain equivalent fractions. Simplification involves lowering terms - cancelling common factors or divisors on top and bottom. Addition & subtraction of fractions may involve raising terms to obtain a common denominators. See below.
   
3. [Play Video] 2-3 minutes A few examples of Simplifying Fractions - lowering terms by canceling common factors until there are no more common factors, so that the numerator and denominator are relatively prime, that is there prime decompositions have no primes in common.
   
4. [Play Video] 2 minutes - Fraction Simplification using Prime Decomposition (factorization) to identify common factors for  cancellations.
   
5. [Play Video] 5 minutes - Product Simplification using Prime Decomposition by Canceling Common Primes, thus avoiding some denominator and numerator multiplication. An alternative common factors as they appear, more opportunistic, is given and is to be recommended.

A justification of  

is part of the treatment of fractions in the rest of this area.  Read the justification alone or with help to improve and perfect your fraction and algebra skills.  Good luck. 

Rule for Division of Fractions

 This rule converts division by a fraction C/D into multiplication by its reciprocal D/C. Then you apply the previous rule for multiplication of fractions.  For Example

In the previous example, there is a compound fractions

Division of one fraction by another, that is can be written as compound fraction.

A compound fraction has a numerators and denominator given by a proper or improper fraction. The latter may be written as a mixed number.

Remember: A mixed number is given by a whole number and a fractional part. The fractional part is proper. The above rules for multiplication and division of fractions can also be applied to mixed numbers after converting the latter into improper fractions with the same value. With time and experience, you may also use the distributive law a(b+c) = ab +ac in order to modify the above rules and do arithmetic more efficiently. But we keep our initial rules as a simple as possible in the first instance.

 www.whyslopes.com > Four Topics & Links Entrance > Fraction How-TOs     [ Next ]
The four topics: [ Fraction Guide ] [ Fractions with Units Guide ] [ Ratios & Fractions Guide ] [ Proportionality Guide ] [ Links ]

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