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Similarities between comparison, addition and subtraction of fractions

Here by putting both fractions over the common denominator 4×6= 24, we see that 

Therefore 

and we can calculate how much more - the smaller subtracted from the larger is 

By putting both fractions over the common denominator, the original comparison can be decided by comparing the over 24 = 4×6  numerators 

Easy Consequence: The raising terms work we done in the comparison and subtraction can be used to calculate the sum. 

We are calculating the sum to show that comparison, subtraction and addition operations are made possible for a pair of fractions (or several) by raising to the case of like denominators.  Remember if you are asked to do one of the operations, you do not have to do the others, unless directly asked. 

Example 2:  The question which is greater

This can be answered by seeing how  (13×17)ths there are in each fraction. We see that

So the first fraction is greater. It provides 153- 143 = 10 more (13×17)ths than the second.

For those of you who insist on knowing, 13×17 =221, a number whose existence we need, but whose value is not required for the comparison. But it is required for the subtraction

9  13	-	11  17	=	9×17 13×17	-	13×11 13×17
			=	153    13×17	-	143    13×17
			=	10  221

Easy Consequence: The raising terms work we done in the comparison and subtraction can be used to calculate the sum. 

9  13	+	11  17	=	9×17 13×17	+	13×11 13×17
			=	153    13 ×17	+	143    13×17
			=	296   221
			=	1 +	75  221
			=	1  75  221

Here 75 = 3 × 25 = 3 × 5^2 has only two prime factors, namely 3 and 5. Neither factor divides exactly in 221.  So the proper fraction part of the mixed number cannot be simplified further.  It just an accident in this and the previous example that the whole number part of the mixed number is the number 1.

Again, we are calculating the sum to show that comparison, subtraction and addition operations are made possible for a pair of fractions (or several) by raising to the case of like denominators.  Remember if you are asked to do one of the operations, you do not have to do the others, unless directly asked. 

the use of common denominator 24 = 4×6 = 6×4 leads to 

and  use of common denominator 36 = 6×6 = 9×4 leads to 

For all three  choices of common denominators, the least and others, conversion to a like denominator, addition and simplification all lead to the result 4¼ . But the use of smaller common denominators leads to smaller numbers in the computation and hence less simplification work in the end.  The use of the smallest or  least common denominators usually gives the most efficient way to add and subtract fractions with unlike denominators. So try to use the least common denominator. But some work may be required to find it. 

Easy Consequences:  Since 

we see that the first fraction 15/6  is more than the second fraction  7/4 by 9 twelfths. We also see that 

Advice and Directions:

In practice, additions, subtractions and comparison may be done with any convenient common denominator.  In the case of comparison, the product of the common denominators may be best (or not) - it does not have to be computed - recall example 1. 
In the case of addition and subtraction, the use of the smallest or least common denominator (some work may be needed to find it) leads to smaller numerators to add or subtract after the original fractions to be added or subtracted are expressed over a common or like denominator.  Beyond that, when we add or subtract fractions by raising terms to a common denominator, the resulting fraction with that common denominators is simplified by lowering terms and/or expressing the result as a mixed number.  We may follow that cosmetic convention because fractions with smallest possible denominators are supposedly easier to comprehend or digest than other fractions.   For example, we may say one half instead of three sixths.  The efficient calculation of sums and differences (the result of additions and subtraction) is described in a following page. 

Algebraic Viewpoint/Description

 for reading as part of algebra skill development - optional reading for now

Addition and Subtraction with like denominators - Algebraic Shorthand Pattern 

The algebraic pattern for addition of fractions with like denominators  i 

A N

+

B N

=

A+B  N

For addition of fractions with the same denominator,  add numerators and keep the denominator

The general pattern or rule for subtraction  in terms of shorthand letters is as follows:

A N

-

B N

=

A-B  N

For subtract of fractions with the same denominator,  subtract numerators and keep the denominator

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