YOU are better than YOU think. Show yourself  how:  

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 Logic chapters 1 to 5  re- appear not in sequence, as is or longer,  in  Volume 1A,  Pattern Based Reason, Bon Appetite.

Logic Mastery
 Amazing, Amusing, Amorous,  Delicious, Delightful, Edifying, Strengthening Elixir. 
It eases work & learning difficulties Makes the hard easier. Opens eyes. Leads to greater precision.
in reading and
writing

Logic mastery makes the hard, easier. Logic mastery  leads to better, stronger and richer comprehension.  Logic mastery  improves reading and writing.  Logic mastery ease learning difficulties.  Logic mastery gives a headstart.  In sum, logic mastery  will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.

After logic,   (a) continue reading Three Skills for Algebra, chapters 8 to 14  and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes  & More Math, chapters 2 to 6;

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What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts.

Try the Twiddla Whiteboard. In principle, it  allows to people to draw and chat together online on a copy of this webpage or a clean sheet. The chat may be via text or audio.  Visit www.twiddla.com to set up whiteboards to work with the webpage of your choice.

For online automated help in senior high school maths & calculus, visit  quickmath.com  For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com  With  overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck.

[RealPlayer Video] 3 minutes - Comparison of Fractions Size or Magnitude, and more examples of the use of common denominators in addition and subtraction. Theory follows.

7. Comparison of Fractions

Justification of a cross-multiplication rule

Example 1. The question which is greater

is often answer by comparing 5*4 = 20 with 6*3 =18. Let use look at this in more detail.  The least common denominator is 12. The following diagram show both fractions in terms of twenty-fourths: Here 24 = 2* 12

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

Therefore 

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

(i)   20 = 5*4 = (first numerator)*(second denominator) 

with

 (ii) 18 =6*3 = (first denominator)*(second denominator). These 

These over 6*4 = 24 numerators indicate how many (6*4)ths there are in the original fractions. 

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 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.

Algebraic Shorthand Description of Ideas- the General Case: The key to comparing fractions is to put them over a common denominator.

Read the following with literals  (a,b,c,d) = (8, 5, 7, 11) in the first instance. Then let the literals or letters a, b, c and d be any whole number you like.

To compare 

we put them over a common denominator b*d.   (Here we assume b and d are non-negative - why?)

Now we need to compare numerators a*d and b*c.

There are three possibilities:

The foregoing explains and justifies the so called cross-multiplication rule for the comparison of fractions with non-negative denominators. The product of the denominators b*d gives a common denominator, most likely not  the least common, but it will do. The use of the least common denominator is optional in the case of comparison.

Instructors: Students may also compare mixed numbers. The comparison there begins with comparison of the whole number parts. If those parts are equal, comparison then proceeds with the (proper)  fraction parts. The cross-multiplication or common denominator method then applies.  Note: there is no need to convert the mixed number into an improper fraction.

6. Addition (and Subtraction) with like Denominators

Instead of counting how much or how is present in terms of whole units we may count in terms of unit fractions. 

Example 1. Counting thirds

The foregoing says 2 thirds plus 4 thirds (of a unit of measure)  gives six thirds  (of the unit of measure) and the latter is equivalent to 2 (of the unit of measure).

Example 2. Counting Tenths

since 33 = 3*10+3.  The foregoing says 18 tenths plus 15 tenths (of a unit of measure)  gives 33 tenths (of the unit of measure) and the latter can be regrouped in to 3 units (as ten tenths equal 1) plus 3 tenths.

Example 3. Subtracting 12ths.

since 33 = 3*12+3.  The foregoing says 11 twelfths minus 5 twelfths is 6  twelfths (of a unit of measure). The latter gives a half. The following diagram attempts to illustrate the subtraction and the fact that 6 twelfths is a half. 

Algebraic Shorthand Pattern or Rule for Addition and Subtraction with like denominators

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

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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