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.

12. Division of Fractions and Compound Fractions

Division

The following diagram indicates that the fraction ¾ goes into 3½ units, 4 full times with ½ left over. The ½ is two-thirds of ¾.

We see that 

So we put

We say 3½ divided by ¾ is 

We also say 3½  is ¾ is  of 

Algebraic Shorthand Description of Ideas

Read ÷ as divide by

when  and only when 

Here the reciprocal 

works.  

Check: 

First Example Revisited:  How many times does   ¾ goes into 3½ = (7/2)?

Our conclusion is that division by a fraction is computed by multiplying by its reciprocal. 

Another Examples:

Check:  

The foregoing says (¹³/8) is exactly (²/3)rds of (³⁹/16).

One More Example:

Check: 

Remember: division by a fraction is computed by multiplying by its reciprocal. 

Compound Fractions

 Instead of writing 

we may write 

Expressions of the form

where the numerator and denominators are given by  fractions or mixed numbers provide compound fractions. They are evaluated by multiplying the numerator by the reciprocal of the denominator in accordance with the rule or pattern for division of one fraction by another. In mathematics, an expression is defined by saying how to evaluate it. The foregoing tells us how to evaluate compound fractions.  

Notation: In compound fractions, the division bar between the fraction numerator (top) and the fraction giving the denominator (bottom) should be longer and thicker than the fraction bars in the numerator and denominator. Use parentheses to indicate the order of operations if you wish to depart from this convention. 

Shorthand Description 

Remark: 

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