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Employ an online or offline tutor at your own risk from
AU:
tutorfinder.com.au
CDN :
findatutor.ca
CDN: .i-tutor.ca
CDN: Montreal
Tutors
NZ: findatutor.co.nz
UK:
tutorhunt.com
UK: tutors4me.co.uk
USA: wiziq.com
USA: ziizoo.com
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YOU are better than YOU think. Show
yourself how:
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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
Do not leave here without it - Logic
mastery will develops critical thinking, improve reading and
writing, and give a firmer base for work and studies at many levels.
Good luck.
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Caution: Site advice is
approximately correct, for some circumstances, not all. Site How-TOs
are logically developed, but not tried and tested. That leaves
room for thought and refinement.. |
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After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside site
area on solving
linear2007 Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
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.
|
Explore collaborative whiteboards from groupboard,
twiddla or
scriblink.
|
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Simple & Compound Fractions
A simple fraction is given by a pair of whole or natural numbers
M
N
where N denotes the denominator and M denotes the numerator.
A ratio of simple fractions is a compound fraction
A
B
in which the numerator and denominator are both decimal
fractions.
|
If |
|
|
| A |
= |
C
D |
|
and |
|
|
| B |
= |
E
F |
| then |
|
|
A
B |
= |
C
D |
* |
F
E |
In other words, the compound fraction
There is a rule in developing mathematics: Saying how to compute a quantity
defines it. The latter expression may be taken as the definition of the
compound fraction
We see that a compound fraction formed from a pair of fractions gives a
fraction.
Special Case:
|
1
---
E
F |
= |
1 |
* |
F
E |
= |
F
E |
That is 1 divided by a fraction equals the reciprocal of the fraction.
Multiplying by the reciprocal
|
If |
|
|
| A |
= |
C
D |
|
and |
|
|
| B |
= |
E
F |
| then |
|
|
A
B |
= |
A |
* |
1
B |
This is another way of saying division of a first fraction by a second
fraction has the same result as multiplying the first fraction by the reciprocal
of the second. The case where A and B are decimal fractions will appear
below.
Theorem: If M and N are given by decimal fractions
then M has a or few places after the decimal point, N has b or
fewer places after the decimal point, and
Moreover if a = b then
Note: We can extend the decimal expansion of M or N with zeroes
after the decimal point to make a = b. The replacement of the decimal
compound fraction M/N by P/Q where P and Q are both whole numbers provides a
rule for moving the decimal point or eliminating it in fractions involving
decimals.
Note: The rule
M
N |
= |
M |
* |
10b
Q |
= |
M*10b
Q |
provides a method for shifting the decimal point in the dividend M and
divisor N to obtain a whole number divisor Q. Then the long division algorithm
can be applied as is, or in continued form.
Division of One Fraction by Another
The question find a number or fraction T such that
has a unique solution
|
T |
= |
C
D
---
E
F |
= |
C
D |
* |
F
E |
Check:
| T |
* |
E
F |
= |
( |
C
D |
* |
F
E |
) |
* |
E
F |
= |
C
D |
* |
( |
F
E |
* |
E
F |
) |
= |
C
D |
* |
1 |
= |
C
D |
Exercise: Explain why the solution is unique.
See the discussion of division in the fraction section of this site. The
solution T above
Here
|
T |
= |
C
D
---
E
F |
= |
C
D |
divided by |
E
F |
is the fractional number of times that the denominator
E
F
goes into the numerator
C
D
of the compound fraction
Note: Compound fractions of the form M/N where M and N are real
numbers will be treated later.
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www.whyslopes.com
Number Theory
[ Back ] [ Up ] [ Next ]
Start of Number Theory
Origins of Counting or Tallying
Adding Wholes
Multipling Wholes
Distributive Law Preamble
Distributive Law for Wholes
Consequences
More Consequences
What is a Fraction
Compound Fractions
Number Theory
Continued
Decimal Place Value
Place Value Reinforcement
Comparison Method
Addition Method
Subtraction Methods
Multiplication Methods
Division Methods
Remainder Arithmetic I
Primes & Composites
Primes Factorization Theorem
Primes & Composites
Prime Factorization Aids
Prime Factorization Examples
Counting Whole No. Factors
Arithmetic Videos
Square Roots & Primes
Long Division Continued
Fractions & Decimals
Fractions as Decimals
1 = 0.999 Recurring
Infinite Decimals Expansion Arithmetic
Ratio of Simple Fractions
Ratio of Decimal Fractions
Unsigned Reals Numbers
Signed Coordinates
Plane Vectors
Horizontal Vectors
Adding Vector Multiplies
Adding Signed Numbers
Multiplying Signed Numbers
Distributive Law for Reals
Real Numbers Axioms
Remainder Arithmetic II
Related Site Pages:
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
Food for thought: 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..
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