YOU are better than YOU think. Show yourself how:
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Logic
Mastery 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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-/[]\- 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. |
[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
How raising fractions over a common denominator leads to direct comparison (and justifies cross multiplication rule methods for comparison
Example 1. The question which is greater
|
5 |
or |
3 |
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
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
| 5 6 |
1 |
||||||||||||||||||||||
|
3 |
1 |
||||||||||||||||||||||
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
Here by putting both fractions over the common denominator 4*6= 24, we see that
|
5 |
= |
20 24 |
is more than | 18 24 |
= |
3 4 |
Therefore
|
5 |
> |
3 |
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
|
9 |
or |
11 |
This can be answered by seeing how (13*17)ths there are in each fraction. We see that
|
9 |
= |
9*17 13*17 |
= | 153 13*17 |
while | 11 17 |
= |
13*11 13*17 |
= | 143 13*17 |
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
|
a |
with |
c |
we put them over a common denominator b*d. (Here we assume b and d are non-negative - why?)
|
a |
= |
a*d b*d |
needs to be compared with | b*c b*d |
= |
c d |
Now we need to compare numerators a*d and b*c.
There are three possibilities:
| (i) if a*d > b*c then |
a |
> |
c |
| (ii) if a*d = b*c then |
a |
= |
c |
| (i) if a*d < b*c then |
a |
< |
c |
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.
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www.whyslopes.com
Fractions, Ratios, Units, Rates
& Proportionality
Fraction
Starter Lesson
(simplify, multiply, divide & then add or subtract)
An Alternative Starter
Lesson
(take your pick, or try both)
Arithmetic Videos - Real Player Format
Decimal Addition
Methods
Decimal
Subtraction Methods
Decimal
Multiplication Methods
Decimal Division
Methods
Fractions
Primes
Greatest Common
Divisors
Least Common Multiples
Square Root
Simplification
Area Content Summary
- Fraction Starter Lesson
- Real Player Videos on Operations with Primes and Fractions
- Continuous Ruler & Line Segment
model for fractions and operations on fractions - Number Theory Area points to the general model. - Distinction between Ratios and Fractions, a nuance: While binary ratios a:b may be identified with a fraction, triple ratios a:b:c and further multiple ratios cannot.
- Saying how to add and subtract like monomials in units and their powers, and saying how multiply and divide like and unlike monomials leads to fraction like expressions involving units and a framework for discussion rates - ratios of quantities - a framework for handling proportionality constants, and framework for carrying units through calculation in quantitative disciplines
Hint: See site area on solving linear equations to strengthen fraction sense and algebra skills together. Good luck.