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. |
13. Fractions versus Ratios
Direct Proportionality: A number or quantity Z is directly proportional to another quantity X in several circumstances when and only when the quotient Z ÷ X = Z/X has a constant value k,.or equivalently, there is a constant k such that Z = k X. That is in each instance where we find or measure the value of X, the value of Z will be kX.
Equivalent Fractions:
A fraction M/N equals 3/4, that is
| M N |
= | 3 4 |
when and only when
| 4M 4N |
= | 3N 4N |
when and only when
4M = 3N or the numerator M = (¾ ) N is proportional to the numerator N.
In general a pair of fractions
| M N |
= | A B |
are equal when and only when the numerator M of the first
| M | = | ( | A B |
) | N | = | kN |
is proportional to the denominator M of the first with proportionality constant k = the second fraction.A/B
Ratios and Fractions:
We read A:B as the ratio A to B. We say one ratio A:B is the same as another ratio C:D when and only when the cross products AD = BC. We write A:B :: C:D when and only when AD = CD and read A:B :: C:D as the ratio A:C and C:D are equal. We could use the equal sign = in place of the old fashioned four dot symbol ::.
Now the equality condition for ratios AD = BC holds when and only when
| AD BD |
= | BC BD |
which in turn holds when and when only
| A B |
= | C D |
So two ratios A:B and C:D are equal or equivalent when and only when the corresponding fraction (or compound fractions)
| A B |
and | C D |
are equal or equivalent.
Scaling Ratios: From the equivalent fraction property that
| A B |
= | nA nB |
we observe A: B = nA : nB when ever the first and second terms in a ratio A:B are multiplied by the same whole number n.
Compound fractions have a similar property:
| A B |
= | qA qB |
whenever q is a fraction (or real number). So A: B = qA : qB when ever the first and second terms in a ratio A:B are multiplied by the same fraction or real number q.
Identification of Fractions and Binary Ratios
In many places around the world, the fraction
| A B |
is called a ratio, and no difference is emphasized between the concept of a ratio A:B and the concept of a fraction. Even I will call a fraction a ratio, or vice-versa. Reasoning involving equivalent ratios written as A:B can also be done with equivalent fractions written as
| A B |
Fractions and Ratios scale in the same way. Therefore A:B = M:N when and only when
| M N |
= | A B |
are equal when and only when the first term M of the ratio M:N
| M | = | ( | A B |
) | N | = | kN |
is proportional to the second term N in the ratio M:N
Differences between fractions A/B and ratios A:B
We can add, subtract, multiply and divide fractions written as
| A B |
But these arithmetic operations are not (to the best of my knowledge) defined for the ratios written as A:B.
We may also identify a fraction written as
| A B |
with a percentage or real number
Convention:
The ratio notation A:B appears when and only when the scaling properties of the first and second term are important.
Ratios of a part to the whole
Earlier writers identify a ratio m: n (read m to n) of a pair of numbers with the fraction
|
m |
That makes sense when considering m parts of equal value out of n parts of equal value. With this identification two ratios a:b and c:d are equal when and only when the corresponding fractions are equivalent
| a b |
= | c d |
(1) |
or have equal values. Here a and d are called the extremes of the ratio;
Therefore a:b = c:d implies c:d = a:b. Therefore a:b = c:d implies b:a = c:d (extremes swapped with means) and d:c = b:a as reciprocals of both sides in (1) must be equal.
Algebraic forward and backward views of the latter equation implies the following when two ratios a:b and c:d are equal.
| ad | = | cb | (2) | clear denominators in (1) by multiplying by bd. So product of extremes a and d equals the product of means |
| a c |
= | b d |
(3) | introduce denominators in (3) by
dividing by cd. So a:c = b:d. Swapping the means preserves equality. |
| d b |
= | c a |
(3) | introduce denominators in (2) by
dividing by ba. So d:c = b:a Swapping the extremes preserves equality. |
Ratios of complementary parts
Imagine a collection of q = m + n objects divided into disjoint subsets of m and n objects, respectively. Here the identification of the ratio m:n with the fraction
m
n
is problematic. The ratio may be identified, if we must with the compound fraction
|
m |
|
|
|
m |
All this is to suggest that a distinction needs to be made or simply known between the ratio written as m:n and the fraction m/n. The question is how. The ratio notation does not distinguish between the ratio of a part to a whole and the ratio of complimentary parts.
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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.