17 Proportional Situations -- Fractions, Ratios, Proportions

Proportionality

This pages describes rates and proportionality constants with units. Rates are in fact proportionality constants. Fractions with units may appears in calculations below as proportionality constants, and as the given or to be found values of variables or quantities below.

Different kinds of Proportionality

There might be more

Direct Proportionality (Direct Variation): A number or quantity y is directly proportional to another quantity x in several circumstances when and only when the quotient y ÷ x = z/x has a constant value k,. or equivalently, there is a constant k, a number or quantity, such that y = k x. That is, in each instance where we find or measure the value of x, the value of y will be kx.

The proportionality constant k here is called a rate if it is a fraction with units or if it equals an irrational number. It is called a ratio if it equals (possible after cancellation of units), a rational number or fraction.

That is, a single quantity y is directly proportional to a second quantity x when and only when there is a non-zero constant k such that y = k x -the proportionality relation.

The direct use of the proportionality relation y = kx is to calculate the value of y from those of k and x. But tin practice, the a problem gives the values (x₁, y₁) first, from which the value of the proportionality k can be computed via a backward use of the formula. And after k is known, the formula y = k x can be used directly or indirectly to compute y or x respectively. In practice, the problem may ask for the latter value of y and x in some circumstance, while including values (x₁, y₁) which can be used to find k. The typical proportional relation problem involves two steps. The first is to find the proportionality constant k. The second is to use proportionality relation forwards or backwards, taking advantage of the just calculated value of k.

Joint Proportionality (Joint Variation). A number or quantity z is directly proportional to quantities x and y in several circumstances when and only when there is a constant k, a number or quantity, such that z = k xy That is, in each instance where we find or measure the value of x and y, the value of z will be kxy

That is, a single quantity z is jointly proportional to a second quantity x and a third quantity y when and only when there is a non-zero constant k such that z = kxy = the product of k, x and y. The backward use of the equation z = k x y may give the value of the proportionality constant k in terms of the quantities x, y and z: Clearly k = z/(xy). Three given values of x, y and z are thus sufficient to calculate k.

Inverse Proportionality (Inverse Variation): A number or quantity z is inversly proportional to another quantity x in several circumstances when and only when the product z×x has a constant value k,.or equivalently, there is a constant k, a number or quantity, 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 k/x. Here z is directly proportional to the reciprocal of x = the multiplicative inverse of x.

New Proportional Relations from old: Suppose quantity z is directly proportional to quantities x and y. Then The backward use of the equation z = k x y may give the value of the proportionality constant k in terms of the quantities x, y and z, or it may give the value of one of the two variables x and y, say y in terms of x, z and k. Clearly z = kxy implies

y	=	z kx
	=	1 k	×	z x
	=	K	×	z x

Thus y is jointly proportional to z and inversely proportional to x. There-in lies another kind joint direct and inverse proportionality. In the one just obtained, the proportionality constant for the new relation is K = 1/k.

More generally, when product of one group of quantities may be proportional to the product of another group of quantities, multiple proportionality relations are implied, one for each quantity or variable that may appear in either of the products. We assume in the first instance that the products have no variables in common.

Algebraic Perspective of Proportionality.

In situations involving multiple proportionalities, amounts are proportional to each other and to any linear function of the amounts in questions (in which the coefficients are fixed and positive).

Example: The sides of similar triangles and of similar polygons are proportional to each other. There are multiple proportions here.

Now the simultaneous equations x =kx, y=ky and z =kz can be used backwards and forwards. If k is unknown, it can be found given from one of the equations Then it can be use in the other two.

Assume x, y and z are all nonzero. In 3D projective geometry, the point (x,y,z) with is equivalent to another point (X,Y,Z) when and only when x:y:z = X: Y: Z when and only when (i) the three ratios x/X y/Y and z/Z have a common value k, when and only when (ii) there is a constant k such that x =kX, y=kY and z =kZ. See the earlier discussion of multiple rations.

A. More Arithmetic a must for algebra etc

D. Logic In Mathematics

G. Algebra with Take Home Value

I. Vectors & Functions

Decimal Lesson - Reference
Counting & Addition   (8 lessons)
Comparison to Subtraction (9 lessons)
Multiplication ( 11 lessons)
Long Division (12 lessons)
Decimals and Primes (8 lessons)
-Primes & Composites
-Primes Factorization
-Greatest Common Divisors & Multiples.
-Prime Factorization Aids (Learn how to find factors quickly)
-Prime Factorization Examples
-Counting & Generating. Factors.
-Divisibility Rules and Remainders for Division by 2, 3, 5, 9 and 11.
Integers (12 lessons) Intro to Signed Numbers
Fractions (< 20 lessons)  Essential Skills & Concepts
Ratios & Fractions (3 lessons): Similarities & Differences
Units in calculations Fractions with Units

B. Basic Algebra

Solving Linear Equations
- in one unknown. Intro with stick diagrams?
the normal way & with good nttn. (the nttn that reappears in Gaussian Elimination. |
-in more unknowns: simultaneous equations essentially one unknown. the let algebra do the work view of word problems.
- still in more unknowns: Gaussian Elimination via substitution, by equality or comparison, by operations on equations.

C. More Algebra

Words before symbols: See if U like the lengthy chapters 8 to 12 in Volume 2, Three Skills for Algebra
What is a Variable. The answer here is a simple prequel to the modern mathematics viewpoint.

First, every rule & pattern U meet in math, logic & science will be used forwards and backwards. Get a head start with this theme by reading Chapter 14 in Three Skills for Algebra. Second, in the study of Proportionality Relations (3 dense lessons here) finding the proportionality constant gives an initial backward use of the proportionality formula.

Talking about words before symbols and the forward and backward use of formulas gives words to make algebra simpler & clearer.

If you can not read or write precisely, you will have difficulty in following instructions. One wordy remedy is given by chapters 2 to 5 in Three Skills for Algebra. Where does Logic or a geometric model for reason Appear in Mathematics? The answer lies in Euclidean-Geometry In North America, Euclidean Geometry disappeared from high school mathematics as it was too hard. The light treatment here is a possible remedy.

E. More Geometry

The Pythagorean Theorem. Chapter 17 from in Three Skills for Algebra uses algebra and geometry   to show why the Pythagorean equation for right triangles holds. Its forward and backward use is common exercise.. At a more theoretical level, the Pythagorean theorem leads the discovery that not all lengths can be fractional multiples of a unit length. That geometrically implies a need for and even existence of irrational numbers.
Analytic Geometry: Common Practices with Maps and Plans drawn to scale give coordinate-dependent base for senior high school development of similarity, trig, vectors and straight lines.
Complex Numbers: This lesson on Complex Numbers draws on Euclidean and Analytic geometry. Sbortcuts simplifiy trig identities, the cosine law; and   trig formulas for 2D dot- and cross-products.

F. Logarithms, Exponentials,
Roots & Powers

Logarithms, exponentials, rational and real powers for secondary students. This complete Operational Viewpoint. (Sufficient for the precalculus forward and backward use of compound growth and decay formulas in biology, physics, chemistry, personal finance, and calculus. To learn more, if you study calculus, see chapter 19 of Volume 3, Why Slopes and More.Math

In Volume 2, Three Skills for Algebra, chapters

identify methods useful in money computations, methods needed for calculus. Your teachers or other writer may present the same ideas with greater clarity and detail - A site to do.

H. Polynomial & Quadratics

Analytic Geometry: - Slopes and Lines - Take 1.   Take 2 appears in site section Maps and Plans.   Two views are better than one. I may combine them later. -In my school days, slopes appeared year after year.   This Why Slopes calculus preview on graphs of functions y = f(x) explains why. Enjoy.
Quadratics and Polynomials: - Operations on Polynomials: Meet a light and ultraquick geometric introduction to multiplication, addition and subtraction of polynomials. Then see how the foregoing combine to permit long division of polynomials. Compare Fractions with Units. Enrichment: A Plus: The Geometric introduction here gives or is almost identical to a justification for column methods in decimal arithmetic.
Geometric Derivation of the Quadratic Formula:   The account here gives a starter lesson for the more algebraically harder geometric-free derivation. If you study physics, chemistry or trigonometry, you will need to know about quadratics, their factorization and the quadratic formula.
Technical Value: The study of polynomials high school mathematics has technical value as part of the senior high school mathematics preparation for calculus. This simple account of Why Factor Polynomials   (Chapters 2 to 6 in Volume 3 .Why.Slopes.&.More.Math.) will give a context for the study of polynomials, their factorization, and sign analysis of functions, all in a way that should improve your algebraic thinking and reasoning skills.

Vectors in the Plane (2 simple lessons)
- Navigation with vectors or arrows
- Sum of Motions
- more lessons to be added later.
Operations on movement or vectors along the line and in the plane have value in mathematics in defining and implying the properties of real and complex numbers before the assumption of those properties as axioms. Vectors and their properties appear in physics, its mathematical description and formulation.
- Functions - Forwards & Backwards. Here is a full technical reference (24 lessons) for use in a calculus or precalculus course as needed. In it, the set viewpoint of functions expression of modern pure mathematics. comes from the set-based codification and
In the mathematics education reforms of the 1960s in North America, primary and secondary school mathematics were expressed in terms of sets. That expression has now retreated from primary and secondary school texts. But it still lingers on, and can be very useful, a source of clarity and precision, in the situations where it should be retained: Counting with the aid of sets and functions; the description of functions; the high school account of probability theory; and in the discussion or illustration of ideas in logic.

J. Pre-Calculus Skill Check

- Arithmetic Skill Check. In the calculus courses I taught 1983-89, too many students had weak skills in arithmetic. I would give and carefully correct these exercises to tell students what they needed to review and master.
- All the skills and concepts in Chapters 1 to 24 or Volume 2, Three Skills for Algebra: Look for those you do not understand and fill the gaps. Do so quickly while balancing this advice with your other duties. Good luck.

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