Examples of Proportionality and Multiple Ratios or Proportions

Examples of Proportionality and
Multiple Ratios or Proportions

Success in high school mathematics and science requires mastery of proportionality relations, forwards and backwards, in both numerical and algebraic ways. See Chapter 14 in Three Skills for Algebra to understand or explain the forward & backward use of formulas. Good luck.

In following a recipe for a cake or a meal, to the amount required of each ingredients is proportional to the number N of serving. The number N is the proportionality constant or scale factor. The ingredients ratios for to serve N are equivalent to ingredient ratios to serve one or another number M.

In mixing concrete from sand, water and powdered cement, the amount of each ingredient required is proportional to the total amount of concrete needed. Here again for different quantities of concrete produced, the multiple ratio of ingredients should be equivalent.

Consumption: The amount of material (cloth) needed to provide students with a with a school uniform is proportional to their number

Consumption: In building homes and towns in similar environment, the amount of resources (Food, Water, Electricity, Oil, Gas, Materials) for construction and maintenance of the homes and then the people there-in is most likely proportional to the number of homes or the number of people to be housed.

In mass balance equations in chemistry and physics, the amount of product is proportional to the amount of reactants.

The perimeter of a circle is proportional to its radius, The proportionality constant is given by the real number k=2p.

The area of a circle is proportional to the square of its radius, The proportionality constant is given by the real number p.

The area of a rectangle is proportional to its width and length. The proportionality constant has value 1.

The area of a triangle is proportional to its base and its height. The proportionality constant has value ½

For sectors of circles, area is proportional to the measure of the central angle. and the perimeter of the section, the portion on the circle, is also proportional to the central angle.

Exercise: Find k given s = 2pr when the angle q = 360 degrees
Reference: Chapter 20, Degrees and Radians (etc) in Volume 2, Three Skills for Algebra .

Linear production models: after fixed costs, the amount of production is proportional to the ingredients consumed. (I once had a job in operations research running computers models in which every thing was linear, and I was not believed when I indicated that the result would increase by 10% if we increase all the inputs or constraints by 10%. So I had to run the cases overnight.

Construction - scaling up: The amount of material to build houses, chairs, cars, boats repeated is often proportional to the number of house, chairs, cars and boats to be constructed.

Shares in a Company and Division of Profits per share: For a person owning shares in a company, the dividend (profit) received in proportional to the number of shares. In forming a company, a person may get a number of shares proportional to his or her contribution to the company formation. Later, when profits are distributed, each shareholder gets a proportion of the dividend (distributed profit) proportional to the number of shares he or she owns. Some textbook problems may say or imply that so and so owns N% of a business (with out mentioning shares) and ask how much of its distributed profit, should the person get. The answer is N%. Unless, there are special rules in place, when profit is distributed, ownership of N% of a business (or N% of the outstanding shares) results in N% of the distributed profit (or dividends paid).

Construction and/or Work: Work done (amount constructed) is proportional to people P working and duration T of work when all people present are equally productive. So W = KPT. The units of K will be the units of work (eg houses constructed) divided by the units of Time. Example K = 0.10 house/ (person-week) = a tenth of a house per person working for a week. Typically, K is given or implied by given values of W, P and T. Then in another situation, the values of one of W, P and T is wanted given the other two and the previously calculated value of K.

Hours worked is proportional to the how many are working and how long each works (when alll work the same number of hours. Here again W = KPT where W = hours worked, P = N persons = the measure of people present and T = the length of time, each one worked.

Work Done: The amount of work W done by N people in a time T taken is proportional to the number N of people and the time T taken. That is, W = k N T for some constant k.

Given any three of the four quantities W, k, N and T in the equation W= k NT, the fourth can be found. The typical work problem begins by saying work W₀ can be done in time T₀ by N₀ people and ends by giving two of the three quantities W, T and N and asking for the third. Do not panic. The solution begins by finding k from the equation W = k N T given W₀N₀and T₀. Here W₀ = k N₀ T₀ gives k = W₀ /(N₀ T₀). So k becomes known. Now we solving W = k N T for the missing quantity W, N or T.

Scale factors in 1, 2 and 3 D. In Maps, Plans, Models and Images drawn to scale 1: 100 say: the length scale factor (or proportionality constant0 is K = 1:100 = 0.01 = 1/100. More over area scale factor is K² and volume scale factor is K³. So in producing a 1:10 scale model of a vehicle, the lengths are reduced by one tenth (K), surface areas are reduced by one hundredth (K².) and volume is reduced by one thousandth (K³.) However, angles are preserved. And in constructing scale models, costs proportional to length, areas and volume are also reduced by the scale factor K, K².and K³. respectively. It could exercise by yourself or for a class of students to see why a scale factor of K for length leads to the other scale factors for area and volume.

Senior High School, Proportionality Example From Geometry: For similar plane figures, the ratio of corresponding lengths and areas (absolute measures) equals a scale factor K or its square K². For similar 3D figures, the ratio of corresponding lengths, areas and volumes equals a scale factor K, its square K² or its cube K³. Student may be asked to find and/or use the length, area and/or volume scale directly or indirectly. From the algebraic viewpoint, the corresponding proportionality equations, relations or formulas (whatever you would like to call them) are being used forwards and backwards.

In simple interest computations, the amount of interest I = P r t where P in the principal (initial amount invested or borrows), t = number of periods invested, and r is the interest rate = the proportionality constant here. The foregoing says or makes the interest jointly proportional to the principal P and the number of periods.

When one quantity Y is directly proportional to another quantity X, we have Y = K X. Further when Y-values are plotted against X values, they fall on straight line for which the rise over run ratio or fraction (rise/run) = K the proportionality constant. When X and Y have the same units of measurement, the proportionality constant K is a pure number and it provides the slope of the line. Why X and Y have different units of measurement, the proportionality constant K = a rate. All rates come from such proportionality constants.

When distance plotted against time falls on a straight line then the change in distance d over change in time t fraction equal a rate and proportionality constant, the speed s. Here s=d/t or equivalent d = st. The latter equation can be used backwards and forwards. Given any two of three quantities d, s and t in it, the value of the third can be found.

Speed. If the ratio (distance traveled)/(Time taken) is constant for a journey then we say the traveler has gone at a constant speed

s = (distance traveled)/( time taken) = d/t

For constant speed journeys, the distance traveled d = st is proportional to elapsed time t and vice versa: t = (1/s) d. The speed with units of length over time provides a constant of proportionality, the rate of change of distance with respect to time.

In equivalent fractions, simple or compound, the numerators are proportional to the denominator with proportional constant be given by the fraction as is or in equivalent form. See the next lesson.

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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