YOU are better than YOU think. Show yourself how:
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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work |
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Logic chapters 1 to 5 re- appear not in sequence, as is or longer, in Volume 1A, Pattern Based Reason, Bon Appetite.
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
Logic mastery makes the hard, easier. Logic mastery leads to better, stronger and richer comprehension. Logic mastery improves reading and writing. Logic mastery ease learning difficulties. Logic mastery gives a headstart. In sum, logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.
After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6;
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Caution: Site advice is approximately correct, for some circumstances, not all. That leaves room for thought |
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What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts.
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.
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.
Examples of Proportionality and
Multiple Ratios or Proportions
Examples
A few examples may illustrate and clarify the above.
- 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.
- 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
- 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. What a waste of time.
- 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.
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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 W0 N0 and 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.
- 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.
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Speed. If the ratio (distance traveled)/(Time taken) is constant for a journey then we say the traveller 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. |
Area Content Summary
Hint: See site area on solving linear equations to strengthen fraction sense and algebra skills together. Good luck. |