Section Topics
Fraction,
Fraction with Units, Fractions & Ratios; and Proportionality
forwards & backwards.
Section Pages
Would you like to show yourself or others how to be algebra power users?
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.
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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.
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.