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Constant Rate Meaning and Concise Solution

A rate is a division relationship of one non-zero numbered word divided by another non-zero numbered word where the words are different. The words can be units of measurement or types of items counted.

Such a rate becomes a constant rate when the same non-zero number multiplies both numbered words to get a correct result in terms of physical reality.

The constant rate becomes a unit rate when the number of the word in the bottom of the fraction is a one if given that way or through division.

Given: 2 ips (n) = 5 aps (n): where n ≠ 0 (ips and aps can be any different words)

Find: 1) Number of aps from (x) ips

        2) Number of ips from (y) aps

Solution: 1) (x) ips (5 aps / 2 ips) = (5 x / 2) aps

              2) (y) aps ( 2 ips / 5 aps) = (2 y / 5) ips

The solutions are clear, concise and precise as is possible. One can also consider and check if the rate is constant. The given rate and found rate can be easily determined from the solution. From just one point condition on a graph (X1, Y1) and the origin (0, 0) a straight line can be drawn from which all other point conditions (X2, Y2) can be easily calculated as follows: (Y1 / X1) X2 = Y2

An enormous number of useful problems can be solved using just the simple form shown above. For example exchange rates established by Government, science, business and barter or physical structure rates between the atoms in molecules or macroscopically or physical property rates like density, pressure and the spring constant or process rates like speed and flow.

In the same system two constant rates can create a third constant rate. For example:

Speed / Efficiency = Flow as follows: (miles / hour) (gallons / mile) = gallons / hour.

The mass of an atom can be calculated from atomic mass divided by the Avogadro number as follows:

(26.98 grams Al / mole) (mole / 6.022 E+23 atoms) = 4.338 E-23 grams / Al atom

1).Exchange Rates:

A) Government:

Given: 1 $ (n) = 100 cents (n): where n ≠ 0

Find: 1) Number of $ from (x) cents

        2) Number of cents from (y) $

Solution: 1) (x) cents (1 $ / 100 cents) = (1 x / 100) $

              2) (y) $ ( 100 cents / 1 $) = (100 y / 1) cents

B) Science

Given: 2.54 cm (n) = 1 inch (n): where n ≠ 0

Find: 1) Number of inches from 10 centimeters

     2) Number of cm from 15 inches

Solution: 1) 10 cm (1 in / 2.54 cm) = (10 / 2.54) in = 3.937 in

         2) 15 in (2.54 cm / 1 in) = (15 * 2.54 / 1) cm = 38.1 cm

C) Business

(Usually when many items are purchased the price per item is decreased. Thus as n is increased the unit price may decrease.)

Given: $ 2 (n) = 1 item (n): where n ≠ 0

Find: 1) Number of items from $10

        2) Number of $ for 15 items

Solution: 1) $10 (1 item / $2) = (10 / 2) items = 5 items

              2) 15 items ($2 / item) = $(15 * 2 / 1) = $30

D) Barter

Given: 2 Apples (n) = 3 peaches (n): where n ≠ 0

Find: 1) Number of Apples from 9 peaches

        2) Number of peaches for 10 apples

Solution: 1) 9 peaches (2 apples / 3 peaches) = 6 apples

              2) 10 apples (3peaches / 2 apples) = 15 peaches 

2) Physical Structure Rates

The Molecule water (H2O) has 4 variables (2 H, 1 O, 3 atoms, 1 molecule) that can form 6 different pairs of 2 items and thus 12 different constant (or unit) rates.

2 boys can play the hydrogen atom and 1 girl can play the oxygen atom and thus 6 students can interact to show how water is formed as follows. 2 H2 + O2 > 2 H2O

Given: 2 H (n) = 3 atoms (n): where n ≠ 0

Find: 1) Number of H from 6 atoms

        2) Number of atoms from 4 H

Solution: 1) 6 atoms (2 H / 3 atoms) = 4 H

              2) 4 H (3 atoms / 2 H) = 6 atoms 

Ideas to develop follows:

3) Physical Property Rates A) Density = mass / volume B) Pressure = force / area

4) Process Rates A) Speed = distance / time B) Flow = volume / time

Next: From the 4 variables of distance, time, cost, and volume in the 3 constant rates of speed, efficiency and unit price, find the other 3 pairs of constant rates and develop the algebra of the system.

Then: Show the volume of a cube as 3 different constant rates on a macroscopic level and then on an the atomic level to calculate the number of atoms in a given volume for the FCC, body centered and simple cube structure.

Show how every number as a series of digits is a constant rate.

Show how every division is either a constant rate or an average rate.

Show constant rates for 3 variable problems like (robots * time / product) can be related to Newton's equation: (F t / m v).

Relevant topics in Wikipedia:

Unit conversion by factor-label

Proportionality (mathematics)

Rate (mathematics)

References

www.chem.tamu.edu/class/fyp/mathrev/mr-da.html -

www.alysion.org/dimensional/fun.htm -

www4.ncsu.edu/unity/lockers/users/f/felder/public/.../units.html -

"Principles of Modern Chemistry" by David W.Oxtoby and Norman H. Nachtrieb

External links

• example.com