0.6 Phy1030: scale factors, ratios, and proportions  (Page 2/5)

General background information

Mathematical expressions are used in physics to describe relationships that are difficult to express in words. The expressions use algebraic symbols to representquantities that consist of numbers and units.

Measurements are important

Conclusions that are drawn in physics and other sciences ranging from chemistry to the social sciences are often based on measurements such as length,width, weight, salinity, population, density, etc.

Each number in an equation often represents the results of a measurement, which is made in terms of a standard. The units indicate which standard was usedto make the measurements.

Knowledge of units is critical

A number that is used to indicate the result of a measurement is of little value unless we know the units in which the measurement was made. For example,it isn't very useful to know that the length of an object is 125 unless we know whether the units are meters, centimeters, millimeters, or miles.

The Google calculator

Although I won't go into detail in this module, I will tell you that the Google calculator is very good at helping you to keep track of units. Forexample, if you enter the following expression in the Google search box,

3 ft + 1 yd + 36 inches

The following result will be displayed immediately below the search box:

(3 ft) + (1 yd) + (36 inches) = 2.7432 meters

Discussion

We often express the relationship between two items using a scale factor.

Scale factors

For example, we might say that that a colt doubled its weight in one year. Thismeans that the colt's weight after one year was equal to the colt's weight at birth multiplied by a factor of two. The factor is the number by which aquantity is multiplied or divided when it is changed from one value to another.

Ratios

The factor is the ratio of the new value to the old value. Regarding the colt mentioned above, we might write:

NewWeight / BirthWeight = 2

where the slash character "/" indicates division.

Percentage increases

It is also common for us to talk about something increasing or decreasing by a given percentage value. For example, we might say that a colt increased itsweight by 50-percent in one year. This is equivalent to saying:

NewWeight = BirthWeight * (1 + 50/100)

This assumes that when evaluating a mathematical expression:

• The asterisk character " * " indicates multiplication.
• Computations begin inside of the inner-most pair of matching parentheses and work outward from there.
• Multiplication and division are performed before addition and subtraction are performed.

This is typical behavior for computer programs and spreadsheets, but not necessarily for hand calculators.

Percentage decreases

In my dreams, I might say that I went on a diet and my weight decreased by 25-percent in one year. This would be equivalent to saying:

NewWeight = OldWeight * (1 - 25/100)

Exercise on scale factors

Write a script that has the following behavior. Given a chalk line that is 100 inches long, draw other chalk lines that are:

• A. Twice the length of the original line.
• B. Twenty-five percent of the length of the original line.
• C. Twenty-five percent greater than the length of the original line.
• D. Twenty-five percent less than the length of the original line.