RATIO
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CPI was 177. This means that the ratio of costs for goods and services in 2000
was 1.77 times as high as the costs for the same items during 1982–1984. The
COL is computed for almost two hundred metropolitan areas.
It reflects the ratio
of costs of goods and services in a specific area to the average for the country as
a whole. The COL is expressed as a percent. At the beginning of 2001, the COL
for San Francisco was 179.8, and for Houston, 95.8. Those ratios mean that it
costs 1.798 times the U.S. average to live in San Francisco, but 95.8 percent of
the national average to live in Houston. Stockbrokers use the price-earnings ratio
(P/E) as a way of evaluating stocks. This ratio is defined as the market value per
share divided by the earnings per share. If a company has stock valued at $40 per
share, and has earned a net of $2
per share over the last year, the P/E ratio for the
company would be $40/$2 = 20. Most stocks traded on the major exchanges have
P/E ratios between 15 and 25. Riskier stocks that have potential for rapid growth
are likely to have P/E ratios above 25, if any at all. (If a company has not pro-
duced any earnings, then its P/E ratio is reported as 0.) In these circumstances,
people invest their money in companies that they think will have low P/E ratios
or a high demand in the future. The P/E ratio is only one of many ratios routinely
reported for stocks.
Percents are based on ratios. If a taxpayer pays $3,000
on an income of
$20,000, then the tax ratio is 3000/20000 = 15 percent. The federal government
refers to this as a tax rate. If an investment of $500 is now worth $550, the per-
cent increase is the ratio of absolute change to starting value, or 50/500 = 10 per-
cent. When you specify how long it took for this increase, you express the change
as a percent per year, or interest rate.
Test scores are often reported as
percentile ranks. A student with a percentile
rank of 60 on a college placement test achieved
a score that was equal to or
higher than the scores of 60 percent of the students taking the test. Therefore the
percentile rank is a ratio of counts of students.
Body mass index (BMI) is the quotient of your weight in kilograms divided
by your squared height, where height is measured in meters. Although this meas-
ure is a rate (kilograms/m
2
), the units are not reported and are not used in com-
putations of other health measures. BMI values from 20 to 25 are associated with
the lowest health risks; values above 30 are associated with the highest. Weight-
to-hip ratio (WHR) is a true ratio that indicates whether an adult carries weight
around the waist or hips. Weight carried around the middle (higher ratio) is asso-
ciated with more health problems. Many ratios developed
for human physiology
are applied to other animals. The ratio of an animal’s surface area to its volume
measures how much energy the animal has to produce to counter the heat lost
through the skin. (See
Inverse (Multiplicative) for an additional explanation.) A
mouse has a surface-area-to-body ratio that is about ten times that of a human,
so the mouse has to eat almost all day long to maintain its body heat, while a
human does quite well with three meals per day.
The modern musical scale is based on a consistent ratio of frequencies from
note to next note for the twelve notes of an octave. Middle C-sharp (275 cycles
per second [cps]) is about 1.059 times middle C (260 cps); D (292 cps) is about
1.059 times C-sharp. This sequence continues to high C (520 cps), which is twice
the frequency of middle C. Pythagoras (as later corrected by Galileo) tried to
identify simple integer ratios for what would correspond
to the white keys of a
piano. The Pythagorean scale uses 9/8 for the ratio that would compute D from
middle C (9/8 of 260 = 292.5).
Time signatures found at the beginning of a piece of music look like fractions
without the fraction bar. They represent beat patterns for the measure. The nota-
tion
3
4
means that there are three beats per measure, and a quarter note (1/4)
receives one beat. This sets the ratio of note values to measures for the piece of
music.
Almost all mechanical objects in your house use gears. A videotape machine
uses gears to control tape motion. Windup and pendulum clocks use gears. Bi-
cycles use gears. Gear ratios tell how rotational motion changes when you con-
nect gears with different numbers of teeth. When a small gear with forty-seven
teeth connects with a larger gear with seventy teeth,
the gear ratio is
47
60
. The gear
ratio can be used to compute how many times the larger gear will rotate com-
pared to the smaller gear. (See
Rotations for additional information about gears.)
Ratios that express mixtures are often written with a colon. When a garden-
ing expert recommends two parts of sand, five parts of potting soil, and one part
perlite for the soil mixture in a window box, the ratios can be written in one
expression as 2:5:1. A fertilizer that is labeled as 25-5-5
represents the percents
of nitrogen, phosphate, and potassium. The high ratio of nitrogen to the other
substances means that this fertilizer is probably for the quick development of
lawns, which need nitrogen. A fertilizer with a lower ratio of nitrogen like 10-20-
20 would be good for a garden. Directions for recipes and household products
are often given in ratios of parts. A wedding punch is two parts orange juice, two
parts lemonade, one part pineapple juice, and one part grapefruit juice. The
juices
are in ratio of 2:2:1:1. (See
Proportions and
Similarity for additional applica-
tions of ratios in this form.)
The
golden ratio or
golden section is based on a rectangle that can be split
into a square and a smaller rectangle that is similar to the original rectangle. The
ratio of length to width of the original rectangle is
1+
√
5
2
≈ 1.61803. The ancient
Greeks believed that this rectangle embodied the most satisfying proportions.
The Parthenon in Athens fits the golden ratio. Some sociologists have argued that
people who have certain facial features close to the golden ratio are judged by
others as being more beautiful or handsome. The golden ratio expresses many
patterns in plant and animal structures. (See
Fibonacci Sequence for more infor-
mation about applications of the golden ratio.)
Measures in science and engineering that produce extremely large numbers
are simplified by ratio measures. In aviation, the Mach number indicates the ratio
of the plane’s speed to the speed of sound. Mach 1 is a critical value for airplanes.
Below the speed of sound, a plane pushes air aside like a boat traveling through
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