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Rate and Ratio
Scale diagrams are a special case. In direct proportion the two quantities are not
usually the same type. In the case of constant speed, the quantities are the
distance travelled and the time taken. Distance and time are different quantities.
The units, such as miles and hours, are different types of unit. For a car
travelling at 50mph, the ratio of distance to time is 50miles:1hour and so the
ratio includes units. You must not write the ratio as 50:1 . That would be like
writing the speed as 50 and forgetting to include the mph.
Note that 50mph and the ratio 50miles:1hour are both measures of speed. Of
course, you would not normally say “the speed of the car is 50 miles to 1 hour”.
It is conventional to say “the speed is 50 miles per hour”. To distinguish
between these alternatives, the 50mph is called a rate whereas 50miles:1hour is
called a ratio. The word ratio describes speed as the ratio of the distance
travelled to the time taken. The word rate describes speed as the rate of travel,
that you calculate by dividing the distance travelled by the time taken. Those
two descriptions may seem different but in mathematics they are equivalent.
Some people object to the idea that a quantity, such as speed, is a ratio. They
prefer to use ratio only when comparing two quantities that are the same type
and not when comparing distance and time. In other words it is correct to use a
ratio to compare the weight of a dog and a cat, or a man’s height with his waist
measurement, but not his height with his weight. They say that the ratio of
height to weight is a rate and not a ratio, because the quantities are different
types. My view is that rates and ratios are just two ways of describing the same
thing; any disagreement about the use of the terms is really a problem of
language and not a problem in mathematics.
According to http://www.eduplace.com/math/mathsteps/6/e/, a rate is a ratio
(although you could easily find a different view from another web-site):
“A ratio is a comparison of two numbers or measurements. The numbers or
measurements being compared are called the terms of the ratio. A rate is a special ratio
in which the two terms are in different units. For example, if a 12-ounce can of corn costs
69 cents, the rate is 69 cents for 12 ounces.”
In the wine example, the ratio (£8:2 bottles) is really the same as the rate (the
cost per bottle of wine):
£8:2 bottles = £4:1 bottle = £4 per bottle of wine
The cost per bottle is constant (a constant ratio) and so the cost and the number
of bottles are directly proportional.
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