Ratio definition

Question

My teacher gave me the definition of ratios as "the comparison between two like quantities"

And one of the example he gave for a popular ratio was that of Body mass index (BMI).

My question is that in the calculation of body mass index we use height and weight as the quantities to arrive at the number , however both of them are not "like" quantities, then why is this called a ratio ?

In that case Is there any better way to define ratio ?

Answer 1

My teacher gave me the definition of ratios as "the comparison between two like quantities"

The ratio of any number of quantities is a quantitative comparison of their measures. There is no need for the quantities to be alike or to have the same units. An example is 2 boys : 1 girl : 3 cars : 0 chauffeur.

While a ratio is a comparison of quantities, a proportion on the other hand is a comparison of ratios.

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Addendum to comment on the other answer

For numbers a ratio is a fraction.

So if $a$ and $b$ are non-zero numbers then the ratio $a:b$ is the same thing as the fraction $\frac{a}{b}$.

It's hard to argue that 2 : 1 : 3 is a fraction or that -2 : 3 and 2 : -3 are equivalent objects.

Answer 2

I have always been uneasy about defining a ratio as a comparison, because a difference is also a type of comparison. For numbers a ratio is a fraction. So if $a$ and $b$ are non-zero numbers then the ratio $a:b$ is the same thing as the fraction $\frac{a}{b}$. Many people are quite happy dividing quantities, but some may argue that only numbers can be divided. In either case we can say that

If you have two like quantities, their ratio is the number that you multiply one by to get the other one.

where two like quantities are two quantities of the same thing, e.g. mass, distance, or pure numbers.

With this definition a ratio is a "pure number" and BMI is indeed not a ratio.

Some books distinguish between a ratio and a rate. A rate is similar to a ratio but has units. With this understanding the Body Mass Index is a rate, which is is defined to be in $kg/m^2$. If you know your measurements in pounds and inches you must convert before calculating, or (equivalently) multiply by a conversion constant.

In practice one can use ratio form or fraction form for numbers and quantities, as long as care is taken with units.