1

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 ?

ryang
  • 38,879
  • 14
  • 81
  • 179
Fin27
  • 958
  • 2
    BMI may not be a good example: it is weight (kg) divided by the square of height (m^2) so is more of a quotient than a ratio. – Henry Apr 11 '23 at 08:00
  • See also this: https://en.wikipedia.org/wiki/Dimensionless_quantity (And the main articles linked therein, like "list of..." and "dimensionless quantities in physics") – student91 Apr 11 '23 at 08:12

2 Answers2

1

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.

Peter
  • 1,701
  • A pedantic mathematician would say that a ratio $a:b$ is not the same as the quotient $\frac ab$, because you can have $1:0$ but you cannot have $\frac10$. – student91 Apr 11 '23 at 09:42
  • @student91, thanks. Answer amended to avoid zero (anywhere). – Peter Apr 11 '23 at 12:17
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.

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.

ryang
  • 38,879
  • 14
  • 81
  • 179
  • If a quantity is a count I would agree the items need not be the same. Though one could argue that boys, girls, cars and chauffeurs are alike because they are all elements of finite sets. But if a quantity is a measure we have a problem. What is the ratio of 1 inch : 1 foot? 1:1? 1:12? – Peter Apr 11 '23 at 12:31
  • @Peter 1 inch : 1 foot clearly equals 1:12 (the two quantities have the same base unit, metre). On the other hand, the ratio that I gave as example cannot be simplified or its units dropped. – ryang Apr 11 '23 at 16:32
  • Interestingly, the article you referenced notes the use of rate for ratios that are not dimensionless. Clearly nomenclature varies. – Peter Apr 12 '23 at 04:09
  • @Peter As pointed out by the comment under your answer, a ratio can accommodate zero anywhere whereas a rate cannot have a zero denominator; moreover, a ratio (3 men : 4 men : 1 man = 3:4:1) need not be binary whereas a rate can deal with only two quantities. All of this is already indicated in my above answer. -) – ryang Apr 12 '23 at 04:20
  • I would regard 3 : 4 : 1 as two ratios, just as a < b < c is two inequalities. – Peter Apr 12 '23 at 05:26
  • @Peter In which case related ratios can be condensed as one, whereas a rate can never be extended to contain more than two entries. $\quad$ In conclusion: for special cases, a ratio and a rate are equivalent, but even then they aren't the same object. – ryang Apr 12 '23 at 06:00
  • So, .what is your mathematical definition of a ratio? Your original statement does not work without defining the word "comparison". – Peter Apr 13 '23 at 01:11
  • @Peter You are cherry-picking again; the requisite phrase is "quantitative comparison" and it's up to the reader to use their mathematical insight, approximation, simple calculation or common sense to make any inference they need. $\quad$ Anyhow, a formal definition will also display the object's form and note the fact that its entries are ordered and scalable in correspondence. $\quad$ Alright, let's move on! – ryang Apr 13 '23 at 04:25
  • @ryang what is you final take on BMI then ? is it a rate or ratio ? I had asked my teacher as well regarding the same and he stated that since BMI is measured in kg/m^2 and while calculating your BMI , you have to convert the weight and height into kg and m respectively , which makes the units redundant in this case and you have just the number in numerator and a number in denominator ? what is your take on this ? – Fin27 Apr 15 '23 at 15:51
  • @Fin27 BMI is not a rate, because it is unitless (the same way that the $θ$ in sector area's $\frac12r^2θ$ is unitless: they are numbers that correspond to kg/m$^2$ and rad, respectively), which is why there is a requisite $×703$ conversion factor if the raw data is in pounds & inches instead of kilograms & metres. $\quad$ BMI is not a ratio either, because it is a single number or fraction. – ryang Jun 05 '23 at 03:58