I recently came across the quantity (a+c)/(b+d), which combines two fractions a/b and c/d. I’m not sure what it’s called, but I found that it has some interesting properties.
Assume x = a/b and y = c/d, and that x <= y and a,b,c,d are all positive.
Define z = (a+b)/(c+d)
Then
[ol]
[li] x <= z <= y[/li][li] If the denominators of x and y are equal, then z is just the mean of x and y[/li][li] If the numerators of x and y are equal, then z is the inverse of the mean of the inverses of x and y (i.e. z = 1/mean(1/x, 1/y))[/li][/ol]
Have you seen this quantity before?
Does this quantity have a name?
Does it have a known list of properties?
I was going to suggest “weighted average” as the name of this quantity.
But then I realized: Suppose x is 1/2, and y is 2/3. z is not one specific number, because a and b might be 1 and 2 (in which case z is 3/5), but a and b might be 1000 and 2000 (in which case z is 1002/2003).
This can be handy in some computation methods where you’re expressing numbers as rationals and want to find a number intermediate between two other numbers, using only “easy” operations.
For instance, suppose I want to find the square root of 2. As a first approximation, I might take, say, 3/2. But if 3/2 is an approximation of sqrt(2), then so is 2/(3/2), or 4/3. The true square root of 2 must be somewhere in between these two, so a better approximation is 7/5. 7/5 means that 10/7 is also an approximation, so my next approximation is 17/12.
Taking this method iteratively gives a better and better approximation. After you’ve taken enough steps, you can then just do one division, if you want a decimal value.
Okay, thanks. So what you’re asking about is NOT the mediant. Instead, as I wrote, z can have any of an infinite number of values, depending on what you chose for a and b: 1/2, 2/4, 3/6, 4/8, etc etc etc. “a/b” will always be the same, but “(a+c)/(b+d)” will vary greatly.
If you’re considering a/b and c/d as numbers, irrespective of representation, then, as Keeve pointed out, (a+c)/(b+d) is not well-defined since it depends on how the number a/b is represented. (For example, 1/2 is the same number as 5/10, and 1/3 is the same number as 2/6, but 2/5 is not the same as 7/16.)
On the other hand, if you’re considering a/b and c/d as ordered pairs of numbers, then (a+c)/(b+d) is really just vector addition. Although it’s reasonable to consider, as you have done, the relationship between vector addition and the function f((a, b))=a/b.
We could make it well-defined if we added the restriction that a and b should be co-prime (and so should c and d). So, in your examples above, we would use 1/2 and not 5/10, and we would use 1/3 and not 2/6, when constructing (a+c)/(b+d)
I use it for weighted averages all the time in grading. For example, if a student has quiz grades of 28/35 and 16/20, I calculate the quiz average as (28+16)/(35+20).