So I was on amazon looking for measurement cups and stumbled upon this set that had a measuring table for converting between one measurement and another. Here is the link for reference Amazon Measurement Cups with Table. I didn't think much of it until I saw something weird.
1oz = 30ml 2oz = 59ml 4oz = 118ml I assumed that 2oz would be 60ml, not 59. This told me that the value of 1oz = 30ml must be an approximation and that in fact, they rounded the number up from the true value X otherwise 2oz should be 60ml. I sought out to find what this true value was using the information I had before me but I have been stumped how.
What I've done so far: I thought maybe it was a linear algebra problem. I use the fact that I know we rounded up to rewrite X ml = 30 - $ \epsilon_1 $ (ml) ,where $\epsilon_1$ is the amount rounded up by. So 1 oz = X ml = 30 - $ \epsilon_1 $ (ml). Similarly 2oz = 59 + $\epsilon_2$. Since there is two unknowns with two equations I thought I could solve it this way, but I had been having some trouble getting epsilons that are to small.
I then tried to use what I know about epsilon, that if we are rounding up it must be between 0 and $\frac 12$ (since we only round up if it ends in a 0.5 or greater) and that it must be greater than zero since we can tell the value is approximated. I then wrote this out as $\frac 12 \ge \epsilon_i \gt 0$ by rewriting: X = 30 - $\epsilon_1$ into $\epsilon_1$ = 30 - X I get $\frac 12 \ge 30 - X \gt 0$ into $30 \gt X \ge 29.5 $
Repeating this with 2X, 4X I get $29.75 \gt X \ge 29.5$ and $29.625 \gt X \ge 29.5$. I note how the upper bound is shrinking and we had a decrease from 30 to 29.75 (-1/4) then 29.75 to 29.625 (-1/8) and so I attempted to extrapolate this out and guess that we'd continue to subtract shrinking powers of two from the upper bound. This is equivalent to 30 - $\sum_{n=1}^\infty \frac 1{2^n} - \frac 12$ which equals 29.5.
However, this is incorrect, so now I am stumped where to go to solve this.
