1

Context: let's say we want to compute the total, with tip, approximately, and in our head. What is the best method you can construct to find the following.

Given $x \in \mathbb{R}^+$, construct $y$ such that $1.14x \leq y \leq 1.21x$ as simply$^1$ as possible.

  1. "Simply" meaning that the goal is to find a method for transforming $x$ into a satisfactory $y$ in an "optimal" way for mental computation. The width of the target codomain should be exploited to facilitate this.

I suspect that the use of some Floor and Ceiling operations interspersed with the necessary multiplication may be beneficial, as well as perhaps log_10 or other operations.

Feel free to generalize to a parameterized window (tip percentages) if you wish. Also, note that we are not computing the tip itself and then adding it to the total, but rather simply skipping to the final result of total+tip.

Steve
  • 189

1 Answers1

1

Mental arithmetic is mainly concerned with small numbers, so to approximate y such that $ax\le y\le bx$ for a given x we should try to find some rational number $a\le\frac mn\le b$ with m and n small. n should preferably be of the form $2^a5^b$ (i.e. $\frac1n$ should have a finite decimal expansion).

We should then divide x by n and multiply the result by m to get y. Having n in the form stated earlier allows us to reduce the division to shifting decimal points, doubling and halving – the easiest of all mental operations. For the same reason, while multiplying these easy operations should be left to the last.

In our case, with $a=1.14$ and $b=1.21$, we note that $\frac65=1.2$ lies between a and b, so we can use that multiplier. Suppose $x=147$. This is what we do:

  • Divide by 10 and multiply by 2: 147 → 14.7 → 29.4. This is the same as dividing by 5.
  • Multiply by 3: 29.4 → 88.2.
  • Multiply by 2: 88.2 → 176.4. We have multiplied by 6.

1.2 is also close to the upper bound of 1.21, so this result is a conservative estimate of what we must pay.

The same idea can be used for other tip percentages, but note that for smaller numbers it is easier to compute the tip separately and add that to the total, as the number to be multiplied by tends to be larger in that case. For example, if $1.03x\le y\le1.07x$, the most appropriate multiplier would be $\frac{21}{20}=1.05$, but multiplying by 3 and then by 7 (to get the numerator of 21) is not very easy for an ordinary person…

Parcly Taxel
  • 103,344
  • An interesting non-standard solution! Thanks. I like the idea of finding a rational multiplier (like 6/5) instead of thinking of the process as decimal multiplication (like multiplying by 1.2). However, we can improve your result by more approximation, right? Divide 147/10 to get 14.7. Make that 14, then double to get 28, make that 30, then multiply by 6 to get 180.
    In this case, 180=1.224*147, so we actually overestimated. Since 1.2 was so close to our upper bound, we should have rounded down every time.     – Steve Aug 20 '16 at 16:30
  • In general, I wonder if we can determine how much these "roundings" will throw off the result. Depending on x, can we determine which way to round up and down to still hit our window? – Steve Aug 20 '16 at 16:31
  • If we rounded down, we'd get 147/10=14.7, make it 14, double to 28, make that 25, multiply by 6, get 150=1.02*147, so the tip is too low. Our approximation can't be this severe either, it seems. – Steve Aug 20 '16 at 16:32