1
$\begingroup$

At work we're often given measurements in inches, specifically "ruler measurements," which are all numbers of the form $\frac{m}{2^n}$ for positive integers $m, n$. For example, we encounter fractions like $\frac{3}{8}, \frac{1}{2}, \frac{11}{16}$ regularly. These are also given to us as decimals, e.g. $0.375, 0.5, 0.6875$ for the previous.

Is there a mental shortcut that I could use to tell, given a decimal expansion, whether it corresponds to a ruler measurement? It doesn't need to give me $m, n$ for any particular expansion, and if helpful I really only need something that works from halves to sixteenths.

(This question is largely hoping there may be something in the same spirit as the universal divisibility test.)

  • 0
    Depends on how good your mental math is - but just keep doubling the number until it either becomes an integer, or its last digit isn't a 5. – 2017-01-02

1 Answers 1

1

I assume you have no problem with $0.2500=\frac{1}{4}$, $0.5000=\frac{1}{2}$ and $0.7500=\frac{3}{4}$.

The fractions of $8$ and $16$ may be less familiar.

Here is a table which can be reconstructed from memory by a few simple rules.

The first row begins with the decimal equivalents of $\frac{1}{16}$, $\frac{2}{16}$ and $\frac{3}{16}$.

The first two digits of the decimal fractions on the first row are multiples of $0.06$ and the last two digits are multiples of $0.0025$.

In rows $2$, $3$ and $4$ add $0.2500$ to the decimal fraction above it and add $\frac{4}{16}$ to the ordinary fraction above it.

The middle column fractions can be reduced to fractions of $\frac{1}{8}$

$\begin{Bmatrix} 0.0625=\frac{1}{16}&0.1250=\frac{2}{16}&0.1875=\frac{3}{16}\\ &&\\ 0.3125=\frac{5}{16}&0.3750=\frac{6}{16}&0.4375=\frac{7}{16}\\ &&\\ 0.5625=\frac{9}{16}&0.6250=\frac{10}{16}&0.6875=\frac{11}{16}\\ &&\\ 0.8125=\frac{13}{16}&0.8750=\frac{14}{16}&0.9375=\frac{15}{16} \end{Bmatrix}$