Henry Ernest Dudeney/Puzzles and Curious Problems/105 - Equal Fractions/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $105$

Can you construct three ordinary vulgar fractions

(say, $\tfrac 1 2$, $\tfrac 1 3$, or $\tfrac 1 4$, or anything up to $\tfrac 1 9$ inclusive)

all of the same value, using in every group all the nine digits once, and once only?

The fractions may be formed in one of the following ways:

$\dfrac a b = \dfrac c d = \dfrac {e f} {g h j}$, or $\dfrac a b = \dfrac c {d e} = \dfrac {f g} {h j}$.

We have only found five cases, but the fifth contains a simple little trick that may escape the reader.

The examples given by Dudeney are:

$\dfrac 2 4 = \dfrac 3 6 = \dfrac {79} {158}$

$\dfrac 3 6 = \dfrac 7 {14} = \dfrac {29} {58}$

$\dfrac 3 6 = \dfrac 9 {18} = \dfrac {27} {54}$

$\dfrac 2 6 = \dfrac 3 9 = \dfrac {58} {174}$

The fifth case which he offers up is:

$\dfrac {\cdotp 2} 1 = \dfrac {\cdotp 6} 3 = \dfrac {97} {485}$

where it is noted that the first two fractions are not actually proper fractions.

It is worth calling into question Dudeney's definition of a vulgar fraction, as he is unclear.