Elementary arithmetic, with brief notices of its history... by Robert Potts.

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3s., then the product is 3s. x 5 = 15s., the same as before; and conversely
15s. - 5 = 3s.; also 15s. 3s. = 5, an abstract number, denoting
that 3s. is contained 5 times in 15s.
Hence since 5s. x 3 = 15s. and 3s. x 5 = 15s., it follows that in
the product of two factors, one of which is concrete and the other abstract, the abstract and concrete factor may be interchanged without
affecting the meaning of the product.'
If both factors be concrete of the same kind, as 3s. and 5s., the
product 3s. x 5s. appears not to admit of an intelligible meaning,
unless 3s. and 5s. each denote 3 and 5 pieces of coin. In that case,
5 shilling coins may be considered as 5 times Is., and 3 shilling coins
3 times ls., and the product 3s. x 5s. may be taken to mean 5 times
3 times Is. or 15 shillings, which admit of being arranged in 3 rows,
with 5 in each row; or in 5 rows with 3 in each row.
If 3 and 5 denote units of different kinds, as 3s. and 5 yards, the
product, 3s. x 5 yards, is apparently absurd.2 But 5 yards is nothing
more than 5 times 1 yard; and since 3s. corresponds to 1 yard, if 3s. be
taken 5 times, or 3s. be multiplied by 5, the result, 15s., corresponds to
5 yards, which affords an intelligible meaning to the product.
And conversely, 15s.-. 3 yards may be explained in the same
manner as giving the quotient 3s. In the same manner other products
and quotients of two concrete factors of the same or different units may
be considered and interpreted.
4. PROB. To multiply and divide any sum of money by an abstract
number.
1. When the multiplier consists of a single figure.
The product of a sum of money by an abstract number can be
found by taking the several units"of farthings, pence, shillings, pounds
as many times as the multiplier denotes, considering at each step how
many of any number of units make one of the -next greater, and
beginning with the smallest unit.3
1 The interchange of the abstract units for the concrete, or the concrete for the
-abstract in the product of two factors, can sometimes be made with great advantage
in the performance of computations.
2 The ordinary forms in which such multiplications and divisions occur are in,such questions as the following:-If one yard of cloth cost 3s., what is the cost of
5 yards? and, If 5 yards of cloth cost 15s., what is the cost of one yard?
In the former question there is given the fact of 3 shillings corresponding to 1
yard, to find the number of shillings which correspond to 5 yards.
In the second question there is given the fact of 15 shillings corresponding to 5
yards, to find the number of shillings which correspond to 1 yard.
3 Ex. To multiply ~5 14s.. d. by 6.
~ s. d.
5 14 74
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~34 7 10, product.