⁵¹/₄₀ × 415 francs, or (117) 51 × 415 francs.
40

239. Such questions as the last two belong to the most extensive rule in Commercial Arithmetic, which is called the Rule of Three, because in it three quantities are given, and a fourth is required to be found. From both the preceding examples the following rule may be deduced, which the same reasoning will shew to apply to all similar cases.

It must be observed, that in these questions there are two quantities which are of the same sort, and a third of another sort, of which last the answer must be. Thus, in the first question there are 22 and 156 yards and 208 pence, and the thing required to be found is a number of pence. In the second question there are 20 and 415 shillings and 25½ francs, and what is to be found is a number of francs. Write the three quantities in a line, putting that one last which is the only one of its kind, and that one first which is connected with the last in the question.[49] Put the third quantity in the middle. In the first question the quantities will be placed thus:

22 yds. 156 yds. 17s. 4d.

In the second question they will be placed thus:

20s. £20 15s. 25½ francs.

Reduce the first and second quantities, if necessary, to quantities of the same denomination. Thus, in the second question, £20 15s. must be reduced to shillings (219). The third quantity may also be reduced to any other denomination, if convenient; or the first and third may be multiplied by any quantity we please, as was done in the second question; and, on looking at the answer in (238), and at (108), it will be seen that no change is made by that multiplication. Multiply the second and third quantities together, and divide by the first. The result is a quantity of the same sort as the third in the line, and is the answer required. Thus, to the first question the answer is (238)

208 × 156 pence, or, which is the same thing, 17s. 4d. × 156 .
2222

240. The whole process in the first question is as follows:[50]