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dered above, they were called continued proportionals ; they may however exist otherwise, as 4 : 12 :: 24; in which case they are called interrupted proportionals.
Let us now examine how the ratios of numbers are discovered. We take the two first of the series we have chosen, 4 and 16; we remember that 16 is 4 times 4; but suppose we had any other two numbers which our memory would not serve us equally well to remember, how ought we to proceed? By dividing the larger by the smaller, the quotient is the ratio. Suppose now we wanted some number that would bear the same proportion to 244, as sixteen bears to four ; we would multiply 244 by 4, and the product 976 exceeds 244 as much as 16 does 4. If we wanted a descending proportional, we would divide 244 by the ratio 4, and the quotient 61 would bear the same proportion; so that the cases would stand thus:
It is now time to explain the dots used to divide the terms.
: denotes, is to, :: denotes, so is.
So that translating the dots (used in the foregoing proportionals) into words, they read thus :
4 is to 16 so is 244 to 976. In our experiments hitherto, where we had three terms, and wished to find the fourth, that would bear the same proportion to the third as the first did to the second, we first found the ratio or rate of progression, by dividing the greater of the two first numbers by the lesser, and then multiplied the number for which we wanted a proportional by the quotient or ratio that existed between the first and second terms.
Now taking the first proposition, what would be the consequence if
4 : 16 :: 244 we multiplied the second and third numbers together? Why, as the ratio was only four, we would have a number four times larger than that required; but this would be no difficulty, our end would at once be gained by dividing the product by four; as I said before, we only wanted a fourth part of it. If you turn to the example you will see that the sum required for a fourth proportional was 976. Let us now work the sum as I have described.
4 : 16 :: 244
976 the number required.
Now that above is a question where less requires more; let us take the other, where more required less.
4 16 ) 976
Here we produce 61, which is also the proportional required. Now in this progression and proportion the whole secret of calculation of numbers consists, and with a due knowledge of it, by carefully studying what I have now written, the pupil will easily work any rule arising out of it, whether is called the Rule of Three Direct, Inverse, the Double Rule of Three, the Chain Rule,
Barter, Interest, or any of the many names invented by schoolmen, not to instruct, but to puzzle, their readers.
Now to continue the Rule of Three or Proportion, although I trust the learner has acquired a knowledge of its operations from what I have said on Progression, yet as he has seen in the compound rules that we use figures to express quantities, whether of weight, measure, money, or time-indeed, that everything amongst us requires different denominations to express them—I feel that it will be necessary by a few examples, in this, the most useful of all rules, to shew how these mixed proportionals are found.
Erample. If two pounds of tea cost nine shillings and sixpence, what will twenty pounds cost?
(3.) lb. d. Ib.
Ib. 2 : 9 6 :: 20 2 :91 :: 20 2 :91 :: 20 12 1 2 10
1 10 10 114
Ans. 95s. 20
10 2) 2280
2) 190 12) 1140
Ans. 95s. 20) 95 Ans. £4 158.
Here, as we laid down in Progression, we have merely to find a sum of money that will bear the same proportion to twenty that nine and six pence does to two. In the first example, the nine and six pence is reduced to pence, the second and third terms are then multiplied together, and divided by the first. (See Progression.) The quotient is the answer in pence. In the second and third, we have not reduced the middle term below shillings, consequently our answer was produced in shil
6 :: 31
lings; and as I have taught in working fractions, that the proportion that two numbers bear to one another is not altered by dividing both by some other number, we here take advantage of it to shorten the work; for it is clear that one bears the same proportion to ten that two does to twenty.
Example. If 1 cwt. 2 qrs. 10lb. of cheese cost £2. 14s.6d., what will 3 lb. cost?
£ s. d. lbs. Reduce the first
1 2 10 2 14
2 and last terms to half
6 pounds, and the mid
12 dle term to pence. Multiply the second
7 and third together,
356 and divide by the
356) 4578 (12153
356 first. The answer,
1018 12153d. will be the
712 proportional requir. ed.
(1.) A draper bought 290 yards of cloth at 10s. 6d. per yard; what was the cost?
Ans, £152 5s. (2.) If a draper bought 290 yards of cloth for £152 5s. at what rate per yard must he sell it to gain £20 on the whole piece ?
Add the proposed gain to the price, and proceed as before.
If 290 : 172 5 :: 1 Ans. 11s. 1029d. (3.) A piece of work can be performed by 60 men in 20 days; in how many days can the same work be done if only 40 men should set about it? Ans. 30 days.
In the three or four sums that
da. we have set down previous to this,
60 : 20 :: 40 the statement was in the natural
20 manner in which the questions pre
40) 1200 sented themselves. In the first,
30 days. more required more; in the second, less required less; in the third, more required more ; for of course 290 yards must cost more than one yard; in the fourth, even although the gain of £20 was to be made, yet one yard must cost less than 290. So far these four are direct proportions; but in the above example, less requires more, for of course forty men must take a longer time than sixty men, and to keep, as we always do in proportion, the term requiring the proportional in the third place, another mode of proceeding becomes necessary, and we accordingly multiply the first and second terms together, and divide by the third ; and to distinguish this mode of proceeding from the other, we call the questions so worked the Rule of Three Inverse, from the Latin word inversus, turned upside down.
As a proof of the correctness of the mode of proceeding adopted, let the pupil imagine that if instead of forty men he only had one to do the work, it is then clear that it would take him 60 times as long to perform it, or 1200 days. Now if he sets forty about it, it will only take a fortieth part of the time, or 30 days.
Again, let us invert the question, and as the one worked was less requiring more, we shall now have it more requiring less.
If 40 men can do a piece of work in 30 days, in how many days can 60 men do the same work?