Thus: 1. A certain quantity of goods cost a certain sum. Then, at the same rate,

Twice the quantity will cost twice the sum.

Half the quantity will cost half the sum.

And so on. Any multiple or fraction of the said quantity of goods will cost the same multiple or fraction of the value of that quantity; so long as the rate remains unchanged, or constant."

66

. If any number of workers complete a certain amount of work, then, in the same time,

Thrice as many such workers will do thrice as much work.
One-fifth as many
one-fifth as much work.
From these and similar examples, we see that

I. Quantity of goods and Gross Value are in direct Proportion; When rate is constant.

11. No. of workers and Work done, are in Direct Proportion; When time, and quality of workers, are constant.

III. Time of working and Work done, are in Direct Proportion; When number and quality of workers are constant.

IV. No. of eaters and Food consumed are in Direct Proportion; Time, and daily allowance for each, being constant.

But, when two Concrete Quantities are in such relation that as one increases the other decreases in an exactly opposite (or reciprocal) ratio, so that, for example, doubling either of them compels us to halve the other, and so on, those two quantities are in Inverse Proportion.

The following cases will illustrate this.

1. If, at a given rate each, a certain No. of articles cost So much," it is evident that, if the price of each be doubled, only half as many can be bought for the same amount of money.

I. Supposing a cask of wine to supply a certain number of men with one pint each, only one-eighth of that number could receive eight pints each from the same quantity of wine.

III. The same quantity of food which would furnish a certain number of men with a certain daily allowance for 20 days, would give to each of them four times as large a daily ration, if they were only to be fed for one fourth of the time.

IV. If a sack of corn last any number of horses one week, it would suffice for seven times as many for one seventh of the time, each receiving the same daily allowance as before.

v. If a certain weight can be carried "so far" for " so much," only half the weight could be carried twice the distance, for the same sum, and at the same rate.

VI. But, twice the weight could be carried the same distance for the same sum, at half the rate.

VII. Also, the whole original weight could be conveyed twice the distance for the same sum, at half the rate.

vш. If 50 workers complete a task in any given time, 300 (six times 50) similar workers should do it in one-sixth of the time.

IX. The Principal requisite to gain a certain sum of Interest in ten months is only one-tenth of the Principal necessary to gain the same Interest in one month, at the same rate.

x. If two rectangles be of equal Area, but the second twice as long as the first, it can only be half as wide.

XI. If two rectangular prisms be of equal solidity and thickness, but the first five times as long as the second, it can only be one-fifth as wide.

XII. If a certain rectangular prism have to sink twice as deep for support in one fluid as it does in another, the Specific Gravity of the first fluid is only half that of the other.

&c.

&c.

From the foregoing examples we see that,

&c.

1. No. of articles and Price of each are in Inverse Proportion; When Gross value is constant.

II. No. fed and allowance to each are in Inverse Proportion; Gross amount of food being constant.

III. Ration and Time of Feeding are in Inverse Proportion; Gross amount of food and No. fed being constant.

IV. No. fed and Time of maintenance are in Inverse Proportion; Gross amount of food and ration being constant.

v. Weight conveyed and Distance are in Inverse Proportion; Gross cost and rate of carriage being constant.

VI. Weight conveyed and Rate are in Inverse Proportion;
Gross cost and distance being constant.

VII. Distance and Rate are in Inverse Proportion;
Weight and gross cost of carriage being constant.

VIII. No. of workers and time occupied are in Inverse Proportn.;
Amount of work remaining constant.

IX. Principal and Time are in Inverse Proportion;
When Interest and Rate are constant.

x. Lengths (of Rectangles) and Breadths are in Inv. Proportion; Areas being constant, but dimensions unlike.

XI. Lengths and Breadths (of sq. prisms) are in Inv. Proportion, Only Solidities and Thicknesses being constant.

XII. Bulks and Specific Gravities are in Inverse Proportion; Weights remaining constant.

Direct Proportion. Quantities are in Direct Proportion when multiplying one of them by any number or fraction causes the other of them to be multiplied by the same number or fraction.

Inverse Proportion. Quantities are in Inverse Proportion, when multiplying either of them by any number or fraction causes the other of them to be multiplied by the reciprocal of that number or fraction. Pages 99 and 145. This latter definition is frequently, but with less precision, given thus:

Two quantities are in Inverse Proportion when multiplying either of them by any number or fraction causes the other of them to be divided by the same number or fraction.

In every Simple Proportion (or Rule of Three) question, three Concrete terms are given. Two of these are of the same kind, so that there exists a ratio between them. PRINCIPLE V.

The odd term is of the same kind as the answer sought: so that there exists a ratio between them. PRINCIPLE V. And these two ratios being equal, the four terms are Proportionals. Page 146.

GENERAL RULE FOR SIMPLE PROPORTION, DIRECT OR
INVERSE.

First. Write the question in an abbreviated form.
See Examples worked out.

Second. State the Proportion, thus:

a. Write the word "As" and the signs of Proportion, leaving spaces for the three given terms, and putting the letter x in the fourth place, to represent the required term (or fourth proportional) until found, thus:

b. Put in the third place the term which is of the same kind as the answer sought.

This, as said above, is the odd term, and, in the abbreviated question, has a note of interrogation under it.

c. Consider whether, from the nature of the question, the answer will be greater or less than the third term.

If the answer will be greater than the third term, put the greater of the other two in the second place.

But, if the answer will be less than the third term, put the less of the other two terms in the second place.

d. Put remaining term in the first place.

And now, if the Proportion be correctly stated:

The first and second terms will be of the same kind. The third term will be of the same kind as the answer. The second term will be greater or less than the first, according as the answer is to be greater or less than the third. Note, also, that

In Direct Proportion, the term whose value is sought stands second.

In Inverse Proportion, the term whose value is given stands second.

Third. Work for the Fourth Proportional, thus: a. If either of the first two terms be a Compound Quantity, or if they be of unlike denominations, reduce both to the same name.

b. If the third term be a Compound Quantity, reduce it to its lowest denomination, or leave it in the Compound form, as may be most convenient.* c. Cancel Common Factors in first and second, and in first and third terms. Page 149.

d. Multiply second and third terms together, and Divide by first.

The result is the fourth proportional, or answer sought, in the same name or names as that or those in which the third term was left.

FIRST EXAMPLE WORKED QUT. A DIRECT PROPORTION. REDUCTION UNNECESSARY. No CANCELLING.

If 21 cwt. cost £53′ 3′s. 1 d., what will 11∙ cwt. cost.?

*This reduction, although only occasionally serviceable, is, in many Arithmetics, enjoined as an essential part of the process.

II.

The same Example worked out without reference to ratio.

This simple and natural method is much more frequently employed than "The Rule of Three," in actual business.

GENERAL RULE FOR WORKING WITHOUT REFERENCE TO RATIO.

First. Find the value of one.

Second. Find, from that, the value of the number required.

When the value of one is either given or demanded, only Multiplication or Division will be requisite, according as the quantities move in Direct or Inverse Proportion. Hence,

When either of the three given terms is Unity, reference to Ratio, or Statement of Proportion, is of no utility whatever.

III.

The Example may also be neatly solved as an Equation.

Thus:

£. 8. d.

Ans. sought cost of 11; and 53′ 3′s. 14 = cost of 21..of Answer sought

cost of 1.

= one another.

of £53′ 3′s. 1d.

Then, multiplying both sides by 11. (PRINCIPLE XXIV.), we have, Answer sought of £53. 3's. 14d. £27⋅ 16's. 10 d.

=

=

In this book, a large number of questions which would ordinarily be placed under "The Rule of Three," have been already given under Compound Multiplication and Division. For instance, in Exercise

In addition to these two hundred and sixty-eight, there are several Examples in Exercise civ., and elsewhere, which would generally be included under "The Rule of Three."