We may divide the first couplet by 12, the second by 40, and we have 9: 1 :: 9: 1, a proportion manifestly true. We may multiply or divide either antecedents or consequents by the same number, and we still have a proportion.

(ART. 75.) By retaining strong hold of the fact, that the products of the extremes and means are equal, in every geometrical proportion, we can find any term, or factor of a term, that may be wanting or lost. Take the proportion, 2 : 4 :: 5 : 10. If the fourth term, the 10, be wanting, we still have the product of the means. We have also one factor of the extremes, which, used as a divisor to divide the common product, will give the other factor, or the fourth term: thus,

When the fourth term of any proportion is wanting, we then have three terms, and the application of Proportion then becomes the Rule of Three, or Golden Rule. The first technicality arose from the fact of there being three given terms to find a fourth, the last from the great utility and beauty of its application.

Resume the proportion, 2 : 4 :: 5 : 10, and suppose the second term wanting; we then have 2 : [] :: 5 10 only. The extremes now give us the common product, 20; which, divided by 5, gives 4, the term wanting. On this principle, if only one term is wanting, no matter which one, it can be found. When the fourth term is wanting, it is the rule of three direct. When the first term is wanting, it is the rule of three inverse.

(ART. 76.) In a direct proportion, we can obtain the fourth term, by multiplying the third term by the ratio.

Example. 5: 15 :: 11 : to the fourth term. What is that term? 5: 15 are as 1 to 3. Then 3 is the ratio. Hence, 11×3=33, Ans. In an inverse proportion, the first term is found by dividing the second term by the ratio.

Example. []: 50: 24: 4. What part of 24 is 4? Ans. the ratio. Hence, divide 50 by 1300 Ans.

L

Or, in consideration of the common product, we have,

for the first term,

50 X 24

4

=300 as before.

(ART. 77.) In any proportion, suppose 300: 50 :: 24: 4, we may conceive any term as composed of two or more factors, and one of them wanting. It can be restored by means of the common product, divided by the known factors, in the deficient extremes or means, as the case may be. Let us suppose the second term, 50, to be separated into factors, 5× 10, and the 5 a term sought. We then have 300 : []×10 :: 24: 4; the extremes being perfect, gives the common product. The means are imperfect; but if we divide the common product by the factor we have in the means, the quotient will give the lost or sought factor. These multiplications and divisions should be performed as directed in Art. 24.

101300 30 6 24 4

6)30

5 Ans.

If we have 300 50 :: 2×[]: 4 to find the lost factor to the third term, the process is the same as before. If we have 3×[]: 50 :: 24: 4, to find the lost or sought factor, we observe that the means are now perfect, and they give the common product, and the extreme factors are divisors.

4150
3242

Thus:

Ans. 100.

APPLICATION OF THE FOREGOING PRINCIPLES.

(ART. 78.) Questions which should fall under proportion, or the rule of three, have two terms of comparison of the same name; one in the supposition, the other in the demand. The third term is of the same name as the answer, or the term required. To state a question, is to arrange the terms in order, agreeable to the laws of proportion. When the question is direct-that is, more requiring more, or less requiring less-state the question by the following

RULE. Of the two terms of comparison, place that of supposition first, and that of the demand second, and that of the same name as the answer required, the third. Then find the fourth term or answer, as di

rected in Art. 75.

Example 1. If 12 bushels of wheat are worth 16 dollars, what is the value of 48 bushels?

Here are two terms of the same name, (bushels) which we call terms of comparison, and one term (dollars) of the same name as the answer required. Of these terms of comparison, 12 is that of supposition. We know it by the word if, or one of like import, that may stand before it, the 48 is that of demand, as the language most clearly implies. Hence, by the rule,

12: 48: 16: to the fourth term.

In former times, when mechanical labor was considered no task, we would multiply 48 by 16, and divide the product by 12; we now do it by canceling, according to Art. 24.

Or, we may say, 12 to 48 as 1 to 4; therefore 16X4 =64 answer, which is canceling without the form, and which is generally most convenient in the rule of three.

Example 2. If 7 pounds of sugar cost 75 cents, how many pounds can be bought for 9 dollars?

Two of the given terms in this question are of the same kind (money); but not of the same denomination. These are the terms of comparison, after one of them is reduced to the denomination of the other (Art. 70).

It is obvious that 75 cents is the term of supposition; but the if or term of condition is not directly before it; however, a little change in the phraseology will bring it so.

If 75 cents will purchase 7 pounds of sugar, how many pounds can be bought for 9 dollars?

Statement, 75: 900 :: 7 to the answer.

Divide the couplet or terms of comparison, by 25,

Same question again, 3
Divide by 3
Mult. 1 & 2 by 4

lb.

: 9 :: 7 : the answer.

1 : 3 :: 7 : the answer.

• 4

1: 12 :: 7: 84 the answer.

As a specimen of the manner of doing things in former days, we cut the following problem and solution, from a well-known book.

Example. If 48 yards of cloth cost $67,25, what will 144 yards cost at the same rate?

Now let it be observed once for all, that these operations are those of comparison or proportion, and the actual numbers given need not be used, provided we use their proportion.

Now 48 is to 144, obviously, as 4 to 12; or, as 1 to 3. Hence 67,25×3=$201,75, Ans.

(ART. 79.) When the terms are of various denominations of weights, measures, &c., as a general rule, reduce them all to the lowest denomination mentioned after

statement; but by making use of fractions, we may very often abridge.

Example. If 3 pounds 12 ounces of tea cost $3,50, what will 11 pounds 4 ounces cost?

lb. oz. lb. oz.

$

Statement,. 3 12 11 4 :: 3,50

In place of reducing the weights to ounces, we observe 12 oz.=3 lbs.

lb. lb. money.

32: 111 :: 3,50 : Ans.

Multiply the terms of comparison by 4, and

The following problem is commonly wrought by a very tedious process, and, indeed, must be, if rules rather than principles are to be our guide.

If 2 hundred weight 3 quarters 21 pounds of sugar cost £6 1 s. 8 d., what cost 35 hundred weight 1 quarter? Ans. £73. By the general rules, (which, as general rules, are good,) we must reduce the quantities to pounds, and the money to pence, either after or before the statement.

cwt. qr. lb.

We proceed thus: 2 3 21 : 35 1

Reduce to qrs.

Multiply by 4,.

Divide by 47,

Hence,

£. s. d. 618

cwt. gr.

Ans. £73 0 0

For further illustration, we extract the following: If 7 pounds of coffee cost 87 cents, what must I pay for 244 pounds? Ans. $30,50.

This is worked out, in the book from which we extracted it, in the longest and most formal manner, thus teaching bad habits at first, which are difficult to eradicate; and until such are eradicated, it is impossible to be skillful