&

3. How many feet of cord wood, 4. How many feet of cord wood, in a load 3ft. 9in. high, and 2ft. in a load 4ft. 7in. high, and 3ft. 10in. wide?

8in. wide?

In the above examples, the 7, which occupies the place of inches in the one, I call of a foot, although it is in reality one twenty-fourth part of a foot more than a quarter. The 9, which occupies the place of inches, in the other example, is exactly & of a foot.

The 6, which possesses the third place, in one example, and the 8, in the other, are not reckoned into the quantity.

5. How many feet of cord wood in a load 4ft. 4in. high, and 3ft. lin. wide?

6. How many feet of cord wood are there in a load which is 3ft. 11in. high, and 3ft. 10in. wide?

Ans. 7.

NOTE 4.-When wood is cut less, or more, than 4ft. long, find the contents of the pad by the foregoing examples; then deduct or add, as the case may require, so many forty-eighths of a foot, as the number of feet in the load will produce when multiplied by the number of inches it falls short, or overruns.

7. How many feet of cord wood are there in a load 4ft. high, and 3ft. wide, and cat only 3ft. 9in. long?

227.

a load?

When wood is cut more or less than 4 feet long, how do you find the contents of

R

8. How many feet of cord wood in a load, 4ft. 3in. high, and 3ft. 6in. wide, and cut 4ft. 7in. long?

Ans. 88.

9. How many feet of cord wood in a load, 3ft. 7in. high, and 3ft. 8in. wide, and cut 3ft. 6in. long?

Ans. 58.

10. How many feet of cord wood in a load, 4ft, 2in. high, and 3ft. 8in. wide ?

Ans. 75.

THE SINGLE RULE OF THREE.

THE SINGLE ROLE OF THREE teaches how to find a fourth number, proportional to three numbers given; for which reason it is sometimes called the RULE OF PROPORTION. It is called the Rule of Three, because three terms or numbers are given, to find a fourth. And because of its great and extensive usefulness, it is often called the GOLDEN RULE.

This Rule is usually considered as of two kinds, viz. Direct and Inverse a distinction, however, which is totally useless, and which has been avoided by some of the best writers. It may not be amiss, however, to explain the difference usually understood between the two parts of this rule.

The Rule of Three Direct is that in which more requires more, or less requires less. As in this; if 3 men dig 21 yards of trench in a certain time, how much will 6 men dig in the same time? Here more requires more, that is, 6 men, which are more than 3

228. What are we to understand by the Single Rule of Three?—229. Why is it called the Rule of Three, or the Golden Rule ?- -230. Is it of more kinds than one?

men, will also perform more work in the same time.

Or, when

it is thus if 6 men dig 42 yards, how much will 3 men dig in the same time? Here less requires less, for 3 men will perform proportionably less work than 6 men, in the same time. In both these cases, then, the Rule, or the Proportion, is Direct: and the stating must be

thus, As 3 21 :: 6 : 42,

or thus, As 6 : 42 : : 3:21.

But the Rule of Three Inverse, is when more requires less, or less requires more. As in this: if 3 men dig a certain quantity of trench in 14 hours, in how many hours will 6 men dig the like quantity? Here it is evident that 6 men, being more than 3, will perform an equal quantity of work in less time or fewer hours.— Or thus: if 6 men perform a certain piece of work in 7 hours, in how many hours will 3 men perform the same? Here less requires more, for 3 men will take more hours than 6 to perform the same work. In both these cases then the Rule, or the Proportion, is Inverse; and the stating must be

And in all these statings, the fourth term is found, by multiplying the 2d and 3d terms together, and dividing the product by the 1st term.

Of the three given numbers, two of them contain the supposition, and the third a demand. And for stating and working questions of these kinds, observe the following

GENERAL RULE.

1. Write that number which is of the same name or kind with the answer or number required, for the second term.

2. Then consider whether the answer must be greater or less than the second term. If the answer must be greater than the second term, write the greater of the two remaining numbers on the right for the third term, and the other on the left for the first term; but if the answer must be less than the second term, write the less of the two remaining numbers on the right for the third term, and the other on the left for the first term.

3. Multiply the second and third terms together, divide by the first, and the quotient will be the answer to the question, which,

231. What is the nature of these useless distinctions ?- -232. What is the general rule for stating and performing all questions in simple proportion?

(as also the remainder) will be in the same denomination in which you left the second term, and may be brought into any other denomination required.*

NOTE--The chief difficulty that occurs in the Rule of Three, is the right placing of the numbers, or stating of the question; this being accomplished, there is nothing to do, but to multiply and divide, and the work is done.

To this end the nature of every question must be considered, and the circumstances on which the proportion depends, observed, and common sense will direct this if the terms of the question be understood.

1. The fourth number is always found in the same name in which the second is given, or reduced to; which, if it be not the highest denomination of its kind, reduce to the highest, when it can be done.

2. When the second number is of divers denominations, bring it to the lowest mentioned, and the fourth will be found in the same name

*This rule is founded on the obvious principle, that the magnitude or quantity of any effect varies constantly in proportion to the varying part of the cause: Thus, the quantity of goods bought, is in proportion to the money laid out;--the space gone over by an uniform motion, is in proportion to the time, &c.

It has been shewn in Multiplication of Money, that the price of one multiplied by the quantity, is the price of the whole; and in Division, that the price of the whole divided by the quantity, is the price of one. Now, in all cases of valuing goods, &c. where one is the first term of the proportion, it is plain, that the answer found by this rule, will be the same as that found by Multiplication of Money; and where one is the last term of the proportion, it will be the same as that found by Division of Money.

In like manner, if the first term be any number whatever, it is plain that the product of the second and third terms will be greater than the true answer required, by as much as the price in the second term exceeds the price of one, or as the first term exceeds an unit; consequently this product, divided by the first term, will give the true answer required.

NOTE 1.- When it can be done, multiply and divide as in Compound Multiplication, and Compound Division.

2. If the first term, and either the second or third can be divided by any nuinber without a remainder, let them be divided, and the quotient used instead of them.

The following methods of operation, when they can be used, perform the work in a much shorter manner than the general rule.

1. Divide the second term by the first; multiply the quotient into the third, and the product will be the answer.

2. Divide the third term by the first; multiply the quotient into the second, and the product will be the answer.

3 Divide the first term by the second, and the third by that quotient, and the last quotient will be the answer.

4. Divide the first term by the third, and the second by that quotient, and the last quotient will be the answer.

233. What is the reason of this rule; or why will multiplying the second and third terms together, &c. produce the true answer?

to which the second is reduced, which reduce back to the highest possible.

3. If the first and third be of different names, or one or both of divers denominations, reduce them both to the lowest denomination mentioned in either.

4. When the product of the second and third is divided by the first; if there be a remainder, after the division, and the quotient be not the least denomination of its kind; then multiply the remainder by that number, which one of the same denomination with the quotient contains of the next less, and divide this product again by the first number; and thus proceed till the least denomination be found, or till nothing remain.

5. If the first number be greater than the product of the second and third; then bring the second to a lower denomination.

6.

When any number of barrels, bales, packages or pieces are given, each containing an equal quantity, let the contents of one be reduced to the lowest name, and then multiplied by the given number of packages or pieces.

7. If the given barrels, bales, pieces, &c. be of unequal contents, (as it most generally happens) put the separate content of each properly under one another, then add them together and you will have the whole quantity.

8.

When any of the terms are given in Federal Money, the operation is conducted in all respects as in simple numbers, observing only to place the point or séparatrix between dollars and cents, to point off the results according to what has been taught already in Federal Money, and Decimal Fractions.

PROOF.-The method of proof is by inverting the question. That is, put the fourth number, or answer, in the first place; the third in the second; and the second in the third; work as before directed, and the quotient will be the first number.

EXAMPLES.

1. If 4 yards of cloth cost 16 dollars, what will 12 yards come to, at the same rate?

234

When the first, second, or third terms are of different denominations, how do -235. What is the method of preof?

you proceed?