ing term of supposition in the second place, which must be of the same name or kind with the answer. Then set the demanding term in the third place, which must be of the same name or quality with the first term.

RULE FOR WORKING.

If the first and third terms are of different denominations, reduce them to the lowest denomination mentioned in either. If the second term stands in different denominations, reduce it to the lowest denomination mentioned in that term: Then multiply the second and third terms together, and divide their product by the first term, and the quotient will be the fourth term or answer in the same denomination. of the second term, in whatever denomination it stands or has been reduced; then if the answer does not stand in the highest denomination, it should be brought to the highest by Reduction.

NOTE 1.-All the following rules strictly belong to the Rule of Three Direct, viz: the Double Rule of Three, Exchange, Interest, Practice, Single and Double Fellowship, Tare and Tret, Barter, Loss and Gain, Alligation, Discount and Annuities.

These Rules have acquired different names from the business to which they are applied; and in treating of them separately, it will be shown that they belong to this rule, and may, like Direct Proportion or the Rule of Three Direct, be reduced back to the fundamental or Simple Rules.

The Double Rule of Three, or Compound Proportion, will in some cases embrace both the Rule of Three Direct and Inverse.

Interest, on account of its importance in business, and its evident relation to the Simple Rules, has been placed before the Rule of Three., though in all cases may be worked by the Rule of Three.

NOTE 2. As the chief difficulty which learners experience in this rule, is in stating questions, Teachers should exercise their pupils in stating fifteen or twenty questions for a few days in succession, and the difficulty will then vanish. They should be in the daily habit of demonstrating to their pupils the rules, and explaining the principles upon which they are founded, tracing them back to the simple rules on which they immediately depend. Instructers by pursuing this course, will present to the young mind this important science free from all its obscurity, and students will soon learn that the whole science of Arithmetick, is nothing more than a proper application of the fundamental ar Simple Rules.

EXAMPLES.

1. If 2 yds. of calico cost 4s. what will 4 cost? Ans. 8s.

yds. 8. yds.

2:4::4
4

2)16

Ans. 8 skillings.

DEM. When we multiply the second term by the third, we are multiplying the price of a quantity, consequently the product must be as many times too great as the first term exceeds a unit; therefore by dividing the product by the first term, the quotient must be the answer, or fourth term,

because the second term is double the price of a unit. N. B. The learner will perceive, that 8, the fourth term or answer, bears the same proportion to 4, the third term, that 4, the second term, bears .to. 2 the first term; because the fourth term is double the third, and the second term is double the first. The 1st and 4th terms are called extremes, and the second and third terms, the means; and the celebrated property of proportional numbers is, that the product of the extremes, is equal to the product of the means; thus, 2: 4 :: 4: 8, then 2X8-4x4-16. The first term of two proportional numbers, is also called the antecedent, and the other, to which it bears proportion, is called its consequent; thus, in the example given, 2, the first term, is the antecedent, and 4, the second term, the consequent; that is, two yards have produced the 4 shillings; and the third term, 4 yards, is also the antecedent which produced 8 shillings its consequent. Now it is evident, that the first antecedent bears the same proportion to its consequent, that the second antecedent bears to its consequent; that is, 2 yards bear the same proportion to the cost, 4 shillings, that 4 yards bear to the cost, 8 shillings; because in each case, the money bears the same proportion to the quantity of cloth purchased.

This rule may be proved by inverting the terms of the question. Or by multiplying the means, and if the product equal the product of the extremes, the work is right; thus, if 4 yds. cost 8s. what will 2 cost? Ans. 4s. This sum serves as a proof of the first example; here we find that 2 yards cost 4s. which agrees with the first supposition.

4yds: 8s. 2yds.

2

4)16

4s.

In order to render this

to the learner, the same

excellent rule as plain as possible example is again repeated and reduced back to the simple rules on which it depends. If 2 yards of calico cost 4 shillings; what will 4 yds.

cost?

DEM. We have learned by division, that dividing the price of a quantity by the quantity gives the price of a unit or 1; then when we di

Ans. 8s.

Reduced to the simple rules by dividing the second term by the first, and multiplying the quotient by the third term; thus, 2)4s.

2s. price of 1 yard.

4

8s. price of 4 yards,

vide 4 shillings, the price of 2 yards, by 2, the quotient, 2 shillings, is the price of 1 yard. And we have found by multiplication, that multiplying the price of 1 yard, by the number of yards, gives the price of

the whole number; then when we multiply 29., the price of 1 yard, by 4, the number of yards, the product 8s. must be the price of 4 yds. Hence it is plain, that Direct Proportion or the Rule of Three Direct, is nothing more than an application of Multiplication and Divis ion, and from Multiplication and Division, it might be reduced back to addition and Subtraction, as those two rules have been reduced back, in their proper places. And it is the better way to divide the second term by the first, and multiply that quotient by the third term, whenever the first term will divide the second without a remainder.

2. If one yard of cloth cost four dollars; what cost eight yards? Ans. $32.

yd. $ yds. 1:4::8

8

$32

DEM. It is plain, when the first term is a unit, that the work is performed by multiplication, because it is only repeating the price of a unit or one, by the whole number, and dividing that product by the first term when it is a unit, can evidently make no difference.

3. If eight yards cost $32; what cost one yard?

yds. $ yd.
8:32:: 1

1

8)32

Ans.

Ans. $4.

DEM. It is plain, when the 3d term is a unit, that the work is performed by division merely, for multiplying by one can make no difference; it is only bringing down the middle or second term, for the convenience of dividing by the first term, and it is evident, if the price of a quantity, be divided by the quantity, that the quotient must be the price of a unit.

4. If 6 yards of cloth cost $24; what cost 12 yards ?

Ans. $48.

6)24 the price of 6 yds. 4 the price of 1 yd.

12

Ans. $48 the price of 12 yds. A. $48 the price of 12 yds.

DEM.-It is evident from the reasoning under the first example, that when we multiply 24 dollars, the price of 6 yards, by 12, the product is 6 times too much; then when we divide the product by 6, the first term, the quotient must be the price of 12 yards. And it is also plain, when we divide $24, the price of 6 yards, by 6, that the quotient must be the price of one yard; and the price of one yard mul tiplied by 12, evidently gives the price of 12 yards.

5. If six pounds of tea cost $3,75cts.; what cost eighteen pounds? Ans. $11,25cts.

6. If 9 bushels of wheat cast $12; what cost 36 bushels? Ans. $48.

7. If 36 bush. cost $48; what will 9 bu. cost? Ans. $12. 8. If 60 bushels of flax seed cost 45 dollars; what will 1 bushel cost? Ans. ,75cts. 9. If one bushel cost 75 cents; what will 60 bushels cost? Ans. $45. 10. If 48 dollars will buy 12 yards of cloth; how many yards will 36 dollars buy? Ans. 9 yards. 11. If 36 yards of broad cloth be worth 144 dollars; what will 9 yards be worth? Ans. $36. 12. If 28 pounds of butter cost $2,52 cents; what will 7 pounds cost? Ans. ,63cts. 13. If 63 bushels of oats cost $20,16cts.; what will 12 bushels cost?" Ans. $3,84cts. 14. At $1,25 per head; what will 40 sheep cost?

Ans. $50. 15. If 50 dollars will buy 40 sheep; how many sheep will 112 dollars, and 50 cents buy?

Ans. 90.

16. What will 300 acres of land cost, at ten dollars and fifty cents per acre? Ans. $3150. 17. If 300 acres of land cost 3150 dollars; what cost 1 acre? Ans. $10,50cts.

Ans. $18,80cts.

18. If lcwt. 2qrs. of beef cost $8,40cts. ; what is the value of 3cwt. 1qr. 12lb.? Cwt. gr. $cts. Cwt. gr. lb.

1

6

28

48

12

168 lb.

28,40: 3 1 12

4

13

28

116

26

376lb.
8,40

150,40
3008 $cts.
168)3158,40(18,80 Ans
168

1478

1344

1844

1344

When the 1st and third terms are of different denominations, the better way is to state the question, and then perform the reduction, because the whole work will then show itself together.

NOTE. The learner must be careful in stating questions, that the first and third terms be alike; that is, if the first term be weight, the third term must be weight; if the first term be money, the third must be money; and the middle or second term

must be the same kind of the answer sought; that is, if weight be required for the answer, the second term must be weight, if money be required for the answer, the second term must be money.

that per day?

19. If a man's yearly income be 438 dollars; what is Ans. $1,20cts. 20. If a man's wages be 162 dollars a year; what is that per calendar month ? Ans. $13,50cts. 21. If 4yds. 2 qrs. cost 19 dollars; what will 3yds. 3qrs. cost? Ans. $15,83cts. 3m. 22. If 2yds. 3qrs. cost $6,60cts. ; what will 11 yards cost? Ans. $26,40cts. 23. If 11 yards, cost 26 dollars 40cts. ; what will 2 yards 3qrs. cost? Ans. $6,60cts. 24. If a man spend 1 shilling and 6 pence a day; how long will £27 7s. 6d. last him? Ans. 1 year or 365d.

25. A merchant failing in trade, compounds with his creditors for 56 cents on the dollar, and at that rate, pays 2840 dollars 12 cents; what was his debt? Ans. $5071,64,2m.

26. Suppose the clothing of a regiment of 750 men, amounts to £2381 5 shillings; what will the clothing of a body of 3400 men amount to? Ans. £10795.

The learner will recollect, that when the second term is expressed in different denominations, it must be reduced to the lowest denomination mentioned, and the quotient or answer will be of the same name of the second term, to whatever it is reduced. NOTE. Whenever there is a remainder, after dividing by the first term, and the second term has not been reduced to the lowest denomination, the remainder must be reduced to the next inferiour denomination, and the division continued, the same as in Compound Division.