third place, and consider each pair of similar terms and this third one, as the terms of a stating in Simple Proportion, and set them severally, in the first and second places, agreeably to the directions under that rule.

When the question is thus stated, reduce the similar terms to like denominations, and then multiply all the terms in the second and third places together, and divide the product by the product of those in the first place: the quotient will be the answer, or term sought.

The above rule is preferred for reasons similar to those which have been given for adopting the new rule for Simple Proportion: the one formerly used is, however, subjoined.

RULE FOR STATING.

Set the two terms of supposition which are of the same name or kind as those of the demand, one under the other, in the first place; that of the same kind as the answer in the second, and those of the demand in the third, with the two corresponding terms of the supposition and demand opposite to each other, and of the same denomination.

When a question is stated, consider the two upper terms with the middle one, as a stating in the Single Rule of Three, and also the two under terms, with the middle one, as a stating in the same rule; if, in both instances, the proportion be direct, the question is in direct proportion; but if in either of them the proportion be inverse, the question is in inverse proportion.

RULE FOR DIRECT PROPORTION.

Multiply the two terms in the third place together, and multiply the product by the middle term; divide the last product by the product of the terms in the first place, and the quotient will be the answer, in the same denomination as the middle term.

EXAMPLE.

If 6 men in 8 days eat 10 lb. of bread, how much will 12 men eat in 24

6 men

8 days S

days?

Ans. 60.

PROOF.

By two statings in the Single Rule of Three.

Note. If either of the two first terms, or both, will divide, or can be divided by any of the three last, or if any other number will divide one of the first and one of the last, without a remainder, the operation may be contracted by using their quotients in their stead.

EXAMPLES.

1. If 6 men in 8 days eat 10 lb. of bread, how much will 12 men eat in 24 days?

men 6: 122 days 8 245

288

Ans. 60.

243: 10lb.

2. If 3 men in 4 days eat 5 lb. of bread, how much will suffice 6 men for 12 days?

Ans. 30 lb.

3. Suppose 4 men in 12 days mow 48 acres, how many acres can 8 men mow in 16 days?

RULE FOR INVERSE PROPORTION.

Ans. 128 A.

Transpose the inverse extremes; that is, set that which is in the first place under the third; and that which is in the third place under the first; then work as in Direct Proportion.

EXAMPLE.

If 7 men reap 84 acres of wheat in 12 days, how many men can reap 100 acres in 5 days?

Ans. 20.

{

100 direct

420

1200

7

420)8400(20

840

0

4. If 10 bushels of oats be sufficient for 18 horses 20 days, how many bushels will serve 60 horses 36 days, at that rate?

Ans. 60 bu. 5. If 7 quarters of malt are sufficient for a family of 7 persons 4 months, how many quarters will 46 persons use in 10 months? Ans. 115. 6. Suppose the wages of 6 persons for 21 weeks be 288 dollars, what must 14 persons receive for 46 weeks? Ans. 1472 dols.

7. If 8 reapers have 3 L. 4 s. for 4 days work, how much will 48 men have for 16 days work? Ans. 76 L. 16s. 8. If 100 L. in 12 months gain 6 L. interest, how much will 75 L. gain in 9 months? Ans. 3L. 7 s. 6d. 9. If 100 L. in 52 weeks gain 6 L. interest, how much will 200 L. gain in 26 weeks?

Ans. 6 L. 10. If the carriage of 8 cwt. 128 miles cost $12.80, what must be paid for the carriage of 4 cwt. 32 miles? Ans. $1.60.

11. If 16 L. 18 s. be the wages of 16 men for 8 days, what sum will 32 men earn in 24 days? Ans. 101 L. 8s.

12. If 350 L. in half a year gain 10 L. 10 s. interest, what will be the interest of 400 L. for 4 years? Ans. 96 L.

INVERSE PROPORTION.

1. If 7 men reap 84 acres of wheat in 12 days, how many men can reap 100 acres in 5 days? Ans. 20 men. 2. If 4 dollars be the hire of 8 men for 3 days, how many days must 20 men work for 40 dollars? Ans. 12. 3. If 4 men have $3.20 for 3 days work, how many men will earn $12.80 in 16 days? Ans. 3 men.

4. If 4 reapers have 12 dollars for 3 days work, how many will earn 48 dollars in 16 days?

Ans. 3.

5. If 100 L. in 12 months gain 6 L. interest, what sum will gain 3 L. 7 s. 6 d. in 9 months? Ans. 75 L.

6. If a footman travel 240 miles in 12 days, when the days are 12 hours long; how many days will he require to travel 720 miles, when the days are 16 hours long? Ans. 27 days. interest, what

7. If 100 L. in 12 months gain 8 L. sum will gain 8 L. 12 s. in 5 months?

8. If 200 lb. be carried 40 miles for 40 may 20200 lb. be carried for $60.60 ?

Ans. 258 L. cents, how far Ans. 60 miles.

PROMISCUOUS EXAMPLES.

1. If 4 men in 5 days eat 7 lb. of bread, how much will suffice 16 men 15 days? Ans. 84 lb. 2. If 100 dols. gain $3.50 interest in one year, what sum will gain $38.50 in 1 year and three months? Ans. 880 dols. 3. If it take 5 men to make 150 pair of shoes in 20 days, how many men can make 1350 pair in 60 days? Ans. 15. 4. If the wages of 6 men for 21 weeks be 120 L., what will be the wages of 14 men for 46 weeks? Ans. 613 L. 6 s. 8 d.

5. If 333 L. 6 s. 8 d gain 15 L. interest in 9 months, what sum will gain 6 L. in 12 months? Ans. 100 L.

6. A wall which is to be built to the height of 27 feet, has been raised 9 feet in 6 days, by 12 men: how many men must be employed to finish the work in 4 days? Ans. 36 men.

PRACTICE.

Practice is a short method of ascertaining the value of any number of articles, or of pounds, yards, &c. by the given price of one article, one pound, or one yard, &c. Practice may be proved by Compound Multiplication, or by the Single Rule of Three Direct.

An aliquot part of a number is any number that will divide it witha remainder; thus 4 is an aliquot part of 20, and 8 of 56. A sum or antity is an aliquot part of a greater sum or quantity, when a certain umber thereof will make the greater: thus a shilling is an aliquot part of a pound, because 20 shillings make one pound.

When the price is less than a penny, work by

RULE 1.

If the price be a farthing, or a halfpenny, set down the value of the given number at a penny, and take such part of that sum as the price is of a penny, for the answer in pence.*

If the price be three farthings, find the value of the given number at a halfpenny, and afterwards at a farthing; then add the two results together, and their amount will be the answer.

*

** If the learner be unable to tell the denomination of a quotient, or how to proceed with remainders, it would be useful to refer him to examples 14, 15, and 16, under Rule 1, and 7, 8, under Rule 3, Compound Division.

EXAMPLES.

1. What is the value of 4528 quills, at 2. What is the value of 4528 quills, at

each?

each?

210)914 4

12)3396 Ans.

The value of any number of articles at a penny, each, is that number of pence: thus, the value of two things at a penny, each, is two pence: of three things, three pence; of twenty things, twenty pence, &c.; and, as a farthing is the fourth part of a penny, the value at a farthing must be a fourth part of the value at a penny; and as two farthings are the half of a penny, the value at two farthings must be half of the value at a penny, &c.

This explanation of the rule, with a little variation, will apply to most of the other rules of Practice.