26. Divide £332. 19s. 9d. by 34.

A. £9. 15s. 10.

A. 2A. 2R. 11rd.

27. Divide 120A. 2R. 37rd. by 47. 28. Divide 485Y. 180d. 15h. 30m. 45sec. into 105 equal periods of time. A. 4Y. 227d. 16h. 8m. 510sec.

RECAPITULATION.

29. COMPOUND DIVISION is the dividing of a compound number by a simple one.

RULE.

30. Begin on the left and divide each denomination separately, as in simple numbers.

31. But if there be a remainder, reduce it to the next denomination, to which add the given number in that denomination, then divide as before.

32. Each quotient will be of the same name with its dividend; and the several quotients taken together will constitute the required quotient or answer.

33. Divide £161. 14s. 4d. by 8.

A. £20. 4s. 31.

34. If a man can earn £3. 14s. 51d. per week, how much can he earn per day?

A. 10s. 7d. 24qr. A. 3lb. 4oz. 17dwt.

35. Divide 30lb. 7oz. 13dwt. by 9. 36. When 7 silver cups weigh 8lb. 9oz., what is the weight of each?

37. Divide 205qr. 19lb. 7oz. 2dr. by 10.

A. 1lb. 3oz.

A. 20qr. 14lb. 7oz. 18 dr. 38. Suppose a poor man labors a month for 149lb. 13oz. of pork; how much does he receive each day, on an average, allowing 26 working days to each month? A. 5lb. 12oz. A. 5yd. 1qr. 3na.

39. Divide 853 yd. 2qr. 3na. by 157.

40. If it take 2,700 yards of broadcloth to clothe a regiment of 800 men, what quantity will each man require? A. 3yd. 1qr. 2na. 41. If a hogshead of wine costs £45. 8s. 3d., what is it worth by the gallon? A. 14s. 5d. 42. Bought 2 dozen (24) silver spoons, which weighed 7lb. 6oz. 13dwt. how much silver did each spoon contain?

A. 3oz. 15dwt. 13gr. 43. Suppose a steamboat, in making 121 trips from Albany to New York, occupies 48d. 17h. 40m.; what will be the average time in which she makes one trip? A. 9h. 40m.

44. How far must I travel each day, to accomplish a journey of 1,400 miles 3fur. 10rd. in 51 days? A. 27m. 3fur. 263rd. 45. Suppose 37 barrels of equal size contain 98bu. 3pk. 2qt. of wheat; what quantity is in each barrel ? A. 2bu 2pk. 517qt.

Q. When there is a remainder, what is to be done with each inferior denomination of the dividend? 16. How do you proceed when the divisor exceeds 12, and is a composite number? 21. How, when it is not a composite number? 24 What is Compound Division? 29. Rule? 30, 31, 32.

46. Suppose a king's salary to be £200,000 per annum; what is A. £547. 18s. 10d. 3145qr.

that per day?

MISCELLANEOUS EXAMPLES.

L. 1. How many farthings are there in £5. 17s. 6d. ? 2. How many pounds are there in 5,640 farthings?

3. How many guineas at 28 shillings each, will pay a debt of £25? A. 17 guineas 24 shillings.

4. A grocer bought 20 hundred weight of sugar for $112, and sold it for 4 d. per pound; what was the gain?

A. $13.

5. A merchant in London borrowed £60 and paid at one time £15. 14s. 6d., and at another £20. 3s. 61d., how much remained unpaid? A. £24. 18. 111⁄2d.

6. From a compound weighing 5łb, an apothecary sold to one man 1lb. 33. 53. 19., and to another 33. 29., how much had he left on hand? A. 3lb. 53. 23. 7. A merchant bought 3 hogsheads of sugar, each weighing 8cwt. 1qr. 20lb., and sold five barrels of the same, each weighing 3cwt. 3qr 17lb. How much had he left? A. 5cwt. 3qr. 8. If you deduct the days in the months of November and December from the year, how many days will there be left in a leapyear? A. 305 days.

9. What is the sum of 30rd. 5yd., 19rd. 4yd., 17rd. 1yd., and 25rd. 4yd.? See XLI. 16, 17. A. 93rd. 3yd. 10. Add together 30A. 2R. 39 sq. r. 30 sq. yd., 29A. 1R. 25 sq. r 23 sq. yd., 16A. 3R. 8 sq. r. 15 sq. yd., and 45A. 27 sq. r. 8 sq. yd. A. 122A. 21 sq. r. 15 sq. yd. 11. Add into one sum 500 sq. r. 272 sq. ft., 450 sq. r. 195 sq. ft., 365 sq. rd. 215 sq. ft., and 985 sq. r. 270 sq. ft.

A. 2,303 sq. r. 1351 sq. ft.

12. If a man travels 25m. 3fur. 15rd. 3yd. a day, for 12 successive days, how far will he go in that time? A. 305m. 26rd. 3yd.

13. From 40rd. 2yd. take 17rd. 4yd. Say 4yd. from 54yd.=fyd. +2yd.=31 and carry 1. A. 22rd. 3 yd. 14. Add together 22rd. 31⁄2yd., and 17rd. 4yd. A. 40rd. 2yd. 15. Suppose a man travels 305m. 26rd. 3yd. in 12 days; what is the average distance per day? A. 25m. 3fur. 15rd. 3yd.

A. 1,600 A. 50bl. 25gal.

16. How many gallons in 50bl. 25gal. 17. How many barrels in 1,600 gallons? 18. How many pint, quart and 2 quart bottles, of each an equal number, can be filled with a hogshead of molasses?

NOTE.-4pt. [=2qt. :] 2pt. [=1qt.] and 1pt. make 7 pints; then divide 63 gallons brought into pints by 7 pints. A. 72 of each.

19. A merchant has 700 quart, 700 two quart, 700 three quart and

700 gallon bottles, and wishes to know how many hogsheads of wine it will take to fill them? A. 27hhd. and 49gal. over..

20. A certain manufacturer employs an equal number of men, boys and girls, to whom he pays daily, as follows, viz: to each man $1, to each boy 50 cents, and to each girl 75 cents, making in all $6 75. How many persons of each class has he in his employ?

A. 300 persons.

21. A merchant has 20 hogsheads of tobacco, each weighing 9cwt. 1qr. 14lb., which he wishes to put into an equal number of small and large boxes, the former to hold 21⁄2lb. and the latter 3 times as much; what number of each must we have? A. 1,878 boxes.

22. How many sheets of paper will it take to make an 18mo. book (VII. 80.) which shall contain 288 pages (=144 leaves?) A. 8 sheets. How many quires to print an edition of only 96 copies? A. 32 quires. How many reams to print an edition of 2,400 copies? A. 40 reams.

23. At $3.50 per ream, what will be the expense of paper for printing an edition of 43,200 copies of a 12mo. work, to consist of 192 pages, making the usual allowance of 2 quires of waste paper in each ream? A. $2,800.

24. How many years of 365 NOTE.-In 2142 days, the

days in 49,000 hours? of a day is of course 3 of 24 hours= hours, the first remainder =34h.=1d. A. 5Y. 215d. 10h.

18 hours, which added to 16 10h. Add the 1 day to the 214 days. 25. "A gentleman in Buffalo has just (Feb. 1838) sold all his real estate for $130,000, payable in instalments at the rate of 1 dollar an hour." What period of time has the purchaser allowed him for the payment of the debt, reckoning 365 days to the year?

A. 14Y. 303d. 4h.

FRACTIONS.

GENERAL PRINCIPLES.

LI. 1. When two numbers are written, one above the other with a line between them, they mean as follows:

(1-half) means 1 of the 2 equal parts of a unit or any thing. (1-third) means 1 of the 3 equal parts of a unit or any thing. (2-thirds) means 2 of the 3 equal parts of a unit or any thing. (1-fourth) means 1 of the 4 equal parts of a unit or any thing. (3-fourths) means 3 of the 4 equal parts of a unit or any thing. (1-fifth) means 1 of the 5 equal parts of a unit or any thing. (2-fifths) means 2 of the 5 equal parts of a unit or any thing. (5-fifths) means 5 of the 5 equal parts, that is, the whole of any thing, and so on in respect to any numbers whatever.

2. Then, or 3, or 4, or 5, or 8, &c. are each equal to 1 unit. LI. Q. What is meant by,,, &c.? 1. What by, or 3, 4,

4. &c.? 2.

3. These expressions are called FRACTIONS (from the Latin fractio signifying broken,) because they stand for numbers broken or divided into parts.

4. The whole unit or thing, of which fractions are broken parts, is called an INTEGER (a Latin word signifying whole,) in order to distinguish it from fractions.

5. Fractions then are the expressions for one or more equal parts of a unit or whole number, called an integer.

6. The number below the line, which shows into how many equal parts the unit or integer is divided, is called a DENOMINATOR'; because it gives the name or denomination to the fraction; as, halves, thirds, &c.

7. The number above the line, which shows the number of parts meant, is, for that reason, called the NUMERATOR.2 The Numerator and Denominator are called the Terms of the Fraction.

3

8. Thus, in 1, 2, 5, 7, the upper terms, 1, 3, 5 and 7 are the numerators, and the lower terms, 2, 4, 6 and 8, are the denominators.

LII. 1. Since the denominator represents all the parts of the integer, therefore,—

2. If we multiply the value of a single part by the denominator, the product will be the entire value of the integer.

3. When of a bushel of rye costs 12 cents, what will 18 or 1 bushel cost? A. $1.20. 4. If of a vessel be valued at $5,000, what is the value of the whole vessel?

5. What is that number of which 36 is 3.

6. 29 is of what number?

7. 75 is of what number?

40

A. $25,000.

A. 1,080.

A. 1,450.

A. 3,000.

8. When of a cask of wine sells for $45, what is the whole cask worth at that rate? Find the value of first, by dividing 45 by 3, then multiply the result by 4?

A. $60. 9. 24 is of what number? The result will be the same, if we multiply by 12 first and divide by 3 afterwards; thus, 24 × 12÷3=96. A. 96.

10. HENCE if we multiply the value of any fraction by its denom

Q. What are such expressions called and why? 3. What then are Fractions? 5. What is an Integer and whence its name? 4. What is the figure below the line called, and why? 6. What, the figure above the line, and why? 7. What do both the numerator and denominator form? 7. Which are the numerators and denominators in and 3? 8.

LII. Q. How may the value of any integer be ascertained from having its fractional part given? 2. Why so? 1. When you pay 3 dollars for of a ton of hay, what would be the price of a whole ton? When of a hogshead of molasses costs 12 dollars, what is the price of a whole hogshead? What is the rule for it? 10.

1 DEMOMINATOR, [L. denomino.] He that names 2 NUMERATOR, [L. numero.] One that numbers.

inator, and divide the result by its numerator, the product will be the entire value of the integer.

11. If of a ship's cargo be valued at $10,000, what is the value of the entire cargo ?

12. 509 is of what number?

A. $15,7142. A. 1,323. 13. The fractional remainders, and above, are, properly speaking, unexecuted divisions; hence fractions are said to have originated in this manner from Division.

14. 815 is of what number?

A. 2,0518.

A. 1,019.

LIII. 1. How many halves are there in 17 dollars? Since 2halves are equal to 1 dollar, there are 2 times as many halves as there are dollars. A. 34 halves=34.

2. Hence multiplying any whole number by a given denominator, shows how many parts are to be taken for the numerator.

3. How many dollars are 34 of a dollar? Evidently as many dollars as there are times 2 in 34, for 2-halves make 1 dollar.

A. 17 dollars. 4. Hence dividing the numerator by the denominator, shows what whole number is contained in the fraction.

5. How many fourths or quarters in $5? Sixths in 118 bushels? Sevenths in 395 barrels ?

6. How many dollars in 20 of a dollar? Bushels in 788 of a bushel? Barrels in 2765 of a barrel?

7. Change 10 to a fraction whose denominator shall be 8. A: 80. How many units in 80 ?

A. 10.

8. Change 625 to a fraction whose denominator shall be 1. How many units are there in 25?

9. Since no number is affected by multiplying or dividing it by 1, therefore,

10. Any whole number becomes a fraction by simply writing 1, før its denominator.

11. What fraction, that has 17 for a denominator, is equal to 365? Or to 415?

179 17

A. 6205. 7055 of a dollar, how many pounds A. 10lb; 3,650lb.

12. When 1 pound of butter costs may be bought for $1? For $365? 13. When 1 gallon of molasses costs of a dollar, what will be the cost of 5gal.? Of 20gal.? Of 1 tierce? A. $1; $4; $83.

14. How many furlongs are equal to 495 fur.? A. 617 furlongs. 15. Hence the value of any fraction, is the quotient arising from dividing the numerator by the denominator.

Q. 20 is of what number? How did fractions originate? 13. LIII. Q. How many halves are there in 17 dollars, and why? 1. How is it ascertained? 2. In 10 minutes how many fourths ?-fifths ?-sixths? How many dollars are there in 34 of a dollar, and why? 3. What is the inference? 4. How many furlongs in of a furlong?-in 121 of a furlong? How does any whole number become a fraction? 10. Why so? 9. Give an example.

60

10