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12. Divide 23 by 1.

Ans. 92. 13. Divide 5 by š.

Ans. 25. 14. Divide 12 by .

Ans. 16. 15. Divide 16 by .

Ans. 32. 16. Divide 100 by ij.

Ans. 11111 17. I have 50 square yards of cloth, how many yards, f of a yard wide, will be sufficient to line it ?

Ans. 834 yards. 18. A. Poor can walk 311 miles in 60 minutes ; Benjamin can walk is as fast as Poor. How long will it take Benjamin to walk the same distance ?

Ans. 731 minutes.

OPERATION.

III. To divide a mixed number by an integer. 19. Divide 173 by 6.

Ans. 274. We divide 17 by 6, and find it is con6) 173 tained 2 times, which we write under the 213 17, and we have 5 remaining, which we

multiply by 8, the denominator of the fraction ; and to the product we add the numerator, 3, and the amount is 43, this we write over the product of 6, the divisor, multiplied by the denominator, 8, = 48. The rationale of the above question is the same as of those in Rule I. of this section. Hence the following

RULE.

Divide the integers as in whole numbers, and if any thing remains, multiply it by the denominator of the fraction, and to the product add the numerator of the fraction, and write it over the product of the divisor, multiplied by the denominator.

20. Divide 17} by 7. 21. Divide 181 by 8. 22. Divide 2711 by 9. 23. Divide 31 ) by 11. 24. Divide 784 by 12. 25. Divide 1891} by 4. 26. Divide 1071, by 3.

Ans. 215. Ans. 212 Ans. 3 U. Ans. 295

2. . Ans. 617 Ans. 4718. Ans. 3536

27. Divide $ 174 among

7 men.

Ans. $249. 28. Divide $ 1067 among 8 boys.

Ans. * 1343. 29. What is the value of of a dollar ?

Ans. $0.341. 30. Divide $ 10711 among 4 boys and 3 girls, and give the girls twice as much as the boys.

Ans. boy's share $ 104. Girl's share $213. 31. If $ 14 will purchase 27 of a ton of copperas, what quantity will $ 1 purchase ? Ans. lcwt. Oqr. 24lbs.

IV. To divide one fraction by another. 32. Divide f by $.

Ans. 111

To understand the raŠ x = $= 131 Ans.

tionale of this process, we

find the two factors of $, which are and }; for 1 multiplied by are $, as is evident from a preceding rule. We now divide ž by 1, which, by case I. of this section, will be { X=32 Again, we wish to divide 32 by s. It is evident, that 32 will contain } nine times as often, as it will a unit, and it contains a unit iz times, therefore it contains į nine times 32 = f x 32==131 Ans. In performing this question, it will be perceived, that the numerator of the dividend has been multiplied by the denominator of the divisor, and the denominator of the dividend by the numerator of the divisor. · Hence the following

OPERATION.

RULE.

Invert the divisor and proceed as in multiplication. If, however, there be mixed numbers in the question, they must be reduced to improper fractions, and compound fractions must be reduced to simple fractions. 33. Divide / by .

OPERATION.

* X1==144 Ang. 34. Divide 71 by 3%.

OPERATION.

7=, 33 = 4, * X = 20 = 2** Ans. 35. Divide f by 1.

Ang. 31

36. Divide 1: by 11.
37. Divide by is
38. Divide by .
39. Divide 4 by it:
40. Divide 73 by 44.
41. Divide 3.1 by 77.
42. Divide 111 by 57.
43. Divide 4 by 17.
44. Divide 1169 by 147.
45. Divide 814 by 9}.
46. Divide of by 1 of 3.

Ans. n. Ans. 23. Ans. 630

Ans. 44. Ans. 19%.

Ans. 15 Ans. 2152 Ans. 22.

Ans. 873. Ans. 81/1 Ans. 183.

Section 25.

EXERCISES IN VULGAR FRACTIONS.

1. What are the contents of a board 9 inches long and 7 inches wide ?

Ans. 63
square

inches. 2. What are the contents of a board 114 inches long, and 44 inches wide ?

Ans. 491. square inches. 3. How many square rods in a garden, which is 189

rods in length and 9% rods wide ? Ans. 17893 rods. 4. What cost 194 acres of land, at $ 174 per acre ?

Ans. $ 35016 5. What cost 1430 tons of coal at $ 74 per ton ?

Ans. $ 11117. 6. What cost 1344 tons of hay at $87 per ton ?

Ans. $ 120405. 17. What cost 13 bushels of corn at $ 1% per bushel ?

Ans. $384. 8. What is the value of of a dollar ? Ans. $ 0.561. 9. What is the value of j7 of a dollar ? Ans. $0.211. 10. What is the value of Art of a dollar ? Ans. $0.25%. 11. What is the value of li of a dollar ? Ans. $ 0 5116 12. Bought a cask of molasses, containing 875 gallons ; ff of it having leaked out, the remainder was sold at 271 cents per gallon ; what was the sum received ?

Ans. $ 15.0333

13. Bought of L. Johnson 7yds. of broadcloth, at $3! per yard, and sold it at $43 per yard; what was gained?

Ans. $ 3.681. 14. Bought a piece of land, that was 47 s rods in length, and 2976 in breadth ; and from this land, there was sold to Abijah Atwood 5 square rods, and to Hazen Webster a piece that was 5 rods square ; how much remains unsold ?

Ans. 136693 square rods. 15. Bought a tract of land that was 97 rods long and 481rods wide ; and from this I sold to John Ayer, a houselot, 181rods long, and 14; rods wide ; and the remainder of my purchase was sold to John Morse, at $ 3.75 per square rod ; what sum shall I receive ?

Ans. $ 16717.3035. 16. What are the contents of a box 8 feet long, 5 feet wide, and 3 feet high ?

Ans. 120 solid feet. 17. What are the contents of 10 boxes, each of which is 7 feet long, 47 feet wide, and 3 feet high ?

Ans. 1812/17 feet. 18. Polly Brown has $ 17.871; half of this sum was given to the missionary society, and of the remainder she gave to the Bible society ; what sum has she left ?

Ans. $3,574. 19. What number shall be taken from 121, and the remainder multiplied by 10% that the product shall be 50 ?

Ans. Us 20. What number must be multiplied by 73, that the product may be 20 ?

Ans. 23%. 21. Bought of John Dow 9% yards of cloth at $ 4.621 per yard ; 'what was the whole cost ? Ans. $ 45.67 16. 22. Bought of John Appleton 47 gallons of molasses for $ 12.371 ; what cost one gallon ? what cost 127 gallons ?

Ans. $3.33911 23. When $ 15.874 are paid for 12; bushels of wheat, what cost one bushel ? what cost 11 bushels ?

Ans. $ 14.115. 24. When $ 19.185 are paid for 3 cords of wood, what cost one cord? what cost of a cord ?

Answer to the last, $? 1376 25. What are the contents of a box 87 feet long, 313 feet wide, and 2 feet high? Ans. 6812 feet.

Section 26.

DECIMAL FRACTIONS.

A DECIMAL FRACTION is that, whose integer is always divided into 10, 100, 1000, &c. equal parts. Its denominator is always an unit, with as many ciphers annexed, as there are places in the given decimal. There is, therefore, no need of having the denominator expressed ; for the value of the fraction is always known by placing a point before it, at the left hand, called the separatrix. Thus, .5 is, 37 is 37,.348 is 348

Ciphers annexed to the right hand of decimals do not increase their value ; for .4 or .40 or .400 are decimals having the same value, each being equal to t or ; but when ciphers are placed on the left hand of a decimal, they decrease the value in a tenfold proportion. Thus .4 is to, or four tenths ; but .04 is to, or four hundredths; and .004 is Tomo, or four thousandths. The figure next the separatrix is reckoned so many tenths ; the next at the right, so many hundredths; the third is so many thousandths; and so on, as may be seen by the following

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From this table it is evident, that in decimals, as well as in whole numbers, each figure takes its value by its distance from the place of units.

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