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

Ans. 83 yards. 18. A. Poor can walk 37 miles in 60 minutes; Benjamin can walk as fast as Poor. How long will it take Benjamin to walk the same distance ?

Ans. 73 minutes.

III. To divide a mixed number by an integer.

19. Divide 17§ by 6.

OPERATION.

Ans. 248.

We divide 17 by 6, and find it is con6) 17 tained 2 times, which we write under the 240 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.

27. Divide $173 among 7 men.

28. Divide $1067 among 8 boys.

29. What is the value of 5 of a dollar?

Ans. $22.

Ans. $135.

Ans. $0.3413.

30. Divide $1077 among 4 boys and 3 girls, and give the girls twice as much as the boys.

31. If 814 will purchase

Ans. boy's share $104. Girl's share $2129. of a ton of copperas, what quantity will $ I purchase? Ans. lcwt. Oqr. 24lbs.

IV. To divide one fraction by another.

32. Divide by .

OPERATION.

X == 13 Ans.

and; for

Ans. 1.

9,

To understand the rationale of this process, we find the two factors of , which are multiplied by are, as is evident from a preceding rule. We now divide by 1, which, by case I. of this section, will be X 32. Again, we wish to divide by . It is evident, that will contain nine times as often, as it will a unit, and it contains a unit times, therefore it contains nine times √ = f× √2 = 133 Ans. In performing this ques32 == tion, 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

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.

73 = 31, 33 = 2, 41 × 7=217=233 Ans.

35. Divide

by 1.

Ans. 3.

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 11

4 inches wide ? 3. How many square rods in a garden, rods in length and 97 rods wide?

4. What cost 19 acres of land, at $17 per acre?

5. What cost 147

6. What cost 13

Ans. 491

inches long, and square inches. which is 189

Ans. 1783 rods.

Ans. $350.

Ans. $1117.

Ans. $120.

tons of coal at $72 per ton?

tons of hay at $83 per ton?

7. What cost 13 bushels of corn at $17 per bushel ?

Ans. $333. Ans. $0.564. 9. What is the value of 17 of a dollar? Ans. $0.214.

8. What is the value of

10. What is the value of

11. What is the value of

of a dollar?

of a dollar? Ans. $0.25§. of a dollar? Ans. $0 51%. 12. Bought a cask of molasses, containing 871 gallons; of it having leaked out, the remainder was sold at 271 cents per gallon; what was the sum received?

Ans. $15.0332.

13. Bought of L. Johnson 73yds. of broadcloth, at $37 per yard, and sold it at $43 per yard; what was gained? Ans. $3.683.

14. Bought a piece of land, that was 47 rods in length, and 29 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. 13668 square rods. 15. Bought a tract of land that was 97 rods long and 481 rods wide; and from this I sold to John Ayer, a houselot, 18 rods 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. $16717.30. 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 72 feet long, 4 feet wide, and 3 feet high?

12

Ans. 1312 feet. 18. Polly Brown has $17.87; half of this sum was given to the missionary society, and g of the remainder she gave to the Bible society; what sum has she left? Ans. $3.57.

19. What number shall be taken from 123, and the remainder multiplied by 10% that the product shall be 50? Ans. 8.13. 20. What number must be multiplied by 73, that the product may be 20? Ans. 24. 21. Bought of John Dow 97 yards of cloth at $4.623 per yard; what was the whole cost? Ans. $45.67. 22. Bought of John Appleton 472 gallons of molasses for $12.37; what cost one gallon? what cost 127 gallons? Ans. $3.33414. 23. When $15.87 are paid for 123 bushels of wheat, what cost one bushel? what cost 11 bushels ?

Ans. $ 14.11. cords of wood,

24. When $19.182 are paid for 3 what cost one cord? what cost & of a cord? Answer to the last, $21336. 25. What are the contents of a box 8 feet wide, and 2 feet high?

feet long, 311 feet.

Ans. 68

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 4.

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 or ; but when ciphers are placed on the left hand of a decimal, they decrease the value in a tenfold proportion. Thus .4 is, or four tenths; but .04 is, or four hundredths; and .004 is Too 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

TABLE.

→ Millions.

Hundreds of Thousands.
Tens of thousands.

Thousands.
co Hundreds.

Thousandths.

Tens.
-Units.

co Hundredths.
Tenths.

Hundred Thousandths. → Millionths.

Ten Thousandths.

1.

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.