8. Divide 98637.42598 by 21.798.

9. Divide 7326.4873 by 86.324.

10. Divide 83465.987 by 4365.3315. Add 1 decimal to dividend.

11. What is the cost of 576 yards at .12?

12. What is the cost of 576 yards at .16?
576 × = $96.

13. What is the cost of 576 yards at .621?
= 576 × 5 ÷ 8 = $360.

576 ×

14. What is the cost of 576 yards at .871? 576 × = 576 × 7 ÷ 8 = $504.

CIRCULATING DECIMALS.

PROBLEMS.

1. In the reduction of common fractions to decimals, when the denominator has no other factor than 2 or 5, or 2 and 5, the decimal will terminate with the number of figures equal to the greatest number of factors of 2 or 5 in the denominator.

2. When the common fraction has any other denominator, the decimal fraction will not terminate; and at some point in the division, the quotient will begin to repeat the same figures; each period of which is called a repetend, and the repetends are called Circulating Decimals.

FORMATION OF REPETENDS.

.111, etc. = .i; } = .222 =

18.1212, etc.=.12.

.2;

REM.-When the repetend has the same figure repeated, a dot is placed over the single figure as above, .i and 2; when the repetend has two or more figures, a dot is placed over the first and last; as .i2.

=999) 237.000 (.237237, etc. = .237

1998

37 20

29 97

7230

6 993

237

Remainder the same as the first dividend; hence, the quotient will repeat.

COR. 1.-A repetend is changed to a common fraction by placing under it, for a denominator, as many nines as there are figures in the repetend.

COR. 2.-When the decimal fraction is partly a common decimal and partly a repetend, it is readily put in the form of a complex fraction, and then may be reduced accordingly; thus,

REM.—Circulating decimals are seldom met with in practice,

and the simplest manner to dispose of them is to reduce them to

common fractions, and then use them as such.

REVIEW.

NUMERATION, ADDITION AND SUBTRACTION,

1. Express in figures six hundred millions twenty-one thousand one hundred and four.

2. Express in figures the number, three billions seven millions and three.

3. Express in figures thirty-one billions three thousand and twenty-three.

4. Express in figures the number represented by four units of the ninth order, six of the eighth, one of the sixth, four of the third, and seven of the second.

5. On what does the local value of a figure depend? 6. Add the following numbers: one billion, thirty-two millions and twenty-five; twenty-one millions eightyfour thousand one hundred and fifteen; three hundred and sixty-five thousand four hundred and seventy-six. Ans. 1053449616.

7. From forty-three millions six hundred and thirtyone thousand four hundred and twenty-one, take sixteen millions two thousand and two. Ans. 27629419.

8. Add one hundred and five thousand three hundred and four; fifty-seven thousand one hundred and two; eighty-six thousand three hundred and ten; thirty-three hundred and nine; one hundred and three.

Ans. 252128.

9. Add two hundred and seven; twenty-one thousand three hundred and one; eight hundred and four

NUMERATION, ADDITION AND SUBTRACTION, 71

thousand and thirty-seven; sixty-one thousand three hundred and four. Ans. 886849. 10. A had five thousand dollars, B two thousand five hundred; A lost twelve hundred and seventy-five and B gained thirteen hundred and forty-three dollars. How much had each ? Ans. A had $3725; B $3843.

11. The minuend is fifty-three thousand two hundred and forty-three; and the difference is seven hundred and eighty-four. What is the subtrahend? Ans. 52459.

12. From 43722110523 take 63110421.

13. Name the three terms used in Subtraction. Which is the greatest?

14. What is the minuend? What is the difference between the minuend and the sum of the other two terms?

15. 3 million 225 thousand and 26 plus 12 hundred plus 1 thousand plus 325, minus 1364 = ?

Ans. 3226187. 16. The subtrahend is 572342, and the remainder is 3642, what is the minuend? Ans. 575984.

17. A has 5021 acres of land, B has 754 acres more than A, and C has 1012 acres more than both the others. How many have they altogether ? Ans. 22604 acres. 18. A commenced business with $5000; the first year he gained $2112, the second year $3743, and the third year he lost $5321; how much had he then ?

Ans. $5534.

19. A borrowed $7000; in one month he paid $2300, and in another month $1000 more; how much did he still owe? Ans. $3700,

20. Two men together have $11364, the one has $3420 ; how much has the other? Ans. $7944.

MULTIPLICATION AND DIVISION.

1. The two factors are 341 and 257; what is the product? Ans. 87637.

2. The divisor is 15, the quotient 31, and the remainder 5; what is the dividend? Ans. 470.

3. One factor of a number is 12 and the other factor is 2602; what is the number ?

Ans. 31224.

4. The remainder is 14, the quotient 407, and the dividend 61878; what is the divisor?

Ans. 152. 5. If a ship sail 185 miles in a day, how far will it sail in 51 days? Ans. 9435 miles. 6. If the earth moves in its orbit 68000 miles in an hour, how far will it move in 24 hours?

Ans. 1632000 miles.

7. James earned $15 in a week and paid $5 for his board; how much did he save in 52 weeks? Ans. $520.

8. A drover bought 45 horses at $84 each, 72 cows at $25 each, 315 sheep at $6 each; what did they all cost? Ans. $7470.

9. Two men start from the same place and travel in opposite directions; the one goes 5 miles an hour and the other 3; how far apart will they be in 12 hours? Ans. 96 miles. 10. If the men mentioned in the last example had traveled the same direction for 12 hours, how far apart would they be? Ans. 24 miles.

11. The divisor is 8, and the quotient 1142; what is the dividend? Ans. 9136. 12. If one sheep cost $5, how many sheep can be bought for $2315 ? Ans. 463 sheep.