18. Vermont contains 247 townships, and is divided into 13 counties, what would be the average number of townships in each county? Ans. 19.

19. Vermont contains 5640000 acres of land, and in 1820 contained 235000 inhabitants, what was the average quantity of land to each person?

Ans. 24 acres.

20. The distance of the moon from the earth is 240000 miles, and the diameter, or

|

distance through the earth, is
8000 miles; how many diame-
ters of the earth will be equal
to the moon's distance from
the earth?
Ans. 30.

21. Divide 17354 by 86.
Quot. 201. Rem. 68.
22. Divide 1044 by 9.
Quot. 116.
23. Divide 34748748 by 24.
Quot. 1447864. Rem. 12.
24. 29702-6-4950 Ans.
25.279060=398651⁄2 Ans.

CONTRACTIONS OF DIVISION.

108. 1. Divide 867 dollars equally among 3 men, what will each

receive?

Divis. 3) 867 Divid.

Here we seek how many times 3 m 8, and finding

it 2 times and 2 over, we write 2 under 8 for the 289 Quot. first figure of the quotient, and suppose the 2, which remains, to be joined to the 6, making 26. Then 3 in 26, 8 times, and 2 over. We write 8 for the next figure of the quotient, and place 2 before the 7, making 27, in which we find 3, 9 times. We therefore place 9 in the unit's place of the quotient, and the work is done. Division performed in this manner, without writing down the whole operation, is called Short Division.

I. When the divisor is a single figure;

RULE.-Perform the operation in the mind, according to the general rule, writing down only the quotient figures.

2. Divide 78904, by 4.

Quot. 19726.

3. Divide 234567 by 9.

Quot. 26063.

-109. 4. Divide 237 dollars into 42 equal shares; how many dollars will there be in each?

42=6X7 7)237-6 rem. 1st.

6)33-3 rem. 2d.

If there were to be kut 7 shares, we should divide by 7, and find the shares to be $33 each, with a remainder of 6 dollars; but as there are to be 6 times 7 shares, each share will be only one sixth In of the above, or a little more than 5 dollars. the example there are two remainders; the first, 6, is evidently 6 units of the given dividend, or 6 7X3+6=27 rem. dollars; but the second, 3, is evidently units of Ans. 527 dolls. the second dividend, which are 7 times as great as those of the first, or equal to 21 units of the first, and 21+6=dollars, the true remainder.

5

II. When the divisor is a composite number. (90)*

RULE.-Divide first by one of the component parts, and that quotient by another, and so on, if there be more than two; the last quotient will be the answer.

5. Divide 31046835 by 56-7 | 6. Divide 84874 by 48=6X8.
X8. Quot. 554407, Rem. 43.
Quot: 176848.

110. 7. Divide 45 apples equally among 10 children, how many will each child receive?

As it ill take 10 apples to give each child 1, each child will evidently receivers many apples as there are 10's in the whole number; but all the figures of any number, taken together, may be regarded as tens, excepting that which is in the unit's place. The 4 then is the quotient, and the 5 is the remainder; that is, 45 apples will give 10 children 4 apples and 5 tenths, or each. And as all the figures of a number, higher than in the ten's place, may be considered hundreds, we may in like manner divide by 100, by cutting off two figures from the right of the dividend; and, generally,

III. To divide by 10, 100, 1000, or 1 with any number of ciphers annexed:

RULE.-Cut off as many figures from the right hand of the dividend as there are ciphers in the divisor; those on the left will be the quotient, and those on the right, the remainder. 8. Divide 46832101 by | mong 100 men, how much 10000. Quot. 4683. will each receive?

9. Divide 1500 dollars a

111. 10. Divide 36556 into 3200 equal parts.

32

45

32

*

Ans. 15 dolls.

Here 3200 is a composite number, whose com32/00)365|56(11 Quot. ponent parts are 100 and 32; we therefore divide by 100, by cutting off the two right hand figures. We then divide the quotient, 365, by 32, and frd the quotient to be 11, and remainder 13; but this remainder is 13 hundred,[109], and is restored to its proper place by bringing down the two figures which remained after dividing by 100, making the whole remainder, 1356. Hence, IV. To divide by any number whose right hand figures are ciphers:

1356 Rem.

RULE.-Cut off the ciphers from the divisor, and as many figures from the right of the dividend; divide the remaining figures of the dividend by the remaining figures of the divisor, and bring down the figures cut off from the dividend to the right of the remainder.

11. Divide 738064 by 2300. | 12. Divide 6095146 by 5600. Quot. 320, Rem. 2064.

Quot. 1088848.

MISCELLANEOUS QUESTIONS.

L. If the minuend be 793, and the subtrahend be 598, what is the remainder?

Ans. 195.

2. If the minuend be 111, and the remainder 63, what is the subtrahend? Áns. 48.

3. If the subtrahend be 645, and the remainder 131, what is the minuend? Ans. 776.

4. The sum of two numbers is 8392, and one of them is 4785, what is the other?

Ans. 3607.

5. The least of two numbers is 77, and their difference is 99, what is the greater? Ans. 176.

6. A certain dividend is 2340, and the quotient is 156, what is the divisor? Ans. 15. 7. If the divisor be 32, and the quotient 204, what is the dividend? Ans. 6528.

8. A certain product is 484848, and the multiplicand is 1036, what is the multiplier? Ans. 468.

9. If a person spend 8 cts. a day, how much will he spend in a year, or 365 days?

Ans. 2920 cts.-$29.20. 10. How many square feet in a piece of ground 17 feet long, 13 ft. wide?(36, 61)

Ans. 221 feet. 11. If a floor containing 242 feet be 22 feet long, how wide is it? Ans. 11 feet.

12. How many rods in a piece of land 40 rods long and 16 broad?

Ans. 640 rods, or 4 acres.

13. The sum of two numbers is 75, and their difference is 15, what are the numbers? the less. 30+15-45, greater. Ans. 75-15-60, 60--30,

14. The difference of two numbers is 723, and their sum is 1111, what are the numbers?

194

917

}

Ans.

.15. If a man travel 35 miles

a day, how far will he travel in 6 weeks and 3 days, allowing 6 days to a week?

Ans. 1365 miles.

must be divided among 18 16. What sum of money men so as to give each man [$112? Ans. $2016.

17. A man raised 64562 bushels of corn on 1565 acres, how many bushels was that to the acre? Ans. 41.

18. If I plant in 14 rows 2072 fruit trees, and set the trees 25 feet asunder, how many feet long are the rows?

Ans. 3675 feet.

19. Subtract 30079 out of ninety-three millions as often as it can be done, and say how much the last remainder exceeds or falls short of 21180?

Ans. 4631 exceeds.

REVIEW.

14. If the subtrahend and remain-kind?[83] der were given, how would you find the minuend?

15. If the sum of two numbers, and one of them were given, how would you find the other?

16. If the greater of two numbers and their difference be given, how would you find the less?

17. If the less of two numbers and their difference be given, how would you find the greater?

18. How would you find how many times one number is contained in another?

DECIMALS and fedERAL MONEY.

DECIMALS.

113. The method of forming numbers, and of expressing them by figures, has been fully explained in the articles on Numeration. (71, 72, 73) But it frequently happens that we have occasion to express quantities, which are less than the one fixed upon for unity. Should we make the foot, for instance, our unit measure, we should often have occasion to express distances which are parts of a foot. This has ordinarily been done by dividing the foot into 12 equal parts, called inches, and each of these again into 3 equal parts, called barley corns. (38) But divisions of this nature, which are not conformable to the general law of Notation, (73) necessarily embarrass calculations, and also encumber books and the memories of pupils, with a great number of irregular and perplexing tables. Now, if the foot, instead of being divided into 12 parts, be divided into 10 parts, or tenths of a foot, and each of these again into 10 parts, which would be tenths of tenths or hundredths of a foot, and so on to any extent found necessary, making the parts 10 times smaller at each division; then in recomposing the larger divisions from the smaller, 10 of the smaller would be required to make one of the next larger, and so on, precisely as in whole numbers. Hence, figures expressing tenths, hundredths, thousandths, &c. may be written towards the right from the place of units, in the same manner that tens, hundreds, thousands, &c. are ranged towards the left; and as the law of increase towards the left, and of decrease towards the right, is the same, those figures which express parts of a unit may obviously be managed precisely in the same manner as those which denote integers, or whole numbers. But to prevent confusion, it is customary to separate the figures expressing parts from the integers by a point, called a separatrix. The points used for this purpose are the period and the comma, the former of which is adopted in this work; thus to express 12 feet and 3 tenths of a foot, we write 12.3 ft. for 8 feet and 46 hundredths, 8.46 feet.

DEFINITIONS.

114. Numbers which diminish in value, from the place of units towards the right hand, in a ten fold proportion, (as