rectly over the other, with a short line between them, showing that the upper number is to be divided by the lower. The upper number, or dividend, is, in fractions, called the numerator, and the lower number, or divisor, is called the denominator.

Note. A number like 171, composed of integers (17) and a fraction, (1) is called a mixed number.

In the preceding example, the one apple, which was left after carrying the division as far as could be by whole numbers, is called the remainder, and is evidently a part of the dividend yet undivided. In order to complete the division, this remainder, as we before remarked, must be divided into 5 equal parts; but the divisor itself expresses the number of parts. If, now, we examine the fraction, we shall see that it consists of the remainder (1) for its numerator, and the divisor (5) for its denominator.

Therefore, if there be a remainder, set it down at the right hand of the quotient for the numerator of a fraction, under which write the divisor for its denominator.

Proof of the last example.

171

5

In proving this example, we find it necessary to multiply our fraction by 5; but this is easily done, if we consider that the fraction expresses one part of an apple divided into five equal parts; hence, 5 times is = 1, that is, one whole apple, which we reserve to be added to the units, saying, 5 times 7 are 35, and one we reserved makes 36, &c.

86

30. Eight men drew a prize of 453 dollars in a lottery; how many dollars did each receive?

Dividend, Divisor, 8) 453

Quotient, 56.

Here, after carrying the division as far as possible by whole numbers, we have a remainder of 5 dollars, which, written as above directed, gives for the answer 56 dollars and (five eights) of another dollar to each man.

T18. Here we may notice, that the eighth part of 5 dollars is the same as 5 times the eighth part of 1 dollar, that is, the eighth part of 5 dollars is g of a dollar. Hence, expresses the quotient of 5 divided by 8.

is 5 parts, and 8 times 5 is 40, that is, 45, which reserved and added to the product of 8 times 6, makes 53, &c. Hence, to multiply a fraction, we may multiply the numerator, and divide the product by the denomina

tor.

Or, in proving division, we may multiply the whole number in the quotient only, and to the product add the remainder; and this, till the pupil shall be more particularly taught in fractions, will be more easy in practice. Thus, 56 X 8448, and 448 +5, the remainder, 453, as before.

31. There are 7 days in a week; how many weeks in 365 days? Ans. 524 weeks.

32. When flour is worth 6 dollars a barrel, how many barrels may be bought for 25 dollars? how many for 50 dollars? for 487 dollars? for 7631 dollars? 33. Divide 640 dollars among 4 men.

6404, or £49=160 dollars, Ans. Ans. 1.3.

34. 678÷6, or 678 = how many?

35. 5040 = how many? 36. 72,34 = how many? 37. 34.64 = how many?

=

38. 2764 how many?

39. 40801 = how many?

40. 2014012 how many?

12

Ans. 8848

T19. 41. Divide 4370 dollars equally among 21 men.

When, as in this example, the divisor exceeds 12, it is evident that the computation cannot be readily carried on in the mind, as in the foregoing examples. Wherefore, it is more convenient to write down the computation at length, in the following manner:

OPERATION.

Divisor. Dividend. Quotient. 21) 4370 (208

42

170

168

2 Remainder.

We may write the divisor and dividend as in short division, but, instead of writing the quotient under the dividend, it will be found more convenient to set it to the right hand.

Taking the dividend by parts, we seek how often we can have 21 in 43, (hundreds ;) finding it to be 2 times, we set down 2 on the right hand of the dividend for the highest figure in the quotient. The 43 being hundreds, it follows that the 2 must also be hundreds. This, however, we need not regard, for it is to be followed by tens and units, obtained from the tens and units of the dividend, and will therefore, at the end of the operation, be in the place of hundreds, as it should be.

It is plain that 2 (hundred) times 21 dollars ought now to be taken out of the dividend; therefore, we multiply the divisor (21) by the quotient figure 2 (hundred) now found, making 42, (hundred,) which, written under the 43 in the dividend, we subtract, and to the rema der, 1, (hundred,) bring down the 7, (tens,) making 17 tens. We then seek how often the divisor is contained in 17, (tens ;) finding that it will not go, we write a cipher in the quotient, and bring down the next figure, making the whole 170. We then seek how often 21 can be contained in 170, and, finding it to be 8 times, we write 8 in the quotient, and, multiplying the divisor by this number, we set the product, 168, under the 170; then, subtracting, we find the remainder to be 2, which, written as a fraction on the right hand of the quotient, as already explained, gives 2082 2 dollars, for

the answer.

This manner of performing the operation is called Long Division. 't consists in writing down the whole computation.

From the above example, we derive the following

RULE.

I. Place the divisor on the left of the dividend, separate them by a line, and draw another ne on the right of the dividend to separate it from the quotient.

II. Take as many figures, on the left of the dividend, as containthe divisor once or more; seek how many times they contain it, and place the answer on the right hand of the dividend for the first figure in the quotient.

III. Multiply the divisor by this quotient figure, and write the product under that part of the dividend taken.

IV. Subtract the product from the figures above, and to the remainder bring down the next figure in the dividend, and divide the number it makes up, as before. So continue to do, till all the figures in the dividend shall have been brought down and divided.

Note 1. Having brought down a figure to the remainder, if the number it makes up he less than the divisor, write a cipher in the quotient, and bring down the next figure.

Note 2. If the product of the divisor, by any quotient figure, he greater than the part of the dividend taken, it is an evidence that the quotient figure is too large, and must be diminished. If the remain der at any time be greater than the divisor, or equal to it, the quotient figure is too small, and must be increased.

EXAMPLES FOR PRACTICE.

1. How many hogsheads of molasses, at 27 dollars a hogshead, may be bought for 6318 dollars? Ans. 234 hogsheads. 2. If a man's income be 1248 dollars a year, how much is that per week, there being 52 weeks in a year? Ans. 24 dollars per week. 3. What will be the quotient of 153598, divided by 29?

4. How many times is 63 contained in 30131 ? Ans. 478 times; that is; 478 times, and

63

Ans. 52961.

of another time.

5. What will be the several quotients of 7652, divided by 16, 23, 34, 86, and 92 ?

6. If a farm, containing 256 acres, be worth 7168 dollars, what is that per acre?

7. What will be the quotient of 974932, divided by 365?

Ans. 267135. 8. Divide 3228242 dollars equally among 563 men; how many dollars must each man receive?

Ans. 5734 dollars.

9. If 57624 he divided into 216, 586, and 976 equal parts, what will be the magnitude of one of each of these equal parts?

Ans. The magnitude of one of the last of these equal parts will be 598406.

10. How many times does 1030603615 contain 3215? Ans. 320561 times. 11. The earth, in its annual revolution round the sun, is said to travel 596088000 miles; what is that per hour, there being 8766 hours in a year?

190= how many?

12. 1234567890

13. 40783020=

how many?

14. 987649031= how many?

9124

CONTRACTIONS IN DIVISION.

I. When the DIVISOR is a COMPOSITE NUMBER.

T20. 1. Bought 15 yards of cloth for 60 dollars; how much was that per yard?

15 yards are 3 X 5 yards. If there had been but 5 yards, the cost of one yard would be 60 12 dollars; but, as there are 3 times 5 yards, the cost of one yard will evidently be but one third part 4 dollars, Ans.

of 12 dollars; that is, 12

Hence, when the divisor is a composite number, we may, if we please, divide the dividend by one of the component parts, and the quotient, arising from that division, by the other; the last quotient will be the answer.

2. If a man can travel 24 miles a day, how many days will it take him to travel 264 miles?

It will evidently take him as many days as 264 contains 24.

II. To divide by 10, 100, 1000, &c.

T 21. 1. A prize of 2478 dollars is owned by 10 men; what is each man's share?

Each man's share will be equal to the number of tens contained in the whole sum, and, if one of the figures be cut off at the right hand, all the figures to the left may be considered so many tens; therefore, each man's share will be 247 8 dollars.

It is evident, also, that if 2 figures had been cut off from the right, all the remaining figures would have been so many hundreds; if 3 figures, so many thousands, &c. Hence, we derive this general RULE for dividing by 10, 100, 1000, &c.: Cut off from the right of the dividend so many figures as there are ciphers in the divisor; the figes to the left of the point will express the quotient, and those to the right, the remainder.

2. In one dollar are 100 cents; how many dollars in 42400 cents? Ans. 424 dollars.

424/00

Ans. 34567 dollars.

Here the divisor is 100; we therefore cut off two figures on the right hand, and all the figures to the left (424) express the dollars. 3. How many dollars in 34567 cents? 4. How many dollars in 4567840 cents? 5. How many dollars in 345600 cents? 6. How many dollars in 42604 cents? 7. 1000 mills make one dollar; how many in 25000 mills? in 845000? 8. How many dollars in 6487 mills?

9. How many dollars in 42863 mills?

in 96842378 mills?

Ans. 426100 dollars in 4000 mills?

Ans. 6487 dollars.

1000

in 368456 mills?

10. In one cent are 10 mills; how many cents in 40 mills?

in 400 mills?

4784 mills?

in 20 mills?

in 34640 mills?

in 468 mills ?

in

III. When there are ciphers on the right hand of the divisor. ¶ 22. 1. Divide 480 dollars among 40 men.

OPERATION.

410) 4810

12 dollars, Ans.

In this example, our divisor (40) is a composite number, (10 X 4 = 40;) we may, therefore, divide by one component part, (10,) and that quotient by the other, (4;) but to divide by 10, we have seen, but to cut off the right hand figure, leaving the figures to the left of the point for the quotient, which we divide by 4, and the work is done. It is evident that, if our divisor had been 400, we should have cut off 2 figures, and have divided in the same manner; if 4000, 3 figures, &c. Hence this general RULE: When there are ciphers at the right hand of the divisor, cut them off, and also as many places in the dividend; divide the remaining figures in the dividend by the remaining figures in the divisor; then annex the figures, cut off from the dividend, to the remainder.

4. How many yards of cloth can be bought for 346500 dollars, at

20 dollars per yard?

5. Divide 76428400 by 900000.

6. Divide 345006000 by 84000.

7. Divide 4680000 by 20, 200, 2000, 20000, 300, 4000, 50, 600, 70000 and 80.