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parts, and give one of these parts to each of the boys. Then each boy's share would be 17 apples, and one fifth part of another apple; which is written thus, 174 apples.
Ans. 17 apples each, The 17, expressing whole apples, are called integers, (that is, whole numbers.) The (one fifth) of an apple, expressing part of a broken or divided apple, is called a fraction, (that is, a broken number.)
Fractions, as we here see, are written with two numbers, one directly 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 17, composed of integers (17) and a fraction, (,) 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.
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 5 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.
Proof of the last example.
30. Eight men drew a prize of 453 dollars in a lottery; how many dollars did each receive?
Divisor, 8) 453
answer 56 dollars
¶ 18. Here we may notice, that the eighth part of 5 dol lars is the same as 5 times the eighth part of 1 dollar, that is, the eighth part of 5 dollars is § of a dollar. Hence, expresses the quotient of 5 divided by 8.
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 and § (five eighths) of another dollar,
is 5 parts, and 8 times 5 is 40, that is, 40 5, 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 denominator.
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 fractions, will be more easy in practice. Thus, 56 x 8= 448, and 448 +5, the remainder, 453, as before. 31. There are 7 days in a week; how many weeks in 865 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 dob Lars? for 487 dollars? for 7631 dollars? 33. Divide 640 dollars among 4 men.
640 ÷ 4, or 64o 160 dollars, Ans. 34. 6786, or 678 how many ? Ans. 113. 35. 1940 how many? 36. 1234 how many? 37. 3464-how many? 38. 2164 how many? 39. 40301-how many? 40. 2014012 how many?
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.
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 ia the dividend, we subtract, and to the remainder, 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 208 dollars, for the answer.
This manner of performing the operation is called Long Division. It consists in writing down the whole computation From the above example, we derive the following
I. Place the divisor on the left of the dividend, separate Chem by a line, and draw another line on the right of the dividend to separate it from the quotient.
II. Take as many figures, on the left of the dividend, as
contain the 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.be 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, be 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 remainder 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? Ans. 52961
4. How many times is 63 contained in 30131 ? Ans. 47813 times; that is, 478 times, and 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. 2671
8. Divide 3228242 dollars equally among 563 men; how many dollars must each man receive? Ans. 5734 dollars.
9. If 57624 be 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 59.
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? 12. 1234567890 — how many? 13. 40703020-how many? 14. 987649031 = how many?
CONTRACTIONS IN DIVISION.
I. When the DIVISOR is a COMPOSITE NUMBER.
↑ 20. 1. Bought 15 yards of cloth for 60 dollars; how mach was that per yard?
15 yards are 3 × 5 yards. If there had been but 5 yards, the cost of one yard would be 50: 12 dollars; but, as there are 3 times 5 yards, the cost of one yard will evidently be but one third part of 12 dollars; that is, 24 dollars. Ans.
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
¶ 21. 1. A prize of 2478 dollars is owned by 10 men, what is each man's share?