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DRY MEASURE.

$1. In 30 chaldrons of coal, how

many

bushels?

32. In 24 bushels, how many pints?
33. At 3 cents a quart, what will 5 bus.

of salt come to ?

Ans. 960. Ans. 1536.

2 pks. 3 qts. Ans. $5,37.

SOLID MEASURE.

1

34. In 12 tons of round timber, how many cubic inches ? Ans. 829440. 35. In 12 tons of hewn timber, how many cubic inches ? Ans. 1036800. 36. How many solid inches in a cord of wood?

Ans. 221184.

37. What will 587 firkins of butter come to at 7 dollars per firkin P Ans. 4109 dollars. 38. A man had a farm on which he raised 360 bushels of wheat; and another on which he raised 6 times as much; what quantity did he raise on both of them ? Ans. 2520 bushels.

39. A man had an estate which he divided among 9 sons as follows, viz. to the first eight, he gave 333 dollars each; to the ninth he gave as much wanting 1000 dollars as to the other eight; what was the value of the estate, and also of the ninth son's share?

Ans.

S4328 dollars value of the estate, 21664 dollars ninth son's share. 40. A farmer sheared 364 sheep, six years successively; each sheep yielded 3 pounds of wool per year. How much wool had he yearly; and how much in six years? S1092 pounds yearly, 26552 pounds in 6 years. pounds at 142 dollars per ton; what

Ans.

41. Sold 342 tons of iron was the price of the whole ?

42. In the West Indies 742 sold at 62 dollars per thousand;

43. What will 6422 quintals lars per quintal

Ans. 48564 dollars. thousand of boards were what did they come to?

Ans. 46004 dollars.. of fish come to at 6 dol-" Ans. 38532 dollars.

44. A merchant at the expiration of the ten years he had been in trade, found himself worth 13000 pounds. His books showed that for the last three years he had cleared 875 pounds a year; the three preceding, 586 pounds a year; and for the time previous to that, 64 pounds a year. With what sum did he commence busiAns. £7167.

ness ?

45. The bridge, which Trajan constructed over the Danube, is said to have had its arches supported by twenty piers. The piers were 60 feet thick, and placed 170 feet distant from each other. How wide was the Danube at that place? and how much did the length of the bridge exceed that of Westminster bridge, which is 1200 feet long? Ans. 4770 feet wide, and 3570 feet longer than Westminster bridge.

46. If a man can travel 7 miles in an hour, how far would he travel in 8 days, when the days are 9 hours long? Ans. 504 miles. 47. If 10 men can do a piece of work in 7 days, how many days will it take one man to do it? Ans. 70 days.

NOTE. Multiplication by 10, 100, 1000, &c. is performed by annexing 1, 2, 3, &c. ciphers to the multiplicand. See (44) and (52).

48. If 100 men receive 8 dollars apiece, how many dollars do they all receive? Ans. 800 dollars. 49. What would an ox weighing 873 pounds, come to t 10 cents a pound ? Ans. $87,30 cts.

SIMPLE DIVISION.

59. The division of one number by another consists in finding how often the first of two numbers contains the

second.

The number to be divided is called the dividend; the number by which division is performed, is called the divisor; and that which expresses the number of times the dividend contains the divisor, is called the quotient.

It is not always the object of division to know how often one of two proposed numbers contains the other, but

E

the operation in all cases, is performed as if that were the end in view. Hence it follows, that if the divisor be multiplied by the quotient, their product will be the dividend; because the divisor is taken as many times as it is contained in the dividend. And this is a general principle, whether the quotient be a whole number or a fraction.

The kind of units contained in the quotient is not determined by those of the dividend or the divisor; because the dividend and the divisor remaining the same, the quotient will always be numerically the same, although the nature of its units may be very different, according to the nature of the question which occasions the division. For example, if it is required to know how often £8 contains £4, the quotient will be an abstract number which expresses 2 times. But if it is demanded how many rods of wall may be built for £8 at the price of £4 for a rod, the quotient will be 2 rods, which is a concrete number that has no relation either to the units in the dividend or the divisor.

Hence it is evident, that the question, which gives rise to the division, can alone determine the nature of the units in the quotient.

DIVISION OF A NUMBER COMPOSED OF SEVERAL FIGURES, BY A NUMBER EXPRESSED BY ONE FIGURE.

60. The operation about to be described, supposes the learner to be already acquainted with the method of finding how often a number, expressed by one or two figures, contains an other expressed by one figure. This knowledge is acquired when the products of numbers expressed by one figure are known by memory. The Table given in article (48) is used for this purpose. For example, if it is required to know how many times 74 contains 9, the divisor 9 is sought in the upper line, and directly under it, the dividend or number nearest to it, which in this case is 72; then the number 8 standing opposite in the first column is the number of times, or the quotient sought.

We will now explain the method of dividing a number composed of several figures.

Place the divisor at the right of the dividend and draw a line between them; draw a line also under the divisor, below which write the figures of the quotient when they are found.

Take the first figure upon the left of the dividend, or the two first figures, if the first does not contain the divisor. Seek how often the first, or the two first figures, contain the divisor, and write the number of times underneath. Multiply the divisor by this quotient figure, and carry the product under the part of the dividend which has been employed. Subtract this product from the part of the divid end, to which it corresponds; and to the remainder, bring down the next figure of the principal dividend. Perform the same operation upon this second partial dividend, which was performed upon the first; place the quotient, to the right of the quotient figure already found, multiply the divisor by the quotient, write and subtract the product as before. In like manner bring down to the side of the remainder of this division the figure in the dividend following the one already brought down, and continue thus to the last inclusively.

This rule is explained by the following example.

EXAMPLE.

Divide 8769 by 7. The numbers are thus written;

[blocks in formation]

The operation should be commenced at the left of the dividend by saying, in 8 thousand how many times 7 ¿

but we simply say, in 8 how many times 7? It is comtained once. This 1 is naturally one thousand, but the figures which are to come after, will give to it its true value; 1 is therefore written under the divisor. We then multiply 7 the divisor by 1 the quotient, and carry the product 7, under 8 the part of the dividend to be divided; after subtraction, 1 is found for the remainder.

The remainder 1 is the part of 8 which has not been divided, and it has the value of 10 when compared with 7 the figure that follows it; the figure 7 is therefore brought down by the side of 1, and the operation continued by saying, in 17 how many times 7? It is contained twice. The quotient figure 2 is then written to the right of 1, the first quotient figure. As in the preceding operation, the divisor 7 is then multiplied by 2, the last quotient figure; the product 14 is then carried under 17 the partial dividend, and after subtraction from it, there remains 3 for the part which cannot be divided. By the side of the remainder 3, 6 the next figure in the dividend is brought down, and the operation continued by saying, in 36 how many times 7? It is contained 5 times; and 5 is written in the quotient. The divisor 7 is then multiplied by 5, and 35 the product is subtracted from 36 the new partial dividend, and 1 remains. By the side of 1, 9 the last figure in the dividend is brought down, and we say in 19 how many times 7? It is contained twice; and 2 is written in the quotient. The divisor is then multiplied by this last quotient figure, and the product 14 subtracted from 19 the partial dividend, and 5 remains. Hence it appears that 8769 contains 7, the number of times denoted by the quotient, or 1252 times with 5 remainder.

It will suffice for the present to observe, that the remainder is written in the quotient over a line, under which the divisor is written, and pronounced five sevenths. An explanation will be given hereafter of this kind of numbers.

61. If in course of the operation any partial dividend does not contain the divisor, a cipher is written in the quo tient; and omitting multiplication, another figure is brought down by the side of the partial dividend, and the division continued.

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