DIVISION OF A COMPOUND NUMBER BY A SIMPLE NUMBER. 124. If the dividend alone be compound, and if, at the same time, the 'dividend and divisor consist of units of different kinds, we first divide the principal units of the dividend according to the ordinary rule: what remains after this division we reduce to units of the second order (57) these we add to those of the same kind which may be found in the dividend, and we divide the whole as usual: in like manner, we reduce the remainder of this division to units of the third kind, to which we add those of the same kind which may be found in the dividend, and we divide the whole as before: we continue to reduce the remainders to units of the next kind, as long as any inferiour kind is found in the dividend. EXAMPLE. We have given 47831. 3s. 9d. for the payment of 87 sq. yds. of work; we demand how much this comes to per square yard. We divide 47831. 3s. 9d. by 87, in commencing with the pounds. The 47831. divided by 87, according to the ordinary rule, gives 547. for the quotient, and 85 pounds for the remainder; these 85 pounds, reduced to shillings (57,) together with the 3 shillings of the dividend, give 1703 shillings, which, divided by 87, give 19 shillings for the quotient, and 50 shillings for the remainder; these 50 shillings, reduced to pence, give, with 9 pence of the dividend, 609 pence; which, divided by 87, give, finally, 7 pence for the quotient. We have, therefore, 54l. 19s. 7d. for the answer. 125. But if the dividend and the divisor consist of units of the same kind, we must, before we make the division, examine whether the quotient should or should not be of the same kind, which the nature of the question always decides. 126. In the case where the dividend and the divisor being of the same kind, the quotient ought also to be of the same kind, the division is performed precisely as in the preceding case: for example, if we proposed this question-1243 pounds have produced a benefice of 7254 pounds, how much does this come to per pound? It is evident that the quotient should consist of units of the same kind with those of the dividend and the divisor; that is to say, should be pounds; and that we should divide 7254 pounds by 1243; reducing the remainder of this division to shillings, as in the preceding example, and the second remainder to pence, we have 57. 16s. 87d. for the answer to the question. 127. But when the dividend and the divisor being of the same kind, the quotient should be of a different kind, we commence by reducing (57) the dividend and divisor each to the least kind of units mentioned in the dividend; after which we perform the division as in the preceding case, and we consider the units of the dividend as if they were of the same kind as those of which the quotient should consist: for example, if this question be proposed, how much work should be done for 7954/. 11s. 7d. at the rate of 721. per square yard? It is evident from the nature of the question, that the quotient should be square yards and parts of a square yard; therefore, according to rule, we reduce 75947. 11s. 7d. to pence, which gives 1909099d.; we also reduce 721. to pence, which gives 17280d.; we then divide 1909099, considered as square yards, by 17280, and we have for the quotient 110 sq. yds. 40 sq. ft. for the answer. 619 1920 DIVISION OF A COMPOUND NUMBER BY A COMPOUND NUMBER. 128. When the divisor is also a compound number, we must reduce it to its lowest denomination (57), and multiply the dividend by the number which shows how many there must be of the smallest units in the divisor to compose the principal unit of that same divisor; the division will then be reduced to the preceding case, where the divisor was simple. EXAMPLE. If 53 cubic yards and 16 cubic feet of excavation cost 8541. 17s. 4d. what is the cost of one cubic yard? We must divide 8547. 17s. 4d. by 53 c. yd. 16 c. ft.; for this purpose, we reduce the 53 c. yd. 16 c. ft. to feet, which gives 1447 for the new divisor, and as there must be 27 feet to make a yard, which is the principal unit of the divisor, we multiply the proposed dividend 854l. 17s. 4d. by 27 (121), which gives 230817. 8s. for the new dividend, so that we divide as follows: Dividing 230817. by 1447, we have 157. for the quotient, and 13767. for the remainder. Then 13761. reduced to shillings, together with the 8s. of the dividend, gives 27528s. which, divided by 1447, gives 19s. for the quotient and 35 shillings for the remainder. Reducing 35 shillings to pence, we have 420 pence; but as these do not contain the divisor 1447, we place O in the quotient, and signify the quotient of 420 divided by 1447 by placing these numbers in the form of a fraction as usual: so that the cost of one cubic yard is 157. 19s. 0-420d. 47 To understand the reason of this rule, we must observe that the 53 c. yd. 16 c. ft. are equal to 1447 cubic feet, and that the foot being the of a yard, the divisor is 1447 cubic yards: now to divide by a fraction, we must (109) invert the divisor fraction, and multiply by this fraction thus inverted; 27 we must then multiply by, which requires that we should multiply by 27, and divide by 1447, as is also prescribed by the rule which we have given. As the division of a compound number by a compound number is reduced, as we have seen, to the division of a compound number by a simple number, we should have the same regard to the nature of the units as in the articles (126) and (127.) EXAMPLES FOR PRACTICE. 1. If 216 bushels of potatoes cost 271. what is that a bushel ? Ans. 2s. 6d. 2. If 103 bushels of apples cost 177. 3s. 4d. what is that a bushel? Ans. 3s. 4d. 3. Bought 177 yards of cloth for 597. what is that a yard? Ans. 6s. 8d. 4. Bought 20 T. 15 cut. of hay for $342,371, what is the cost of a ton? Ans. $16,50. 5. Sold 6374 yards of duck for 11221. 1s. 91d. what was the price of a yard? Ans. 3s. 6d. 6. If a parcel of land containing 156 A. 3 R. worth $4116,984, what is the worth of one acre ? 14 P. be Ans. $26,925 nearly. OF THE FORMATION OF SQUARE NUMBERS, AND THE 129. We call the square of a number the product which results from the multiplication of that number into itself; thus 25 is the square of 5, because 25 is the result of the multiplication of 5 by 5. 130. The square root of a proposed number is the number which, multiplied by itself, would reproduce that same proposed number: thus 5 is the square root of 25; 7 is the square root of 49. ✔ is the radical sign, and when placed before a number signifies that the square root of it is to be extracted. Thus ✔497 is read the square root of forty-nine equal to seven. 131. A number which we square is then at the same time both multiplicand and multiplier; it is then twice factor (42) of the product; for this reason we also call this product or square the second power of that number. Note. The number 2 is placed over a number to indicate its square or second power: thus 52 is the square or second power of 5. Hence 2 is called the index. Το square a number, no other art is required than to multiply it by itself, according to the ordinary rules of multiplication: but the extraction of the square root requires a method, at least, when the number or proposed square consists of more than two figures. When the proposed number has only one or two figures, its root, in integers, is some one of the numbers the 1, 2, 3, 4, 5, 6, 7, 8, 9, squares of which are 1, 4, 9, 16, 25, 36, 49, 64, 81. The square root of 72, for example, is 8 for the whole number, because 72 being between 64 and 81, its root is between the roots of these numbers, that is to say, between 8 and 9, it is therefore 8 and a fraction, which fraction indeed we cannot exactly assign, but to which we can approach continually, as we shall see shortly. 132. The square root of a number which is not a perfect square, is called a surd, irrational, or incommensurable number, 133. Let us proceed to numbers which have more than two figures, It is by observing what takes place in the formation of a square that we shall find the method which we should follow to obtain its root. To square a number such as 54, for example: 54 54 216 270 2916 Having written the multiplicand and multiplier, as we see |