that the day is composed of 24 hours, we mean that the unit of time is the duration of one hour, and that 24 of these hours are equal in duration to one day. Numbers of this kind, composed of a particular unit, which is repeated as many times as are indicated by an abstract number, are termed concrete numbers, and are consequently products of which the multiplicand is the unit, and the multiplier an abstract number. Hence 24 hours means 24 times one hour, and 36 miles signifies 36 times one mile. Now suppose a distance is to be measured, we may take a mile as the unit, and the distance may be represented as nearly as we please, and sufficiently accurate for all practical purposes, either by a certain number of miles or a certain number of parts of a mile, and may therefore be expressed either by a whole number or a fraction. If one distance be represented by the one hundred and fiftyfourth part of a mile, and another by the one hundred and forty-fourth part of a mile, we have but an imperfect notion as to how much the second distance is longer than the first. It is necessary to have some smaller measure, and if a mile be divided into 1760 equal parts, and each of these parts be called a yard; then the first distance will be 1760154 or 11 yards and three-sevenths of a yard, and the second distance will be 1760 144 or 12 yards and two-ninths of a yard. We have now a better notion of these different distances, and if the yard be supposed to be divided into 3 equal parts, and each of these parts be called a foot, a still clearer notion of the two distances would be obtained. Hence large measures are convenient for measuring large quantities, while smaller measures are necessary and more convenient for measuring smaller quantities. 33. A compound quantity is one consisting of several others, expressed in different units, as 17 miles, 57 yards, 2 feet, or £3. 17s. 6d. ; and the different denominations into which money, time, weight, length, or distance, etc., are divided, constitute so many scales or systems of numeration, by means of which operations on concrete and compound numbers are assimilated to those on abstract and simple numbers. The usual tables of the different divisions of money, weights, and measures will be found in the following article, and the determination of the standard weights and measures will be found in a subsequent part of the Arithmetic. 34. TABLES OF MONEY, WEIGHTS, AND MEASURES. £. S. d. 421 15 71 20 8435 shillings. 12 35. Reduction is to change a concrete number, consisting of one or more denominations, into another, without altering its value. *Ex. 1. Let it be required to reduce £421. 15s. 74d. to farthings. Since there are 20 shillings in 1 pound, it is obvious that in 421 pounds there will be 421 times 20 or, which comes to the same thing, 20 × 421, or 8420 shillings. To this product we must add 15 shillings, which may be done mentally while the multiplication by 20 is being made, and the entire number of shillings in £421. 15s. is 8435. Again, since there are 12 pence in 1 shilling, there will be in 8435 shillings 8435 times 12 pence, that is, 12 x 8435, or 101220 pence. Adding to this number 7 pence, we have £421. 15s. 7d., equal to 101227 pence. Lastly, since 4 farthings are equal to 1 penny, 101227 pence will be equal to 101227 times 4 farthings, or 101227 × 4=404908 farthings; hence, in £421. 15s. 74d. there are 404908 +2, or 404910 farthings. 101227 pence. 4 404910 farthings. Since there are 960 farthings in 1 pound, 48 in a shilling, and 4 in a penny, the previous example may be performed in the following 4 X 7 2 x 1 = = 404160 farthings in 421 pounds 720 farthings in 15 shillings 7 pence 2 farthings in 1 halfpenny. hence there are 404910 farthings in £421. 15s. 74d. Ex. 2. How many pounds, shillings, pence, and farthings are in 337587 farthings? VOL. I. C 4337587 farthings. 12 84396 pence. 2,0 703,3s. 04d. £351. 13s. 0fd. Here the operation must be the reverse of the former, and dividing the number of farthings by 4, the quotient is 84396 pence, and the remainder is 3 farthings. Dividing the number of pence by 12 gives 7033 shillings, and no remainder, and dividing the number of shillings. by 20, gives 351 pounds, and a remainder of 13 shillings; hence, 337587 farthings are equal to 84396 pence and 3 farthings, or equal to 7033 shillings and 3 farthings, or equal to £351. 13s. Old. £. S. d. 227 12 1 20 4552 shillings. Ex. 3. How many half-crowns are equivalent to £227. 12s. Id.? The given sum is first reduced to pence by multiplying by 20 and 12, adding the shillings and pence in succession; then since there are 30 pence in 1 half-crown, the number of pence is divided by 30, which gives 1820 half-crowns and 25 pence remaining, or 1820 hf.-cr. 2s. id. These three examples are sufficient to illustrate the principles of Reduction, as, whatever be the denominations, the operations are performed in a similar manner. 3,0)5462,5 pence. 1820 half-crowns 25 pence. COMPOUND ADDITION. 36. Compound Addition is the collecting into one sum two or more numbers expressed in different denominations; and the process is precisely similar to that for the addition of simple numbers, with this difference, that the numbers connecting the different denominations must be employed instead of ten. Ex. Find the sum of £73. 2s. 94d., £ 25. 8s. 4 d., £68. 3s. 114d., £28. 11s. 7 d., and £17. 14s. 114d. Here the numbers are arranged so that those of the same denomination are in the same vertical column; then beginning at the lowest denomination, viz., farthings, the sum is 12 farthings, which are equivalent to 3 pence. Then 3 pence are carried to the next column, and added thereto, making the entire number of pence 45. But 45 pence are 3 shillings and 9 pence, and writing 9 under the column of pence, the 3 shillings are added with the numbers in the column of shillings, making 41 shillings, which are equivalent to 2 pounds I shilling; writing I under the column of shillings, and carrying the 2 pounds to the left column, the entire number of pounds is found to be 213; consequently, the sum of the whole is £213. 1s. 9d. The sum may also be obtained in the ordinary manner by reducing each of the numbers to the denomination of farthings, as has been done above, and then reducing the sum, viz., 204564 farthings, to pounds, shillings, and pence, in the usual manner. In practice, it is more useful to adopt a separate scale of notation for each case, as in the first method, without changing the denominations into the lowest. COMPOUND SUBTRACTION. 37. Compound Subtraction is the taking a less number from a greater, when both numbers are composed of different denominations. Ex. Find the difference between 35 yards 2 feet 8 inches and 52 yards 1 foot 4 inches. Since the difference of two quantities is not altered by adding the same quantity to both, we must first add 12 inches to the upper line, and 12 inches or 1 foot to the lower line, making 16 inches in the one and 3 feet in the other. Again, as 3 feet cannot be subtracted from 1 foot, we must add 3 feet to the upper line, and 3 feet or 1 yard to the lower; making 4 feet in the former, and 36 yards in the latter. The subtraction can now be effected, the difference being 16 yds. 1 ft. 8 in. Reducing both numbers to inches, the difference (596 inches) is obtained in the ordinary manner, and then reduced to yards, feet, and inches, as in the example above. COMPOUND MULTIPLICATION. 38. Compound Multiplication is the finding the amount of a number consisting of different denominations, repeated any number of times. If a quantity consists of several parts, and each of these parts be multiplied by a number, and the products be added, the result is the same as would arise from multiplying the quantity by that number. Ex. 1. Multiply £24. 17s. 8d. by 23. £. S. d. The sum of all these is £ 572 7 91 This product may be obtained in another manner; for since 23 = 4 × 61, we may multiply £ 24. 17s. 8d. first by 4, then the product by 6, and from this last product subtract £24. 178. 8d. Or, since 23 2 x 11+ 1, we may first multiply by 2, then the product by 11, and add to this last product £24. 17s. 8 d. = 39. When the multiplier is a large number, we may reduce the multiplicand to the lowest demonination included, and proceed in the ordinary way. Or we may multiply by the separate units, tens, hundreds, etc., of the multiplier. Ex. 2. Multiply £3. 15s. 6d. by 327. 40. Compound Division is the dividing a number consisting of several denominations into as many equal parts as there are units in the divisor; or it is the finding how many times one compound number is contained in another consisting of like denominations. Ex. 1. Divide £172. 11s. 51d. by 5. Dividing 172 pounds by 5, gives a quotient of 34 pounds, and 2 pounds remain to be divided by 5. In £. S. d. 5/172 11 5吋 34 10 3 2 pounds there are 40 shillings, and 40+ 11, or 51 shillings, divided by 5, gives 10 shillings, and 1 shilling remains. But in 1 shilling there are 12 pence, and 12+ 5, or 17 pence, divided by 5, gives 3 pence, and 2 pence remain; then 2 pence are 8 farthings, and 8+2, or 10 farthings, divided by 5, gives 2 farthings, and £34. 10s. 34d. is the quotient required. Ex. 2. Divide £629. 16s. 51d. by 91. This may be effected either by reducing the dividend to the lowest denomination included, and then dividing in the ordinary way, or by dividing as in Division. 91 |