=,142857 ž=,428571 consequently, in dividing by 7, any number to which it is prime, the decimal period, which (260) begins at the comma, will (251 ) always consist of the same 6 figures; and these will succeed each other in the same order, although the period will not always commence with the same figure. Thus we find that 특 =,714285 ž= ,285714 =,857142 =1,142857 4=,571428 =1,285714 Hence, the student will perceive that the twelfth example (317) may be performed as follows: Because ,285714 t. = of a ton, or (158) the seventh part of 2 tons, we divide 2 tons by 7, thus : T. cwt. qrs. lbs. 7) 2 0 0 0 0 5 2 24 Wherefore ,285714 t. = 5 cwt. 2 qrs. 24 lbs. By operating decimally, thus : ,285714 20 4 28 23,999999 we find the same result. For, as ,999999 is the same as its denominator, it is equal to a unit; therefore 23,999999 = 24. SECTION XVI. ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION OF COMPOUND NUMBERS. Addition of Compound Numbers. 319. We write the given numbers under each other, so that units of the same kind may be in the same column, and commence by adding the units of the lowest denomination. If their sum does not contain a sufficient number of units to compose a unit of the next higher denomination, we write it under its kind : but if it contains as many units as will make one or more, we reduce it (313) to that denomination, writing the remainder, if any, under the units of its kind, and adding the figures of the quotient to those of the same order in the next higher denomination, with which we proceed in like manner. 481 17 8 As the sum of the pence is 32, which (313) ĝives 2s. 8d., we write 8 under pence, and add 2 to the unit column of shillings : then as the sum of this is 27, we write 7, and carry 2 to the column of tens. The sum of this is 5, and as each unit in this 5 is 10 s., two of these will make £1. We therefore take the half of 5 for pounds, which gives 2 and 1 over. This 1, which is 10 s., we place on the left of 7, and carry the £2 to the column of pounds. Example 2. A. R. P. 8. yds. 3 3 36 26 8 36 2 27 19 7 46 3 13 25 5 16 3 35 21 6 25 1 12 12 4 129 3 6 15 51 8. f. Here, the sum of the s. f. is 30, which, reduced (313) to 8. yds., gives 3 s. yds. 3 s. f. We therefore write 3 s. f. under the column of feet, and carry 3 s. yds. to the column of yards. The sum of the yards is 106; and, as 304 make a pole, we divide 106 by 301 = 131; that is, we multiply 106 by Tipe which gives 124, or 36.1 P. Reducing this fraction of a pole to s. yds., we have 11 X 121=1=154 s. yds. We have, then, for the sum of the yards, 3 P. 154 s. yds. We, therefore, write 15 under s. yds. ; but, as it would be awkward to leave a fraction in connection with the s. yds. while we have a lower denomination in the question, we reduce 4 s. yd. to feet, in multiplying by 9, which gives =24 s. f.; and adding this to the 3 s. f. first written, we have 54 s. f., which we leave under that column. Then, carrying 3 to the column of poles, we find the sum 126, and, reducing this to roods, we have 3 R. 6 P. Writing 6 under poles, and carrying 3 to the roods, we find the sum 15, which, reduced to acres, gives 3 A. 3 R. Lastly, we write 3 under roods, and carry 3 to the column of acres, the sum of which is 129. Having completed the addition, the sum total is 129 A. 3 R. 6 P. 15 s. yds. 54 s. f. 3. Required the sum of the four last numbers of the preceding example. Ans. 125 A. 3 R. 9 P. 18 s. yds. 8.1 s. f. By adding this sum to the first number of Example 2, we shall again have 129 A. 3 R. 6 P. 15 s. yds. 51 s. f., and this serves as proof of the correctness of the first addition. In the same manner the scholar may prove the following Examples. d. 11 11 93 79 12 78 16 21 32 14 7 89 12 111 59 17 8 65 19 7 43 10 23 14 13 6 86 4 11 33 15 9 97 13 63 8. d. Subtraction of Compound Numbers. 320. We place the less number under the greater in the same manner as for addition; and, commencing with the lowest order, we subtract the units of each order in the less number from the corresponding units of the greater, writing each remainder under the order which gave it. But, if the number of units of any inferior order of the less number is greater than the number above it, we add as many units to the upper number as will make a unit of the next higher order, and, having subtracted the lower number from the sum, we add a unit to the next higher order of the lower number. The numbers are thus both increased by the same quantity, which (84) does not affect their difference. Examples. A. R. P. S. yds. s. f. 15 51 3 3 36 26 8 Here, as we cannot subtract 1=from , we borrow a unit from the 5 s. f., which is equal to 4 : then 4+1=, and - =, which we write underneath. Having borrowed a unit from the 5 s. f., instead of diminishing this by a unit, we add a unit (84) to the 8, which makes 9 : then, as 9 is greater than 5, we borrow a s. yd., reducing this to s. f., and adding it to 5, we have 14: then 14 9=5, which we write under s. f. Having borrowed a s. yd., we add 1 to 18, which makes 19: then as 19 is greater than 15, we borrow a pole, which, reduced to yards, and added to 15, makes 451 : subtracting 19, there remains 261: wherefore, we write 26 underneath, and reducing 1 s. yd. to s. f. we have 24, which added to the 54 s. f. already found, gives 8. This we leave under the column of s. f. Having borrowed a pole, we add 1 to 9, which makes 10: we then say, not 10 from 6, but borrowing a rood or 40 poles, 10 from 46, leaves 36, which we write underneath. We then add 1 to 3, which makes 4, and borrowing an acre, or 4 R., we say, 4 from 7 leaves 3, which we write underneath. Lastly, having borrowed an acre, we add 1 to 5, which makes 6, and 6 from 9 leaves 3, which, when written underneath, completes the operation. This example is a proof of the second example, (319,) the greater number being the sum total of the five numbers added, the less, the sum of the four last numbers, and the remainder, the first number. Note. We may, in all cases, reduce both numbers to a vulgar or decimal fraction of the principal unit, and find, after subtraction, the value of the result. The student may prove the following examples by addition and subtraction : 2. From į of of £1, take of g of 1 s. Answer, 1 s. 103 d. 3. From 3 t. O cwt. 2 qrs. 5 lbs. 13 oz., take of of 1 cwt. Answer, 2 t. 19 cwt. 3 qrs. 27 Ibs. 9 oz. 73 dr. 4. From 3 A. O R. 4 P. 9 s. yds. 3 s. f., take of 1 A. Answer, 2 A. 2 R. 30 P. 29 s. yds. 4 s. f. 72 s. in. 5. From ,428571 of a league, take off of a mile. Answer, 5 fur. 5 p. 3 yds. 2 ft. 93 in. 6. From , of a square mile, take 100; acres. Answer, 356 A. 1 R. 36 P. 5 s. yds. 69 s. f. P. 8. yds. s. f. (7.) 23 23 2 21 19 6 (8.) 396 81 15 3 35 24 7 13 17 93 A. R. £ 8. d. (9.) 22 1 cwt. qrs. lbs. oz. drs. 22 1 13 7 8 11 3 17 8 11 bar. gal. qt. (10.) 18 19 2 7 29 3 Multiplication of Compound Numbers. To multiply a compound number by a simple or homogeneous one: 321. A familiar instance, requiring this multiplication iswhen a number of units of the same kind is given, and the |