Or we may carry as we go along, and get rid of this second addition; thus, 43 50 51 43 65 6, 5, being the first two digits of the sum, and 3, 1, 0, 3, the others. Or, simpler yet, we may write the last digit of each partial sum in its proper place, and put underneath what there is to carry; thus, 526953 - 30 - 3 5 4 We have seen (page 17) that any number may be resolved into 9's and the sum of its digits. 66745 9's + 28 (6 +6 +7+4 + 5 = 28). 289's 10 (2810). 109+1. So, casting out all the 9's from 66745, leaves a remainder of 1. Casting out 48-12-3 the 9's from 48371, leaves a remainder of 5. The sum of all these remainders is 48-9's+12=9+3. The sum 526953 = 9's +30=9's+3, and as this result agrees with the other, the sum 526953 is in all probability correct. We may add the remainder 1 with the digits of the second number, the remainder thus obtained with the digits of the third number, and so on. Also, we may omit 9's as we go along; thus, 12, 191 (6+6=12,+7=19. Drop the 9 of 19, also drop 459). 5, 13, 246 (1+4=5,+8=13, +11=24. 2+4=6). 12, 13, 235 (6+6=12, etc.). 10, 12, 15-6 9, 8, 145 12, 19, 9-0 (5+7=12, +7=19. Drop the 9 of 19. 9's +r Any number may be resolved into 9's a certain remainder. Let r represent this remainder. Any other number may be resolved into 9's a remainder. Let (read r prime) represent this remainder. The product of the two numbers is made up of 9's + a remainder of r'r; for multiplying out, we have r'×r r+ 9's × 9's, each term except the first one Product = 9's + r'r +' x 9's + 9's 9's 3. The inference is 2316656167293.. 3 r'r = 12=9+3. The product that the product obtained is correct. To apply the principle to division, we have only to remember that a dividend == quotient × divisor + remainder. Perform the following problems, and test the results by casting out the 9's. FROM PRACTICE TABLE NO. 2. (PAGE 53.) 178. Add 1-13; a-e, f-j, k-o, p-t, u-z. 179. Add 14-25; a-m, n−z. 180. Add 1-25; a-h, i-q, r-z. 181. Add the whole of Practice Table No. 3. Keep the record of what there is to carry each time. Test the result by adding downward instead of upward, and also by casting out the 9's. 182. Multiply a-j in each of the first eight lines of Table No. 2 by k-m in the same line. 183. Multiply a-h in each line from 9 to 16 by i-m in the same line. 184. Multiply a-o in each line from 17 to 25 by p-u in the same line. 185. Divide aj in each of the first eight lines by k-m in the same line. 186. Divide a-h in each line from 9 to 16 by i-m in the same line, to 3 decimal places. 187. Divide a-o in each line from 17 to 25 by p-u in the same line, to 2 decimal places. 188. What is the square of 61341? of 4231406? 189. What is the cube of 627? of 4585? 190. What is the fourth power of 243? of 621 ? 191. What is the fifth power of 47? of 65? 192. Extract the square root of 828051; of 227834749; of 534533475567. 193. Extract the cube root of the same three numbers. 194. Extract the fourth root, to 2 decimal places, of 407458; of 3259210133. 195. Extract the sixth root, to 1 decimal place, of 23140625; of 75567287960. Add the denominators for the numerator of the sum; multiply the denominators for the denominator of the sum. To subtract, take the difference between the denominators for the numerator; the product for the denominator. 196. What is the sum and also the difference of By annexing two O's and subtracting, we do the work with 6 figures less than is required by the ordinary method. |