Multiplication Table. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 1 21 22 14 21 28 35 42 49 24 32 40 27 36 28 30 32 48 51 68 38 57 60 14 16 18 20 22 24 27 30 28 32 36 40 44 40 45 50 55 42 48 54 60 66 56 63 70 64 72 80 72 81 90 99 234388 24 36 39 42 48 52 72 77 84 26 28 56 60 64 68 72 76 80 84 88 38 40 42 44 46 48 50 66 234 65 70 75 80 85 90 95 100 105 110 115 120 125 78 96 102 108 114 120 126 132 138 144 150 98 105 112 119 126 133 140 147 154 161 168 175 78 91 104 117 130 143 156 169 182 195 208 221 234 247 260 273 286 299 312 325 13 84 98 112 126 140 154 168 182 196 210 224 238 252 266 280 294 308 322 336 350 14 90105 120 135 150 165 180 195 210 225 240 255 270 285 300 315 330 345 360 375 15 64 80 96 | 112 | 128 | 144 | 160 | 176 | 192 | 208 | 224 | 240 | 256 272 298 304 320 336 352 368 384 400 16 42 56 45 60 75 96 104 112 120 128 136 144 152 160 168 176 184 192 200 108 117 126 135 144 153 162 171 180 189 198 207 216 225 9 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 99|110| 121 132 143 154 165 176 187 198 209 220 231 242 253 264 275 11 72 84 96 108 120 132 144 156 168 180 192 204 216 228 240 252 264 276 288 300 85 102 119 136 153 170 187 204 221 238 255 272 289 306 323 340 357 374 391 408 425 17 72 90 108 126 144 162 180 198 216 234 252 270 288 306 324 342 360 378 396 414 432 450 18 76 95 114 133 152 171 190 209 228 247 266 285 304 323 342 361 380 399 418 437 456 475 80 100 100 140 160 180 200 220 240 260 280 300 220 340 360 380 400 420 440 460 480 500 105 | 126 | 147 | 168 | 189 | 210 | 231 | 252 | 273 | 294 | 15 | 336 | 357 378 399 | 420, 441 462 483 504 525 21 88 110 132 154 176 198 220 242 264 286 308 330 352 374 396 418 440 462 484 506 528 550 22 92 115 138 161 184 207 230 253 276 299 322 345 368 391 414 437 460 483 506 529 552 575 23 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360 384 408 432 456 480 504 528 552 576 600 24 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 625 25 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 NOTE.-It will be of great advantage to the student to fully master the above table. Any delay caused by following this suggestion will be offset by time gained in subsequent work; such mastery will so increase the rapidity of work in business applications as to greatly lessen the labor of accounting. 92. When either Factor is a Number within one's thorough knowledge of the Multiplication Table. EXAMPLE.-1. Multiply 417 by 5. 417 5 OPERATION. EXPLANATION. -Write the multiplier 5 below the unit figure of the multiplicand, and multiply each figure of the multiplicand by the multiplier, thus; 5 times 735, or, 3 tens + 5 units; write the 5 units in units' place and reserve the 3 tens to add to the next product; next, 5 times 1 ten are 5 tens, and adding the 3 tens reserved gives 8 tens, which write in tens' place; next, 5 times 4 hundreds are 20 hundreds, or 2 thousands; write a naught, or cipher, in the hundreds' place and the 2 in the thousands' place, thus completing the multiplication and obtaining 2085 as the product of 417 multiplied by 5. 2085 EXAMPLE.-2. Multiply 123 by 12. OPERATION. 123 12 EXPLANATION.-Multiply each figure of the multiplicand by the multiplier, 12; 12 times 3 = 36, or, 3 tens +6 units; write the 6 in units' place and reserve the 3 tens to add to the next product; next, 12 times 2 tens are 24 tens, and adding the 3 tens reserved gives 27 tens, or 2 hundreds +7 tens; write the 7 in tens' place and reserve the 2 hundreds to add to the next product; next, 12 times 1 hundred are 12 hundreds, and adding the 2 hundreds reserved gives 14 hundreds, or, 1 thousand 4 hundreds, which write in hundreds' and thousands' places, thus completing the multiplication and obtaining 1476 as the product of 123 multiplied by 12. 1476 Rule. Write the factors one below the other, arranged so that figures of like orders will stand in the same vertical line. Multiply each figure of the upper factor, beginning at the right, by the lower factor, placing in order the last figure of the product so obtained, and carrying to the next product all figures except the last; continue so doing until the last product is found, which write in full. 95. When the Multiplier consists of two or more figures. EXAMPLE.-Multiply 417 by 352. OPERATION. Hundreds. Tens. Units. 352 Multiplicand. 417 Multiplier. Units. 834 Tens. 2085 Hundreds. 1251 = EXPLANATION.-Write the numbers one below the other in the same unit order from the right. Then, beginning with the unit figure of the lower factor multiply; 2 times 7 units are 14 units 1 ten + 4 units; write the 4 units in units' column and add the 1 ten to the next product; next, 2 times 1 ten are 2 tens and the 1 ten added makes 3 tens, which write in tens' place; next, 2 times 4 hundreds are 8 hundreds, which write in hundreds' place, giving 834 as the first partial product, or the product of the upper factor multiplicd by the unit figure of the lower factor. Next take the tens' figure of the lower number as a multiplier; 7 taken 5 tens or 50 times = 35 tens, or 350; write the 5 of the number 35 in tens' column, or below the 3 tens of the first partial product, and carry the 3 of the 35 to the next product; next, 5 times 1 are 5 and the 3 to carry added make 8, which write under the 8 of the first partial product; then, 5 times 4 are 20, which write still to the left, making the second partial product 2085 tens. Next, take the third figure, or hundreds, of the lower factor, as a multiplier; 3 times 7 hundreds are 21 hundreds; write the 1 in the hundreds' place and reserve the 2 for the next product; then, 3 times 1 are 3 and 2 to carry makes 5, which write in its order; then, 3 times 4 are 12, which write still to the left, having as a product 1251. Since, however, the several figures of the factor taken as a multiplier were of successive orders of units, 146784 Therefore, 146784 is found to be the product of the numbers 417 and 352. Rule.-I. Place the multiplier below the multiplicand, the unit figures in the same vertical line. II. Beginning with the unit figure, multiply all the figures of the multiplicand by each successive figure of the multiplier, writing the first figure obtained in each partial product directly below the figure by which it was multiplied. Add the partial products. REMARK.-The object of writing each succeeding partial product below and one place to the left of its predecessor, is that units of the same grade, or order, may, for convenience in adding, be found in the same vertical line; this arrangement precludes the necessity of filling the vacant orders with ciphers. As before shown, the arrangement of factors will not vary the result; as, 4 x 5 = 20, also 5 X 420; therefore, in business or school practice, arrange X factors in such order as to save time and space; by so doing, problems otherwise long and difficult, may be solved by mental processes. EXAMPLE.-Multiply 120000 by 7256. EXPLANATION. - Consider the factors as reversed in order; thus, 7256 X 120000. Then multiply the 7256 mentally by 12, and to the product, 87072, annex four ciphers, because the 12 was not 12 simple units but 12 units of the fifth order, or tens of thousands. 96. When one Factor is 10, 100, 1000, 10000, or 1 with any number of ciphers annexed. EXAMPLE.-Multiply 324 by 1000. EXPLANATION.-Since there are three ciphers in the multiplier, annex three to the multiplicand, 324, thus obtaining the product, 324000. Rule. To the one factor annex as many ciphers as there are ciphers in the other factor. 97. A Composite Number is a number that may be resolved or separated into integral factors; or, it is a number that may be formed by multiplying together two or more numbers; thus, 12 = 4 x 3; or 12 = 2 x 2 x 3 ; or 4 × 3 = 12; or 2 × 2 × 3 = 12. = 98. When the Multiplier is a Composite Number. Whenever it is required to find the product of numbers one or more of which is composite, the result may be obtained by using as multipliers the factors of such composite number or numbers; thus, 6 x 4 = 24, or 6 × (2 × 2) = 24. × Rule.-Separate the multiplier into its factors. Multiply the multiplicand by one of these factors, that product by another factor, and so on, using in succession all the factors; the last product will be the result required. REMARK.- Since the order in which factors are used will not vary the product, the student is recommended to seek the simplest number—the one most easily factored — as a multiplier. EXAMPLES FOR PRACTICE. 99. 1. Multiply 41 by 15, using as factors 3 and 5. 13. Multiply 81 by 64, using as factors 8 and 8. 14. Multiply 64 by 81, using as factors 9 and 9. 15. Multiply 81 by 64, using as factors of 81, 9 and 9, and as factors of 64, 8 and 8. 11. A clerk sold 9 shirts at 80 cents each, 2 neck-ties at 35 cents each, 10 collars at 25 cents each, a pair of gloves for 75 cents, and two suits of underwear at 95 cents per suit. What was the price of all? 12. I bought 15 cows at 32 dollars per head, a pair of horses for 245 dollars, a harness for 22 dollars, and 81 sheep at 3 dollars per head; what was the total cost of my purchases? 13. The cost of furnishing a house was, for parlor and library furniture 762 dollars, halls 150 dollars, dining room and kitchen 295 dollars, chambers 648 dollars, stoves and furnace 350 dollars, carpets and curtains 825 dollars, what was the total cost? 14. 14250 dollars was paid for four houses, the first costing 2750 dollars, the second 400 dollars more than the first, the third 250 dollars less than the first and second together, and the fourth the remainder. Find the cost of the fourth house? 15. Find the difference between the continued products of 91 × 4×3×11x9 and 5 X 5 X 12 X 4 X 6 X 7. 16. Find the difference between seven units of the sixth order and the continued product of 15 x 6 x 5 x 12 x 4 x 7 x 11 x 8 X 2 X 9. 17. A merchant having 17462 dollars to his credit in a bank, gave checks as follows: for dry goods 5416 dollars, groceries 5995 dollars, boots and shoes 1416 dollars, hardware 1850 dollars, and drew out 500 dollars for family expenses; what amount was left in the bank? 18. Exchanged a city block valued at 35000 dollars, for a farm of 175 acres valued at 95 dollars per acre, eight horses at 110 dollars each, 14 cows at 28 dollars each, 225 sheep at 4 dollars each, farm machinery valued at 825 dollars, and received the balance in cash. How much cash was received? 19. A drover bought 135 horses at an average price of 115 dollars for 100 of them, and 125 dollars per head for the remainder; he sold 25 at 100 dollars per head, twice that number at twice the price per head, and the remainder at 67 dollars per head. How much was gained or lost? 20. A ranchman sold to a trader, 46 ponies at 60 dollars per pair, 116 calves at 9 dollars per head, 41 cows at 35 dollars per head, and a pair of mules for 375 dollars. He received in part payment, 15 barrels of flour at 9 dollars per barrel, 11 hundred weight of bacon at 12 dollars per hundred weight, 4 suits of clothes at 22 dollars per suit, 2 saddles at 13 dollars each, a wagon at 75 dollars, a set of furniture for 58 dollars, and the remainder in cash. What amount of cash did the trader pay? |