it he bought coffee at $.15 per lb. He sold $.16 per lb., and the rest for $.17 per lb. receive for the coffee? of the coffee at How much did he 32. When sugar was worth 9 cents per lb., a man gave 8766 lb. of it for 81 acres of land. How much did the land cost per acre? 33. If of a ton of hay cost as much as of a ton of straw, and 36 tons of straw cost $405.72, how much will a ton of hay cost? 34. A man sold a pile of wood, 96 ft. long, 8 ft. wide, and 7 ft. high, for $190.554. At what rate did he sell it per cord? 35. A furniture dealer bought 57 bedsteads at $4.73 apiece, which he sold so as to gain $44.64 by the transaction. He immediately invested the money he thus received for chairs at $.75 apiece. How many chairs did he buy? 36. A fruit dealer paid $169.75 for apples at $1.75 per barrel, $80.625 for apples at $1.875 per barrel, and $92.64 for apples at $1.93 per barrel. He sold them all at the rate of $2.068 per barrel. What did he receive for them? How much did he gain on them? SECTION XXIV. It is frequently the case that by carefully examining the numbers we are to operate upon, we can discover some abbreviated method of performing the work. We have already suggested several such methods, and it is the design of this section to suggest others. A. It is often advantageous to add on the principles involved in the following equations: 1. 786 +99 =786 +100 1. 2. 4783457994 = 47834 +58000 6. . 4. 16£, 13s. 3d. +7£, 19s. 8d. = 16£, 13s. 3d. +8£, — 4d. 5. 14 H, 5 3, 2 3 29, 13 gr.+4 H, 113,73,2 3, 16 gr. 14 tb, 5 3,2 3, 2, 13 gr. 5b, 4 gr. 6. How many are 3476437 +1499992 ? 7. How many are 732475 +9999387 ? 8. What is the value of 48£, 13s. 5d. 2qr. + 99£, 19s. 11d. 1qr.? 9. What is the value of 8 cwt. 1 qr. 16 lb. 13 oz. 5 dr. cwt. 2 qr. 27 lb. 15 oz. 8 dr.? +5 10. What is the value of 19 rd. 4 yd. 2 ft. 4 in. + 18 rd. 5 yd. 0 ft. 6 in. ? B. Principles similar to those under A, may often be applied in subtraction as illustrated below: 1. 3786437-5799933786437 · 580000 +7. 2. 864379 298994864379 3. 18 lb. 6 oz. 14 dwt. 7 gr. = 18 lb. 6 oz. 14 dwt. 7 gr. 3000001006. 12 lb. 11 oz. 19 dwt. 23 gr. 13 lb. + 1 gr. 4. 28 yd. 1 qr. 3 na. 1 in. 21 yd. 3 qr. yd. 1 qr. 3 na. Î in. 22 yd. + 14 in. 3 na. 1 in. 28 7. What is the value of 17 T. 13 cwt. 5 lb. 4 dr. 3 T. 19 cwt. 3 qr. 27 lb. 15 oz. 11 dr.? 8. What is the value of 9 w. 3 da. 15 h. 23 m. 16 sec. 4 w. 6 da. 23 h. 59 m. 47 sec.? = C. To multiply by 99, observe that since 99 100 — 1, 99 times a number must equal 100 times the number minus once the number. For example, 99 times 837 100 times 837 837 83700 837= 82,863. To multiply by 999, observe that since 999 1000 - 1,999 times a number must equal 1000 times the number minus once the number. For example, 999 times 14.67 = 14.67 14655.33. 1000 times 14.67, — 14.67 — 14670 By observing how in examples like the above the figures of the minuend and subtrahend must compare with each other, the results can be written at once. 100 1. Multiply 18 by 99; 25 by 99. 2. Multiply 8105 by 99; 71.15 by 99. 3. Multiply 75.074 by 99; 706 by 99. 4. Multiply 31748 by 9; 31748 by 99. 5. Multiply 31748 by 999; 31748 by 9999. 6. How many are 98 times 3587? (Observe that 98 - 2.) 7. How many are 994 times 7387? (Observe that 994 — 1000 6.) 8. How many are 997 times 7943? 9. How many are 9992 times 543876 ? 10. How many are 99993 times 6854297? D. When the multiplier is a convenient fractional part of 10, 100, 1000, or a unit of any higher denomination, the following principles may be advantageously applied. 1. How many are 25 times 679 ? Solution. Since 25 equals one fourth of 100, 25 times 679 must equal one fourth of 100 times 679, or, of 67900, which equals 16975. 2. How many are 25 times 657 ? 3. How many are 50 times 657 ? 4. How many are 12 times 834? (Observe that 121 of 100.) 5. How many are 16 times 957? (Observe that 16% of 100.) 6. How many are 334 times 871? (Observe that 331: of 100.) 7. How many are 142 times 249? (Observe that 144: of 100.) = = = 3 8. How many are 3 times 249? (Observe that 3 = of 10.) 9. How many are 2 times 822? (Observe that 2 10.) = 10. How many are 64 times 944? (Observe that 6д of 100.) of 11. How many are 3333 times 9478? (Observe that 333 of 1000.) 12. How many are 125 times 8767? (Observe that 125 of 1000.) = = 13. How many are 250 times 6894? (Observe that 250 = of 1000.) E. It frequently happens that one part of the multiplier is a factor of another part. When this is the case, labor can often be saved by applying the principles illustrated in the following examples: 1. What is the product of 7678 multiplied by 427? By examining the multiplier we perceive that its tens equal 60 times its units, and, therefore, that the desired product can be obtained by adding to the product of the multiplicand multiplied by the units' figure of the multiplier, 60 times this product. The following method of writing the numbers used will exhibit the process: 7,678. 2. How many are 72144 times 874369 ? By examining the multiplier, we observe that the number expressed by the figures in the three places nearest to the point, is .02 of the number expressed by the other figures. We may, therefore, perform the work by the method indicated below: 874,369. 4. What is the product of 76437 multiplied by 14874? 5. What is the product of 879438 multiplied by 19899? 6. What is the product of 43678 multiplied by 17525? 7. What is the product of 476437 multiplied by 83415? 8. What is the product of 9763278 multiplied by 43821973 ? 9. What is the product of 5428973 multiplied by 36324? 10. What is the product of 463829 multiplied by 1998999? F. 1. How many times is 99 contained in 437 ? Solution. 100 is contained 4 times in 437, with a remainder of 37; but as 99 is contained in 100 once, with a remainder of 1, it must be contained 4 times in 400, with a remainder of 4, which remainder added to 37, the remainder by the first division, gives 41 as the remainder of the division by 99. The quotient must therefore be 4, and the remainder 41. 400+37; and since 100 2d Solution. 437 99 +1, 400 must equal 4 times 99+ 4. Adding 4 to 37, we have 41 as the remainder of the division by 99. Therefore, 437 = 4 times 99, with a remainder of 41. 2. How many times is 99 contained in 15378? We separate the hundreds from the tens and units of the dividend by a vertical line, thus: 153 78. 231. 155 33. Then, since 100—99 + 1, 15300 must equal 153 times 99, 153. Adding 153 to 78, gives 231 as the remainder of the first division by 99. This remainder being greater than 99, we divide it by 99, as we did the original dividend. 200 2 times 99 plus 2, and adding the 2 to 31, we have 33 as the remainder of the second division by 99; this being less than 99, the division is completed. Now, by adding 153 and 2 together, we obtain 155 as the quotient of the division by 99, and we have a remainder of 33. In a similar manner we can divide by 999, 9999, &c. Let the pupil find, if he can, the application of a similar principle to the division by 98, 97, 96, &c.; also to 998, 997, &c., and afterward perform the following examples : 3. 186738 how many times 99 ? 4. 49763842 = how many times 999 ? 6. 9842987483 = how many times 99999? 9. 248763 10. 69874325 how many times 96 ? how many times 997? how many times 9998? Since in 100, 25 is con G. 1. 7987 how many times 25? Solution. 79877900 + 87. tained 4 times, in 7900 it must be contained 79 times 4 times, |