Art. 27. When the same factor occurs in a dividend and in its divisor, it may be omitted from both without changing their ratio. Hence all the factors that are common to a dividend and its divisor may be stricken out (cancelled) and the quotient (ratio) be unchanged. NOTE. In this discussion and in the solution of problems in this connection, all the numbers are regarded as abstract. 12 2 42 7 Observe that the striking out of the factors 2 and 3 from the dividend and its divisor does not change their ratio—the quotient. 70 70 12 = 1 Observe that if all the factors of one of the numbers are cancelled the number becomes 1 and not 0. The factor 5 is 5 times 1; the factor 7, 7 times 1. Hence in the above problem there really remain in the divisor, after the cancellation, the factors 1 and 1=1X1=1. III. Divide 48 × 8 × 4 = 1536 by 8 × 4 × 4 = 128. Operation No. 1. 128)1536 (12 128 256 Observe that it is not necessary to obtain the prime factors of a dividend and its divisor to employ cancellation in finding the quotient. In the above the composite factor 8 is stricken out of the divisor and out of the 48 of the dividend. IV. Divide 56 × 35 = 1960 by 15 × 8 = 120. NOTE.-Employ "Short Methods" in the solution of the follow ing problems. How many cords of wood 1. In a pile 32 feet by 8 feet by 4 feet?* 2. In a pile 40 feet by 16 feet by 6 feet? 3. In a pile 32 feet by 30 feet by 10 feet? (aa) Find the sum of the three results. How many acres of land 4. In a piece 180 rods by 28 rods? † (bb) Find the sum of the three results. 7. Multiply 64 by 96 and divide the product by 16× 24 × 2. 8. Multiply 250 by 72 and divide the product by 163 × 3 x 24. (cc) Find the sum of the two results. *Think of a cord as 8 feet by 4 feet by 4 feet. + Think of an acre as 40 rods by 4 rods Find the cost 9. Of 346 acres of land at $50 per acre. 10. Of 346 acres of land at $51 per acre. 11. Of 346 acres of land at $52 per acre. 12. Of 346 acres of land at $49 per acre. 13. Of 254 acres of land at $51 per acre. (dd) Find the sum of the five results. 14. Of 243 ft. iron pipe at 3314 a foot. 15. Of 243 ft. iron pipe at 34¢ a foot. 16. Of 243 ft. iron pipe at 35¢ a foot. 17. Of 243 ft. iron pipe at 323¢ a foot. 18. Of 156 ft. iron pipe at 35¢ a foot. (ee) Find the sum of the five results. 19. Of 260 lb. butter at 25¢ a pound. (ff) Find the sum of the five results. 24. Of 350 lb. coffee at 1214 a pound. 25. Of 350 lb. coffee at 13¢ a pound. 26. Of 350 lb. coffee at 14¢ a pound. 27. Of 350 lb. coffee at 1114 a pound. 28. Of 330 lb. coffee at 1634 a pound. 29. Of 330 lb. coffee at 173¢ a pound. coffee at 1534 a pound. coffee at 25¢ a pound. 32. Of 240 lb. coffee at 26¢ a pound. 33. Of 240 lb. coffee at 27¢ a pound. 30. Of 330 lb. (gg) Find the sum of the ten results. (hh) Find the sum of the three results. 37. Of 25 tons of meal at $25 per ton. 40. Of 23 yd. cloth at 27¢ a yard. (jj) Find the sum of the three results. 43. Of 3240 ft. lumber at $15 per M. 46. Of 99 lb. butter at 23¢ a pound. (11) Find the sum of the three results. 49. Paid $15.50 for ribbon at 1634 a yard. yards did I buy? How many 50. Paid $24.75 for ribbon at 1214 a yard. How many yards did I buy? (mm) Find the sum of the two results. EXAMINATION PROBLEMS. Miscellaneous Problems. NOTE.-The following problems are selected mainly from sets of examination questions supplied to the author for this purpose by one hundred school principals and superintendents. 1. Each edge of a cube is diminished by of its length: (a) By what fraction of itself is the volume diminished? (b) By what fraction of itself is the surface diminished? 2. How many cubical blocks, each edge of which is ft., are equivalent to a block 8 ft. long, 4 ft. wide, and 2 ft. thick? 3. A ladder 78 ft. long stands perpendicularly against a building. How far must it be pulled out at the foot that the top may be lowered 6 ft.? 4. A merchant sold of a quantity of cloth at a gain of 20% and the remainder at cost: (a) His gain was what per cent of the cost? (b) If he gained $7.29 what was the cost of the goods? 5. What must I pay for 4% stock to get 5% on the investment? 6. The cubical content of one cube is eight times that of another: (a) How does an edge of the first compare with an edge of the second? (b) How does the surface of the first compare with the surface of the second? 7. A creditor receives $1.50 for every $4.00 that is due him and thereby loses $301.05: |