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 (canceled) and the quotient (ratio) be unchanged. Divide 180 by 42. Operation No. 1. Operation No. 2. 42)180(4? 180 = ax2x3x3x5 = 30 2 168 42 2x8x1 ~ 7 ~ 7 12 2 — = — Observe that the striking out of the 4.9 7 • *-^' factors 2 and 3 from the dividend and its divisor does not change their ratio—the quotient. II. Divide 420 by 35. Operation No. 1. Operation No. 2. 35)420(12 420 = 2 x 2 x 3 x g X ? _ 12 35 35" Six? 1 70 nr. Observe that if all the factors of one of the numbers are canceled, the number becomes 1 and not 0. The factor 5 is 5 t>mes 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 x 8 x 4 = 1536 by 8 x 4 x 4 = 128. pre 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 S is stricken out of the divisor and out of the 48 of the dividend. IV. Divide 56 x 35 = 1960 by 15 x 8 = 120. Operation No. 1. Operation No. 2. 120)1960(164, ixi_49 1 ~760 »xf 8~ 3 720 Observe that in the above the factor 5 4Q \ is stricken out of 15 and 35, and the factor 8 is stricken out of the divisor and out of the 56 of the dividend. 120 3 Miscellaneous Problems. Note.—Employ " Short Methods" in the solution of the following 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? f 5. In a piece 64 rods by 96 rods? 6. In a piece 136 rods by 32 rods? (bb) Find the sum of the three results. 7. Multiply 64 by 96 and divide tbe product by 16 x 24 x 2. 8. Multiply 250 by 72 and divide the product by 16| x 3 x 24. (cc) Find the sum of the two results. * Think of a cord as 8 feet by 4 feet by 4 feet, t Think of an acre aa 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 33^0 a foot. 15. Of 243 ft. iron pipe at 34^ a foot. 16. Of 243 ft. iron pipe at 35$0 a foot. 17. Of 243 ft. iron pipe at 32^0 a foot. 18. Of 156 ft. iron pipe at 35}0 a foot. (ee) Find the sum of the five results. 19. Of 260 lb. butter at 250 a pound. 20. Of 260 lb. butter at 260 a pound. 21. Of 260 lb. butter at 270 a pound. 22. Of 260 lb. butter at 240 a pound. 23. Of 184 lb. butter at 270 a pound, 24. Of 350 lb. coffee at 12£0 a pound 25. Of 350 lb. coffee at 13|-0 a pound. 26. Of 350 lb. coffee at 14£0 a pound. 27. Of 350 lb. coffee at 11^0 a pound. 28. Of 330 lb. coffee at 16 §0 a pound. 29. Of 330 lb. coffee at 1710 a pound. 30. Of 330 lb. coffee at 15|0 a pound. 31. Of 240 lb. coffee at 250 a pound. 32. Of 240 lb. .coffee at 260 a pound. 33. Of 240 lb. coffee at 270 a pound, Find the cost— 34. Of 2\ tons coal at \ per ton. 36. Of 4|- tons coal at $4|- per ton. 37. Of 25 tons of meal at $25 per ton. 38. Of 35 acres of land at $35 per acre. 39. Of 45 M. ft. of lumber at $45 per M. 40. Of 23 yd. cloth at 27^ a yard. 41. Of 36 yd. cloth at 34^ a yard. 42. Of 42 yd. cloth at 48^ a yard, 43. Of 3240 ft. lumber at $15 per M; 44. Of 2460 ft. lumber at $15 per M. 45. Of 1620 ft. lumber at $16 per M. 46. Of 99 lb. butter at 230 a pound. 47. Of 99 lb. butter at 280 a pound. 48. Of 98 lb. butter at 24^ a pound. 49. Paid $15.50 for ribbon at 16J-0 a yard. How many yards did I buy? 50. Paid $24.75 for ribbon at 121^ a yard. How many yards did I buy? (mm) Find the sum of the two results. PRACTICAL APPROXIMATIONS. So far as practicable, solve the following problems without the aid of a pencil. At least, exercise the Judgment on every problem before making any figures. 1. The specific gravity* of iron being about 7^, how much does a cubic foot of it weigh? How much does a cubic inch of iron weigh? 2. A 4-inch iron ball weighs about pounds. A 2 inch iron ball weighs about pounds. 3. An iron rod, 1 inch in diameter and 12 feet long, weighs about pounds. An iron rod 2 inches in diameter and 12 feet long weighs about pounds. 4. A sheet of boiler iron, 8 feet square and f of an inch thick, weighs about . . pounds. 5. What is the weight of the water that will fill a tank 2 feet wide, 2 feet deep, and 10 feet long? 6. The specific gravity of limestone is about 2|. What is the weight of a piece of limestone that is 4 feet square and 3 inches thick? 7. The specific gravity of seasoned white pine is about .5; that is, a piece of white pine weighs about 5 tenths as much as the same bulk of water weighs. How much does a pine board 1 foot wide, 1 inch thick, and 12 feet long, weigh? 8. What is the weight of a stick of timber 12 inches by 12 inches and 20 feet long, if its specific gravity is .7? 9. What is the weight of 1000 feet of green lumber if its specific gravity is .9? *When we say that the specific gravity of iron is about 71, we mean that it weighs about 7} times as much as water, the same bulk being considered. |