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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 diridend and its divisor may be stricken out (canceled) and the quotient (ratio) be unchanged.

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2

27

12

42

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Observe that the striking out of the factors 2 and 3 from the dividend and its

divisor does not change their ratio-the quotient.

II. Divide 420 by 35.

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70

comes 1 and not 0.

Observe that if all the factors of one of

the numbers are canceled, the number beThe 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

III. Divide 48 × 8 × 4

Operation No. 1.

128)1536(12

128

256

256

=

1 x 1 = 1.

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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.

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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? †
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 the product by 16 × 24 x 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 331¢ a foot.
15. Of 243 ft. iron pipe at 34¢ a foot.
16. Of 243 ft. iron pipe at 3514 a foot.
17. Of 243 ft. iron pipe at 321¢ a foot.
18. Of 156 ft. iron pipe at 351¢ a foot.
(ee) Find the sum of the five results.

19. Of 260 lb. butter at 25¢ a pound.
20. Of 260 lb. butter at 26¢ a pound.
21. Of 260 lb. butter at 27¢ a pound.
22. Of 260 lb. butter at 244 a pound.
23. Of 184 lb. butter at 27¢ 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 1314 a pound.
26. Of 350 lb. coffee at 141⁄2¢ a pound.
27. Of 350 lb. coffee at 111¢ a pound.
28. Of 330 lb. coffee at 1634 a pound.
29. Of 330 lb. coffee at 173¢ a pound.
30. Of 330 lb. coffee at 153¢ a pound.
31. Of 240 lb. coffee at 25¢ a pound.
32. Of 240 lb. coffee at 26¢ a pound.
33. Of 240 lb. coffee at 274 a pound.
(gg) Find the sum of the ten results.

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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.
(ii) Find the sum of the three results.

40. Of 23 yd. cloth at 274 a yard.
41. Of 36 yd. cloth at 34¢ a yard.
42. Of 42 yd. cloth at 48¢ a yard.

(jj) Find the sum of the three results.

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.
(kk) Find the sum of the three results.

46. Of 99 lb. butter at 23¢ a pound.
47. Of 99 lb. butter at 28¢ a pound.
48. Of 98 lb. butter at 24¢ a pound.

(11) Find the sum of the three results.

49. Paid $15.50 for ribbon at 1634 a yard. How many yards did I buy?

50. Paid $24.75 for ribbon at 1214 a yard. How many yards did I buy?

(mm) Find the sum of the two results.

PRACTICAL APPROXIMATIONS.

aid of a pencil.

So far as practicable, solve the following problems without the At least, exercise the judgment on every problem before making any figures.

1. The specific gravity* of iron being about 74, 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

inch iron ball weighs about

pounds.

pounds. A 2

3. An iron rod, 1 inch in diameter and 12 feet long, weighs about pounds. An iron rod 2 inches in diam

eter and 12 feet long weighs about

pounds.

4. A sheet of boiler iron, 8 feet square and 3 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 21. 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.

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