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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 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.

I. Divide 180 by 42.
Operation No. 1.

Operation No. 2.
42)180 (44
180 2 x 2 x 3 x 3 x 5 30 2

4
168

42 2 x B x 7 7 7 12 2

Observe that the striking out of the 427

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 $ x 7

5 12

12
35

35
$ x 7

1
70

Observe that if all the factors of one of the 70

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.

6

2

III. Divide 48 x 8 x 4 = 1536 by 8 x 4 x 4 = 128.
Operation No. 1.

Operation No. 2.
128)1536 (12
128

12
48 x 8 x 4

12

8 X 4 X 4 1 256 256 Observe that it is not necessary to obtain

the prime factors of a dividend and its di. visor 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.

7

7

IV. Divide 56 x 35 = 1960 by 15 x 8 = 120.
Operation No. 1.

Operation No. 2.
120) 1960 (16)
120

56 x 35 49 1

16

15 x 8 3
760
720

Observe that in the above the factor 40 1

5 is stricken out of 15 and 35, and the 120 3

factor 8 is stricken out of the divisor and out of the 56 of the dividend.

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? +
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 x 24 x 2.

8. Multiply 250 by 72 and divide the product by 163 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. + Think of an acre as 40 rods hy 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}¢ a foot. 15. Of 243 ft. iron pipe at 343¢ a foot. 16. Of 243 ft. iron pipe at 35$$ a foot. 17. Of 243 ft. iron pipe at 3234 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.
20. Of 260 lb. butter at 26¢ a pound.
21. Of 260 lb. butter at 274 a pound.
22. Of 260 lb. butter at 24¢ 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 12¢ a pound.
25. Of 350 lb. coffee at 131¢ a pound.
26. Of 350 lb. coffee at 1434 a pound.
27. Of 350 lb. coffee at 1134 a pound.
28. Of 330 lb. coffee at 1634 a pound.
29. Of 330 lb. coffee at 1734 a pound.
30. Of 330 lb. coffee at 1534 a pound.
31. Of 240 lb. coffee at 25° a pound.
32. Of 240 lb, coffee at 264 a pound.
33. Of 240 lb. coffee at 27% a pound.

(gg) Find the sum of the ten results.

Find the cost

34. Of 24 tons coal at $24 per ton."
35. Of 34 tons coal at $3} per ton.
36. Of 44 tons coal at $44 per ton.
(hh) Find the sum of the three results.

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 27¢ 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 124¢ a yard. How many yards did I buy?

(min) 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 1 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:

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