28. If the dividend is and the quotient 61, find the divisor.

29. The dividend is 1217, quotient 3, remainder 1; find the divisor.

30. Find the G. C. M. and the L. C. M. of 833, 1127, 1421,

343.

31. Arrange in order of magnitude, 4, 14, 17, 18.

65

32. Find the L. C. M. of 15, 1, 102.

33. Find the G. C. M. of 85, 2, 14, and 61.

34. Convert into common fractions in lowest terms: 7.2011; 6.954; 5.303; 21.396.

35. Simplify 31 +41-3

51-772827+

62 34

36. Simplify 6+5×31-7

3+21-4

.24+.53

40. Simplify 1 of 24 + 67 ÷ 23 +(51+

2.2.64

41. Simplify .9 of § of 4 of 154.

42. What part of is?

43. What part of .390625 is .05?

44. .09 is what fraction of .2045 ?

45. Convert into decimals 4; 14; 9.

46. The G. C. M. of three numbers is 15, and their L. C. M.

is 450. What are the numbers?

CHAPTER XII.

MEASURES IN COMMON USE.

285. The unit of measure for lengths is a yard; and from this are derived the units of surface and volume.

286. The standard yard of Great Britain, as defined by act of Parliament, is the distance between the centres of two cylindrical holes in a certain bar of gun-metal, when the metal has a temperature of 62 degrees Fahrenheit. The standard yard of the United States conforms as nearly as possible to that of Great Britain.

320 rods, 1760 yards, or 5280 feet = 1 mile (mi.).

288. A quantity expressed with reference to a single unit is called a simple quantity; but a quantity expressed with reference to different units is called a compound quantity. Thus, 20 miles is a simple quantity; but 20 mi. 70 rds. 5 ft. is a compound quantity.

NOTE. A number expressed with reference to a particular unit is sometimes called a denominate, or concrete, number; and, in distinction, a number expressed without reference to a unit is called an abstract number. But it is to be observed that all numbers are abstract.

(1) Change 20 mi. to rods.

320 rds. 20 6400 rds.

Since a mile contains 320 rds., 20 miles contain twenty times 320 rds.

(2) Change 20 mi. 70 rds. 5 ft. to feet.

20 mi. 70 rds. 5 ft.

=

6400 rds.

320

6400

70

6470 rds.

161

3235

38820

6470

106755

5

=

70 rds. are added. Again, 6470 times 16 ft. 106,755 ft., to which the 5 ft. are added. Observe that multiplicand and multiplier are interchanged in the operation.

106760 ft.

(3) Change 6400 rds. to miles.

320) 6400

Since there are 320 rds. in 1 mi., in 6400

20 mi. rds. there are (6400 ÷ 320) mi. = 20 mi.

There are 161⁄2 ft., or 33 half-feet, in a rod; so the 106,760 ft. are changed to half-feet, and these to rods, by dividing by 33. The remainder is 10 half-feet

=

5 ft.

6470 rds. are changed to miles by dividing by 320, the number of rods in a mile. The remainder is 70 rds.

289. The process of changing the unit in which a quantity is expressed, without changing the value of the quantity, is called reduction.

290. If the change be from a higher denomination to a lower, it is called reduction descending; if from a lower to a higher, it is called reduction ascending.

291. To reduce a higher denomination to a lower,

Multiply by the number required of the lower denomination to make one of the higher.

If there be intermediate denominations, reduce step by step, and add to each result the given number of the denomination reached.

292. To reduce a lower denomination to a higher, Divide by the number required of the lower denomination to make one of the higher.

If there be intermediate denominations, reduce step by step.

15. If the height of a horse be 16 hands, how many feet is

his height?

16. If a train move 40 ft. in a second, what is its rate in miles per hour? (One hour = 3600 seconds.)

NOTE. The units of surface measure, except the acre, are squares, the sides of which are linear units. Thus, a square inch is a square each side of which is one inch in length.

Hence, in surface measurement, the scale ascends and descends by squares of the linear units; 144-122; 9 = 32; 304 = (5+)2; 2724=(16+)2,