lb.

Troy Weight.

13. Reduce 3lb. 10 oz. 12pwt. to penny-weights.

oz. pwt. 3.. 10 .. 12 12 oz. 1 lb.

46 oz.

20 pwt.=1 oz.

932 pwt. Ans.

Here, in multiplying the 3lb. by 12, to reduce them to ounces, I add in the 10 ounces; and in multiplying the 46 oz. by 20, to reduce them to penny-weights, I add in the 12 penny-weights.

14. Reduce 4lb. 7 oz. Opwt. 8gr. to grains.

15. In 8oz. how many grains?

Ans. 26408 gr.

1 oz.=480 grains; therefore 480×8=3840 gr. the Ans.

17. In 9 lb. 8oz. 1 dr. 2sc. 19 gr. how many grains?

Long Measure.

Ans. 55799 gr.

19. Reduce 5 leagues, 2 miles, 4 furlongs, to rods.

Ans, 5600 rd,

20. In 18 rods, 2 yards, how many yards?

rd. yd. 2)18.. 2

5 yd. 1 rd.

9=1 of 18 rd.

90

Here the multiplier contains a fraction, and I perform the multiplication according to the rule for Prob. 2d in the Supplement to Simple Multiplication.

99

+2

101 yd. Ans.

21. Reduce 22 rods to feet.

Ans. 363 ft.

22. Reduce 2 miles, 7 fur. 30 rd. to feet. Ans. 15675 ft. 23. Reduce 1520 miles, 1 fur. 36 rd. 1 yd. to yards. Ans. 2675619 yd. 24. How many barley-corns will reach round the earth, supposing its circumference to be 24912 miles?

1 mile 190080 b. c.; therefore, 24912×190080= 4735272960 b. c. the Ans.

Square Measure.

Ans. 19 qr.

Ans. 1500 na.

Ans. 2106 in.

Ans. 4779.

28. In 29 acres, 3 roods, 19 square rods, how many square rods? 29. In 5 square yards, 8 sq. feet, 140 sq. inches, how many square inches? Ans. 7772.

30. The area of the State of Connecticut is about 4764 square miles reduce these to acres. Ans. 3048960 A.

31. St. Peter's church, (so called,) in the city of Rome, is 730 feet long, 520 wide, and 450 high; and it occupies about 8 acres and 2 roods of ground: Reduce these acres and roods to square feet.

8 acres, 2 roods,=34 roods; and' 1 rood=10890 square feet: Hence, 10890×34=370260 sq. ft. the Ans.

Cubic Measure.

32. In 5 cords and 47 cubic feet of wood, how many cubic feet?

Ans. 687.

33. Reduce 2 cords to cubic inches.

Ans. 442368.

34. Reduce 15 tons of round timber to feet. Ans. 600.

Wine Measure.

35. In 5 gallons, 2 qt. 1 pt. of wine, how many gills? Ans. 180.

36. It is said that 20000 pipes of Port wine are annually exported from Oporto in Portugal: how many pints are in that quantity? Ans. 20160000. 37. How many pint bottles may be filled from a hogshead of cider?

38. How many half gills in a gallon?

Dry Measure.

39. Reduce 75 bushels, 3 pecks, to quarts. 40. Reduce 4 bushels and 1 quart to pints. 41. Reduce 10 chaldrons to bushels.

Time.

Ans. 504.

Ans. 64.

Ans. 2424.

Ans. 258. Ans. 360.

42. How many seconds in a solar year; it being 365 days, 5 hours, 48 min. 48 sec.? Ans. 31556928.

43. The time which elapses between two successive changes of the Moon, (called a synodic revolution of the Moon,) is, at an average, 29 days, 12 hours, 44 min. 3 sec.: How many seconds are in that time? Ans. 2551443. 44. Reduce 52 weeks to days.

45. Reduce 8 years to calendar months.

Ans. 364.

Ans. 96.

46. In 18 centuries and 31 years, how many years?

Ans. 1831.

47. How many days from the birth of Christ to Christmas, A. D. 1832, allowing each of the 1832 years to consist of 3654 days? Ans. 669138.

Circular Measure.

48. The Moon moves each day, at an average, through 13° .. 10., 35" of the Zodiac: Reduce these degrees, &c. to seconds. Ans. 47435.

49. Reduce 9 signs, 13° .. 25' to seconds.

Ans. 1020300.

50. Reduce 5 degrees and 48 sec. to seconds.

Ans. 18048.

REDUCTION ASCENDING.

RULE.

Divide the given quantity by the number which it takes of that denomination to make 1 of the next higher, and the quotient will be of the said higher denomination, and the remainder, if any, will be of the same denomination as the dividend. In like manner reduce the quotient, thus found, to the next higher denomination, setting down the remainder, if any; and so proceed until the quantity is reduced to the denomination required: then, the last quotient, together with the several remainders, (if any,) of the lower denominations, will be the answer.

Or, instead of reducing the given quantity gradually from one denomination to another, as directed above, you may divide it by a number which will reduce it at once to the denomination required; that is, divide by the number which it takes of the denomination given to make 1 of that required. The divisors to be used in working by this last rule, will be found in the Supplemental Tables.

Note 1.-The remainder is always of the same denomination as the dividend.

Note 2.-When the divisor contains a fraction, the division must be performed according to the rule given in the Supplement to Simple Division. If, after dividing in this manner, there is a remainder, it must be divided by the denominator of the fraction belonging to the given divisor, and the quotient, thus found, will be the true remainder, or number of that denomination, to be set down in the answer.

PROOF. Reduction Ascending and Descending are the reverse of each other, and hence they reciprocally prove each other. The following questions are the reverse of the corresponding questions in Reduction Descending, and the answers to the latter may serve as proofs of those to the former.

EXAMPLES.

Money.

1. Reduce 26616 farthings to pounds.

Operation.
4)26616 q.
12)6654 d.
2,0)55,4 s. 6d.

Explanation. Here, because 4 farthings make 1 penny, I divide the 26616 farthings by 4, to bring them into pence; and the quotient is 6654 pence. Then, because 12 pence make 1 shilling, I divide those 6654 pence by 12, to reduce them to shillings; and the quotient is 554 shillings, and 6 pence remain, which I set down a little to the right hand of the Ans. 277. 14s. 6d. quotient. Then, because, 20 shillings make 1 pound, I divide the 554 shillings by 20, to reduce them to pounds; and the quotient is 27 pounds, and 14 shillings remain. So, the answer is 277. 14s. 6d.-The operation may be proved by reducing the answer to farthings, by Reduction Descending.

277. 14s.

7. In 12096 pence, how many guineas at 21 shillings each?

Ans. 48.

Note. From the foregoing Rule, and the 1st Contraction in Simple Division, it appears that Federal Money is reduced from lower to higher denominations by merely cutting off as many places as the given denomination stands to the right of that required; the figures cut off belonging to their respective denominations.

8. How many dollars are equal to 500 cents?

Here, 500-100-5 dollars, the Ans.

Or, what amounts to the same thing, by cutting, off the two ciphers at the right hand, 5 remains, which is the number of dollars, as before.

9. Reduce 48 dimes to dollars. 10. Reduce 7050 cents to eagles. 11. Reduce 14200 mills to dollars. 12. Reduce 804 cents to dollars.

Ans. 4D. 8 dm. Ans. 7E. OD. 5dm. Ans. 14D. 2dm. Ans. 8D. 4c.