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8. Reduce 7 of a mile to integers of lower denominations.

Process.
i mi. = 11 of 320 rd. = 2440 rd. = 20311 rd.
1 rd. = 1 of 53 yd. = ; yd. = 34 yd.
yd. = 4 of 3 ft.

= ft.
= 1 of 12 in.

- 6 in.

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- 14 ft.

3 ft.

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Analysis. Since there are 320 rd. in 1 mi., in fr mi. there are Ir of 320 rd. = 2037 rd. Since there are 54 yd. in 1 rd., in Ir rd. there are 1 of 5} yd. = 3} yd. Since there are 3 ft. in i yd., in tyd. there is į of 3 ft. = 14 ft. Since there are 12 in. in 1 ft., in

of a ft. there is ļ of 12 in. = 6 in.
Therefore, I of a mile = 203 rd. 3 yd. 1 ft. 6 in.
Reduce to integers of lower denominations :
9. } of a ton.

12. is of lb. Troy. 10. is of a hhd.

13. i of a sovereign. 11. f of a yr.

14. 4 of an acre. 15. Reduce 264 rd. 3 yd. 1 ft. 6 in. to the decimal of a mile. Process.

Analysis.-Since 1 in. is te of 12)6.0 in.

a ft., 6 in. are ft. = .5 ft., to

which the 1 ft. given is added 53 3.5 yd.

= 1.5 ft. Since 1 ft. is j of a 2 2

yd., 1.5 ft. is * of a yd. 11 17.0

to which the 3 yd. given are ad320264.631

ded 3.5 yd. Since 1 yd. is i 11 11

of a rd., 3.5 yd. are i rd. = 3520) 2911.00.82687 mi.

.6311 rd., to which the 264 rd. 28160

given are added = 264.631 rd. 9500

Since 1 rd. is to of a mile, 7040

264.637 rd. are 38-40 of a mile 24600

.82687 mi. Therefore, 264 rd. 3 21120

yd. 1 ft. 6 in. = .82687 mi. Rem. 3480

3)1.5 ft.

1.5 yd.,

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lb.,

Reduce to a decimal of the highest denomination : 16. 14 cwt. 9 lb. 12 oz. 19. 124 rd. 4 yd. 2 ft. 8 in. 17. 3 pk. 7 qts. 1 pt. 20. 56 gal. 3 qts. 1 pt. 2 gi. 18. 89 sq. rd. 8 sq. yd. 8 sq. ft. 21. 87° 29' 52". 22. What part of a ton is 14 cwt. 72 lb. 12 oz. ? Process.

Analysis.-Since 1 12 oz. = 12 * to lb. = { lb.

oz. is it of a lb., 12 72 lb. + 3 lb. = 723 lb.

oz. are ft lb.

to which the 72 lb. 724 lb. = 724 X ito cwt. = 281 cwt. 14 cwt. + 387 cwt. = 14781 cwt.

given are added. Since

1 lb. is ito of a cwt., 14187 cwt. = 200 x T. = $881 T.

728 lb. are 491 cwt., to which the 14 cwt. given are added = 14481 cwt. Since 1 cwt. is zt of a ton, 14430 cwt. is 1884 of a ton. Therefore, 14 cwt, 72 lb. 12 oz. =

$881 T.
Reduce to a fraction of the highest denomination :

23. 3 pk. q qts. 1 pt. 26. 185 rd. 4 yd. 2 ft.
24. 80° 14' 18".

27. 16 gal. 3 qts. 1 pt. 3 gi. 25. 36 sq. rd. 18 sq. yd. 28. 18s. 6d. 3 far.

OZ.

Addition of Denominate Numbers. 1.—What is the sum of 12 lb. 8 oz. 11 pwt. 9 gr.; 8 lb. 7 oz. 8 pwt. 8 gr.; 21 lb. 6 oz. 19 pwt.; 27 lb. 5 oz. 21 gr.? Process.

Analysis.—We write like de

nominations in the same column, lb.

pwts. gr. and begin with the lowest de12 8 11

9

nomination to add. 8

8

8

21 gr. 8 gr. and 9 gr. are 38 21 6 19 0

gr., equal to 1 pwt. and 14 gr. 27 5 0 21

We write 14 under the column of 70 3 19 14 gr. and add 1 pwt. to the column

of pwt. 20 pwt. 8 pwt. 11 pwt. are 39 pwt., equal to 1 oz. and 19 pwt. We write 19 under the column of pwt., and add 1 oz. to

the column of oz. 6 oz. 6 oz. 7 oz. and 8 oz. are 27 oz., equal to 2 lb. and 3 oz. We write 3 under the column of oz., and add 2 to the column of lb. 29 lb. 21 lb. 8 lb, and 12 lb. are 70 lb., which we write under the column of pounds.

2. Find the sum of £18 11s. 7.; £110 18s. 9d.; £188 13s. 1d.; £99 2s. 5d. 3 far.; £90 8d. 2 far.

3. What is the sum of 18 mi. 88 rds. 5 yds. 1 ft. 9 in.; 19 mi. 5 yds. 2 ft. 11 in.; 74 mi. 200 rds. 4 yds. 2 ft. 6 in.; 144 rds. 5 yds. v in. ?

4. What is the sum of 18 bush. 3 pks. 6 qts.; 23 bush. 4 qts.; 11 bush. 1 pk. 7 qts. 1 pt.; 3 pks. 1 pt.; 40 bush. 4 qts. 1 pt.; 29 bush. 6 qts. 13 pts.; 77 bush. 2 pks. 4 qts. ?

5. Add 15° 28' 47"; 24° 36' 52" ; 5 S. 20° 51' 11"; 9 S. 15° 34' 54".

6. Add 3 A. 110 sq. rds. 10 sq. yds.; 4 A. 3 sq. rds. 11 sq. yds.; 120 sq. rds. 20 sq. yds.

7. Add 4 of a mile and of a rod.

Process.

455 rd.

= 214 ft.

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7 mi. = 1 of 320 rd. =
$rd. = of 4 yd. = 314 yd.

Analysis.—Reduce the de1 yd. = 14 of 3 ft. 1 ft. = 17 of 12 in. = 93 in.

nominate fractions to inte

gers .of lower denominations, of u yd. 41. yd. and add. If a fraction of a tyd. iš of 3 ft.

= 1ft.

higher denomination occurs, ft. = of 12 in. = 9 in.

as the } yd. in this example, rd. yd.

ft. in. it must be reduced to inte45 3 2 9gers of lower denominations

4 1 9 and the result must be

46 2) 1 63 changed accordingly. 1 yd. =

1 6 4 yd. = 46

3 0

8. Add of a ton to # of a pound. 9. Add & A., & sq. rd., 4 sq. yd.

10. Illustrate addition of denominate numbers by a problem in long measure.

11. Illustrate addition of denominate numbers by a problem in cubic measure.

Subtraction of Denominate Numbers.

ART. 235.-1. From 33 gal. 2 qts. 1 pt. 3 gi. take 27 gal. 3 qts. 1 pt. and 2 gi.

Process.

Analysis.-We write like de

nominations in the same colgal. qts. pts. gi. 33 2 1 3

umn and begin with the lowest 27 3 1 2

to subtract. 2 gi. from 3 gi.

leaves 1 gi., which we write 5 3 0 1

under the column of gills. 1 pt. from 1 pt. leaves 0 pts., which we write under the column of pts. 3 qts. cannot be taken from 2 qts., so we increase 2 qts. by 1 gal., making 6 qts. 3 qts. from 6 qts. leaves 3 qts., which we write under the column of qts. Since we increased the minuend by 1 gal., we increase the subtrahend by 1 gal. 28 gal. from 33 gal. leaves 5 gal., which we write under the column of gallons.

Subtract:

2. 121 lb. 1 oz. 11 pwt. 17 gr. from 144 lb. 2 oz. 10 pwt. 18 gr.

3. 99 hhd. 24 gal. 3 qts. from 100 hhd. 20 gal. 1 pt.

4. 180 mi. 220 rds. 4 yds. 1 ft. 9 in. from 202 mi. 218 rds. 3 yds. 1 ft. 8 in.

5. A jeweler having 11 lb. of gold, used 1 lb. 2 oz. 5 pwt. 5 gr. in making watch chains, and y lb. 9 oz. 9 pwt. 20 gr. in other articles : how much gold was left ?

6. General U. S. Grant died July 23, 1885, and General G. B. McClellan, October 29 of the same year : what is the difference between the two dates ?

7. Sir Moses Montefiore was born October 24, 1784, and died July 28, 1885 : what was his age when he died ?

8. The steamship Alarm left New York at 4.15 P.M. December 19, 1885, and reached the Galveston bar at midnight December 25 : what was the time occupied in the passage ?

9. Alaska was ceded by Russia to the United States, June 20, 1867: how long was that after the discovery of America by Columbus, October 12, 1492 ?

10. How many years between the breaking out of the American Revolution, April 19, 1775, and the declaration of war against Great Britain, June 19, 1812 ?

11. War was declared against Mexico, May 13, 1846, and closed February 2, 1848 : how long did the war last ?

12. The war for the Union began at Fort Sumter, April 14, 1861, and General Lee surrendered April 12, 1865 : how long did the war last ?

13. The Military Academy at West Point was founded by Congress, March 16, 1802, and the Naval Academy at Annapolis was opened October 10, 1845 : how much time between the dates ?

14. The first passenger railroad was opened in England, September 27, 1825 : how long was that after the arrival of the Savannah (the first steamer which crossed the Atlantic) at Liverpool, May 24, 1819 ?

15. The first public telegram that ever passed over a wire was sent from Baltimore to Washington, May 27, 1844 : how long ago was it ?

16. Washington was inaugurated first President April 30, 1789 : how long was that previous to the first success of the telephone, May 14, 1877 ?

17. How long after the surrender of Cornwallis at Yorktown, October 19, 1781, was fought the battle of Waterloo, June 18, 1815 ?

18. Illustrate by an original problem subtraction of denominate numbers.

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