Hence, to determine the denomination of the product of two factors in duodecimals,

RULE. Add the indices of the two factors together, and the sum will be the index of the product.

Ex. 1. A board is 6ft. 7′ 9′′ in length and 2ft. 7′ 5′′ in breadth; what is its area?

ten down, and the 1ft. is carried to the product of the feet, making 13ft. In like manner we multiply by the 7′ and then by the 5", setting the partial products as in the margin. Finally, the sum of these partial products is the product sought. Hence, 219. To perform Multiplication of Duodecimals,

RULE. By the rule for multiplication of compound numbers, multiply the multiplicand by each term in the multiplier, and write the terms of the several partial products in the order of their values, so that similar terms shall stand in a column together; the sum of the partial products will be the entire product.

4. What quantity of boards will be required to lay a floor 12ft. 6′ 4′′ long and 8ft. 3′ 6′′ wide? Ans. 103ft. 10′ 6′′ 2′′".

218. Rule for determining the denomination of a product? Explain philosophically and familiarly. 219. Rule for multiplication of duodecimals?

5. What are the contents of a granite block that is 6ft. 3′ long, 2ft. 4' wide, and 1ft. 3' thick?

Ans. 18ft. 2" 9". (See Art. 104). 6. How many feet of flag-stone in a walk 15ft. 6' long and 3ft. 4' wide?

7. How many solid feet of marble in a block that is 8ft. 3' long, 3ft. 6′ wide, and 1ft. 4′ thick?

8. How many cubic feet of earth must be removed in digging a cellar 15ft. 6′ long, 12ft. 8′ wide, and 6ft. 8′ deep?

9. How many feet in a stock of 8 boards, that are 10ft. 8′ long and 10' wide? Ans. 71ft. 1′ 4′′. 10. How many feet of boards 1' thick can be sawed from a a stick of timber that is 12ft. 8′ long, 10′ wide, and 8′ 4′′ thick, provided no timber is destroyed by the saw-cut?

11. How many cords of wood in a pile that is 18ft. 6′ long, 6ft. 8′ high, and 4ft. wide?

12. How many square yards of carpeting will cover a room that is 18ft. long and 16ft. 6′ wide?

13. Multiply 3ft. 6' 4" by 8ft. 9' 6".

DIVISION.

220. Division of duodecimals is like division of other compound numbers.

Ex. 1. Divide 24ft. 10′ 10′′′ 4′′′′ by 7. Also by 9.

NOTE. When both dividend and divisor are expressed as compound numbers, they may be reduced to the smallest denomination in either; after which divide, and the quotient will be units, i. e. feet; thus, 68ft. 10/8/ divided by 2ft. 8' equals 9920 384′′ = 251, i. e. 25ft. 10′, Ans.

220. How is division of duodecimals performed? How when the divisor is compound?

4. The area of a floor is 197ft. 1′ 8′′, and the length of the floor is 15ft. 8'; what is its width? Ans. 12ft. 7'.

5. The area of a garden walk is 89ft. 4' and its width is 2ft. 8' what is its length?

MISCELLANEOUS EXAMPLES IN COMPOUND NUMBERS.

1. If 152bush. 3pk. 3qt. 1pt. of wheat grow on 9 acres of land, how many bushels grow on 7 acres?

2. A man having 207m. 4fur. 25rd. lyd. to travel in 6 days, goes 30m. 3fur. 25rd. 5yd. on the first day, and 33m. 4fur. 20rd. 4yd. on the second day; how far per day must he travel to finish the journey in the remaining 4 days?

3. Multiply 3£ 15s. 6d. 1qr. by 857, and divide the product by 157.

4. I have a stock of 9 boards, which are 12ft. 8′ long and 10' wide. With these boards I wish to lay a floor 15ft. in length; how wide can I make it?

5. If 1 cubic foot of foot of granite weighs 2

water weighs 62 lb. 8oz., and if a cubie

times as much, what is the weight of a

block of granite 12ft. long, 1ft. 8′ wide, and 9' thick?

6. From the sum of 3wk. 6d. 16h. 20m. 18sec. and 2wk. 3d. 18h. 50m. 40sec. take the difference between 6wk. 5d. 8h. 25m. 30sec. and 5wk. 2d. 22h. 18m. 15sec.

7. What is the difference in time between Amsterdam 4° 44' east longitude, and Annapolis 76° 43′ west longitude?

8. When it is noon in Dublin, 6° 7′ 13′′ west longitude, it is 10m. and 161 sec. past 8 o'clock in the evening in Peking; what is the longitude of Peking?

9. How many days, hours, etc., from 30m. 20sec. past 3 o'clock, P. M., Feb. 8, 1864, to 40m. 25sec. past 8 o'clock, a. M., July 4, 1864, reckoning each month at its actual length?

10. Bought 3cwt. 2qr. 18 lb. of sugar at 8c. per pound, and sold of it at 8c. and the remainder at 94c. per pound; what was gained by the transactions?

11. What is the value in Avoirdupois Weight of 24lb. 6oz. 12dwt. 20gr. Troy Weight?

12. How long a time will be required for one of the heavenly bodies to move through a quadrant of a circle, if it moves at the rate of 1′ 3′′ per minute?

13. The distance from Eastport, Maine, to San Francisco, California, is about 2760 miles. If a man, starting from Eastport, travel toward San Francisco for 75 days, at the rate of 24m. 3fur. 20rd. per day, how far will he then be from San Francisco?

14. A certain island is 75 miles in circumference. A and B, starting at the same time, and from the same point, and going in the same direction, travel round this island, A at the rate of 24m. 3fur. 10rd., and B at the rate of 15m. 6fur. 20rd. per day; how far apart are A and B at the end of five days?

15. A merchant bought 125 barrels of flour, at 1£ 15s. 6d. per barrel, and afterward exchanged the flour for 260 yards of broadcloth, which he sold at 18s. 9d. 3qr. per yard; did he gain or lose, and how much?

16. How many feet of boards will be required to make 12 boxes whose interior dimensions are 5ft. 6', 4ft. 9', and 3ft. 8', the boards being 1' in thickness?

17. How many feet less are required to make 12 boxes whose exterior dimensions are like the interior of those in Ex. 16, the boards being of the same thickness? Ans. 111ft. 4'.

18. What is the difference of the capacities of the two sets of boxes described in Ex. 16 and 17? Ans. 122ft. 10'.

19. How many times will a wheel 9ft. 8in. in circumference turn round in running from Boston to Worcester, a distance of 44m. 4fur.?

20. How many gallons, wine measure, in a water tank 4ft. 6in. long, 3ft. 8in. wide, and 3ft. 9in. deep?

21. If a teacher devote 5h. 30m. per day to 50 pupils, what is the average time for each pupil ?

22. If a man, employed in counting money from a heap, count 75 silver dollars each minute, and continue at the work 12 hours each day, in how many days will he count a million of dollars?

23. How many pounds of iron in one scale of a balance, will poise 75 pounds of gold in the other scale?

PERCENTAGE.

221. PER CENT. is a contraction of per centum, a Latin phrase which means by the hundred; thus, ten per cent. of a bushel of corn means ten one-hundredths of it; i. e. ten parts out of every hundred parts; six per cent. of a sum of money, is six one-hundredths of the sum, i. e. $6 out of every $100; etc. NOTE. Instead of the words per cent., it is customary to use this sign, %; thus, 6 per cent. is written 6%; 43 per cent., 44%

222. The RATE PER CENT. is the number for each hundred; thus, 6% is 180,or .06, i. e. 6 parts for each hundred parts.

223. The PERCENTAGE is the sum computed on the given number; thus, the percentage on $200 at 6 per cent. is $12.

224. The BASE of percentage is the number on which the percentage is computed; thus, $200 is the base on which the percentage is computed in Art. 223; a bushel of corn is the first base mentioned in Art. 221.

225. The rate per cent., being a certain number of hundredths, may be expressed either decimally, or by a common fraction, as in the following