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4. In 11151360 square feet, how many acres ? 2721=1089 fourths, the divisor ; then 11151360=
44605440 fourths, the dividend, the quotient is 40960 square rods; then 40960 square rods
256 acres, the answer. 5. Reduce 1 square mile, or 640 sq. acres to rods. 6. Reduce 102400 square rods to acres. X.
Solid Measure. 1. In 25 tons of hewn timber, how many solid inches? 2. In 2160000 solid in., how many tons of hewn timber? 3. In 25 cords of wood, how many solid inches ? 4. Reduce 5529600 inches to cords. 5. In 98 tons of round timber, how many solid feet? 6. In 3920 feet of round timber, how many tons ? XI.
Time. 1. In 18d. 5h. 41m. 56s., how many seconds ? 2. In 1575716 sec., how many days ? 3. In 41 weeks, how many minutes ? 4. Reduce 413280 minutes to weeks. 5. In 33h. 5m. 12s., how many seconds ? 6. Reduce 119112s. to hours. XII.
Circular Motion. 1. In 6 signs 13deg. 45m. 21 sec., how many seconds ? 2. Reduce 697521 seconds to signş. 3. In 45° 56' 30'',
seconds ? 4. Reduce 165390% to degrees.
Questions. What does Reduction teach?
How are lower denominations re Of how many kinds is Reduction ? duced to higher ? What is Reduction Descending? By what would you multiply to reWhat is Reduction Ascending?
duce cwts. to lbs.? -days to hours ? &c. How are higher denominations re By what would you divide to reduce duced to lower?
lbs. to cwts. ? &c.
EXAMPLES FOR EXERCISE. 1. In 20144 farthings, how many English guineas, at 28 shillings?
Ans. 14 guineas, 27s. 8d. 2. A man borrowed 10 English guineas, and 24 English crowns, at 6s. 8d. ; how many dollars at 6s., will
3. In £48, how many shillings, nine-pences, six-pences, four-pences, and pence, and an equal number of each?
Direction. ----First reduce £48 to pence for a dividend, then add together 12d., (1s.) 9d., 6d., 4d., and id., for a divisor. Divide, and the quotient will be the number of each?
Ans. 369. 4. A goldsmith received three ingots of silver, each weighing 270z., which he was directed to make into spoons of 20z., cups of 5oz., salts of loz., and snuff-boxes of 2oz., and to have an equal number of each. What was the num ber?
Ans. 8 of each, and loz. over. 5. How many rings, each weighing 5pwt. 7grs., may be made of 3lb. 5oz. 16pwt. 2grs. of gold.
Ans. 158. 6. What will 5cwt. 2qrs. 181bs. of sugar come to, at 12 cents per pound?
Ans. $76,08. 7. What will 3 tons of iron come to, at 6 cts. per pound
Ans. $403,20. 8. A merchant soli 12hhds. of brandy, at 16 cents a · pint ; what did it amount to?
Ans. $1008. 9. In 12 bales of cloth, each 20 pieces, and each piece 25 Eng. 'Ells, how many yards ? Ans. 7500 yards.
10. At 20 cents a nail, what is the value of 3 pieces of broadcloth, each 25 yards?
Ans. $240. 11. How many times will a carriage wheel, 15ft. Sin. in circumference, turn round in going from Baltimore to Washington, it being 38 miles ?
Ans. 12739+times. 12. The forward wheels of a carriage are 12ft. hin. in circumference, and the hind wheels 15 feet 9 inches ; in running from New York to Boston, how many more times will the forward wheels turn round than the hind wheels, it being 223 miles ?
Ans. 19437. 13. One field contains 7 acres 2qrs., another 10 acres, a third 4 acres lqr. 30rd., which are to be divided into 15 shares ; how
rods into each share ? Ans. 234, 14. In 29 acres 1 rood of land, how many shares of 76 rods each?
Ans. 61 shares, and 44rds. over. 15. Admit a ship's cargo from Bordeaux to be 250 pipes, 130 hogsheads and 150 barrels : how many gallons in all, and allowing every pint to be a pound, what was the burden of the ship? Ans. 44415gals., and the ship's burden was
158 tons 12cwt. 2qrs.
16. How many days from the birth of Christ to Christe mas, 1834, allowing the year to contain 365 days 6 hours, or 3654 days?
Ans. 669868d. ==12h. 17. From March 2d to November 19th following, inclusive how many days ?
Ans. 262. 18. Suppose a rail-way car to run a mile in 3 minutes, how long would it be at that rate in running 1280 miles ?
Ans. 3d. 2h, 40m. 19. How many strokes does a regular clock strike in 365 days, or a year ?
Ans, 56940. 20. How long would it take to count the national debt of England, which is not less than $1900,000,000, at the rate of 50 dollars a minute, réckoning, without intermission, 12 hours a day, and 365 days to the year ?
Ans. 144 years 217d. 9h. 20m. 21. There is a certain piece of land 4 rods square ; how many rods does it contain?
Ans. 16 sq., rods. 4 rods.
Illustration.- A piece of land 1 rod long and 1 rod wide, contains just 1x1=l square rod. A piece 2 rods long and i rod wide contains 2x1=2 sq. rods, &c.; and a piece 2 rods long and 2 rods wide contains 2x2=4sq. rods. So a piece of land 4 rods long and 4 rods wide contains just 4X4=16 square rods. See
the annexed figure. Therefore, to find the number of square feet, rods, acres, miles, &c., in any figure that has four right angles, (corpers,) multiply the length and breadth together, and the product will be the answer.
22. How many square feet in a floor that is 14ft. long and 12 feet wide ?
Ans. 14 x 12=168. 23. How many acres in a piece of land 40 rods long and 20 rods wide ?
Ans. 800 rods =5 acres. 24. How many square feet in a floor that is 4 yards long and 4 yards wide, or 4 yards square ? Ans. 144 sq. ft.
Note. From the foregoing it will be seen that a square foot is a space a foot long and a foot wide, but of no thickA solid or cubic foot, is a foot long, a foot wide and
a foot thick. Thus, a block 2 feet long, 2 feet wide and 2 feet thick, contains 2 X2X2=8 cubic feet, &c.
25. How many solid feet in a pile of wood 8 feet long, 4 feet wide and 4 feet high?
Ans. 8 X 4X4=128. 26. How many cords of wood in a pile 8 feet long, 8 feet wide and 8 feet high?
Ans. 4 cords. 27. How many cords of bark in a pile 9 feet long, 8 feet wide and 4 feet high ?—Thus, 9 X8 X4=288 feet, and 288 solid feet=2 cords 32 feet. Ans.
28. How many cubic inches are contained in a room 4 yards long, 4 yards wide, and 3 yards high? Ans. 2239488.
29. How much wood is contained in a load 8 feet long, 4 feet 6 inches high, and 3 feet 9 inches wide ?
Ans. 1 cord 7 feet.
FRACTIONS. 1. Fractions or broken numbers are expressions for any part of a unit or whole number.
All fractions arise from the division of an integer or a unit into parts ; thus, when a whole thing is divided into 2 equal parts, these parts are halves; when into 3 equal parts, they are thirds; when into 4 equal parts, they are fourths; when into 5 equal parts, they are fifths; when into 6 equal parts, they are sixths; when into 7 equal parts, they are sevenths; when into 8 equal parts, they are eighths ; when divided into 9 equal parts, they are ninths, and so on: that is, the fraction takes its name or denomination from the number of parts into which the unit is divided. Thus, if the unit be divided into 15 equal parts, they are fifteenths ; if into 20 equal parts, they are twentieths, &c.
Fractions are of two kinds, Vulgar, and Decimal. II. A Vulgar Fraction is represented by two numbers placed one above the other, with a line drawn between them.
three five three 11 eleyen Thus, î=half. 7=fourths. 6 sixths. Z eighths. 12 twelfths.
III. The number below the line is called the denomina. tor, because it denominates or shows how many equal parts the integer or whole is divided into.
IV. The number above the line is called the numerator, because it numerates or shows how many of those equal parts are expressed by the fraction.
In division, the denominator is the divisor ; and the numerator the dividend, thus : Dividend=5
5 shows that 5:8=; Divisor=8
8 1. If we divide an apple into 2 equal parts, what part of the whole apple will one of those parts be? If an apple be cut into 4 equal parts, what will one of those parts be? What part of an apple will 2 of those parts be ?-will 3 be?
2. How many halves are equal to the whole of any thing? how many thirds ? how many fourths ? how many fifths ? how many sixths ? how many sevenths ? how many eighths ? how many ninths ? how
twelfths ? &c. 3. If 4 apples be equally divided among 6 boys, what part of 1 apple is each boy's share ?
1 apple among 6 boys would be of an apple apiece ; and 4 apples would be 4 times as much, or, the answer. 4. What part of 5 dollars is 1 dollar ?
Ans. 5. What part of 8 dollars is 5 dollars ?
Ans. 6. If you cut an orange into 8 equal parts, what part of the whole orange will 1 of those parts be?
7. If I divide an orange into 8 equal parts, and give 5 of those parts to Henry, and the other three to Eliza, what part of whole orange will each have ?
How are five eighths expressed?
V. Fractions are either proper, improper, single, compound, or mixed.
1. A simple or proper fraction is when the numerator is less than the denominator, and the fraction is less than a whole thing or unit ; as 1, 1, 1, 11, &c.
2. An improper fraction is when the numerator is equal to, or greater than the denominator; and the fraction is equal to, or greater than a unit or whole 1; thus, j, 1, , 12.
Obs. 3 thirds of any thing are equal to the whole, and 4 halves, 7 fifths, and 12 fourths, are each greater than unity or l; consequently these are called improper fractions.