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RULE.

Multiply each man's stock by the time it continued in trade, and consider each product a numerator, to be written over their sum, as a common denominator ; then multiply the whole gain or loss by each fraction, and the several products will be the gain or loss of each man. 2. A., B., and C. trade in company. A. put in $ 700 for 5 months ; B. put in $ 800 for 6 months ; and C put in $500 for 10 months. They gain $399. What is each man's share of the gain ?

Ans. A.'s gain $ 105, B.'s gain $ 144, C.'s gain $ 150. 3. Leverett Johnson, William Hyde, and William Tyler, formed a connexion in business, under the firm of Johnson, Hyde, and Co. ; Johnson at first put in $ 1000, and, at the end of 6 months, he put in $ 500 more. Hyde at first put in $ 800, and, at the end of 4 months, he put in $ 400 more, but, at the end of 10 months, he withdrew $ 500 from the firm. Tyler at first put in $ 1200, and, at the end of 7 months, he put in $ 300 more, and, at the end of 10 months, he put in $ 200. At the end of the year they found their net gain to be $ 1000. What is each man's share ?

Ans. Johnson's gain $348.02131, Hyde's $273.78731, Tyler's $ 378.1923T 4. George Morse hired of William Hale, of Haverhill, his best horse and chaise for a ride to Newburyport, for $ 3.00, with the privilege of one person's having a seat with him. Having rode 4 miles, he took in John Jones and carried him to Newburyport, and brought him back to the place from which he took him. What share of the expense should each pay, the distance from Haverhill to Newburyport being 15 miles ?

Ans. Morse pays $ 1.90, Jones pays $ 1.10. 5. J. Jones and L. Cotton enter into partnership for one year. January 1, Jones put in $ 1000, but Cotton did not put in any until the first of April. What did he then put in to have an equal share with Jones at the end of the year?

Ans. $ 1333.331.

Section 46.

DUODECIMALS.

Duodecimals are so called because they decrease by twelves, from the place of feet towards the right.

Inches are called primes, and are marked thus'; the next division after is called seconds, marked thus" ; and

So on.

OPERATION

1. Multiply 8 feet 6 inches by 3 feet 7 inches.

As feet are the integers of units, it 8 6 is evident, that feet multiplied by feet 3 7

will produce feet; and, as inches are 25 6 twelfths of a foot, the product of inches

4 11 6" by feet will be twelfths of a foot. For 30 56' 6"

the same reason, inches multiplied by

inches will produce twelfths of an inch, or one hundred and forty-fourths of a foot. Hence we deduce the following

RULE.

Under the multiplicand write the same names or denominations of the multiplier; that is, feet under feet, inches under inches, fc. Multiply each term in the multiplicand, beginning at the lowest, by the feet of the multiplier, and write each result under its respective term, observing to carry a unit for every 12 from each denomination to its next superior. In the same manner the multiplicand by the inches of the multiplier, and write the result of each term one place further towards the right of those in the multiplicand. Proceed in the same manner with the seconds, and all the rest of the denominations, and the sum of all the lines will be the product required. 2. Multiply eft. 3in. by 7ft. 9in. Ans. 63ft. 11' 3". 3. Multiply 12ft. 9' by 9ft. 11'. Ans. 126ft. 5' 3". 4. Multiply 14ft. 911" by 6ft. 11' 8".

Ans. 103ft. 4' 5" 84".

5. Multiply 161ft. 8'6" by 7ft. 10. Ans. 1266ft. 87". 6. Multiply 87ft. 1' 11" by 5ft. 7' 5".

Ans. 489ft. 8' 0" 21"q". 7. What are the contents of a board 18ft. long and ift. 10in. wide ?

Ans. 33ft. 8. What are the contents of a board 19ft. 8in. long and 2ft. llin. wide ?

Ans. 57ft. 4' 4". 9. What are the contents of a floor 18ft. 9in. long and 10ft. 6in wide ?

Ans. 196ft. 10' 6'. 10. How many square feet of surface are there in a room 14ft. 9in. long, 12ft. 6in. wide, and 7ft. Sin. high ?

Ans. 791ft. 1' 6". 11. John Carpenter has agreed to make 12 shoe-boxes of boards that are one inch thick. The boxes are to be 3ft. 8in. long, ift. 9in. wide, and ift. 2in. high. How many square feet of boards will it require to make the boxes, and how many cubic feet will they contain ? Ans. 280

square feet ; 66 cubic feet, 864 inches. 12. My garden is 18 rods long and 10 rods wide ; a ditch is dug round it two feet wide and three feet deep, but the ditch not being of a sufficient breadth and depth, I have caused it to be dug one foot deeper and lft. Ein. wider. How many solid feet will it require to be removed ?

Ans. 7540 feet.

Note 1. A pile of wood, that is 8 feet long, 4 feet high, and 4 feet wide, contains 128 cubic feet, or a cord; and every cord contains 8 cord-feet; and, as 8 is to of 128, every cord-foot contains 16 cubic feet; therefore, dividing the cubic feet in a pile of wood by 16, the quotient is the cord-feet; and, if cord-feet be divided by 8, the quotient is cords.

When wood is “corded " in a pile 4 feet wide, by multiplying its length by its height, and dividing the product by 4, the quotient is the cord-feet; and, if a load of wood be 8 feet long, and its height be multiplied by its width, and the product divided by 2, the quotient is the cord-feet.

NOTE 2. Small fractions are rejected in the operation. 13. How

many cords of wood in a pile 56 feet long, 4 feet wide, and 5 feet 6 inches high ? Ans. 9g cords. 14. How many cords of wood in a pile 23 feet 8 incheslong, 4 feet wide, and 3 feet 9 inches high?

Ans. 22 cords

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*

15. How much wood in a pile 97 feet long, 3 feet 8 inches wide, and 7 feet high?

Ans. 19 cords 34 feet. 16. If a pile of wood be 8 feet long, 3 feet 9 inches wide, how high must it be to contain one cord ?

Ans. 445 feet. 17. If a board be 1 foot 7 inches wide, how long must it be to contain 20 square feet?

Ans. 12 feet 715 inches. 18. From a board 19 feet 7 inches long, I wish to slit off one square yard; how far from the edge must the line be drawn ?

Ans. 51; } inches. 19. I have a shed 19 feet 8 inches long, 14 feet 6 inches wide, and 7 feet 6 inches high ; how many cords will it contain ?

Ans. 16 cords 5£ feet +. 20. I have a room 12 feet long, 11 feet wide, and 71 feet high; in it are 2 doors, 6 feet 6 inches high, and 30 inches wide, and the mop-boards are 8 inches high ; there are 3 windows, 3 feet 6 inches wide, and 5 feet 6 inches high ; how many square yards of paper will it require to cover the walls ?

Ans. 25208 square yards.

Section 47.

INVOLUTION.

INVOLUTION is the raising of powers from any given number, as a root.

A power is a quantity produced by multiplying any given number, called a root, a certain number of times continually by itself; thus,

3= 3 is the first power of 3 = 31.
3x3 = 9 is the second power of 3 = 32.
3 X3 X 3 27 is the third

power of 3
3x3 x3 x3=81 is the fourth power of 3 = 34.

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The number denoting the power is called the index, or exponent, of the power. Thus, the fifth power of 2 is 32, or 25; the third power of 4 is 64, or 48.

To raise any number to any power required, we adopt the following

RULE.

Multiply the given number continually by itself, till the number of multiplications be one less, than the index of the power to be found, and the last product will be the power required. 1. What is the 3rd power of 5 ? 5x5x5= 125 Ans. 2. What is the 6th power of 4 ?

Ans. 4096. 3. What is the 4th power of 3 ?

Ans. 81. 4. What is the 1st power of 17 ?

Ans. 17. 5. What is the 0 power of 63 ?

Ans. 1.

Section 48.

EVOLUTION,

OR THE EXTRACTION OF ROOTS.

EVOLUTION, or the reverse of involution, is the extraction or finding the roots of any given power.

The root is a number, whose continued multiplication into itself produces the power, and is denominated the square, cube, biquadrate, or second, third, fourth, &c., power, equal to that power.

Thus, 4 is the square root of 16, because, 4 x 4 = 16; and 3 is the cube root of 27, because, 3x3 x3= 27; and so on.

Roots, which approximate, are surd roots; and those, which are perfectly accurate, are called rational roots.

EXTRACTION OF THE SQUARE ROOT.

1. What is the square root of 625 ?

To illustrate this question, we will suppose, that we

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