Exercises for the Slate or Black-board. 1. Multiply 17 ft. 7' by 1 ft. 5', and prove by resolution into its original elements by division. 2. Multiply 4 ft. 6' by 3 ft. 10', and prove. 3. How many cubic feet in a block 2 ft. 3′; by 6 ft. 5′; by 8 ft. 4'? Prove. 4. How many cubic feet in a block whose dimensions are 3 ft. 6', 2 ft. 1', and 1 ft. 2′? Prove. 2. Practice. Practice will be sufficiently understood from a few illustrations. 1. What will 6 cwt. 2 qr. 12 lb. of sugar cost, at £3 15s. 6d. per cwt. 2. What will be the cost of 55 bushels 3 pecks 5 quarts of wheat, at 10s. 2d. 3q. per bushel ? 3. What will 24 lb. of sugar cost, at $11.25 per cwt. [2 divisors] 25 5)11.25 2′25 price of 20 lb.= { cwt. 45 price of 4 lb= 265 price of 24 lb. cwt. To some students the last operation may appear more like division than multiplication. And, in effect, multiplying by or by, &c., really is division. For multiplication, it will be remembered, is taking the multiplicand as many times as there are units in the multiplier. . Exercises for the Slate and Black-board. : 1. What will 7 yds. 3 qr. 2 na. of cloth come to, at £2 2s. 6d. per yard? Ans. £16 14s. 8d. 1q. 2. What is the value of 6 cwt. 3 qr. 12 lb. of sugar, at £3 7s. 8d. per cwt. ? Ans. £23 4s. 1011d. 3. What would 37 T. 14 cwt. 2 qr. iron cost, at £5 14s. 8d. per ton? Ans. £216 5s. 9ąd. 4. What will 20 a. 2 r. 25 sq. rd. of land cost, at $29 per Ans. $603-5625. 5. What will 75 yd. 2 qr. of broadcloth cost, at $4.75 per yard? acre? 6. What is the value of 13 lb. ver, at £4 17s. 6d. per pound? Ans. 358 625. 10 oz. 12 dwt. 16 gr. of silAns. £67 13s. 10d. 3q. 7. What will 4 bu. 2 pk. 3 qt. of beans cost, at $1.121 per Ans. $5 166+. bushel ? 8. What is the cost of 7 hhd. 7 gal. 2 qt. of molasses, at Ans. £15 9s. 84d. £2 3s. 6d. per hhd. ? 9. What will 1 cwt. 3 qr. 12 lb. of raisins cost, at £2 11s. 8d. per cwt.? Ans. £4 16s. 7ęd. 10. What will 57 cwt. 3 qr. 8 lb. of cordage cost, at £3 Ans. £224 1s. 9d. 3q. 17s. 6d. per cwt.? 11. What will 14 gal. 2 qt. 1 pt. of milk cost, at 2s. 6d. per gallon? £1 16s. 6d. 3q. 12. What will 32 bu. 2 qt, 1 pt. of rye cost, at 2s. 3d. per bushel ? 13. What will 25 bu. 3 pk. 2 qt. bushel ? 14. What is the value of 3 cwt. £2 14s. per cwt.? Ans. £3 12s. 2d. of oats cost, at 1s. 6d. per Ans. £1 18s. 8d. 2q. 2 qr. 10 lb. of raisins, at Âns. £9 14s. 4d. 2q, 15. If 1 cwt. of rice cost $9.30, what is the value of 144 cwt. 2 qr. 21 lb. Ans. $1345'8. 16. What is the value of a silver tankard, weighing 1 lb. 7 oz. 14 dwt., at £3 16s. per lb.? Ans. £6 4s. 9d.+ CHAPTER IV. PRACTICAL APPLICATIONS OF THE METHODS OF INCREASE AND DECREASE, PROMISCUOUSLY ARRANGED. THE different modes of increasing and decreasing numbers, whether integral or fractional, having now been fully developed and illustrated, it will be proper to furnish the pupil with a variety of questions for practice, promiscuously arranged, to accustom him quickly to decide as to the appropriate mode of solution in every case likely to occur in practical business. The following general principles will aid in forming this decision. Still, however, much must be left to his own judgment in the application of the various modes of solving questions with which he has become familiar. I. All questions in which quantities of the same kind are to be counted together are solved by ADDITION; it being always remembered that quantities of different kinds cannot be numbered or added together, unless, by changing their denomination, we bring them to the same name. Thus, although a farmer may enumerate together 2 horses, 18 cows, 2 oxen, 4 calves, 75 sheep, and 8 pigs, their denomination must first be changed to some common term, such as live stock, &c. The same principle applies to the case of fractions, whether common or determinate. Those of different denominations cannot be added together without a change to one common denomination. Thus and cannot be added; but by changing to and to 19, they become capable of union, forming together 19. Again, 5 lb. and 12 oz. cannot be added. But the de 3 T nomination pounds may be changed to ounces, when the 5 lb. being equal to 80 oz., the whole forms 92 oz. II. When we wish to ascertain the difference between two numbers of the same kind we have recourse to SUBTRACTION. The same observations apply to this as to ADDITION, namely: that the numbers, whether integer or fractional, must be of the same kind or denomination before their difference can be ascertained. III. MULTIPLICATION applies to cases where a quantity occurs repeatedly; the number called the multiplier showing how often the repetition occurs. IV. DIVISION applies to cases where a quantity or number is to be divided equally among a number of persons, or into a number of equal portions. It is also applied to find the price of a single piece of which a number has been purchased for a certain price. In MULTIPLICATION and DIVISION it is not necessary that the separate numbers be of the same denomination, either in the case of integers or fractions. In the former case, the question is what is the amount of a certain number taken a certain number of times; in the latter, how many times is one number contained in another. Where one or more divisors and multipliers enter into a computation, the same result will follow, in whatever order they are taken; and these numbers may be either used separately, or collected into one product. Thus, if 20 is to be multiplied by 4, and by 5, and by 6, and divided by 3 and by 8, these numbers may be used in the order given, or in any other order whatever; or, to shorten the process, each series may be collected into one product. By this last method the 20 will be multiplied by 120, and divided by 24. The process may be still more abridged by using the quotient of these products in place of the products themselves, considering that quotient as a multiplier or divisor according as the one or the other proves to be the greater. Thus, the product of the multipliers being 120, and that of the divisors 24, the quotient 5 is a multiplier; whereas, had the product of the divisors been 120, and that of the multipliers been 24, the quotient 5 would have been a divisor. All this, however, is but another form of cancellation, as becomes evident when exhibited in a fractional form. Thus, Similar remarks apply when addition and subtraction enter into a computation. The order in which the numbers are taken is indifferent; and, in place of the respective numbers, their difference may be used, adding it when the sum of the additive numbers is the greater, and subtracting it when the sum of the subtractive numbers is the greater. When the number to be divided or diminished will not run into fractions by the process, it will always save time to divide before multiplying, or to subtract before adding. Exercises for the Slate or Black-board. 1. A merchant bought, in the spring, goods to the amount of $106,409, and on the first of January following found he had sold to the amount of $74,326; what amount of goods was left unsold? Ans. $32,083. Suggestive Questions.-What do we want to know here? Is it the sum of these two numbers, or their difference? or is either of those numbers to be taken a certain number of times, or to be divided into a number of equal portions? 2. A merchant bought 340 pieces of cotton. In every piece there were 26 yards. How many yards were there in all? Ans. 8840 yards. Suggestive Questions.-What is required here? In every piece 26 yards. How many in all? Is it the sum, difference, product, or quotient ? 3. A merchant making an inventory of his stock, finds he has cotton cloth to the value of $356, linen $152, broadcloth $575, cassimere $264 75, silk goods $254-25, ginghams $125, calicoes $240, and various small articles to the amount of $336'56. What is the whole value of the stock? Ans. $2303'56. 4. A man died leaving an estate worth equally divided among his six children. they have apiece? 5. A merchant, on New Year's day, sent his clerk to collect debts and make some purchases. He received from John Stokes $265, from William Budd $375, from Jacob Jones $526, and from Thomas Strickland $623. The clerk then bought at one of the cotton mills 360 yards of cotton cloth, at 6 cents per yard; 525 yards of calico, at 10 cents per yard; 240 yards of gingham, at 183 cents. He also bought at a |