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Exemplification for the Black-board. 1. What are the solid contents of a block of marble 6 ft. 4' long, 3 ft. 5' wide, and 2 ft. 6' thick ?
2 7 8 Product of 6 feet 4' by 5', or
2 6 Thickness to opposite side. 43 3 4 Product of 21 ft. 7' 8" by 2 ft.
10 9 10 Product of 21 ft. 7' 8'' by 6'.
ft. 54 1' 2 Solid contents. Suggestive Questions.—Recollecting that a foot is the integer, and primes (1), seconds ("), and thirds ("''),
what are 3 times 4' ?
Equal to how many feet? 3 times 6 feet and 1 foot carried ? 5'X4' (or EX12), 20 of what denomination? Equal to how many primes? 5'X6 ft. (or X 6) 30, and 1 carried, of what denomination ? Equal to how many feet? [The same questions, varying only in the numbers and denomination, may be applied to the rest of the exemplification.]
The result of such computations as these may be restored to its original elements by division; that is, the thickness of a solid
may be found, if its solid. contents and the superficies of one of its sides be given; and one side of a superficies may be found, if its superficies and the other side be given. Thus, in order to prove the above computation, let 54 ft 1' 2" be the solid contents of a block of marble, and 21 ft. 7' 8" the superficies of one of its sides, to find its thickness :
ft. Solid contents, 54 1 2(21 7 8 (divisor) superficies of
10 9 10 2 6(quotient) thickness.
Again, given superficies and length, to find breadth :
ft. " ft. Superficies of a side, 21 7 8(3 5 (divisor) length of side. 1 1 8 6 4 (quotient) breadth of
Exercises for the Slate or Black-board. 1. Multiply 17 ft. 7' by 1 ft. 5ʻ, 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.
1 of 2 qr.
10 lb.= cwt. 1 62 price of
2 lb=} of 10 lb. 25 18 810 price of 6 cwt. 3 gr.
2. What will be the cost of 55 bushels 3 pecks 5 quarts wheat, at 10s. 2d. 39. per bushel ?
11 5 12 6 1
price of 11 bushels.
5 1 11 price of 2 pk.= i bushel.
4 qts.= peck. 3 314 price of
1 qt.=4 of 4 qts. £28 11 10 1jt price of 55 bu. 3pk. 5 qt.
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=z's cwt.
2.65 price of 24 lb. To some students the last operation may appear more like division than multiplication. And, in effect, multiplying by } or by 25, &c., really is division. For multiplication, it will be remembered, is taking the multiplicand as many times as there are units in the multiplier.
per ton ?
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. 12. 2. What is the value of 6 cwt. 3 qr. 12 lb. of sugar, at £3 7s. 8d. per cwt. ?
Ans. £23 4s. 104d. 3. What would 37 T. 14 cwt. 2 qr. iron cost, at £5 14s. 8d.
Ans. £216 5s. 93d. 4. What will 20 a. 2 r. 25 sq. rd. of land cost, at $29 per acre ?
Ans. $603-5625. 5. What will 75 yd. 2 qr. of broadcloth cost, at $4.75 per
Ans. 358-625. 6. What is the value of 13 lb. 10 oz. 12 dwt. 16 gr. of silver, at £4 17s. 6d. per pound ? Ans. £67 13s. 10d. 3q.
7. What will 4 bu. 2 pk. 3 qt. of beans cost, at $1•12) per bushel ?
Ans. $5.166+. 8. What is the cost of 7 hhd. 7 gal. 2 qt. of molasses, at £2 3s. 6d.
Ans. £15 9s. 8 d. 9. What will 1 cwt. 3 gr. 12 lb. of raisins cost, at £2 11s. 8d. per cwt. ?
Ans. £4 16s. 7 d. 10. What will 57 cwt. 3 qr. 8 lb. of cordage cost, at £3 17s. 6d.
Ans. £224 ls. 9d. 39. 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 ?
Ans. £3 12s. 2d. 13. What will 25 bu. 3 pk. 2 qt. of oats cost, at Is. 6d. per bushel ?
Ans. £1 18s. 8d. 29. 14. What is the value of 3 cwt. 2 qr. 10 lb. of raisins, at £2 14s. per cwt. ?
Ans. £9 14s. 4d. 2q.
15. If 1 cwt. of rice cost $9-30, what is the value of 144 cwt. 2
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. +
OF THE METHODS OF INCREASE AND DECREASE, PROMISCUOUSLY
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 ito i's and to 19, they become capable of union, forming together 13. Again, 5 lb. and 12 oz. cannot be added. But the de
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,
4X5 X6 120
=20 X =20X5. 3x8