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Exercises for the Slate or Black-board. 1. How much New York currency is equal to £98 Pennsylvania currency?
2. How much Pennsylvania currency is equal to £104 New York currency?
3. What sum in New England currency will pay a debt of £376 10s. in New York ?
4. What sum in New York currency will pay a debt of £282 7s. 6d. in Boston ?
5. How much New England currency must a Vermont merchant pay to cancel a debt in New York of £144 10s. ?
6. How large a debt in New York currency will be cancelled by the payment of £108 7s. 6d. New England currency?
7. How many dollars will pay a debt of £86 in Philadelphia ?
8. How many pounds Pennsylvania currency will be paid by $229 ?
9. How many Bergonia ducats, at 52d. each, can be had for 2163 Saragossa ducats, at 66d. each?
10. How many Saragossa ducats, at 66d. each, can be had for 275 Bergonia ducats, at 53d. each?
11. When the exchange from Antwerp to London is at £1 4s. 7d. Flemish for £1 sterling, how many pounds sterling must be paid in London to balance £236 Flemish at Antwerp? See “ Provincial currencies,” p. 227, for the manner of solving this.
12. When the exchange from London to Antwerp is at £1 sterling for £1 4s. 7d. Flemish, what must a merchant pay
in Antwerp for a bill of £192 sterling ?
13. In a settlement between C, of Philadelphia, and D, of London, C is found indebted £750 2s. sterling. How many dollars should he remit to pay this balance, when exchange is $4656 per pound sterling ?
14. A merchant of Philadelphia remitted to London property to the amount of $3420:456. For how much sterling money could he draw at the rate of $4.56 per pound sterling ?
b. Addition, Subtraction, Multiplication, and Division, of
Determinate Fractions. The only peculiarity in these processes consists in carrying by the denominator of the several fractions, or by the sub
divisions of the several units (accordingly as we consider the ‘numbers as determinate fractions or as compound numbers), in place of carrying by tens.
Exemplifications for the Black-board. 1. Add together £54 14s. 3d. 29.; £23 5s. 8d.; £65 19s. 6d.; £42 178. 4d. 39.; £36 15s. 8d. 19.; £9 17s. 6d.; and prove by adding from above downwards.
£ s. d. q.
£233 10 0 2
Suggestive Questions.—What is the amount of the column of farthings ? Equal to how many pence? To which column should the penny be carried ? What is the amount of the column of pence? Equal to how many shillings? To what column, then, does the 3 belong? What is the amount of the column of shillings? Equal to how many pounds ? To which column, then, does the 4 belong ?
Take 37 2 16 12 14 Subtrahend.
Proof, 224 0 20 8 12 Amount of Difference
and Subtrahend. Suggestive Questions.-Can 14 drams be taken from 12 drams? How many drams in an ounce? If, then, we change one of the 8 oz. the minuend into drams, how many
drams will there be from which to take the 14 drams? Since we have changed one of the ounces in the minuend into drams, how many ounces remain from which the 12 oz. may be taken ? Is it necessary, then, to change one of the pounds into ounces in the minuend? How many ounces will there then be ? Since one of the pounds has been changed into ounces, how many pounds remain from which to take the 16 pounds ? Is it necessary to change one of the hundredweights into quarters ? How many will then remain from which to take the 37 pounds ?
3. How long a period elapsed from May 3, 1852, to Jan. 15, 1854 ?
d. From 1854 1 15 Take 1852 5 3
Leaves ly. 8m. 12d. Ans.
4. Multiply 1 cwt. 2 qr. 14 lb. 12 oz. 3 dr. by 236, and prove it by division.
lb. OZ. dr.
2 329 2 2 5 8 product of 200. 49 1 17 13 10 product of 3x10=30. 9 3 13 9 2 product of
6. 388 cwt. 3 qr. 8 lb. 12 oz. 4 dr.(236 divisor, 236 Rem'r., 152 x4 Proof, 1 cwt. 2 qr. 14 lb. 12 oz. 3 dr., quot.
611 Rem'r., 139 x 25
1123 Rem'r., 179 X16
2876 Rem'r., 44x16
Suggestive Questions. The multiplication above will be readily understood. The questions that follow relate to the division. What is the quotient of 388 cwt. divided by 236 ? Of what denomination is the remainder, 152 ? How many quarters in a cwt. ? Why, then, is the remainder, 152, multiplied by 4? Why is the product 611 in place of 608 ? What is the quotient of 611 quarters divided by 236 ? Of what denomination is the remainder, 139 ? How many pounds in a quarter? Then why multiplied by 25 ? What is the quotient of 3483 pounds divided by 236 ? Of what denomination is the remainder, 179 ? How many ounces in a pound ? Why, then, is the remainder, 179, multiplied by 16 ? Why is the product 2876 in place of 2864 ? What is the quotient of 2876 divided by 236 ? Of what denomination is the remainder, 44 ? How many drams in an ounce? Why, then, is the remainder, 44, multiplied by 16 ? Why is the product 708 in place of 704? What is the quotient of 708 drams divided by 236 ?
Exercises for the Black-board or Slate. 1. Add together £14 12s. 5d. 29.; 6s. Od. 19.; £33; £67 4s. Od. 1q.; £3 15s. 6d. 29.; £29 19s. 9d.; £55 9d. ; £37 17s.
Ans. £241 16s. Od. 3q. 2. Add together 3 a. 2 r. 29 sq. rd. 6 sq. yd.; 15 a. 3 r. 17 sq. rds. 18 sq. yd.; 5 a. 3r. 6 sq. rd. 3 sq. yd.; 15 a. 1 r. 18 sq. rd. 2 sq. yd. Ans. 40 a. 2 r. 30 sq. rd. 29 sq. yd.
3. What is the difference between 9 cwt. 2 qr. 18 lb., and 6 cwt. 3 qr. 24 lb. ?
Ans. 2 cwt. 2 qr. 191b. 4. What is the difference between 9 m. 6 fur. 15 rods, and 18 m. 3 fur. 12 rods?
Ans. 8 m. 4 fur. 37 rods. 5. Find the time elapsed from March 19, 1838, to Feb. 17, 1852.
Ans. 13 y. 10 m. 28 d. 6. Find the time from Dec. 16, 1853, to Jan. 28, 1854.
Ans. 1 m. 12 d. 7. Find the time from May 21, 1796, to April 15, 1828.
Ans. 31 y. 10 m. 24 d. 8. Find the time from Jan. 1 to Oct. 16, 1853.
Ans. 9 m. 15 d. 9. Multiply 1 lb. 43 33 13 12 gr. by 6, 14, 150, and 325, separately, and prove each product by division.
10. Multiply 42 gal. 2 qt. 1 pt. by 6 and by 72, and prove each separate product by division.
11. Multiply 16 yds. 3 qr. 1 na. by 14, and prove by division.
Besides the multiplication, as above, by integers, there are two
other kinds of multiplication of determinate fractions, namely, that of one lineal fraction by another, to form surfaces and solids; and that of fractional coins by other determinate fractions. The first of these is called Multiplication of Duodecimals, or Cross Multiplication ; the other is called Practice.
1. Multiplication of Duodecimals, or Cross Multiplication.
It was stated above, p. 248, that the only peculiarity in the computation of determinate fractions was that of carrying by their varying denominations, in place of carrying uniformly by tens, as in integers. Cross multiplication, however, may seem an exception to this remark. But it is only a seeming exception, as will presently appear. A surface has been defined as the product of the length and breadth of the sides of any substance or space; solids are estimated by the product of their respective lengths, breadths, and depths. Now, it frequently happens that the extent of these sides consists of more than one denomination. For instance, a substance may be 6 feet 5 inches long, and 4 feet 1 inch broad, and in such a case the pupil might be at a loss how to multiply the one number by the other. But this difficulty vanishes, if we consider these dimensions as mixed numbers in common fractions, the foot representing the integer, and the inches the fraction. The length will then be 61', the breadth 41ż, and the multiplication can be readily performed as has been already shown. But the mode of computing such numbers by Cross Multiplication is shorter than that by common fractions, all changes from one kind of fraction to another being avoided. A single example will make the subject plain and easy. First, however, it is necessary to remark, that inches, in these operations, are called primes, or twelfths of a foot, and that the primes are subdivided into twelfths, called seconds, or 144ths of a foot, and those seconds into twelfths, called thirds. Hence their name, duodecimals, which signifies twelfths. They are respectively marked thus :