Imágenes de páginas
PDF
EPUB

consequently, in dividing by 7, any number to which it is prime, the decimal period, which (260) begins at the comma, will (251) always consist of the same 6 figures; and these will succeed each other in the same order, although the period will not always commence with the same figure. Thus we find that

[ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

Hence, the student will perceive that the twelfth example (317) may be performed as follows:

Because,285714 t. = 2 of a ton, or (158) the seventh part of 2 tons, we divide 2 tons by 7, thus:

T. cwt. qrs. lbs.

7)2 0 0 0

0 5 2 24

Wherefore,285714 t. 5 cwt. 2 qrs. 24 lbs.

By operating decimally, thus:

[blocks in formation]

we find the same result. For, as ,999999 is the same as its denominator, it is equal to a unit; therefore 23,999999=24.

SECTION XVI.

ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION OF COMPOUND NUMBERS.

Addition of Compound Numbers.

319. We write the given numbers under each other, so that units of the same kind may be in the same column, and commence by adding the units of the lowest denomination. If their sum does not contain a sufficient number of units to compose a unit of the next higher denomination, we write it under its kind: but if it contains as many units as will make one or more, we reduce it (313) to that denomination, writing the remainder, if any, under the units of its kind, and adding the figures of the quotient to those of the same order in the next higher denomination, with which we proceed in like manner.

[blocks in formation]

As the sum of the pence is 32, which (313) ĝives 2s. 8d., we write 8 under pence, and add 2 to the unit column of shillings: then as the sum of this is 27, we write 7, and carry 2 to the column of tens. The sum of this is 5, and as each unit in this 5 is 10 s., two of these will make £1. We therefore take the half of 5 for pounds, which gives 2 and 1 over. This 1, which is 10 s., we place on the left of 7, and carry the £2 to the column of pounds.

Example 2.

[blocks in formation]

424

121

Here, the sum of the s. f. is 30, which, reduced (313) to s. yds., gives 3 s. yds. 3 s. f. We therefore write 3 s. f. under the column of feet, and carry 3 s. yds. to the column of yards. The sum of the yards is 106; and, as 301 make a pole, we divide 106 by 301121; that is, we multiply 106 by 11, which gives 11, or 3,61 P. Reducing this fraction of a pole to s. yds., we have 61121-1151 s. yds. We have, then, for the sum of the yards, 3P. 151 s. yds. We, therefore, write 15 under s. yds.; but, as it would be awkward to leave a fraction in connection with the s. yds. while we have a lower denomination in the question, we reduces. yd. to feet, in multiplying by 9, which gives 21 s. f.; and adding this to the 3 s. f. first written, we have 51 s. f., which we leave under that column. Then, carrying 3 to the column of poles, we find the sum 126, and, reducing this to roods, we have 3 R. 6 P. Writing 6 under poles, and carrying 3 to the roods, we find the sum 15, which, reduced to acres, gives 3 A. 3 R. Lastly, we write 3 under roods, and carry 3 to the column of acres, the sum of which is 129. Having completed the addition, the sum total is 129 A. 3 R. 6P. 15 s. yds. 51 s. f.

[ocr errors]

3. Required the sum of the four last numbers of the preceding example. Ans. 125 A. 3 R. 9 P. 18 s. yds. 81 s. f.

By adding this sum to the first number of Example 2, we shall again have 129 A. 3 R. 6 P. 15 s. yds. 51 s. f., and this serves as proof of the correctness of the first addition.

In the same manner the scholar may prove the following

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small]
[blocks in formation]

Subtraction of Compound Numbers.

320. We place the less number under the greater in the same manner as for addition; and, commencing with the lowest order, we subtract the units of each order in the less number from the corresponding units of the greater, writing each remainder under the order which gave it. But, if the number of units of any inferior order of the less number is greater than the number above it, we add as many units to the upper number as will make a unit of the next higher order, and, having subtracted the lower number from the sum, we add a unit to the next higher order of the lower number. The num

bers are thus both increased by the same quantity, which (84) does not affect their difference.

[blocks in formation]

Here, as we cannot subtract 1

=

2 from, we borrow a unit from the 5 s. f., which is equal to : then 1+1=4, and 27 =2, which we write underneath. Having borrowed a unit from the 5 s. f., instead of diminishing this by a unit, we add a unit (84) to the 8, which makes 9: then, as 9 is greater than 5, we borrow a s. yd., reducing this to s. f., and adding it to 5, we have 14: then 14 9 5, which we write under s. f. Having borrowed a s. yd., we add 1 to 18, which makes 19: then as 19 is greater than 15, we borrow a pole, which, reduced to yards, and added to 15, makes 451: subtracting 19, there remains 261: wherefore, we write 26 underneath, and reducing s. yd. to s. f. we have 24, which added to the 5 s. f. already found, gives 8. This we leave under the column of s. f. Having borrowed a pole, we add 1 to 9,

which makes 10: we then say, not 10 from 6, but borrowing a rood or 40 poles, 10 from 46, leaves 36, which we write underneath. We then add 1 to 3, which makes 4, and borrowing an acre, or 4 R., we say, 4 from 7 leaves 3, which we write underneath. Lastly, having borrowed an acre, we add 1 to 5, which makes 6, and 6 from 9 leaves 3, which, when written underneath, completes the operation.

This example is a proof of the second example, (319,) the greater number being the sum total of the five numbers added, the less, the sum of the four last numbers, and the remainder, the first number.

Note. We may, in all cases, reduce both numbers to a vulgar or decimal fraction of the principal unit, and find, after subtraction, the value of the result.

The student may prove the following examples by addition and subtraction :

2. From of 3 of £1, take 1⁄2 of 7 of 1 s.

Answer, 1 s. 101 d. 3. From 3 t. 0 cwt. 2 qrs. 5 lbs. 13 oz., take of § of 1 cwt. Answer, 2 t. 19 cwt. 3 qrs. 27 lbs.

4. From 3 A. 0 R. 4 P. 9 s. yds. 3 s. f., take

9 oz. 71 dr.

of 1 A. Answer, 2 A. 2 R. 30 P. 29 s. yds. 4 s. f. 72 s. in. 5. From ,428571 of a league, take of of a mile.

Answer, 5 fur. 5 p. 3 yds. 2 ft. 9 in.

6. From of a square mile, take 100 acres.

A.

Answer, 356 A. 1 R. 36 P. 5 s. yds. 69 s. f.

R. P. s. yds. s. f.

(7.) 23 2 21 19 6

15 3 35 24 7

£ 8. d.

(8.) 39 6 81
13 17 93

cwt. qrs. lbs. oz. drs. (9.) 22 1 13 7 8

11 3 17 8 11

bar. gal. qt.

(10.) 13 19 2

7 29 3

Multiplication of Compound Numbers.

To multiply a compound number by a simple or homoge

neous one:

321. A familiar instance, requiring this multiplication iswhen a number of units of the same kind is given, and the

« AnteriorContinuar »