the unit column of acres. Having completed the addition, the sum total is 129 a., 3 r., 6 p., 15 sq.yds., 5 sq.f. (106.)

SUBTRACTION OF COMPOUND NUMBERS.

118. Write the proposed numbers under each other, as in addition, and commence the subtraction by the units of the lowest order. If the lower number can be subtracted from the

upper one, write the remainder underneath. If it cannot be subtracted, borrow from the next higher order a unit, which you will reduce to the order upon which you are operating, and add to the number from which you cannot subtract. Do the same for each kind, and, when you have been obliged to borrow, add a unit to the next higher order of the lower number, which is the same thing as if you diminished by a unit the number from which you borrowed. Lastly, write each remainder, as it shall be found, under the number which gave it.

As 9d. cannot be subtracted from 6d. I borrow 1s., which is 12 pence; this added to 6, gives 18, from which taking 9, there remains 9; I then add 1s. to 12, which makes 13, and subtracting this from 17, there remains 4: lastly I subtract 751. from 1431., and there remains 681.

Here, as I cannot subtract from 1, I borrow a square foot, then (86)1+1=5 and §-2=2, which I write underneath. Having borrowed a unit from the 5 S. F., instead of

diminishing this by a unit, I add a unit to the 8, which makes 9, then as I cannot take 9 from 5, I borrow a square yard, reducing this to square feet and adding it to the 5, I have 14, and taking 9 from 14 there remains 5, which I write under 8. Having borrowed a square yard, I add 1 to 18, which makes 19, and as this is greater than 15, I borrow 1 pole, which reduced to yards and added to 15, makes 451, subtracting 19, there remains 261; wherefore I write 26 underneath, and reducing of a square yard to square feet, I have 21 square feet; this added to 53 gives 8 square feet, which I leave under the column of feet. Having borrowed a pole, I add 1 to 9 which makes 10, then I say not 10 from 6, but borrowing a rood or 40 poles, I say 10 from 46 leaves 36, which I write underneath, and having borrowed a rood, I add 1 to 3, which makes 4, then borrowing an acre or 4 roods, 4 from 7 leaves 3, which I write under the roods: lastly I add 1 to 5, which makes 6, and 6 from 9 leaves 3, which being written underneath completes the operation.

This is a proof of the second example (117.) The number subtracted from, being the sum total of the five numbers added, the number subtracted, the sum of the four last, and the remainder the first.

3. Find the sum of 56A. 1R. 19P. 23yds. 8ft.....7A. 2R. 36P. 19yds. 3ft.....49A. 3R. 7P. 14yds. 7ft.....15A. 2R. 32P. 26yds. 8ft., and 233A. 1R. 29P. 12yds. 5ft.; after which prove by subtraction.

MULTIPLICATION OF COMPOUND NUMBERS.

119. We can always reduce the multiplication of compound numbers to the multiplication of a fraction by a fraction; the rule for which we have given (106.) For example, to find the cost of 54 sq. yds. 6 ft. of work, at the rate of 91. 4s. 6d. per square yard; we reduce the multiplicand 91. 4s. 6d. to pence (57,) which gives 2214 pence; and as the penny is the 240th part of a pound, the multiplicand may be represented by 2214 of a pound; in like manner, we reduce the multiplier 54 sq. yds. 6 sq. ft. to feet, which gives 492 square feet; and as the square foot is the ninth part of a square yard, we have for the multiplier 423 of a square

yard; so that the question is reduced to the multiplication of 2214 by 492, or (93) and (94) 369 by 164, which (106) gives 27715043 pounds, or (112) 504/. 6s.

24.0

This method extends to all kinds of compound numbers; but as it generally requires more calculation than that which we are about to explain, we shall not dwell upon it any longer.

120. A number which is contained exactly in another number is called an aliquot part of that other: thus, 3 is an aliquot part of 12; the same may be said of 2, of 4, and of 6.

Let us recollect, that to multiply is nothing else than to take the multiplicand a certain number of times: to multiply by 8, for example, is to take the multiplicand 8 times, and to take it again of a time, or to take the of it. Now, we can take the 3, either by taking the fourth and writing it three times, or by first taking the half, and afterward the half of this half: thus, to multiply 84 by 8, I write,

I

34 8

672

42 half of 84.

21 half of 42, or one-fourth of 34.

735 product.

In multiplying 84 by 8, I have 672. Afterward, to take the of 84, I first take the half, which is 42; after which, for the remaining fourth, I take the half of 42, which is 21; and adding these three products, I have 735 for, the total product.

121. To apply this to compound numbers, we must observe that the different kinds of units, inferiour to the principal unit, are fractions with regard to each other, and with regard to this principal unit; that, consequently, to multiply with facility these kinds of numbers, we must endeavour to divide them into aliquot parts of the principal unit, so that these aliquot parts may be employed conveniently; or to divide them into aliquot parts of each other; and if this division only furnishes aliquot parts which are inconvenient in

the calculation, we supply the defect by false products: this we shall exhibit in the following examples.

EXAMPLE I.

We demand the cost of 54 sq. yds. 4 sq. ft. of work, at the rate of $25 per square yard.

$cts. 25 00 54yds. 4.

ift.

100

125

12 50

1362 50

We first multiply, in the usual manner, $25 by 54. After which, to multiply by 4 feet, which is half a square yard, and which, consequently, ought to give only half the price of a square yard, we take the half of $25, and adding, we have $1362 50cts. for the total product.

We first multiply $25 by 54. Afterward, instead of multiplying by, because 7 feet make of a yard, we divide 78q. ft. into 4 ft. and 3 ft., of which the first is the half, and the second the third of a square yard; we therefore first take the half of $25, and afterward the third of $25, and adding all these products, we have $1370 33 3; that is, one thousand three hundred and seventy dollars, eighty-three cents, three mills, and of a mill, for the total product.

Note. In ordinary business, mills are not noticed unless they equal or exceed half a cent, in which case, we count one

Having multiplied by 5 cwt. we multiply by 3 qr. and to this end we divide this number into 2 qr. and 1 gr.; for 2 qr. we take the half of $72, which is $36; and for 1 qr. we take the half of 36 or the fourth of 72, which is 18. After which, to multiply by 14 lbs. instead of comparing these to the hundred weight, we compare them to the quarter, and as they are half a quarter, we take the half of 18, the price of a quarter, this gives $9. Lastly, having performed the addition, we have $423 for the product.

122. If the multiplicand be also a compound number, we proceed as in the following example :