(39.) SUBTRACTION OF COMPOUND QUANTITIES.

RULE 1. Place the smaller of the two quantities under the greater, so as that the several parts may be under those of the same denomination.

2. Begin with the lowest denomination, and subtract, if the upper number be large enough; if not, increase it by as many as will make one of the next higher denomination, taking care, in this case, to carry 1, after the subtraction, to the next number you subtract: and proceed in this way till the subtraction is finished.

For example, let it be required to subtract £173 17s. 91d. from £241 138. 74d.

£. S. d. 241 13 71 173 17 9/1/

£67 15 92

Placing the quantities, as in the margin, and beginning with the lowest denomination, you see that you cannot subtract 2 farthings from 1; you therefore increase the 1 farthing by 4, because 4 farthings make a penny: you then say, 2 from 5, and 3 remain; 2, and carry 1. 1 and 9 are 10, and, increasing the 7d., which is too small, by 12d., because 12d. make a shilling, you say, 10 from 19, and 9 remain; or, it is a trifle easier to say, 10 from 12, and 2 remain, and 7 are 9: carry 1. 1 and 17 are 18; 18 from 20, and 2 remain, and 13 are 15: carry 1. 1 and 3 are 4; 4 from 11, and 7 remain: carry 1. 8 from 14, and 6 remain: carry 1. 2 from 2, and nothing remains: therefore, the difference between the two sums is £67 158. 93d.

*Ex. 10, is to find the difference of latitude of two places on the earth, north of the equator; Ex. 11, is to find the difference of latitude of two places south of the equator; Ex. 12, is to find the difference of longitude of two places east of the meridian of Greenwich. The difference of latitude of two places, one north and the other south, is found by adding the two latitudes together; and the difference of longitude of two places, one east and the other west, is also found by adding the two together; what is called the difference being, in each case, the interval, in degrees, minutes, and seconds, between the two places.

(40.) MULTIPLICATION OF COMPOUND QUANTITIES.

To multiply a compound quantity by any number, the rule is as follows:

RULE. Place the multiplier under the quantity of lowest denomination. Multiply this quantity by it, divide the product by the number of such quantities contained in the next denomination, put down the remainder, and carry the quotient to the product arising from the next term: and so on to the end.

NOTE. When the multiplier is greater than 12, and can be decomposed into factors, each not greater than 12, use these factors instead of the composite multiplier, and proceed by short multiplication. The table of factors, at the end of the book, will be of great assistance in supplying the proper factors of all composite numbers up to 10000.

£. s. 17 13

d.

4

7

£123 13 7

Ex. 1. Multiply £17 13s. 44d. by 7. Putting the 7 under the lowest denomination, farthings, we multiply the 2 farthings by the 7, the product is 14 farthings; this divided by 4, the number of farthings in a penny, the next denomination, the quotient is 3 pence, and 2 farthings over; we therefore put down the 2 farthings, namely, d., and carry 3: we then multiply the 4 pence by the 7; the product is 28, which, with the 3 carried, make 31 pence; dividing these pence by 12, the number of pence in a shilling, the quotient is 2, with 7 pence for remainder; so we put down the 7 pence, and carry the 2 shillings. Multiplying now the 13s. by the 7, the product is 91, which, with the 2 carried, make 93 shillings; that is, 4 pounds 13s.: 13, and carry 4. And, lastly, multiplying the £17 by 7, and taking in the £4 carried, we have the whole product, £123 138. 74d.

2. Multiply £13 9s. 8d. by 693. By looking at the table at the end of the book, we find the multiplier 693 to be a composite number, of which the factors are 11, 9, and 7, we may therefore use these factors as multipliers, and proceed by short multiplication, as in the margin. You cannot require any explanation of the work after attending to the operations in the example just given, so I leave it for you to carefully look over, and thence to form your own opinion of the usefulness of the table of factors in calculations of this kind.

£9346 2 33

When, however, you have to multiply a compound quantity by a large number, which cannot be decomposed into factors suitable for short multiplication, you may seek in the table for the number nearest to it that can be so decomposed; employ the factors of this number, and note the result: then, multiply the compound quantity by the difference between the given multiplier and that actually used, add the result to the former result if the multiplier used be less than the given one, and subtract if it be greater.

There is another way of proceeding, thus: count the number of figures in the multiplier, disregarding the units figure. Multiply the compound quantity by 10, then the product by 10, and so on, till the number of 10's amount to the same as the number of figures counted: this done, multiply the compound quantity by the units-figure of the given multiplier, the first of the above products by the tens-figure, the next product by the hundreds-figure, and so on, till all the figures of the multiplier have been used: add up all these latter products, and the required product will be obtained. The work of the following example shows both methods.

66

NOTE.-You must always bear in mind, that a compound quantity can never be multiplied by another compound quantity; nor by anything but a mere number, since multiplication means the taking a proposed quantity a certain number of times. You must at once see the absurdity of the following questions, taken from a recent work on what the author calls "Arithmetic;" namely, "Multiply 7 tons by 9 cwt. ;" Multiply of a £ by of a guinea;" "Multiply 4 of an acre by of a rood ;" and so on. As Mr. Walker justly observes (Philosophy of Arithmetic, p. 58), "You might as well be told to multiply 5 lbs. of beef by 3 bars of music." I shall have occasion to direct your attention more fully to matters of this kind hereafter.