£ s. d. 59 17 8 II penny), becomes 5 d., of which we set down the d., and “carry" the 5d. Multiplying the 8d. by II, we obtain 88d., to which we add the carried 5d.: we then have 93 pence, and this amount, when divided by 12 (the number of pence in a shilling), becomes 78. 9d., of which we set down the 9d., and carry the 78. Multiplying the 178. by 11, and adding the carried 78. to the product, we obtain 1948.: this amount, when divided by 20 (the number of shillings in a pound), becomes £9 148., of which we set down the 148., and carry the £9. Lastly, multiplying the £59 by 11, and adding the carried £9 to the product, we obtain £658 So that the required product is £658 148. 9žd. 658 14 9 NOTE. In the absence of Reduction, this result (obtained by means of Simple Multiplication) would appear under the form £649 1778. 88d. And £658 14s. 94d. would be obtained-although in a round-about way-by means of Compound Addition if £59 17s. 8d. were taken as addend 11 times. EXAMPLE II. There are 9 silver spoons, each weighing 1 oz. 6 dwts. II grs.: how much do they all weigh? oz. dwts. grs. I 6 II 9 18 Here we have to multiply 1 oz. 6 dwts. 11 grs. by 9. Multiplying the II grs. by 9, we obtain 99 grs. : this number, when divided by 24 (the number of grains in a pennyweight), becomes 4 dwts. 3 grs., of which we set down the 3 grs., and carry the 4 dwts. Multiplying the 6 dwts. by 9, and adding the carried 4 dwts. to the product, we obtain 58 dwts.: this number, when divided by 20 (the number of pennyweights in an ounce), becomes 2 oz. 18 dwts., of which we set down the 18 dwts., and carry the 2 oz. Multiplying the I oz. by 9, and adding the carried 2 oz. to the product, we obtain II oz. So that the required product is 11 oz. 18 dwts. 3 grs. I I 3 83. Rule for Compound Multiplication: Multiply the several denominations, successively, beginning with the lowest,-by the multiplier, and reduce as in Compound Addition; taking care to add to each product the number "carried" from the preceding product. £ s. d. 3 17 NOTE I. When the multiplier exceeds 12, but is resolvable into a pair of factors, neither of which is greater than 12, we proceed as in Simple Multiplication -first, multiplying the multiplicand by one of the factors, and then multiplying the resulting product by the other factor. Thus, if we wanted to find the rent of a farm containing 72 acres, at £3 178. 5d. per acre, we could imagine the farm divided into 9 fields of 8 acres each, in which case the rent of one field would be £3 178. 5d. x8=£30 19s. 8d., and the rent of the whole 9 fields £30 198. 8d. x9= £278 178. od. NOTE 2. 30 19 8 9 278 17 0 -The multiplier, when greater than 12, and not resolvable into factors, is sometimes broken up (as in Simple Multiplication) into the parts which its digits individully represent the rent of 365 acres of land, for instance, at £3 178. 5d. per acre, being calculated as follows: £ s. d. ΙΟ 38 14 7 1Ο 387 5 10 1161 17 6 = :: 3 In many cases, however, the pupil will some to proceed according to the Rule. the last exercise : 365xd=365 halfpence = 1824d. 365 × 5d.= 365 X 178.= 365 X £3.= 365×£3 178. 54d.= 1825d. *The rent of 60 acres is obtained from the multiplication of £38 148. 7d. (the rent of 10 acres) by 6. COMPOUND DIVISION. 84. Division is called COMPOUND when the dividend, or the divisor, or the resulting quotient is a compound number; or when all three being simple numbers the dividend and the divisor (whilst of the same kind) are of different denominations. We perform an operation in Compound Division when we divide £23 by 8, and write the quotient under the form £2 17s. 6d.; also, when we find how many sums of 158. each are contained in £3. 85. Compound Division is performed by means of Simple Division and Descending Reduction. EXAMPLE I.-If £658 14s. 91d. were divided equally amongst II persons, how much would each receive? Here we have to divide £658 14s. 91d. by 11. The division £ s. d. 11)658 14 91 59 17 8 of £658 by 11 gives £59 for quotient, and £9 for remainder. This remainder, when reduced, becomes (9×20=) 180 shillings, to which we add the 14 shillings in the dividend. Altogether, we then have (180+14=) 194 shillings, the division of which by 11 gives 178 for quotient, and 78. for remainder. This second remainder, when reduced, becomes (7 X 12 =) 84 pence, to which we add the 9 pence in the dividend. We then have, altogether, (84+9=) 93 pence, the division of which by 11 gives 8d. for quotient, and 5d. for remainder. This last remainder, when reduced, becomes (5X4=) 20 farthings, to which we add the 2 farthings (d.) in the dividend. Altogether, there then are (20+2=) 22 farthings, the division of which by II gives 2 farthings or d. for quotient, and no remainder. We thus find that each person's share would be £59 17s. 8d. This result could be obtained, although in a very roundabout way, by Compound Subtraction—in other words, by Simple Subtraction and Descending Reduction. Thus, taking LII as often as possible from £658, we should find that I could be given 59 different times to each of the II persons, and that the number of pounds then remaining would be 9. Taking IIS. as often as possible from 1948. (£9 14s.), we should next find that Is. could be given 17 different times to each person, and that the number of shillings then remaining would be 7. Taking 11d. as often as possible from 93d. (7s. 9d.), we should next find that Id. could be given 8 different times to each person, and that the number of pence then remaining would be 5. Lastly, taking II farthings as often as possible from 22 farthings (5d.), we should find that d. could be given 2 different times to each person, and that nothing would then remain to be distributed: each person's share being thus found to be, as before, £59 178. 8d. EXAMPLE II.-How many lengths of 2 ft. 8 in. each could be cut off a ribbon 17 yds. long? 86. Rule for Compound Division, (1.) when the Divisor is an abstract number: Divide the several denominations successively-beginning with the highest-by the divisor, taking care, when a portion of any denomination is left as "remainder," to include this portion, after having reduced it, in the next (lower) denomination. (2.) When the Divisor is a concrete number, reduce it and the dividend to the lowest denomination which occurs in either, and then find the quotient by Simple Division. NOTE I. When the divisor is an abstract number greater than 12, but resolvable into a pair of factors of which s. d. neither exceeds 12, we 8)365 17 112÷72 can proceed as in Simple Division first dividing the dividend by one of the factors, and afterwards dividing the resulting quotient by the other factor. If, for instance, it were required to divide 9)45 14.81+d. (Ist remainder.) 57+d. (2nd £365 178. 11d. by 72, we could resolve 72 into the factors 8 and 9, and proceed as in the margin. THE METRIC SYSTEM OF WEIGHTS AND MEASURES. "The great diversity of Weights and Measures which has existed in all countries has principally arisen from the lesser communities of which they were originally composed having each adopted its own system. In process of time these lesser communities were amalgamated into separate nations, with whose increase of population and trade the inconvenience of a variety of Weights and Measures soon made itself apparent, and the desire of establishing uniformity arose. France was the first country to relieve itself from its barbarous multiplicity of Weights and Measures, by adopting a uniform system. Louis the XVI., at the recommendation of the Constituent Assembly, invited, by a decree, all the nations of Europe, and particularly the King of Great Britain, to confer respecting the adoption of an international system of Weights and Measures. No response being given to this invitation, France committed the consideration of the subject to some of the most learned men of the age, who devised what is called the Metric system: the most simple, convenient, and scientific system of Weights and Measures in existence." The Metric system is so called from the circumstance that it is based upon the METRE, which is the standard of length. The metre is the ten-millionth (10.000.000), part of the distance -measured upon the meridian of Paris-between the equator and the pole, and is longer by about 33 inches than the English yard. The standard of superficial measure is a square described Report of Parliamentary Committee on Weights and Measures of United Kingdom: 1862. H· |