household furniture, £143. 19s. 7d.; and in recoverable debts, £246. 17s. 7d. What will the creditors lose? 16. I bought two saddles, a black horse and a white horse; the first saddle cost £16. 10s. 8d.; the second, £11. 16s. 10d.; the black horse, £89. 19s. 6d. When the first saddle is put on the black horse, and the second saddle on the white horse, the black horse is of the same value as the white. Find the price of the latter. 17. In 1850, the spring lasted 92 days 21 hrs. 16′ 15′′; the summer, 93 days 13 hrs. 52′ 8′′; autumn, 89 days 17 hrs. 8' 3"; and winter 89 days 1 hr. 31′ 2′′. What was the length of the year, and how much longer were the two first seasons than the two last? MULTIPLICATION OF COMPOUND QUANTITIES. 242. We shall here make three general cases: 1st, to multiply a compound quantity by an integer. 2nd, to multiply a compound quantity by a fraction, or by a mixed quantity. 3rd, to multiply a compound quantity by another. 243. First case: To multiply a compound quantity by an integer. How much will 9 yards of cloth cost, at £2. 17s. 6d. per yard? £. S. d. 9 £25 17 6 It is evident that 9 yards will cost 9 times as much as 1 yard, that is to say, 9× 6d., 9× 17s., and 9× £2. Hence we see that every part of the multiplicand must be multiplied by the multiplier. It is indifferent by which unit we begin, but it is preferable to commence with the least, thus: 9× 6d.=54d.= 4s. 6d., the 6d. are set down under the pence, and the 4s. carried to the shillings; 9×17s.+4s.=157s., or £7. 17s., the 17s. are set down under the shillings, and the £7 are carried to the pounds; 9× £2+£7=£25. Thus, the whole cost of 9 yards is £25. 17s. 6d. EXERCISES. 1. What will 12 loads of corn cost, at £16. 12s. 8d. per load? 2. If one yard cost 18s. 84d., what will 16 yards cost? 3. How many acres are there in 18 farms, each of 56 acs. 3 rds. 28 po. 16 yds. ? 4. What is the duty on 24 gallons of rum, at 10s. 10d. per gallon? lb.; 5. A person bought of a grocer 12 lbs. of coffee, at 1s. 3d. per lb.; 14 lbs. of tea, at 4s. 5d. per lb.; 18 lbs. of lump sugar, at 84d. per lb.; 20 lbs. of brown sugar, 43d. per lb.; 8 lbs. of cocoa, at ls. 2d. per lb. ; 28 lbs. of soap, at 9d. per 10 lbs. of mould candles, at 1s. 1d. per lb.; 16 lbs. of dip candles, at 6ad per lb. What is the amount of the bill? 6. A bought of B 10 pairs of worsted stockings, at 3s. 8d. per pair; 8 pairs of silk, at 12s. 6d. per pair; 24 pairs of cotton, at 2s. 7d. per pair; 18 pairs of gloves, at 4s. 2d. per pair; 30 yards of flannel, at 1s. 8d. per yard; 24 ells of diaper, at 2s. 6d. per ell; 36 yards of Irish linen, at 2s. 8d. per yard. What is the cost of the whole ? 244. The student will have observed that this process becomes laborious when the multiplier exceeds 20; but any of the following methods will facilitate his work. 1st method: To multiply a compound quantity by 36 is the same as to multiply it first by 6, and then the product by 6, for 36=6×6. Likewise, 56 times a quantity is the same as 7x the quantity, and 8 times the product, for 56=7×8. Since 38 6x6+2, we multiply a quantity by 38 if we multiply it by 6 first, then the product by 6, and add twice the original quantity to the last product; or since 38 =4×10-2, multiply the quantity by 4, then the product by 10, and subtract twice the original quantity from the last product. Since 112 4x4x7, to multiply a quantity by 112 we should first multiply it by 4, then the product by 4, and this second result by 7. = Since 325 4x9x9+1, to multiply a quantity by 325, we might proceed to multiply it by 4, the result by 9, and the second result by 9, and to this product add the original quantity. For instance, in this question: what is the value of 273 cwt. of goods, at £2. 16s. 44d. per cwt. ? Since 1 cwt. costs £2. 16s. 44d., evidently 273 cwt. will come to 273 x £2. 16s. 41d., but 273=3x7x13. We shall then multiply the price of 1 cwt. by 13, this product by 7, and the last product by 3, thus :— 1. Find the price of 288 yards of cloth, at £1. 11s. 7d. per yard. 2. Multiply 12 days 15 hrs. 32 min. by 176. 3. What distance will an engine go in 16 days, working 8 hours per day, and going at the rate of 28 miles 6 fur. 19 po. 3 yds. per hour? 4. A chest of tea weighs 1 cwt. 3 qrs. 24 lbs. How much will 96 chests weigh ? 5. Required, the value of 29 quarters of corn, at £2. 5s. 34d. per quarter. 6. If I spend £1. 17s. 5d. a day, how much is that a year of 365 days? 246. 2nd method: Another way of performing questions of this kind would be to reduce the multiplicand to the lowest denomination contained in it, multiply this result by the multiplier, and then reduce the product back again to higher denominations. Thus, if one person receives £2. 14s. 6d. per day, how much will he receive in 164 days? Now, £2. 14s. 6d. 1309 halfpence, the sum received in 1 day; And 164 × 1309 h.p.=214676 h.p. = £447. 4s. 10d., the sum received in 164 days. 247. EXERCISES. 1. How much ditching will a man do in 42 days, at the rate of 3 po. 4 yds. 2 ft per day? 2. A ship's crew, consisting of 450 men, received £16. 17s. 94d. each. How many pounds were distributed ? 3. In a town, the inhabitants consume daily 324 cwt. 2 qrs. 17 lbs. 13 oz. of bread. How much will they use in one year? 4. One piece of cloth measures 28 yds. 3 qrs. yards are there in 97 pieces? How many 5. If 1 bale of cotton weighed 1 cwt. 1 qr. 13 lbs., how much would 272 bales weigh? 6. How much beer would a garrison of 2640 men drink in 97 weeks, allowing each man 1 pint 1 gill per day? 248. 3rd method: The following questions will illustrate this third method: Ex. Find the value of 346 tons, at £7. 10s. per ton. Since 1 ton cost £7. 10s., 346 tons will cost 346 x £7. 10s. Therefore, 346 x £7. 10s. = £2595 The operation is usually arranged in the following way : Ex. What is the value of 109 cwt., at £2. 17s. 9d. per cwt. ? Now, 109 cwt. will cost 109 x the price of 1 cwt., or 109 × £2. 17s. 9d. We may observe, first, that 17s. 9d. 10s.+5s.+2s. 6d. +3d. which = = 109 × 5s. = × 109 × 10s. = × £54. 10s. = 109 × 2s. 6d. = × 109 × 5s. = 4 × £27. 5s.= 249. This convenient method of performing compound multiplication is called Practice, from its daily use amongst merchants and tradesmen. It consists in decomposing the units of the subdivisions into aliquot parts or into fractions, whose numerator is unity, either of the highest denomination or of any other higher. A knowledge of fractions will soon make the pupil expert in finding the aliquot parts in any case. But the following observations will be found useful : 1st. Since 10s. £, to multiply a number by 10s. we have to take the half of the number, and the result is the answer in pounds. Example, 48 x 10s.=48× £1=£48= £24. 2nd. Since 6s. 8d. £, therefore 100 x 6s. 8d. 100x £= £100 £33. 6s. 8d. |