£. S. Here the mo- d. ney must be the 160 : 47 3 21 :: 1 4 6 : 58 13 130 third term, 1 80 4 12 3 acre or 160 per. 12 the first, and 191 the other quan 40 147d. tity the second. This latter must 7661 be reduced to 147 perches, in order that the first 53627 two terms may 30644 be of one com- 7661 mon denomination, the com- 8,0)112616.7 mon denomination (perches) 12) 14077.d. being disregarded after the 2,0) 117,3 1d. reduction. Previously, how- £ 58 138. 135d. Ans. ever, we may simplify the first and third terms, by taking only the balf of each. This done, the money, for convenience of multiplication, is reduced to pence, and the work stands as in the margin. The yearly rent is thus found to be £58 138. 1d.; the fraction, which is only about one-third of a farthing, being disregarded. days. days. 4. If 37 workmen can complete 57 : 37 :: 241 · 15675 a piece of work in 241 days, in 37 how many days would 57 men finish it? 1687 As the answer is to be days, 723 241 days must be the third term; and since the greater the number 57) 8917(15644 of workmen the less the time, the 57 answer must be less than 241 days; therefore the first two terms 321 must be 57 : 37; and the work as 285 in the margin, the number of days being found to be 1567, days. 367 342 5 x 108.=98778.= 5. The following is an example in fractions : If i cwt. of lead cost 36 s. what will 3 cwt. cost ? Here, if we multiply the first 11:3} :: 364.: 98778. and third terms by 8, we shall 10 : 3 :: 2918. get rid of the fractions in those 291 x 17 terms, and the stating will then 5 be as in the margin. The second £4 188. 11 d. term, reduced to an improper fraction, is ¥; hence, multiplying the 2918. by this, and dividing by the 10, we find the answer to be 98878. Exercises. 1. If 17 lbs. cost 11s. 7d., what will 23 lbs. cost ? 2. If 9 lbs. of tea cost £1 188. 6d., what will 1 cwt. cost? 3. What is the price of six cheeses, each weighing 52 lbs., at 5 d. per lb. ? 4. If 28 persons reap a harvest in 36 days, how many will be required to reap it in 21 days? 5. If a garrison of 1000 soldiers have provisions for 9 months, how many must be dismissed in order that the provisions may last 15 months ? 6. A besieged garrison has 5 months' provisions, allowing 12 oz. a day for each man, but finding that it must hold out for 9 months, how much must each man have per day to make the provisions last? 7. What must be paid for 1 cwt. 3 qr. 17 lb. of wool, at 78. 4d. per stone of 14 lbs. ? 8. If 1787 cwt. 2 qr. of lead cost £907 10s., what is that per fother of 191 cwt. ? 9. If the entire rental of a parish amount to £2500, and a poor-rate of £112 28. is to be raised, what must a person pay whose rental is £525 ? 10. If five-eighths of a ship be worth £525, what is the value of three-fourths of three-sevenths ? 11. In a single mass, weighing 3 cwt., found in July 1851, at about 50 miles from Bathurst, in Australia, there were discovered to be 106 lbs. of gold; what would this fetch, at the rate of £3 68. 8d. per ounce? 12. From Sept. 29, 1850, to Sept. 27, 1851, there died of the population of London, within the walls of the city, 2978 persons, giving about 23 deaths for every 1000 persons living at the latter date : what was the amount of population of that part of London in Sept. 1851 ? gr. 13. From 1 lb. of standard gold, 444 guineas used to be coined: how many sovereigns are now coined out of the same weight? 14. 1 lb. avoirdupois is heavier than 1 lb. troy; for 144 lbs. avoirdupois are equal to 175 lbs. troy: what is the troy weight of 1 lb. avoirdupois ? 15. The imperial gallon contains 12 lbs. 1 oz. 16 dwt. 16 g troy weight of distilled water: how many pounds avoir dupois does it contain ? 16. At what time between 7 and 8 o'clock are the hour and minute hands exactly in opposition, or in the same straight line ? * 17. At what time between 5 and 6 o'clock are the hour and minute hands exactly together? 18. Eleven Irish miles are equal to 14 English miles : what is the length in English miles of a road which measures 57 Irish miles ? 19. A ream of paper contains 20 quires, and a quire contains 24 sheets: what would be the cost for paper for 2500 copies of a book containing 7 sheets, at 158. 6d. per ream ? 20. The average price of wheat for the year 1830 was 648. 3d. per quarter; and for the year 1850 it was 40s. 3d. : the sixpenny loaf in the latter year weighed 4 lb., what did it weigh in 1830 ? 21. The shadow of a cloud was observed to move 36 yards in 5 seconds : what was the hourly motion of the wind ? 22. If a person pays £22 78. 5d. for income-tax, at the rate of 7d. in the £, what is his income? 23. There are 18 dwt. of alloy in 1 lb. of standard silver; this 1 lb. is coined into 66 shillings: how much pure silver is there in 20s. ? 24. What was the weight of the £275000 taken in silver coin at the doors of the Great Exhibition of 1851, in tons, cwts., &c. avoirdupois ? * In order to work this exercise, the learner must remember that the minute-hand moves 12 times as fast as the hour-hand, so that while the minute-hand goes over any space, the hour-hand gains upon it 11 times that space. Now, it is plain, that under the conditions of the question, the gain of the hour-hand will be 1 hour-space, the space from XII to I; and it is required to determine what advance the minute-hand must make to allow of this gain, the space gone over by the minute-hand being always to the gain of the hour-band as 1 to 11. By the same considerations the next question may be easily answered. [For additional exercises in this rule, the learner may take the examples already given under the head of Practice.] (75.) DOUBLE RULE OF THREE, OR COMPOUND PROPORTION. The Double Rule of Three is so called, because it implies, at least, tro single Rule-of-Three statings, every question coming under this rule being resolvable into, at least, two questions, each of which may be worked by the single Rule of Three. This will be best understood by an example. Suppose 6 men can mow 9 acres of grass in 4 days, how many men will be required to mow 27 acres in 3 days? This question may be divided into two, thus : 1st. If 6 men can inow 9 acres in 4 days, how many men can mow the same in 3 days? Here, 3 : 4 :: 6 men : 8 men. Consequently, 8 men can mow the 9 acres in 3 days. 2nd. If 8 men can mow 9 acres in 3 days, how many will be required to mow 27 acres in the same time? Here, 9 : 27 :: 8 men : 24 men. It is plain, therefore, that the answer to the question is 24 men. In the first of these proportions, the fourth term is got by dividing the third term (6 men) by the ratio 3 : 4; that is, by multiplying by the fraction in the second, the fourth term is got by dividing what has just been found (8 men) by the ratio 9 : 27; that is, by multiplying by the fraction The answer to the original question is, therefore, obtained by dividing the third term in the first stating by the product of the ratios 3 : 4 and 9 : 27, that is, by 3x9 : 4x27, or, by 4 x 27 multiplying by 3 x 9 The product of two ratios is called their compound ratio, and this is why the Double Rule of Three is called also compound proportion ; and examples in it are usually worked, not by working with the several ratios singly, as above, but by taking the compound ratio, at once. In this way, the stating of the above question would have been written as follows: 3 : 4 :: 6 men; that is 3x9 : 4x27 :: 6 men; the 9 : 27 4 x 27 x 6 fourth term of which proportion is men = 4X3 x 2 3x9 = 24 men. There are sometimes more than two simple ratios to be compounded: thus, in the present example, it might have been a condition, that the 4 days occupied in mowing the 9 acres were 8 hours long, and that the days occupied in mowing the 27 acres were to be 12 hours long; then, as the longer days would require fewer men, a third ratio would have been 12 : 8; so that only iss, or ģ, of 24 men would have sufficed ; that is, 16 men. I think you will now be prepared for the following general rule for all questions of this kind. RULE 1. As in the Single Rule of Three, put, for the third term, that one of the given quantities which is of the same kind as the quantity sought. 2. Then, selecting any pair of the remaining quantities, like in kind, complete the stating, just as if these three were the only quantities given in the question, disregarding all the others. 3. In like manner, take another pair, like in kind, from the given quantities, and place them under the former pair, and so on, till all the pairs are used ; two dots, to signify ratio, being put between the terms of each pair; and these terms being arranged, as to first and second, just as you would arrange them if they and the third term were the only quantities concerned, 4. Multiply the third term by the product of all the consequents of these ratios, and divide the result by the product of all the antecedents: the quotient will be the answer. Note. The terms of each of the given ratios, together with the common third term, may be reduced, when possible, to smaller numbers, just as if they were the three terms of a simple Rule-of-Three stating; or the multiplications and divisions, implied in the rule, may be indicated by the signs for these operations; and factors, common to multipliers and divisors, struck out before the operations are actually performed; the factors that may be struck out, will often be discovered, by merely inspecting the simple ratios as they stand: thus, in the example worked above, the stating 3: 4 1:4 :: 6 : may be replaced by i: 3 9 : 27 ) :: 2 :; therefore, 1 : 12 :: 2 : 24. It is plain, that a factor common to any antecedent and a consequent, may always be struck out, since the consequents are all so many multipliers, and the antecedents so many divisors. You will, of course, understand, that what is called the common third } } |