Ex. 4. If 1500 navvies working 10 hours a day can cut a canal 20 miles long, 20 feet broad, and 6 feet deep, how many soldiers will it take to make a trench 10 miles long, 30 feet wide, and 5 feet deep, supposing that they only work 6 hours a day, and that in the same space of time 7 navvies can do the work of 10 soldiers? Ex. 5. If 10 men dig a trench 80 yards long, 6 yards wide, and 2 feet 8 inches deep, in 6 days, how deep a trench can 96 men dig in 3 days if it be 240 yards long and 16 yards wide? Ex. 6. A can do a piece of work in 20 days, B in 12 days; A and B work at it together for 6 days, and then C joins it in 2 days; in how many days could C have done it? (1) In 20 days A can do all the work. .. in 1 day A can do of the work. .. in 6 days A does, ie., of the work. (2) In 12 days B can do all the work. .. in 1 day B can do of the work. .. in 6 days B does, i.e., of the work. Therefore, (3) In 6 days A and B, working together, do (1%) of the work, i.e., of the work. And, therefore, when A and B cease working, of the work remains to be done. The remaining fifth is done by C in 2 days. Therefore C could do all the work in 2 days X 5, i.e., in 10 days. Answer, 10 days. 201. The questions given in proportion are not always set in these perfectly simple terms; sometimes both care and ingenuity are required to prepare the question before it is stated as a proportion. We shall accordingly now proceed to give examples where the questions are not put exactly in the straightforward manner in which the previous examples have been given; yet we shall show how a little previous consideration will enable us to reduce such question to a plain proportion. Ex. 7. If 12 oxen and 35 sheep eat 6 tons 7 cwt. of hay in 4 days, how much will it cost per week to feed 4 oxen and 6 sheep, the price of hay being £3 158. per ton, and 2 oxen being supposed to eat as much as 5 sheep? If the consumption of 2 oxen that of 5 sheep, Hence the question becomes-"If it cost to feed 30 sheep + 35 sheep for 4 days, how much will it cost to feed 10 sheep + 6 sheep for 7 days? Ex. 8. If 6 horses are worth 14 cows, and 10 cows cost as much as 60 sheep, and 32 sheep cost £330, what is the value of 16 horses? Such problems may more easily be solved by the following process: (1) State the several equations under each other, as below Let the number of £ in the Answer. Then the value of 6 horses the value of 14 cows, =the value of 60 sheep, £330, and 2 the value of 16 horses. (2) Divide the product of the numbers in the right hand column by the product of the numbers in the left-hand column. I. 2. EXAMPLES FOR PRACTICE. If ro men can do a piece of work in 6 days, working 10 hours a day, how many hours a day must 8 men work to complete the same undertaking in 15 days? If the carriage of 8 cwt. for 60 miles be 198., what ought to be the charge for sending 36 cwt. a distance of 38 miles? 3. A railway company charges £1 88. 6d. for the carriage of 19 cwt. for 30 miles, what weight ought to be carried 21 miles for 10s. 6d. at the same rate? 4. If the sixpenny loaf weighs 36 oz. when wheat is at 558. per quarter, what would be the weight of an eightpenny loaf when wheat is 60s. per quarter? 5. If 3000 copies of a book of 11 sheets require 66 reams of paper, how much paper will be required for 5000 copies of 12 sheets? 6. Some repairs to an engine have occupied 5 men during 3 weeks, and have cost £41 10s. wages; some larger repairs are necessary, which it is expected will occupy 22 men 27 days: how much will be required to pay them at the same rate of wages? 7. The repairs to an engine have taken 12 men 5 weeks and 3 days, and have cost £46 155. wages; some larger repairs are necessary, which it is expected will occupy 16 men 7 weeks; how much will be required to pay them at the same rate? 8. If 6 iron bars 4 ft. long, 3 in. broad, and 2 in. thick, weigh 288 lbs., how much will 15 weigh, cach 6} ft. long, 4 in. broad, and 3 in. thick. 9. If a page of a newspaper, consisting of 6 columns of large print, contain 205.000 letters, how many letters will be contained on the page of another newspaper, which consists of 7 columns each as large as one of the former, there being on an average 48 letters in the second newspaper occupying the same space of 41 of the first. IO. II. 12. A plate of copper 2 ft. long, 9 in. wide, and 2 in. thick, is rolled into a sheet 3 ft. 6 in. wide and 18 ft. long; how thick will it be? Bar iron 1 in. square and 22 ft. long is rolled down to inch square; how long will it be? A horse-power will lift 33,000 pounds one foot high per minute; how many gallons of water per day will an engine of 120 horse-power lift to the height of 140 fathoms ? 13. If 35 men working 8 hours a day can build a wall 1700 yards long, 13 ft. high, and 2 ft. thick, in 130 days; how many hours a day must 40 men work to construct a wall miles long, 17 ft. high, and 2 ft. thick, in 210 days? 14. It has been calculated that a square degree (about 69 X 69 square miles) of water gives off by evaporation 33 mill ons of tons of water per day; how much may be supposed to rise from a square mile in a week? 15. When the mercury in the barometer stands at a height of 30 inches, the pressure of the air on every square inch of surface is 15 lbs.; what will be the pressure on the human body-supposing its whole surface to be 14 square feet, and that the barometer stands at 31 inches? 16. 5 compositors in 16 days of 14 hours long can compose zo sheets of 24 pages, 50 lines in a page, and 40 letters in a line; in how many days of 7 hours long may 10 compositors compose a volume to be printed in the same letter, containing 40 sheets, 16 pages in a sheet, 60 lines in a page, and 50 letters in a line? 17. If 1100 men make 10 miles of railroad in 3 months, how long will it take 2750 men to make 75 miles ? 18. The driving wheel of a locomotive engine 5 feet in diameter turned 2500 times in going 6 miles. Supposing the circumference of a circle to be 31416 times the diameter, find what distance was lost owing to the slipping of the wheel on the rail ? 19. If a man travels 90 miles in 3 days by walking 8 hours a day, in what time will he travel 540 miles by walking 6 hours a day? 20. 21. 22. If 8 men can dig a trench 100 ft. long, 3 ft. broad, and 4 ft. 6 in. deep in 9 days, how many will be required to dig a trench 80 ft. long, 5 ft. broad, and 2 ft. deep in 5 days? If 7 fires burning 10 hours a day consume 4 tons 10 cwt. of coal in 30 days, how much coal will be consumed by 20 fires in 12 days burning for 14 hours a day? If 5 pumps, each having a length of stroke of 3 feet, working 15 hours a day for 5 days, empty the water out of a mine, how many pumps with a length of stroke of 2 feet, working 10 hours a day for 12 days will be required to empty the same mine, the strokes of the former set of pumps being performed four times as fast as those of the latter? PROPORTIONAL PARTS. 202. As in some sort a sequel to the rule of proportion, we place next the method of finding, by proportional parts, into what portions any quantities should be divided, when the ratio which the several required parts bear to one another is given. Most questions under this head might be solved by making several statements in proportion; but the simpler process by which the result may be arrived at will now be explained. EXAMPLES. Ex. 1. Let it be required to divide the number into two parts which shall bear to one another the ratio of 67. We have here to find two numbers which shall together make up 9:, and shall be in the ratio of 6: 7. Now we may either say that the first of the two numbers in the given ratio is to the sum of those two numbers as the first of the required parts is to the whole number 91, which would give us the statement 6 : 13 :: Ans. : 91 Ans. X 91 = 42. Or we may directly apply the following rule : RULE LVIII. "Form fractions which shall have the numbers composing the given ratio as the numerators, and the sum of these numbers as the common denominator, take these fractional parts of the proposed quantity; they will be the parts required." This would give us― of 91 =6X7=42 of 917 X 749 The reason of the above rule may be thus explained:- and are clearly in the ratio of 67, as likewise are of 91 and of 91; for we may multiply and divide both the terms of any ratio by the same quantities without thereby altering the value of the ratio. Again, fs added to makes or 1; therefore of 91 added to 1 of 91 will make 91. But the conditions of the problem before us only required that we should find two numbers which were in the ratio of 6: 7, and which, when added together, would make 91. Hence of 91 and of 91 are the numbers required. Ex. 2. Divide the sum of £95 os. 21. among 3 persons in the ratio of, . . } + {+} = 1 + $ + &= }}. Therefore the fractions are ÷ 11, or; 11, or ; ÷ 13. or Y. and of £95 03. 2d. = 6 × (£7 68. 2d.) = £43 178. od. Ex. 3. Divide £1278 among three persons in the ratio of 13, 17, and 41. Ex. 4. A bonus of £140 is to be divided among 3 engineers in proportion to their wages. The first has received 201, the second £341, and the third 185: what amount of the £140 does each receive? If the terms of the given ratio are expressed as fractions, it is better to reduce them to integers, and then proceed as above. Ex. 5. Divide £1580 into parts proportional to 1, 11, 1. Multiplying by 20 : 1† : 1† : 17 = 30: 25 : 24 and the parts required are 9 of £1580 = £600, of £1580 £500, of £1580 = £480. Ex. 6. Gunpowder is made up of nitre, sulphur, and charcoal, which are mixed in this proportion, 75 parts of nitre to 10 of sulphur and 15 of charcoal; in half a ton of powder how many pounds of each ingredient ? 75 + 10 +15=100, and ton = 1120 lbs. Hence of 1120 of 11202 of 1120 of 1120 840 lbs. nitre. EXAMPLES FOR PRACTICE. 1. Divide tne number 100 into two parts which shall have to one another the ratio of 2: 3. 2. Divide the number 45 into three parts which shall be to one another in the ratio of 7, 5, and 3. 3. Divide the number 2679 into parts which shall be to one another in the ratio of 1, 1, and . 4. Of a certain dynasty Jrd of the kings are of one name, 4th of another name, 4th of another, andth of another, and there are 10 kings besides; what is the number of kings in the dynasty? 5. A bonus of £150 is to be divided among three engineers in proportion to their length of service in the employ. The 1st has served 2 years 2 months, the 2nd has served 1 year 9 months, and the 3rd only 3 months, how much should each receive? 6. A bonus of £120 is to be divided among three engineers in proportion to the pay received. The 1st has received £142, the 2nd has received £116, and the 3rd £42, how much will each get of the bonus ? |