PROP. V.-The fourth term of a proportion equals the third divided by the ratio of the third to the fourth. NOTE. Let the pupils demonstrate the above propositions. They are immediately derived from the first. EXAMPLES FOR PRACTICE. Find the terms not given in each of the following proportions: 251. Simple Proportion is employed for the solution of problems in which three of four quantities are given, so related that the fourth may be determined from them, by equality of ratios. The quantity required must bear the same re.ation to the given quantity of the same kind that one of the other two does to the second. We can then form a proportion and find the unknown term by the principles of proportion. 1. What will 20 yards of cloth cost, if 5 yards cost $15? SOLUTION.-It is evident that the cost of 20yd. bears the same relation to the cost of 5yd. as 20yd. bears to 5yd., hence we have the proportion, cost of 20yd. is to $15 as 20yd. is to 5yd., from which, by Prop. II., we have the cost of 20yd OPERATION. Cost of 20yd.: 15 :: 20:5 yd. yd. 60 5 20 X 15 $60. Hence the 5 RULE.-I. Put the unknown quantity for the first term, and the similar known quantity for the second term, and the other two for the third and fourth terms, according to the nature of the problem. II. Find the first term by multiplying together the second and third and dividing the product by the fourth. NOTES.-I. Other authors place the unknown term last, which would require us to reason in a reverse order. The former method seems more logical, since the law of reasoning is to compare the UNKNOWN to the KNOWN, and not the known to the unknown. II. It will be well for the pupil to place the unknown quantity in different terms, as a test of his understanding of the principles of proportion. III. Compound numbers must be reduced to a single unit value, and those of the same couplet to the same unit value. These problems may also be solved by analysis, and with young pupils it may be preferred to proportion. We will solve the first problem as a model. ANALYSIS.-If 5yd. cost $15, one yard cost of $15, and 20yd. cost 20 times of $15, or X15, which equals $60. OPERATION. Cost of 20yd.=>15=$60. 2. What will 27 yards of cloth cost, if 19 yards cost $57? Ans. $81. 3. What will 38 oranges cost, if 42 oranges cost 63 cents? Ans. 57cts. 4. How many yards of cloth will $144 buy, if 28yd. cost $112? Ans. 36. 5. What cost 78hhd. of molasses, if 13hhd. are worth $250 ? Ans. $1500. 6. What cost 132 acres of land, if 110 acres are worth $8250? Ans. $9900. 7. If $100 gains $6 in a year, how much will $250 gain in a Ans. $15. year? Ans. 58.50. 8. If 16 horses eat 26 bundles of hay in a week, how many will 36 horses eat in the same time? 9. If 75 horses cost $9000, how many horses can be bought for $16200? 10. If there are 84 privates in each panies in a brigade of 3360 men? Ans. 135. company, how many com Ans. 40. 11. If 25 head of cattle eat 36A. of grass in a month, how many cattle would 468A. keep the same time? Ans. 325. 12. If 79 men earn $395 in a week, how many men can earn $675 in the same time? Ans. 135 men. 13. How much will 34lb. of tea cost, if 8lb. 8oz. of the same kind of tea cost $11? Ans. $18. 14. In what time will the cars go from Lancaster to Philadelphia, 68 miles, at the rate of 5 miles in 10h. 45sec.? 15. If 19bu. of rye make 4bar. of flour, how will it require to make 19 barrels ?` Ans. 146th. many bushels Ans. 901bu. 16. How much will 28cwt. 3qr. of sugar cost, at the rate of 7cwt. 2qr. for $40.50? Ans. $155.25. 17. If a man spends $290 in the three spring months, at the same rate per day how much will he spend in a year? 18. If 12 men build a wall in 24 days, how long will it take 60 men to build it at the same rate? Ans. 4tda. NOTE. Here it is evident that the time in which 60 men do it is to 24 days, the time in which 12 men do it, as 12 men is to 60 men. 19. If 28 men mow a field of grass in 12 days, how many men will be required to mow it in 9 days? Ans. 37 men. 20. What is the height of a staff which cast 41ft.-of shadow, if a staff 6ft. 9in. cast a shadow 13ft. 8in.? Ans. 201ft. 21. If 134bu. of corn cost $6.25, what will 16bu. cost at the same price per bushel? Ans. $7.50. 22. If of a barrel of flour cost of an eagle, how many dollars will 15 barrels cost? Ans. $91.20. 23. If a person do a piece of work in 142 days, working 9 hours per day, in what time will he do it working 63 hours a day? Ans. 1891. Ans. 14da. 24. If 96 bushels of oats keep 42 horses 8 days, how long will 168 bushels keep them? 25. If 4A. 3R. of land cost $437, what will 16A. 2R. cost, at a rate of $25 less an acre? Ans. $1105.50. 26. If 17 men can mow a field in 9 days, how long would it take to reap half of it if 5 men refuse to labor? Ans. 63. 27. A failed and could pay only 75cts. on each dollar he owed; how much did C receive, whom he owed $1968? Ans. $1476. 28. If a 3 cent loaf weigh 9 ounces when flour is $6 a barrel, how much should it weigh when flour is $8 a barrel? Ans. 63oz. 29. Two cog-wheels, one having 28 and the other 20 cogs, run together; in how many revolutions of the larger wheel will the smaller gain 12 revolutions? Ans. 30. 30. A lent me $560 for 10 months; how long should I lend him $800 to reciprocate the favor? Ans. 7mo. 31. A and B together have $950, and 3 times A's is to 4 times B's as 5 to 6; required the shares of each. NOTE. Since 3 times A's: 4 times B's :: 5 : 6, 18 times A's equals 20 times B's, or A's of B's, which added to B's equals of B's $950, etc. 32. C and D have $6200, and 4 times C's money is to 5 times D's as 3 to 4; what sum has each? Ans. C, $3000; D, $3200. 33. E and F have $8970, and of E's money is to of F's as 8 to 6; how much money has each? Ans. E, $5382; F, $3588. 34. A man owns 1728 cattle, and of the cows is to % of the horses as is to §; required the number of each. Ans. 864 of each. 35. A man divided 367 acres between a son and daughter so that the son's share, plus 24 acres, was to the daughter's as 8 to 9; how many acres did each receive? Ans. Son, 160; D, 207. 36. A has grain worth $1.12 a bushel, and B has flour worth $6.25 a barrel; now if in an exchange A puts his grain at $1.25 a bushel, what should B charge for his flour? Ans. $6.944. 37. A garrison of 2400 men has provisions sufficient to last them 20 days, at the rate of 14lb. a day; how large a reinforcement could be received for the time if the allowance be reduced to 15oz. a day? Ans. 1440 men. COMPOUND PROPORTION. 252. A Compound Proportion is a proportion in which one or both ratios are compound. 253. A few of the more important principles are stated in the following propositions: PRINCIPLES OF COMPOUND PROPORTION. PROP. I.-The product of the simple ratios of the first couplet equals the product of the simple ratios of the second couplet. ILLUSTRATION. In the second of the above proportions from the principles of compound ratio we have XX, from which by reduction we have, which proves the proposition. PROP. II-The product of all the terms in the extremes equals the product of all the terms in the means. ILLUSTRATION. In the above demonstration we have XX and clearing of fractions we have 2X6X6X148X11X; which, by examining the terms, we see proves the proposition. PROP. III.-Any term in either extreme equals the product of the means divided by the product of the other terms in the extremes. PROP, IV.-Any term in either mean equals the product of the extremes divided by the product of the other terms in the means. 254. Compound Proportion is employed in the solution of problems in which the required term depends on a compound ratio. The subject is rather too difficult for elementary classes; the problems under it should therefore be solved by Analysis. We will solve a problem by both methods, suggesting, however, that young pupils employ the analytical method, which is clear and simple. |