(middle term or means.) Therefore, it is evident that any one of the four terms is readily obtained, when we have the other three given. Thus if 4 lbs. of tea cost 8 dollars, what will 12 lbs. cost? 8 4)96 D24 cost. quently, 4 pounds will cost 8 dollars, 12 pounds will cost 24 dollars, &c. Again: If 24 : 12 8 8 4 lbs. Ans. Hence we find, to obtain either the first or fourth term, divide the product 24)96(4lb. of the second and third (middle) terms, by the given extreme, and the quotient will be the other extreme, or term sought. And to obtain either of the middle terms, divide the product of the extremes by the given middle terms, and the quotient will be the middle term required. The first and second terms of a proportion always express quantities of the same kind, and so likewise do the third and fourth terms. Thus :: That is, if 2 pounds cost 6 dollars, what will 12 pounds cost? Ans. 36 dollars; it is 3 dollars per pound, and 3×12=36 dollars. This last example is the modern method of stating Proportion, and now considered the most correct. The old method was thus: If 2lb. cost 6D. :: what cost 12lb. : 36D.; 12× 6=72-2=36 as above. The following short rules can often be used to advantage :1. Divide the second term by the first, multiply the quotient by the third, and the product will be the answer. 2. Divide the third term by the first, multiply the quotient by the second, and the product will be the answer. lbs. 5: 10 :: 15 : 30 3. Divide the first term by the second, and the third term by that quotient, and the last quotient will be the answer. D. lbs. D. 4. Divide the first term by the third, and the second term by that quotient, and the last quotient will be the answer. Rule for stating.-1. Place that term in the third place, which is of the same name or kind, with that in which the answer is required; and if the answer requires to be greater than the third term, set the greater of the two remaining numbers for the second term, and the other number for the first terin. 2. But if the answer requires to be less than the third term, set the less of the two remaining numbers in the second place, and the other in the first place. Rule for working.-1. Reduce the third term to the lowest denomination mentioned in it. 2. Reduce the first and second terms to the lowest denomination mentioned in either of them. 3. Multiply the second and third terms together, and divide by the first, and the quotient will be the answer, in the same name to which the third was reduced; then bring this denomination into the answer required. When 1, or unity, is one of the terms, a formal statement is not required; it can be done by multiplication or division PROOF. 1. Multiply the first and fourth terms together (extremes). Multiply the second and third terms together (means). If the four numbers are proportional, these products will be *equal. 2. Divide the larger of the first and second terms by the less; divide the larger of the third and fourth terms by the less, and the two ratios will be equal. 3. Invert the question. The terms may be distinguished generally in the following manner the first term is preceded with an if, or supposition, and the second by a demand, "what will," "what cost," &c. 7. If 4 yards of cloth cost D9.50, what will 18 yards cost? cost? Ans. D42.75. 8. If 1 bushel of clover-seed cost D3.25, what will 7 bushels cost? Ans. D22.75. 9. If 6 bushels of turnips cost D1.25, what will 30 bushels Ans. D6.25. 10. When eggs are 18 cents a dozen, what will 47 dozen cost? Ans. D8.46.' 11. When eggs are 12 cents per dozen, how many can you have for 11.50? Ans. 92 dozen. 12. Purchased a barrel of mackerel for D11.50, on counting them there were 210 fish; what was the cost of 1? Ans. 5 cents, 4 mills. 13. Purchased a box containing 70 hats, for which I paid D280, what was the cost of 1 hat? Ans. D4 00. 14. If 1 pound of sugar cost 11 cents, what will 1200 pounds cost? Ans. D132.00. 15. If 5 bushels of salt cost D3.12.5, what will 50 bushels cost? Ans. D31.25. 16. If 4 men can perform a piece of work in 20 days, how many days will it require for 3 men to do the same? Ans. 26.666. 17. If 7 bushels of corn cost 13D., what will 40 bushels cost? Ans. D74.28.5. 18. When hay is 68 cents per cwt. what is it per ton? Ans. D13.60. 19. When sugar is 11.5 cents per pound how much is it per cwt. ? Ans. D12.88. 20. What costs a hogshead of molasses, when it is selling at 52 cents per gallon? Ans. D32.76. 21. If a barrel of wine cost 50 dollars, how much is that per quart? Ans. 39.7 cents nearly. 22. If a bushel of cloverseed cost 6 dollars, what is that per pint? Ans. 9.37 cents. 23. If 30 men can perform a piece of work in 11 days, how many men will it require to perform a piece of work 4 times as large in 12 days? Ans. 110 men. 24. How many men must be employed to finish a piece of work in 15 days which 5 men can do in 24 days? Ans. 8 men. 25. If 6 men can harvest a piece of grain in 12 days, in what time will 24 men do it? Ans. 3 days. 26. How long must a board be, that is 9 inches wide, to make 12 square feet? 9 in.: 12 :: 12 ft. Ans. 16 feet long. 27. If a pound of sugar cost 9.5 cents, what will be the price of a hogshead weighing 5 cwt. 2 qrs. 17 lbs.? Ans. D60.13.5. 28. If a person's daily expenses be D2.25, and his annual income 1250D, what sum will he save at the end of the year? Ans. 428.75. 29. How much tea can you purchase with 500D., when it is selling at D1.25 per pound? Ans. 400 pounds. 30. If a man is paying D2.12.5 per week for board, how much is it for 1 year? (52 weeks=1 year.) Ans. D110.50. 31. If 1 cwt. of sugar cost D11.50, what will 15 cwt. 2 qrs. 18 lbs. cost? Ans. D180.09.8. 32. If the annual income of a person is 800D., how much may he use daily, and have 300D. remaining at the expiration of the year? Ans. D1.37 nearly. 33. If 4.25 yards of cloth cost D16.25, what will 16.5 yar ls cost? Ans. D63.08.8. 34. If 1.25 yards of sheeting cost 12.5 cents, what will 50.5 yards cost? Ans. D5.05. 35. If 1.5 bushels of wheat cost D2.25, what will 20.5 bushels cost? Ans. D30.75. 36. If 20.5 bushels of wheat cost D30.75, how much is it per bushel? Ans. D1.50. 37. How much will 20.5 bushels of wheat cost, at D1.50 per bushel? Ans: D30.75. 38. A farmer sold 6 bags of wheat at 1.12 dollars per bushel of 60 pounds; the bags on being weighed, contained the following number of pounds of grain, namely: No. 1, 186 pounds; No. 2, 182 pounds; No. 3, 176 pounds; No. 4, 181 pounds; No. 5, 190 pounds; No. 6, 185 pounds; it is required to know how many bushels he had, and how much money he received. lbs. lbs. lbs. D. lbs. or 60: 1100 :: 1.12 then 60)1100(18 bushels. 186 182 176 |