LECTURE IX. DIRECT PROPORTION. (Sign) PROPORTION shews the direct relation of one object or thing to another, as to comparison or symmetry, viz: form, size, length, breadth, depth, rate, price, &c. But in relation to Arithmetic or Geometry, it has a determinate meaning. EUCLID has proven his Theory of proportions in the fifth book of his Elements by demonstrating its principles, relations, and application to lines; therefore, it is apparent, that we thence derive the name of Geometrical Proportion. In Direct Proportion, the first term is to the second as the third term is to the fourth, that is, as 2 is to 4, so is 8 to 16, written thus: 2:4 :: 816, these numbers are proportional, because, agreeably to the 16th proposition of Euclid, Lib. 6, the rectangle or product of the extremes is equal to the product of the means; hence, 2 x 16 = 4 x 8 32 or = 1o 2. From this it is plain, that Proportion is the combination of two equal ratios, and that there are two antecedents and two consequents. = = Note to Teachers.-Require the learner to recite the answers to the following questions. Q. Why is this called the rule of Proportion? A. Because it shews the combination of two equal ratios. Q. What is the first term of a proportion called? Q. What is the second term called? A. The consequent. Q. If 2 yards of muslin cost 25 cents, what will 6 yards cost? Ans. 75c. Q. Why is the second term (25 cts.) called the first consequent? A. Because it is the value or cost of the antecedent. Q. How are the terms arranged? A. As the first term (or antecedent) is to the second (its consequent) so is the third, (or antecedent) to the fourth, (or its consequent.) Q. How is the operation performed? A. Multiply the second and third terms together, and divide the product by the first. Q. What is the first antecedent called? A. When the first and third terms can be brought to the same denomination. Q. What is the general rule for stating? A. When a question is written thus: for instance, at 10 cts. per lb., what will 10 lbs. cost? or if 1 lb. cost 10 cts. what will 10 lbs. cost? In either case, the first term is 1 lb., then say as the first antecedent is to its consequent, so is the second antecedent to its consequent. Q. When the commencement of a question is written with the words, what is the value; how much, bought, or sold; what is the rule? A. In either case, the first object or thing mentioned in the question will be the third term. Q. What terms of a proportion can be contracted or reduced? A. The first and third, or first and second. Q. Why except the second and third terms? A. Because the means do not shew a proper relation to each other as to cost or value; hence, it is evident, that the second or middle term must be of the same name or denomination with the answer (or 'demand.') Q. What is meant by relation as applied to proportion? A. Nothing more than the quotient of a division. 1. If 3 lbs. of sugar cost 24 cents, what will 10 lbs. cost? Ans. 80c. lbs. cts. As 3 24 lbs. 10 80c. or by contraction, As 1 : 8: : 10 = 80c. 1 lb. and 10 lbs. are the antecedents. 8 cts. and 80 cts. the consequents. The relative proportion of the antecedents and consequents is the same. Thus, fcts. = cts. = 8. 2. If 3 yards of cloth cost $9, what will 12 yards cost? Ans. $36. 3. What is the value of 9.7 Ibs. of silver at $1.5 per ounce? Ans. $174.60. 4. What will 240 bushels of wheat amount to at the rate of $6 for 5 bushels? Ans. $288. 5. How much will 17 cwt. 3 qrs. 14 lbs. of iron cost at $4.75 per cwt.? Ans. $84.90§. 6. Sold 120 bushels of corn for $54, how much did it cost per bushel? Ans. 45c. 7. Bought 29 yds. of muslin for $10.873, how much is it per yard? Ans. 37 c. 8. Bought 2 loads of corn, one containing 75 bushels and the other 87 bushels, at 52 cents per bushel, what is the amount? Ans. $84.24. 9. Bought 3 pipes of brandy containing 120, 1241, and 123 gallons at 433 cts. per gallon, how much is the amount' Ans. $161.217. 10. If a staff 4 feet long cast a shade on level ground 7 feet long, what is the height of a steeple whose shade at the same time is 198 feet? Ans. 1134. 11. An express who rides from Washington city had been dispatched at the rate of 60 miles per day for five days, when a second was sent to overtake him, in order to do which, he must travel 75 miles a day, in what time will he overtake the former? Ans. in 20 days. 12. A prize of $2,329 was divided between two persons, A. and B. whose shares therein were in proportion as 5 to 12, what was the share of each? Ans. A. $685, and B. $1644. 13. When a man's yearly income is $949, how much is it per day? Ans. $2.60. 3 qrs. 24 lbs. at 183 cts. per lb. 14. Bought a stove weighing 4 cwt. $2.10 per cwt. and 27 lbs. of pipe at with 2 elbows at 50 cts. each, what is the stove pipe and elbows? price of the Ans. $16.483. 15. Bought 4 pieces of linen, viz: No. 1 and 2, each contained 27 yards, No. 3 and 4, contained each 25 yards at 62 cts. per yard. what was the cost? Ans. $66.561. 16. If a person's salary be $1,333 per annum, and his daily expences $2.14, how much will he save? Ans. $551.90. 17. What will 4 pieces of sattinet, containing 23, 24, 25, and 27 yards come to at 72 cents per yard? Ans. $71.28. 18. A farmer upon measuring his corn produced by a certain field, found he had but 48 bushels. It appeared that it yielded one third more than was sown, how much was that? Ans. 36. 19. A bookseller sold 10 books at a certain price, and afterwards 15 more at the same rate; now at the latter sale he received $2.25 more than at the former, what did he receive for each book? Ans. 45c. CONTRACTED OPERATIONS. CASE I. If the first term be a multiple or part of the second, the third will be a multiple or part of the fourth. Example 1.-If 6 yards cost $18, how much will 24 yards cost? Ans. $72. CASE II. If a part of the first be added to or subtracted from the first so as to be equal to the second, a like multiple must be added to or subtracted from the third. Example 1.-If 12 yards of cloth cost $15, what will 20 yards cost? Ans. $25. 2. If 6 yards cost $9, what will 24 yards cost? Ans. $36. SOLUTION.-Agreeably to Case II. the difference between the first and second terms is 3, and as 3, is the of the third term must be added to it. half of 6, Thus, 24 + 2 3. If 8 yards cost $12, what will 48 yards cost? Ans. $72. 4. If 45 yards cost $30, what will 165 yards cost? Ans. $110. 5. If 3 yards cost $2, what will 27 yards cost? Ans. $18. 6. If 9 yards cost $6, what will 30 yards come to? 7. If 3 yds cost $6, what will 10 yds. cost?" $20. 8. Bought 6 yards for $8, how much did 30 yards cost? Ans. $40, INVERSE PROPORTION. It is generally admitted by writers on Arithmetic that more requires less, or less requires more; more requires less, when the third term is greater than the first, and the fourth less than the second. 2d. Less requires more when the third term is less than the first, and the fourth |