men. term greater than the second. These definitions may be illustrated by the following questions, viz: 1. If in 10 days 8 men can perform a piece of work, in what time could 40 men do the same. men. days. ī 5 5 Illustration-8 men at work 10 days uniformly would perform as much as 80 men in 1 day. Consequently, if 80 men perform the same in 1 day, 40 men will do it in 2 days. Q. ED. 2. If 8 men can perform a piece of work in 10 days, how many men would be required to perform the same in 2 days? Ans. 40. days. men. days. Solution-As 10 : 8. : 2 : 40. Notwithstanding, these statements are made to correspond with the above definitions, and bring correct answers. Yet, it is manifest, they are wrong !!! if the doctrine of proportion be recognized, for agreeably, to Prop. B. book 5 Euclid, in Inverted Proportion, the third is to the second, as the first is to the fourth. In solution 1st, we find that more requires less for 10 days require 2 days, and in solution 2d, less requires more--for 8 men require 40 men. Now the proper method of stating is thus: as 40 men, : 10 days : : 8 men = 2 days. Solution 20—2 days : 8 men : : 10 days = 40 men. Hence, 40 x 2 = 10 X 8, And 2 x 40 = 8 X 10—for by Prop. 16, 6 Euclid, the rectangle of the extremes will be equal to that of the means, which is not the case with the first and second statements; therefore, it follows, that although custom and the opinion of most writers, sanction the truth of the first and second statements they are evidently mistaken. 3. Suppose 450 soldiers are in a garrison, and their provisions are calculated to last them but 5 months, how many must leave the garrison, that the same allowance may be sufficient for those who remain 9 months. Ans. 200. 4. If a man perform a piece of work in 15 days of 12 hours long, in how many days of 10 hours long can he perform the same! Ans. 18 days. 5. There is a cistern having a pipe that will empty it in 6 hours, how many pipes of the same capacity will empty it in 20 minutes? Ans. 18 pipes. COMPOUND PROPORTION. Is that, in which 5 terms are given to find the sixth. Three of those terms are a supposition and two a demand. RULE. Arrange the terms so that two terins, one side of the statement shall be of the same name and denomination with two terms on the other side. Consequently, the demand will be in the middle term. Example.—If 10 men in 18 days earn $56, how many dollars can 20 men earn in 36 days? demand. Mode of operation 10 $56 ( 1020 2 by contraction days 18 ) 4 1836 2 4 1)224 men. men. } { $224.00 Answer. 2. If 7 reapers get $21 for 3 days work, how many reapers will earn $96 in 32 days? Ans. 3 reapers. 3. If a family of 8 persons in 24 months spend $480, how much would 16 persons spend in 8 months. Ans. $320.00. 4. If 4 men mow 96 acres in 12 days, how many acres can 8 men mow in 16 days? Ans. 256 acres. 5. If 4 men receive $24 for 6 days work, how much will 8 men receive for 12 days work. Ans. $96. 66 66 66 Measures. Measures. | Measures. Measures. Measures. Parts of a yd. Paris of a bu. Parts of a gal. Parts of an ac Parts of an acre. 4 qr 1 yd 4 p. 1 bu. 4 qts. 1 gl 160 ps 1 a. 4 rds. 1 a. 266 1 66 12 16 66 2 1 66 80 66 2 1 16 1 66 1 1 66 40 66 1 nails quarts. pecks pints. 20 66 perches, 4 t of ydj 8 1 2 - 520 rood. 16 2 j 4 11 5 6646 116 66 32 2 32 66 166 85 : 1 qt. 10 10 CASE I. How to take parts for î of a cent, i of a yard, i of a ton, i of a cwt. of a year, &c. RULE.—Take the aliquot parts for { first, and then for i, and add the product together; or, multiply the given quantity by 3, and divide by 4. If it be , divide by 4, for t, divide by 2. CASE II. When the given price is dollars and cents. Rule.—Multiply the given quantity by the dollars, and take aliquot parts of the cents, and add the products together. CASE III. When the given quantities are of various denominations, as cwts. qrs. and lbs.; yards, quarters, nails; or bushels, pecks, and quarts, &c. RULE.—Set down the price per cwt., multiply it by the number of cwts. given, and take aliquot parts, for the quarters and lbs. &c. Q. What is the amount of 15 cwt., 2 qrs, 7 lbs. of iron at $10 per cwt? Operation $10.00 x 15 2 qrs. $150.00 qrs. 0.93% price of 7 lbs. 7 lbs. } 66 $158.434 1. What will 36 yards of tape come to at ļof a ct. per yd. 2. 4 3. 4. 5. 6. 66 66 96 66 295 66 320 .66 440 66 896 66 66 66 66 cim ini 66 66 66 66 66 CLOTH MEASURE. $ cts. 1. What is the value of 27 3 at $9.65 per yd. 267.79 yds. gr. nls. 2. 860 1 at 84 cts. $722.61 3. 126 2 256 $4.75 601.47 4. 7 3 2 " 5.60 44.10 110.39 $ cts. 1. What is the value of 120 2 at 35 cts. per 42.17 bush. pks. qts. pts. 780 3 2 O at $1.17 per bush. 913.55 1354 1 5 1 66 25 338.60į 4. 325 3 623 203.591 5. 94 2 871 82.688 LIQUID MEASURE. 1. What will 784 gals come to at 84 cts pergal. $658.56 gals. qts. pts. Ans. $ cts. 2. 765 3 1 at $2.184 per gal. 1675.35 3. 5 hhds. 31 ) gals. at $47.12 per hhd. 259.16 4. 17 154 or 3 qts. 63.75 1116.93 5. " 428 gals. 3 qts. at 1.40 600.25 cts. 1. What is the value of 35 2 9 at $54.35 1935.54 A. R. P. 5153 12 3 750 9190.87 9160.274 196 2 32 - 44.10 8674.47 6. 189 3 398 49.99 9999.42 EXERCISES FOR THE SLATE. Tons. cwt. gr. lbs. 19 19 3 271 at $99.994 per ton, Ans. $1999.93 66 66 66 466 ai ni i ho 66 Now 20 tons at $100 per ton is $2000.00 |