term greater than the second. These definitions illustrated by the following questions, viz: may be 1. If in 10 days 8 men can perform a piece of work, in what time could 40 men do the same. men. days. men. Solution.-As 88: 10 :: 40 = 1 x 10 Illustration-8 men at work 10 days perform as much as 80 men in 1 day. 80 men perform the same in 1 day, 40 2 days. 5 = 2. uniformly would Consequently, if men will do it in Q. E. D. 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 2d-2 days: 8 men : Hence, 40 x 2 = 10 x 8, = 10 days =40 men. And 2 × 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 perform the same? 5. There is a cistern having a pipe hours long can he Ans. 18 days. 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 terms, one side of the statement shall be of the same name and denomination with two terms on the other side. the demand will be in the middle term. Consequently, Example. If 10 men in 18 days earn $56, how many dollars can 20 men earn in 36 days? 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 Ans. 256 acres. can 8 men mow in 16 days? 5. If 4 men receive $24 for 6 days work, how much will 8 men receive for 12 days work. Ans. $96. quarts. pecks pints. qr 1 yd 4 p. 1 bu. 4 qts. 1 gl 160 ps 1 a. 2 " 66 26 12 14 80 " 16 16 16 1 66 1 66 14 nails OF MEASURES. Measures. Measures. | Measures. Measures. Measures. Parts of a yd. Parts of a bu. Parts of a gal. Parts of an ac 4 Parts of an acre. 4 rds. 1 a. 1" 40" 1 66 1 66 1 66 2 66 66 4 of yd 8 1 2 1 qt. 16 5 66 166 32" 66 20rood. 10 8 66 1 pts. of gal. 2 = CASE I. How to take parts for of a cent, of a yard, of a ton, of a cwt. 4 of a year, &c. RULE.-Take the aliquot parts for first, and then for , and add the product together; or, multiply the given quantity by 3, and divide by 4. If it be 4, divide by 4, be, for, 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 1. What will 36 yards of tape come to at of a ct. per yd. 66 96 66 CLOTH MEASURE. cts. 1. What is the value of 27 3 at $9.65 per yd. 267.79 1. What will 784 gals come to at 84 cts per gal. $658.56 1. What is the value of 35 2 9 at $54.35 1935.54 Tons. cwt. gr. lbs. 19 3 27 at $99.99 per ton, Now 20 tons at $100 per ton is 5153 12 9190.87 9160.274 8674.47 9999.42 Ans. $1999.93 $2000.00 |