LESSON XVIII.

PRACTICAL EXAMPLES IN ADDITION AND SUBTRACTION OF FRACTIONS.*

188. A man bought a horse for $39, and sold it for $146; what was his profit?

189. John bought 3714 lbs. of butter, and then sold 1474 lbs. ; how many pounds had he left?

190. A man bought at one time 343 bbls. of flour, at another 47 bbls., and at another 44 bbls.; how many

barrels did he buy in all?

191. I sold one lot of books for $43.75%, another lot for $37.623, and another for $38.274; how much do my books amount to?

192. If I have $4373, and pay out $341, how much have I left?

193. 11 horses are worth $15013; 41 cows are worth $923; how many dollars are the horses worth more than the cows?

194. A drover bought a yoke of oxen for $8977, and paid $143 for pasture; he afterwards sold them for $89; did he gain, or lose, and how much?

195. A merchant bought 37 bales of cloth for $3473, and 48 bales for $402; he then sold the whole for $1000; how much did he gain?

196. A merchant bought 37 barrels of flour for $157, and 1434 barrels for $597; how many barrels did he buy, and how much did they cost him?

197. A gentleman owns a farm worth $3397, and a store worth $1413.473; how much more is the farm worth than the store?

*NOTE.—The analysis is the same as in the simple rules.

QUESTIONS.-How is the denomination of a fraction determined? (139.) Upon what does the value of a fraction of any denomination depend? (140.) What is the reduction of fractions? (141.) What is reduction ascending? (142.) What is reduction descending? (148.) When are fractions said to be expressed in their lowest terms? (144.) What is the difference between a common multiple and the least common multiple of two or more numbers? (121.) (122.)

LESSON XIX.

198. A gentleman gave for a house and lot $7384, and for a cotton factory $9503.134; how much more did the factory cost than the house and lot?

199. A man valued a factory at $9410.30, and a farm at $8964.37%; what is the difference of their values?

200. A merchant had 1987 yds. of cloth, and sold 8473 yds. of it; how many yards had he remaining?

201. A speculator sold a quantity of flour for $7897, and, by so doing, gained $332.463; what did it cost him?

202. A man sold an estate for $9873, and thereby gained $3896.46; how much did the estate cost him? 203. A man sold an estate for $9873, and, by so doing, lost $3896.46; how much did the estate cost him?

204. A gentleman having $57897, gave his eldest son $26805; how much had he remaining?

205. Joseph had $473, and Weston as many lacking $8; how many dollars had Weston?

206. The sum of two numbers is 34172; one of them is 2417; what is the other?

207. If I have in different banks $1413, $4678, $417%, and $348, how many dollars have I in all the banks?

208. If I sell 3 horses for $430, 3 cows for $3244, and 55 sheep for $223, for how many dollars do I sell them all?

QUESTIONS.—What is a fraction? (123.) What is the difference between a common fraction and a decimal fraction? (124.) (125.) What is the difference between the uses of the numerator and those of the denominator? (127.)(128.) Are decimal fractions ever used in connection with common fractions? (124.†) Are they written differently? (124.+) Are decimal fractions usually written with a denominator? (125.) What determines the denomination of a common and a decimal fraction? (139.) What is a rule in Arithmetic? (12.) Write, on the blackboard, a rule for the reduction of a fraction to its lowest terms. (145., b.) Write a rule for the reduction of an improper fraction to an integral or a mixed number. (146.)

LESSON XX.

209. A farmer bought 3 loads of hay; the first contained 14733 lbs., the second 1834 lbs., and the third 21711⁄2 lbs.; how many pounds in the three loads?

210. The same farmer afterward sold 3479,3 lbs.; how many had he remaining?

211. James Smith had two lots of wild land; the first contained 387 acres, the second 417 acres; he sold 521 acres; how many acres had he remaining?

212. A lumber merchant bought 43473 feet of pine lumber; he then sold 2417 feet, and afterward bought 3874 feet; how many feet had he then?

213. The hind-quarters of au ox weighed 3753 lbs. each ; the two fore-quarters weighed 621; the hide and tallow weighed 112 lbs.; what was the entire weight?

214. There are two numbers whose difference is 87643; the greater number is 15287; what is the less?

215. What number is that, which, if taken from 37941, will leave 865?

216. The subtrahend is 34174, the difference is 9137; what is the minuend?

QUESTIONS.-Write on the blackboard a rule for the reduction of an integral number to a given denomination. (148.) ANS.—Multiply the number of equal parts of the given denomination in one unit by the integral number, and the product will be the numerator of a fraction of the denomination required. What is the rule for the reduction of a fraction to a given denomination? ANS.-Divide the number of equal parts of the given denomination in one unit by the denominator of the fraction, and multiply the quotient by the numerator, and the product will be the numerator of a fraction of the denomination required. Write a rule for the reduction of a mixed number to a given denomination. (148.) What is the addition of fractions? When is a fraction said to have a common denominator? (150.) What is the subtraction of fractions?

LESSON XXI.

217. The sum of two numbers is 1408; the less is 216: what is the greater?

218. The minuend is 4183, the difference is 421; what is the subtrahend?

219. The minuend is 4 million and 3 seventeenths, the difference is 3 million and 5 thirteenths; what is the subtrahend? 220. A man pays $300 for 100 sheep, $95 for a yoke of oxen, $60 for a horse, and $124, for a buggy; how much do they all cost?

221. A man bought a barrel of sugar for $30, 25 yards of broadcloth for $534, two barrels of molasses for $473; in return for these he gave 100 bushels of oats worth $651, a colt worth $40, and the balance in money; how much money did he pay?

222. A flour merchant, having 700 bbls. of flour on hand, sold 371 bbls. to one man, and 142 bbls. to another; how many barrels had he left?

and 46,?

223. What is the sum of 137,
224. What is the sum of 46 and 37?

225. What is the difference between 38, and 221? 226. What is the difference between 37 and 24'? 227. Give the sum and the difference of 434 and 293. 228. Give the sum and the difference of 383 and 32,. 229. What is the amount of the sum and the difference of 43 and 37?

QUESTIONS.-What are the rules for adding and subtracting fractions which have units for their numerators? (152.) What is a unit? (1.) What is the difference between abstract and concrete numbers? (3.) (4.) What is the difference between simple and complex problems? (8.) (9.) What is an analytical step? (10.) Illustrate the definition of a rule. (12.) Illustrate on the blackboard the definition of a sign. (13.) What is the answer called in Addition? (30.) In Subtraction? (42.) In Multiplication? (49.) In Division? (57.)

LESSON XXII.

153. MULTIPLICATION AND DIVISION OF

FRACTIONS.

154. To multiply a fraction by an integral number.

(a.) What is the product of

(b.) What is the product of

multiplied by 4? multiplied by 4?

ANALYSIS.-(a.) Decreasing the denominator of a fraction in any ratio, increases the value of the fraction in the same ratio.* If, then, the denominator of be divided by 4, the result, , will be 4 times greater than 3.

Therefore, multiplied by 4 equals 11.

*NOTE.-Since the denominator is a divisor, decreasing it in any ratio must increase the quotient in the same ratio.