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158. C borrowed of D, $72.85, agreeing to pay it in 16 months, increased by .08 of itself. What was the amount to be paid ?

159. What will it cost to insure a house, worth $ 2500, against the danger of fire, for one year, the price of insurance being .025 of the value of the house?

160. Suppose I purchase a ship for $ 12900, and sell it at an advance equal to .019 of the cost; for how much do I sell it ?

161. How many gallons of wine can be purchased for

4° 161. 19137 per galands of raisins

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162. How many pounds of raisins can be bought for $9, at 16į cents per pound ?

163. If a man travel 5.335 miles in 1 hour, in how many hours will he travel 166 miles ?

164. If 18 bushels 3 pecks of wheat grow on 1 acre, how many acres will produce 396 bushels ?

165. If 3 shillings will pay for 1 bushel of barley, how many bushels will 26 shillings pay for ?

166. If 5 s. 8 d. will pay for 1 bushel of wheat, how many bushels will £11 pay for ?

167. If 8s. 3d. will pay for 1 gallon of wine, how many gallons will £18 pay for ?

. 168. What is the value, in Federal money, of £3 17s. 8d., of the old currency of New England ?

169. If I buy 230 pelts, in Canada, at 4s. 3 d. apiece, for what amount Federal money must I sell the whole, in the United States, in order to gain $36.15 ?

170. How many square feet in a floor, that is 18.63 feet long, and 14 ft. 3 in. wide ? .

171. How many square feet in a board, that is 16 ft. 5 in. long, and 11 inches wide ?

172. How many cubic feet in a box, that is 4 ft. 6 in. long, 3ft. 2 in. deep, and 2 ft. 9 in. wide ?

173. Goliath is said to have been 6 cubits high, each cubit being 1 foot 7.169 inches. What was his height in feet ?

174. How many square feet of paper will it take to cover the walls of a room, that is 18 ft. 9 in. long, 14 ft. 6 in. wide, and 9ft. 3 in. high?

175 Suppose a man's property to be worth $6520, and his tax to be .02 of the value of his property; how much is his tax ?

176. If a man earn one dollar and one mill per day, how much will he earn in a year ?

177 What is the cost of three hundred seventy-five thousandths of a cord of wood, at four dollars per cord ?

178. A has nine hundred thirty-six dollars, and B has five dollars, three dimes and one mill. How much more money has A than B ?

179. A trader sold 4 pieces of cloth—the first contained 86 and 3-thousandths yards; the second, 47 and 3tenths yards; the third, 91 and 7-hundredths yards; the fourth, 22 and 9-ten-thousandths yards. What did the whole amount to, at $ 7 per yard ?

180. A has $31.32, B has $575, C has $ 104, and D has $ 955 ; and they agree to share their money equally. What must each relinquish, or receive ?

181. Suppose a car wheel to be 2 feet 9 inches in circumference; how many rods will it run, in turning round 800 times ?

182. If a car run 1 mile in 3 minutes and 9 seconds, in what time will it run 18 miles ?

183. Suppose the sum of two certain quantities to be - 1, and one of those quantities to be .8036, what is the other? (See Prob. 1, page 20.)

184. Charles and Joseph together have $ 4.38; of which Charles's share is 17 shillings and 3 pence. What is Joseph's share ?

185. Suppose .08 to be the difference between two quantities, and the greater quantity to be 80; what is the smaller ? (See PROB. II, page 20.)

186. There is a field, 5.864 acres of which is planted with corn, and the rest, with potatoes. There is 2 A. 3R. 10r. more of corn than potatoes. How much is planted with potatoes ?

187. Suppose 7426.1 to be the difference between two . quantities, and the smaller quantity to be .93; what is the greater ? (See PROB. III, page 21.)

188. Henry has $1.355 more money than William;

and William has 19s. 10 d., New England currency. How much has Henry?

189. What are the two quantities whose sum is 290. 009, and whose difference is .99?. (See PROB. IV, page 21.)

190. If a horse and chaise cost $ 437.25, and the chaise cost $67.08 more than the horse, what is the cost of each?

191. Suppose 15675.266547 to be the product of some two factors, one of which is 27.381; what is the other? (See ProB. V, page 21.)

192. If a board be 1 ft. 9 in. wide, how long must it be, to contain 26.5 square feet of surface ?

193. Suppose 566.916128724 to be a dividend, and 108.273 the quotient; what is the divisor? (See PROB. vi, page 21.)

194. 4397.4 pounds of beef was equally divided among a number of soldiers, and each soldier received 3.49 pounds. How many soldiers were there?

195. Suppose .025 to be a divisor, and .045 the quotient; what is the dividend ? (See PROB. VII, page 22.)

196. Such a quantity of bread was divided equally · among 13 sailors, as allowed each sailor 1.236 pounds. How many pounds were divided ?

197. If the product of three factors be 70.4597, the first of those factors being 3.91, and the second 3.5, what is the third ? (See PROB. VIII, page 22.)

198. What must be the depth of a pit, that is 8 ft. 5 in.' long, and 4 ft. 3in. wide, in order that it shall contain 231 cubic feet ? (Consider 231 as a product.)

199. Suppose the bottom of a wagon to be 9 feet long, and 4 ft. 3 in. wide; how many feet high must wood be piled in this wagon, in order that the load shall contain 1 cord ? (View the cubic feet in a cord as a product.)

200. Suppose wood to be piled on a base, 15 ft. 6 in. long, and 7 ft. 9 in. wide, what must be the height of the pile, to contain 16 cords.?

201. If a stick of timber be 1 ft. 9 in. wide, and 1.4 ft. deep, what must be its length, in order that the stick shall contain 1 ton ?

XII.

INFINITE DECIMALS.

Learners, who are preparing for commercial business, and who do not intend to prosecute an extensive course of mathematical studies, may omit this article, and proceed immediately to Art. XIII.

INFINITE DECIMALS are those which are understood to be indefinitely continued; either by one and the same figure perpetually repeated, or, by some number of figures perpetually recurring in the same order. For example, .444444, &c. . 26262626, &c. .057057057, &c. .134913491349, &c. Decimals of this kind result from division, when the divisor and dividend are prime to each other, and the divisor contains prime numbers other than those contained in 10; that is, other than 2 and 5.

An infinite decimal which is continued by the repetition of a single figure, is called a repeating decimal; and the repeated figure is called the repetend.

An infinite decimal which is continued by the repetition of more than one figure, is called a circulating decimal; and the repeated period of figures is called the circulate, or compound repetend.

When other decimal figures precede the repetend or circulate, the decimal is called a mixed infinite decimal. For example, .8476666, &c. .38171717, &c. · A single repetend is distinguished by a point over it, thus, .3, which signifies .33333, &c. A compound repetend is distinguished by a point over its first, and last figure, thus, .849, which signifies .849849949, &c.

Similar repetends—whether single or compoundare those which begin at the same place, either before or after the decimal point. For example, .13 and .72 are similar; also, .264 and .9038 are similar; also, 3.54 and 7.36 are similar.

Dissimilar repetends are those which begin at different places. For example .6127 and .405 are dissimilar.

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Conterminous repetends are those which end at the same place. For example, .749 and .506.

Similar and conterminous repetends are those which begin and end at the same places. For example, .1308 and .4012.

Any quotient continued by annexing decimal ciphers to the dividend, is known to be infinite, whenever a remainder occurs, that has occurred before; and the repetend is known to consist of those quotient figures which succeed the first appearance, and precede the second appearance of the recurring remainder. It may also be observed, that every quotient which does not terminate, must, at some place, repeat or circulate. This truth is evident from the consideration, that the several remainders, which precede their respective quotient figures, must all be within the series of numbers, 1, 2, 3, 4, and so on, up to the number of the divisor. Therefore, it is impossible that the number of partial divisions in any operation shall equal the number indicated by the divisor, without the recurrence of some one of the remainders.

REDUCTION OF INFINITE DECIMALS.

CASE I. To reduce a repetend to a vulgar fraction.

The observations which lead to the rule are as follows. If 1, with ciphers continually annexed, be divided by 9, the quotient will be ls continually; that is, if į be reduced to a decimal, it will produce the repetend .i: and since .i is the decimal equal to 1, 2= ,.= 3,4= , and so on, up to .9=, or unity. Therefore, every single repetend is equal to a vulgar fraction, whose numerator is the repeating figure, and whose denominator is 9. Again, if be reduced to a decimal, it becomes .01; and since .oi is the decimal equal to 9, .02=ing .03=5, and so on, up to .99=;; or unity. Again, if nog, be reduced to a decimal, it becomes .001, and since .ooi is the decimal equal to og, 002=ng,

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