whole distance is to be travelled in 5 hours; how many miles an hour is it? Ans. The distance is 31M. 4F. 20R, and the rate of travelling is 6M. 2F. 20R. 41. There are two numbers, whose dividend is the number of inches in a solid foot, the quotient the square of 8; what is the divisor and the difference between the cube of the quotient, and the sum of the squares of the divisor and dividend. Ans. Divisor 27, and the difference 2724569 42. If in a naval engagement, there is of the men killed } wounded, one tenth driven overboard, and drowned, and the remainder, fit for duty, amounts to 306 men; how many were there on board, how many killed, and how many wounded, and how many drowned? Ans. 680 on board, 170 killed, 136 wounded, and 68 drowned. 43. If out of 4 dolls. per week I lay up 15cts. per day, (sundays excepted,) and have saved 46 dolls. 95cts. how long was I laying it up, and how much have I spent in the time? Ans. 365 days or 1 year in laying up, and spent $161.05cts. 44. Suppose that during the ebb tide, a boat should set out from the battery at New York, for Sandy Hook; taking the distance at 27 miles, and at the same instant another should set cut from Sandy Hook, for New York; the current forwards the one, and retards the other, say 23 miles an hour; suppose the boats to be equally laden, and the rowers equally good, and in the ordinary rate of rowing in still water, would proceed 5 miles an hour, the question is whereabouts in the river I would the boats meet? Ans. They would meet 20.625 miles from New York, and 6.875 miles from Sandy Hook. 45. If a gentleman purchase a span of horses with a harness, the whole of which cost 500 dolls. the harness, if laid on the first horse, then both will be of equal value; but, if the harness be put on the other horse, he will be worth double of the first; how much did the horses and harness cost? Ans. The first horse cost 1663 dolls, the harness 833D. and the second horse 250 dollars. 46. If a person sells oranges at 50s. per hundred, and thereby clears of the money; but the season getting advanced and oranges growing scarce, he raised them to 57s. per hundred; what may he clear per cent. by the last mentioned price? Ans. 90£. per cent. 3 the interest of $25000 for 1 hour at 6 per cent. Ans. 17cts. 1m. rem, 48. If a merchant buys two dozen pieces of muslin, for four times as many pounds, and sells them again for five times as much; but had they cost as much as they were sold for, what must they have been sold for to have gained at the same rate? Ans. 150€. 49. Suppose a merchant buys a quantity of rum and brandy, the whole of which costs 208 dolls. the quantity of rum 80 gallons, at 85cts. per gallon; and for every 5 gallons of rum there were 7 gallons of brandy; how many gallons of brandy were there, and how much per gallon? Ans. 112 gals. of brandy, at 1D. 25cts. per gal. 50. Suppose a man pretty well acquainted with numbers, should agree to work in a nail factory for one year, and to receive for his labour, barely 1 nail for the first week, 2 nails for the second, 4 for the third, &c. doubling each week to the end of the year; what does his wages amount to, supposing it takes 80 nails to make a pound, and the whole quantity to be sold for 8cts. per lb? Ans. 4503599627370D. 48cts. rem. 51. Suppose Leonard, Abraham and Samuel, trade together; Leonard puts in a certain sum; Abraham puts in 247 pieces of muslin; Samuel puts in 650 dolls. and they have gained 725 dolls. whereof Leonard ought to have 220 dolls, and Abraham, 305; what is Samuel's share; what did Leonard put in, and what was the 247 pieces of muslin worth. Ans. Samuel's share is 200 dolls. Leonard laid in 715D. and Abraham's muslin was worth $991.25cts. 52. Find the number whose half being divided by 12, its, by S, its by 6, and its by 4; and each quotient shall be 9. Ans. 216 53. Suppose I have purchased a farm for 3600 dolls. and the accounts stand thus; one third of the purchase money I paid down, one third was to be paid in 18 months, with the interest from the date, at 6 per cent. and the other third was to be paid 12 months after the payment of the second, but no interest allowed; now at the end of 18 months, how much have I to pay in all, if I settle the account; supposing that I am allowed 6 per cent. discount on the last payment for the 12 months. Ans. 2440 dolls. 07cts. 5m. 25 54. As I was beating on the forest grounds, How far they ran before they caught the hare? Ans. The dogs ran 336 rods, and the hare 240 rods. 55. If 600lbs. of beef serve 210 men 6 days, how many lbs. will serve 450 men 5 weeks? Ans. 7500 56. 'If John can build a boat alone in 5 days; David in 10 days; Nathan in 20 days, and Charles in 40 days; how long would it take them to build a boat, if all worked together? Ans. 23 days. 57. Suppose I have an orchard consisting of 3600 trees, the rows being 16 feet a part one way, and 12 feet the other, in a rectangular lot of ground; how much land does the orchard take up? Ans. 15.8677 + acres, or 15A. 3R. 18.8 + P. 58. Suppose that it is between 4 and 5 o'clock, and the minute hand, and hour hand are both together; what o'clock is it? Ans. 21 min. past 4, MISCELLANEOUS. OF LEVERS. There being three orders of levers, or three varieties, wherein the weight, prop, or moving power, may be differently ap plied to the vectis or inflexible bar, in order to effect mechanical operations in a convenient manner. A lever of the first order has the power at one of its ends, and the weight to be raised put at the other; and the fulcrum or prop, somewhere between them. Observe. In this order, the power applied at one end, will be reciprocally proportional to the distance of those ends from the fulcrum,or point supported, or in the steelyards, as the distance of the weight from the point of suspension. EXAMPLES. What weight will a person be able to raise, who presses From the length of the bar 108 inch, take the length of the prop from the end of the bar, viz. 10 inches; then state it, making the remainder the first term, the weight (160lbs.) the second, and the 10 inches the third; performed by inverse proportion. 2. What weight hung 7 feet from the fulcrum of a steelyard will equipoise a hogshead of sugar that weighs 8C. it being freely suspended at 3 inches distant from the other end? Stated thus; 3in.: 8C. :: 7ft. by inverse proportion. Ans. 32lbs. In mechanism,a lever of the second order is where the power acts at one end; the prop fixed directly at the other end, and the weight somewhere between them. In this order of levers, their force is in contra proportion to their length. EXAMPLES. 1. If a lever be 9 feet long, what weight lying 10 inches from the end resting on the ground, may be moved with the force of 160lbs. lifting at the other end of the lever? Ans. 1728lbs. Stated thus; 10in.: 160lbs. :: 9in. done by the Rule of Three Direct. A lever of the third order, is when the prop is planted at one end of the bar, the weight at the other, and the moving force somewhere between them. EFFECTS OF LIGHT, HEAT, AND ATTRACTION. Observe. The effects of light, heat and attraction, are reciprocally proportional to the square of their distance from the centre whence they are propagated. EXAMPLES. 1. Suppose that Solomon and Seth sit in a room where there is a fire, from which Solomon is 4 feet, and Seth 8 feet distant; now, how much hotter is it at Solomon's seat than at Seth's? Ans. 4 times hotter at Solomon's. Square each distance, and the proportion of their products will be the answer; thus, as the square of the greater distance is to 1, so is the square of the lesser distance, &c. by inverse proportion. 2. Suppose a body is placed 8 feet from a source of heat, and it becomes necessary that it should receive double the heat it does at that place, at what distance must it be placed to receive the required heat? Ans. 5,6568+ feet. Square the distance, and say, as 1 is to the square found, so is 2 to the square of the required distance; performed by inverse proportion; extract the square root of the number found, and it will be the answer. VIBRATION OF PENDULUMS. By experiment, it has been found,that a pendulum 39,2 long, vibrates 60 times in one minute, and the length o dulums are reciprocally as the square of the number of their vibrations, made in the same space of time. EXAMPLES. 1. Give the length of a pendulum that will vibrate 120 times in a minute. Ans. 9.8 inches. A pendulum that vibrates seconds, will make 60 vibrations in a minute; accordingly, say, as the square of 60 is to the length of the pendulum, (39.2) so is the square of 120, the given number, to the length of the pendulum required. Inverse proportion. 2. Give the length of a pendulum that will swing, or vibrate, 30 times in a minute. Ans. 156.8 inches. The first and second terms as before, and the square of 30 for the third term. DIVERTING. 1. Suppose a loaf of sugar weighs 11lbs. in one scale, but being put into another, it weighs only 8lbs. what is the true weight? Ans. 8lbs. 2.3oz. Rule. Multiply the two weights together, and extract the square root of the product; said root will be the true weight. 2. Suppose you wish 4 weights made so that you can weigh any number of pounds, from 1 to 40; what must be the weight of each? Ans. 1lb. 3lbs. 9lbs, and 27lbs. Rule. To double of the first or least weight, (which is 1 pound,) add 1, and it gives the second weight; then; to double the weight of these two weights, add 1, and you have the third weight; and to double the sum of these three weights add 1, and it will be the fourth weight. 3. If the reckoning of a certain company amount to 9s. 24dand the number of persons be equal to the farthings each spent, how many were there in the company, and how much did each ¿pend? Ans. 21 men, and each spent 54d. Rule. Reduce the given sum to the lowest name mentioned; and the square root of that number will be the answer. Rules for finding the contents of Superficies and Solids. SUPERFICIAL MEASURE. The superficial area, or content of any plane surface, is considered to be divided into squares, which is either greater or less, according to the different measures by which they are taken. If taken in feet, we say, the number of square feet; it taken in inches, the square inches, &c. Thus suppose a board measures 1 foot, or 12 inches, upon each side; then, 12×12 14, the square inch in a superficial foot. To find the content of a square, having equal sides, or or long square. |