Qu. 7. At what times between 2 and 3 o'clock, are the hour and minute hands of a clock together; at right angles; and in opposite directions? At two o'clock, the hour hand is two of the portions, called hours of one hand and five minutes of the other, in advance of the minute hand; and their rates being as 1 12, the minute hand gains 55 in 60, or 11 in 12, upon the hour hand: whence we have 11 12 2 : 2, the time at which the minute and hour hand are together. Again, when they are at hand must have gained 2 + 3 = right angles, the minute 5 portions; and we have 11 12 5: 5; and therefore at 5 × 5 or 27 hands are at right angles. minutes past two, the Also, if they point in opposite directions, 2 + 6 = 8 portions must be gained by the minute hand; and therefore we have 11 12 8: 8, or, the hands will be in opposite directions at 8 × 5, or 43 minutes past two. When the minute hand has gained 2+9= 11 portions, the two hands will be at right angles again; and 11 12 11 : 12, which shews that this circumstance occurs at 60 minutes past two, or at three o'clock, as we know to be the case. Qu. 8. Two clocks point out 12 at the same instant: one of them gains 7" and the other loses 8" in 12 hours: after what interval will one have gained half an hour of the other, and what o'clock will each then shew? Here, 7"+8" = 15", is the separation which takes place in 12 hours; and hour 30′ 1800′′: whence, = = 15" : 1800" : 12hrs. 1440 hrs.; that is, in 1440 hours or 60 days, they will be separated 30 minutes or half an hour. Also, the first gains 7" in 12 hours, or 14" in 1 day: and 1day: 60 days :: 14" 14′, and therefore it will shew 12 hours 14 minutes. The second loses 8" in 12 hours, or 16" in 1 day; and 1 day 60 days :: 16" : 16′; whence the time pointed out by it will be 12h.-16', or 11 hours 44 minutes: and it will be observed that these times differ by half an hour, as they ought. Examples for Practice. (1) How much cloth at 14s. 6d. a yard, must be given for 3 cwt. 3 qrs. of sugar, at £3. 4s. per cwt.? Answer: 16yds. 2qrs. (2) If 126 yards of cloth be bartered for 3hhds. of brandy, at 6s. 8d. per gallon: what is the price of the cloth per yard? Answer: 10s. (3) If I buy goods at £3. 16s. 8d. per cwt.: how must I retail them per lb., to gain 15 per cent.? Answer: 91d.. (4) If by selling tea at 6s. 4d. per lb., a grocer lose 6 per cent.: what did it cost him per lb.? Answer: 6s. 8d.19. (5) A grocer bought 2tons. 3cwt. 3qrs. of sugar for £120., and paid £2. 10s. for expences: what must he sell it at per cwt. to clear 50 per cent.? Answer: £4. 4s. (6) A person, by disposing of goods for £182., loses at the rate of 9 per cent.: what ought they to have been sold for, to realize a profit of 7 per cent.? Answer: £214. (7) Bought 2688 yards of cambric at 8s. 8d. a yard, and sold at 10s. 2d.; at 10s. 11 d., and the remainder at 11s. 4 d. a yard: what is the whole gain, and also the gain per cent.? Answer: £304. 14s. 8d., and £26. 3s. 23d.. (8) A stationer sold quills at 11s. a thousand, by which he cleared of the money, and he afterwards raised them to 13s. 6d. a thousand: what did he clear per cent. by the latter price? Answer: £96. 7s. 3 d.1⁄2· (9) At what price must a commodity, purchased at the rate of £14. 5s. per cwt., be sold to gain 21 per cent.; and what quantity of it must be sold at that rate to clear £100.? Answer: at £17. 4s. 10}d. per cwt., and the quantity of it, is 33 cwt. 1qr. 18lbs. 117oz. (10) A merchant bought 160 quarters of wheat at 41s. 3d. per quarter, and sold it at 58s. 4d.: what was his gain? At what price ought it to have been sold to gain exactly £100? Answers: £136. 13s. 4d., and 53s. 9d. (11) If a parcel of goods bought for £18., be sold four months afterwards for £25.; what is the gain per cent. per annum? Answer: £116. 13s. 4d. (12) Divide £64. among A, B and C, so that A may have three times as much as B; and C may have one third of what A and B have together. Answer: A has £36., B has £12., and C has £16. (13) A person paid a tax of 10 per cent. upon his income: what must his income have been, when after he had paid the tax, there was £1250. remaining? Answer: £1388. 17s. 91d.. (14) A grocer had 150 lbs. of tea, of which he sold 50lbs. at 9s. per lb., and found that he was thereby gaining 7 per cent.; at what rate must he sell the remaining 100lbs., so as to clear 10 per cent. upon the whole? Answer: 9s. 3d.. (15) A mixture of wine and water of 32 measures contains one measure of wine: how much water must be added to this mixture, that 32 measures of it may contain of a measure of wine? Answer: 224 measures. (16) A hare starts 40 yards before a greyhound, and is not perceived by him till she has been up 40 seconds: she scuds away at the rate of 10 miles an hour, and the dog pursues her at the rate of 18 miles an hour: how long will the course last, and what distance will the hare have run? Answer: 60 seconds, and 490 yards. (17) At what time, between twelve and one o'clock, do the hour and minute hands of a watch point in directions exactly opposite? Answer: 32 min. 437 sec. past 12. (18) If 5 men or 7 women can perform a piece of work in 35 days: in what time can 7 men and 5 women do the same? Answer: 16 days. (19) If 15 men, 12 women, and 9 boys can complete a piece of work in 50 days; what time would 9 men, 15 women, and 18 boys take to do twice as much, the parts done by each in the same time being as the numbers 3, 2 and 1? Answer: 104 days. (20) If A by himself can do a piece of work in 5 days; B twice as much in 7 days, and C four times as much in 11 days: in what time can A, B and C together do three times the said work? Answer: 3days. 12 hrs. 46 min. 109 (21) If A and B together can build a boat in 18 days, and with the assistance of C they can do it in 11 days; in what time can C do it by himself? Answer: 28 days. (22) If A can do a piece of work by himself in 1 hour, B in 3 hours, C in 5 hours, and D in 7 hours: in what time can they do three times as much, all working together? Answer: 1 hour. 47 min. 23 sec. (23) A and B can do a piece of work in 10 days; A and C in 12 days, and B and C in 14 days: in what times can they do it jointly and separately? Answer: All together in 7 days; A in 17 days; (24) If A, B and C could reap a field in 18 days; B, C and D in 20 days; C, D and A in 24 days, and D, A and B in 27 days: in what times would it be reaped by them all together, and by each of them separately? Answer: By them altogether in 16 days: by A in 87 days: by B in 50g days: by C in 41 days, and by D in 170 days. CHAPTER VII. INVOLUTION AND EVOLUTION, WITH THE ARITHMETIC OF SURDS. INVOLUTION. 152. DEF. A Power of any number or quantity is the number or quantity which arises from successive multiplications by itself: the operation by which it is obtained is termed Involution; and the Degree or Order of the power is denoted by the number of equal factors employed. Thus, taking the number 2, we shall have the following powers of it: 2 = 2, the first power of 2: 2 × 2 = 4, the second power of 2: 2 × 2 × 2 × 2 × 2 = of 2: 32, the fifth power of 2: 2 × 2 × 2 × 2 × 2 × 2 = 64, the sixth power of 2: and so on, as far as we please: but instead of expressing these multiplications at length, which would soon become inconvenient, we denote the same operations by means of Indices, or small figures placed a little above the line to the right of the quantities whose powers are intended to be exhibited: thus, what is given above may be denoted by, where it is evident that the Index, sometimes called the Exponent, is equal to the number of factors employed, and greater by one than the number of operations. |