(17) If the carriage of 3 cwt. 2qrs. 14lbs. for 51 miles come to 18s. 5d.; what will be the charge for carrying 10tons. 3 cwt. the same distance? Answer: £51. 12s. 6d. (18) At the rate of 11s. 7d. in the pound, what is the sum paid by a bankrupt for a debt of £2735. 10s. ? Answer: £1590. Os. 21d. (19) If a labourer earn 2s. a day when wheat is at 8s. a bushel, what ought he to earn when wheat is at 6s. a bushel? Answer: 1s. 6d. (20) If a tradesman gain 1s. 4d. on an article which he sells for 5s. 6d., what does he gain on every £100.? Answer: £25. (21) If 15 workmen can do a piece of work in 25 days, in what time can 25 men do the same? Answer: 15 days. (22) How much in length, that is 3ft. 9 in. broad, will be equivalent to 37 ft. 9 in. in length, which is 7 ft. 6 in. broad? Answer: 75ft. 6 in. (23) If 69 yds. of carpet 3qrs. wide, cover a room 8 yds. 2qrs. 2nls. long; find the width of the room. Answer: 6 yards. (24) What would be the purchase-money of an estate producing a rental of £3223., at the rate of £2. 155. per cent? Answer: £117200. (25) What may a person, having an income of £1000. a year, spend daily, so as to lay by £434. 5s. yearly? Answer: £1. 11s. (26) If I lend a friend £250. for 6 months, how long ought he to lend me £187. 10s. to requite the kindness? Answer: 8 months. (27) If the rate levied upon a rental of £763. 15s. amount to £133. 13s. 1d., how much is that in the pound? Answer: 3s. 6d. (28) A person buys 136yds. of cloth for £150., and retails it at £1. 18s. a yard; what does he gain by the transaction? Answer: £108, 88, (29) A person's daily income is £1. 15s. and his quarterly expenditure £135. 10s.: how much will he have saved at the end of 9 years? Answer: £870. 15s. (30) If a gentleman spend £152. 10s. every week; what must be his daily income that in 15 years he may lay by £7522. 10s.? Answer: £23. 2s. 68. Questions frequently occur, in which it is necessary to repeat the process just explained, and they are on this account said to belong to the Double Rule of Three: but we shall here adapt what has already been done, to the solution of a single example, which will be sufficient to point out the steps to be pursued in every other instance. Ex. If a person_travel 300 miles in 10 days, when the day is 12 hours long; how many days will it take him to travel 600 miles, when the day is 15 hours long? We will here give two solutions, each of which produces the same result. Examples for Practice. (1) If the expenses of 7 persons for 3 months amount to 70 guineas; what will be the expenditure of 10 persons for 12 months at the same rate? Answer: £420. (2) If 10 horses consume 7bush. 2pks. of oats in 7 days; in what time will 28 horses consume 3 qrs. 6bush. at the same rate? Answer: 10 days. (3) If 10 men reap 20 acres of corn in 4 days; how many men can reap 70 acres in 10 days, at the same rate of labour? Answer: 14 men. (4) If 48 men can do a of 9 hours each: in how each will 64 men be able to times as great? piece of work in 16 days many days of 12 hours do a piece of work three Answer: 27 days. (5) If the carriage of 13 cwt. 2qrs. 19lbs. for 35 miles come to £4. 17s. 6d.; what must be paid for the conveyance of 41 cwt. 1lb. for 49 miles? Answer: £20. 9s. 6d. (6) If £20. in trade gain £16. in 15 months, what sum will gain £24. in 3 months, at the same rate? Answer: £150. (7) If 12 men can perform a piece of work in 20 days; required the number of men who could perform another piece of work four times as great in a fifth part of the time. Answer: 240 men. (8) If with a capital of £1000., a tradesman gain £100. in 7 months, in what time will he gain £60. 10s., with a capital of £385.? Answer: 11 months. CHAPTER IV. THE DOCTRINE OF FRACTIONS, USUALLY TERMED VULGAR FRACTIONS. 69. DEF. ALL whole numbers, or, as they are generally called, Integers, being supposed to be formed by the repetition of an unit, may therefore be regarded as the result of the multiplication of that element; but if an unit be considered capable of division into any number of equal portions, the quantities thence arising must be viewed in the light of broken magnitudes; and these are therefore termed Fractions, or more generally, Vulgar Fractions, in order to distinguish them from fractions of a different form, whose nature will be discussed in the next chapter. NOTATION, &c. OF FRACTIONS. 70. DEF. 1. If we suppose the unit to be divided into 2, 3, 4, 5, &c., equal portions, one of the portions in each case is represented by,,,, &c., which may be regarded as the primitive Fractions of their respective denominations, and are called the Reciprocals of the natural numbers 2, 3, 4, 5, &c.: also, the fractions, 1, 1, 1, &c., are read one-half, one-third, one-fourth, one-fifth, &c. 71 DEF. 2. If two or more of the equal portions into which an unit is supposed capable of being divided, be taken together, the aggregates thence arising are expressed by repeating the unit as often as such portions are repeated, the number below the line remaining the same. Thus, if the primitive fraction be taken twice, there will arise a new fraction expressed by : if be repeated thrice, there results a new fraction expressed by : again, ifbe taken four times, the new fraction corresponding will be ; and similarly of all the other primitive fractions: also, the fractions,,, &c., are read two-thirds, threefourths, four-fifths, &c.: and all quantities of this form are called Simple Fractions. 72. DEF. 3. Hence, in every simple fraction, the number below the line denotes the number of equal portions into which the unit is supposed to be divided, ‘and is therefore called the Denominator; and the number above the line, expressing the number of such equal portions intended to be taken, is therefore termed the Numerator. Thus, of the fraction, whose Terms are 5 and 7, the denominator 7 below the line implies that the unit is supposed to be divided into seven equal portions; and the numerator 5 above it shews that five of such equal portions are here the object of our consideration: and hence it is also manifest, that the integer 5 is 7 times as great as the fraction; and 5 may therefore be expressed in a fractional form by . 73. From the last article it follows, that if the numerator be less than the denominator, the value of the fraction is less than unity; if the numerator be equal to the denominator, the value of the fraction is unity, and if the numerator be greater than the denominator, the value of the fraction is greater than unity. 74. DEF. 4. If the numerator be less than the denominator, the fraction is termed a Proper Fraction; but if the numerator be greater than the denominator, it is called an Improper Fraction: also, if these two terms be equal to one another, we have merely the representation of the unit in a fractional form. 2 11 Thus, is a proper fraction, an improper fraction, and is merely a representation of the unit in a fractional form, being of the same value as 8, &c. 9 75. From the preceding view of fractions, we are enabled to find those which arise from their multiplication and division by an integer. 4 If the fraction be multiplied by the integer 3, the product is evidently 4 x 3 12 12 ; because in 13' three times as many parts of the unit are implied, as there are in be 4 If the fraction be divided by 3, the quotient will |