(21.) Prove the rule for the transposition of terms in equations. (§ 40.) (22.) Prove the rules for the extraction of the square and cube roots. (§ 46. and 47.) (23.) Find the interest of P pounds for d days, at r per (24.) One merchant owes another A pounds due in a days, B pounds due in b days, C pounds due in c days, &c.; find the equated time of payment. Ans. Aa + Bb + Cc +, &c. (25.) Divide the number n into parts proportional to the numbers a, b, c, &c. an Ans. 1st part = 2d= a+b+c+, &c.' bn &c. a+b+c+, &c.' (26.) A number of merchants, M, N, P, &c., enter into partnership; M puts in A pounds for a days, N puts in B pounds for b days, P puts in C pounds for c days, &c.; they gain g pounds: what is the share of each? (27.) Find A, the amount of P pounds for n years, at r per cent. compound interest; denoting by a the amount of £1 for 1 year, at the same rate. Ans. A Pan. (28.) Find P, the present worth of A pounds due n years hence, at r per cent. compound interest. Ans. P Aa1—n. (29.) In an equation of the form ax = b; suppose that when any number n is substituted for x, the result is an = = c; prove that cn::b:x, and thence derive the common rule for single position. (30.) In an equation of the form ax + b = cx + d, or (a · c)x + b — d = 0; suppose that when any two numbers m and n are substituted successively for x, the results are respectively (31.) Find A, the amount of an annuity of P pounds for n years, at r per cent. compound interest; denoting by a, the amount of £1 for 1 year, at the same rate. Ans. A = 2 ... α, 1 = pan 1 (32.) If A denote the amount of an annuity of P pounds for n years, at r per cent. compound interest, and a the amount of £1 for one year, at the same rate; prove that the present value= A an = 1 -a "P. (33.) If the annuity is to continue for ever, prove that its present value = Р a- 1 or P: 100 present value. (34.) Apply the binomial theorem to obtain the 5th root of 10000 to 20 places of decimals. Ans. 999, 996, 000, 047, 999, 296, 011. (35.) The perimeter of a right angled triangle is 24 feet, and the base is 8 feet; find the other two sides. Ans. 6, and 10. (36.) A besieged place, garrisoned by 10,000 men, was victualled for 27 days; but after 9 days 2500 men cut their way out how long would the provisions last the survivors, supposing the daily rations to remain undiminished? Ans. 24 days. (37.) Prove that every fraction may be expressed either as a decimal with a finite number of places, or else as a recurring decimal. (39.) Given 3x+5y= 171, 7x+2x= 209, and 7y+22 =90; to find x, y, and z. Ans. x 32, y = 15, and ≈ =- 7.5. (40.) Two gunners keep up a fire on a battery from two different guns; the first, who had spent 24 shots before the second opened fire, discharges 8 shots to 7 of those of his comrade, but uses in 4 rounds only the same quantity of powder as the other in 3; how many shots must the second fire before he has consumed as much powder as the first? Ans. 126. (41.) It is required to cut a piece equal to a solid foot from a plank 24 inches thick and 8 inches wide. Ans. Length 86.4 inches. (42.) I make a journey of 3040 miles on horseback, on foot, and by water; 3 times as much is performed on land as by water, and 24 times as much on horseback as on foot. How far did I travel on foot? Ans. 6751⁄2 miles. (43.) Given 3x + 2y = 118; x+5y= 191; to find x and y. Ans. x 16, y = 35. (44.) Given 3/(112 — 8x) = 19+√(3x+7); to find x. Ans. = 6, and 11.8368. (45.) Given xy=8; (3—y)x = 12; (2—x) (4—≈) = 4; to find x, y, and z. Ans. 1.6, y=5, z=-6. (46.) The fore-wheel of a carriage makes six revolutions more than the hind-wheel in going 120 yards; but if the circumference of each wheel be increased 1 yard, the forewheel will make only 4 revolutions more than the hindwheel in the same distance: required the circumference of each wheel. Ans. 4, and 5 yards. (47.) If a cubic foot of metal weighs 4 cwts. 1 qr., and is worth ten guineas per ton; what will be the cost of a mile of piping made out of it with a 9-inch bore and of an inch thick ? Ans. £903, 11s. 10 d. (48.) How many cubic feet of water are contained in a ditch shaped like the frustum of a wedge, 120 yards long, 6 feet deep, 10 yards broad at the top and 4 at the bottom. Ans. 45360 cubic feet. (49.) Three men set out at the same time from three stations in the same straight line, each distant 25 miles from the next. They travel in straight lines at the rates of 21, 3, and 3 miles respectively per hour, and all arrive at the same time at the same point. How far has each travelled? Ans. 125, 150, and 175 miles. (50.) Two trains set out at the same time, one from London to Plymouth, and the other from Plymouth to London. The latter travelled 2 miles an hour faster than the former from Plymouth to Exeter, and 4 miles an hour faster from Exeter to Bristol. They arrived at the same instant at Bristol. At what rate did each travel; the distance from London to Bristol being 118 miles, from Bristol to Exeter 75 miles, and from Exeter to Plymouth 53 miles? Ans. Train from London 37.1 miles per hour nearly. THE END. |