29 yards, which shall take up one fourth of the garWhat must be its width ? den. Ans. 2.554354 yards. 13. Divide the number 56 into two such parts, so as their product will be 559. Ans. 13 and 43. 14. The temple of the three graces at Athens stood upon a rectangular area, the length of which exceeded its breadth by three paces, and the distance from one angle to that which was diametrically opposite to it, exceeded the length also by so many paces. Required its length, breadth, and diametrical distance. Ans. Breadth 9, length 12, and the diagonal 15. 15. A mercer sold a certain quantity of lace for £4. 19s., and found if he had sold 4 yards less for the same money, he would have received a shilling per yard more than he did. Required the number of yards sold. Ans. 22. 16. A farmer bought a flock of sheep, consisting of ewes and lambs; the number of lambs exceeded the number of ewes by 10: for each lamb he gave as many shillings as there were ewes, and for each ewe, as many shillings as there were lambs. He afterwards bought a number of ewes at the same rate, which number is to the number of ewes in the first flock as 7 to 4; he paid £7. 10s. less for the second flock than for the first. Required the number of ewes and lambs in the first flock, also the number of ewes in the second flock. Ans. 20 ewes and 30 lambs in the first flock. and 35 ewes in the second flock. 17. Required that number, which being divided by two thirds of the sum of the digits, the quotient will be 9; but if the square of the right-hand digit be added to the number, the sum will be equal to the digits inverted plus 25. Ans. 54. 18. Required two numbers whose product is 300, and if 10 be added to the less number, and 8 subtracted from the greater, the product of the sum and remainder shall also be equal to 300. Ans. 20 and 15. 19. A grocer sold a quantity of sugar for £30, and by so doing gained four-fifths of the prime cost per cent. profit. Required the prime cost. Ans. £25. 20. A began trade on the first of January, and on the first of May following he took in B as a partner, who brought into stock £200; at the expiration of 12 months from this transaction they dissolved partnership, when it was found that there had been gained, since A commenced, £300. A received for his stock and gain £500. Required the sum with which A commenced, and each person's share of the gain. Ans. A began with £300, and his gain was £200; and B's gain was £100. 21. Two boys, A and B, comparing their marbles, found that the number which they both had, added to its square root, the sum would be 12, and the sum of their squares was 41. How many had each? Ans. A had 4 and B 5. 22. The plate of a looking-glass is 18 inches by 12, is to be framed with a frame of equal width, whose area is to be equal to half that of the glass. Required the width of the frame. Ans. 1.6241438. 23. Find two numbers, the square of whose sum shall exceed twelve times their sum by 325, and that their product may exceed five times their sum by 31. Ans. 12 and 13. 24. A and B set out from two towns, which were 120 miles distant, to meet each other; A travelled five miles a day, and the number of days, at the end of which they met, was greater by three than the number of miles which B went per day. How many miles did each go? Ans. A 50 and B 70. 25. A mercer sold 50 yards of lace and 80 yards of silk for £40, and observed that his customer received 5 yards more of lace for £8, than he did of silk for £3. 15s. Required the price per yard of each. Ans. Lace 8s. per yard, and silk 5s. 26. Two merchants, A and B, entered into partnership, by which they gained £120; A's money was three months in trade, and his share of the gain was £60 less than his stock: B's money, which was £50 more than A's, was in the trade five months. Requir ed each person's stock. Ans. A's £100, and B's £150. 27. It is required to find two numbers such, that if their sum be multiplied by the greater, the product will be 126, and whose difference, multiplied by the less, the product will be 20. Ans. 9 and 5. 28. Bought a number of yards of silk for £7, 4s., and a quantity of Irish cloth, exceeding the silk by 16 yards, for an equal sum, but paid 1s. 6d. per yard less for the cloth than for the silk. Required the number of yards of each. Ans. 32 yards of silk, and 48 of cloth. 29. A person bought four remnants of silk, the number of yards in the first was in the same proportion to the yards in the second, that the third was to the fourth, and the number of yards in the first was to the number of yards in the fourth as 1 to 5, the number of yards in the second was to the number of yards in the third as 2 to 4, and the number of yards in the second and fourth was just 20. Required the length of each remnant. Ans. 3, 5, 9, and 15 yards respectively. 30. Two merchants, A and B, received each £5940 per annum, in their respective trades; A by industry cleared 2 per cent. more than B, by which means he gained £100 per annum more than B did. Required each person's gain per cent. and the sum each realized in 10 years. Ans. A gained 10 per cent. and realized £5400, 31. Two bills of exchange, one £120, payable in 6 months, and the other £150, payable in 9 months, were discounted by a Banker for £8. 10s. Required the rate per cent. he charged. Ans. £5. 1s. 104d. 32. A and B travelled on the same road, at the same rate, from Leeds to London: at the fiftieth mile stone from London, A overtook a flock of geese, which was proceeding at the rate of three miles in two hours, and two hours afterwards met a stage-waggon, which was moving at the rate of nine miles in four hours: B overtook the same geese at the forty-fifth mile stone, and met the same waggon exactly forty minutes before he came to the thirty-first mile stone: where was B when A reached London? Ans. 25 miles from London. ARITHMETICAL AND GEOMETRICAL PROGRESSIONS. For the Rules and Application of Arithmetical and Geometrical Progression the reader is referred to Bonnycastle's Arithmetic, in which he will find these subjects well digested. 1. Three days after a waggon, which travelled 9 miles a-day, was dispatched, a person started to overtake it; in order to do this he was to travel two miles the first day, three the second, and so on. In how many days will he overtake the waggon? Let x the number of days required; then, by Bonnycastle's Arithmetic, problem 3rd, (x-1)+2 the number of miles the man tra velled the last day; and, by problem 4th, (x2+3x)÷2 = the whole distance he travelled. Also, x+3= the number of days the waggon was on the road before the man overtook it ; then, 9x+27 the distance the waggon had travelled; 2. Find three numbers in arithmetical progression such, that the square of the first, plus the product of the other two, shall be 16; and the square of the mean, plus the product of the extremes, shall be 17. Let the second term, and y the common difference; |