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chains from the bridge A, which entered the river at the same point with the sewer, and was equally inclined to the river and sewer. Now it was found that a drain down the middle of each street, at the rate of £9 per chain, would have cost only £54 more than the expense of the sewer. Required the lengths of the streets, and the sewer; and the distance of its mouth from the bridge B.
3. On the institution of Saving Banks, an industrious labourer, with his wife and children, saved each of them a certain number of pence in a decreasing arithmetic progression. The sum saved monthly, was less by 3s. 3d. than would bave purchased one-sixth of as many bushels of wheat, as the seventh ehild saved pence;
the price of wheat being such, that the sum saved by the eldest and the fifth child, augmented by 10s. would buy two bushels. But wheat rising 2s. per bushel, and work being scarce, the family find the sum saved would not buy as much wheat as their former savings, by two bushels; when it appears that at this rate the sum annually saved would be less by five guineas than by the former.-Now the two youngest dying, it is found that if the remaining members of the family saved each one shilling less than the oldest child had done before the rise of wheat, their monthly account with the bank would not be affected by the deaths of the two youngest; but if they saved only 2d. less than the oldest had done, their monthly account would be 2s. 1d. more than it was at the first institution. Of how many did the family consist? What were the sums saved by each ? and the price of wheat?
5. A farmer laid up a stock of corn, expecting to sell it in six months at three shillings per bushel more than he gave for it. But the price of corn falling one shilling per bushel, he found that by selling it, he should lose the price of five bushels. He there
fore kept it till the end of the year, and selling it at two shillings per bushel under prime cost, found his loss to be ten shillings less ihan his expected gain. Required the quantity of corn laid up and price per bushel, allowing five per cent. simple interest.
6. A ship and crew of 175 men set sail with a store of water sufficient to last to the end of the voyage. But in thirty days the scurvy made its appearance, and carried off three men every day, and at the same time a storm arosé, which protracted the voyage three weeks. They were however just enabled to arrive in port, without any diminution in each man's daily allowance of water. Required the time of the passage, and the number of men alive when the vessel reached harbour.
7. The hold of a vessel partly full of water (which is uniformly increased by a leak) is furnished with two pumps worked by A and B, of whom A takes three strokes to two of B's, but four of B's throw out as much water as five of A's. Now B works for the time in which A alone would have emptied the hold, A then pumps out the remainder, and the hold is cleared in thirteen bours twenty minutes. Had they worked together, the hold would have been emptied in three hours forty-five minutes, and A would have pumped out 100 gallons more than he did. Required the quantity of water in the hold at first, and the horary influx at the leak.
Y 5. On January 1, 1799, a certain beggar received from A as many groats as A was years old, who repeated a similar donation every January for the seven following years, during the last of of which A died, his alms to the poor man having in all amounted to £7 185. 8d. Required in what year he was born, and his age his death.
6. A entered into a canal speculation with fourteen others, and the profits of this concern amounted in all to £595, more than five times the price of an original share. Seven of his former partners in this affair joined in a scheme for navigating the said canals with steam-boats, each venturing a sum of money less than his former gains by £173. But the steam-boats unexpectedly blowing up, A found he had lost £419 by them, for the company not only never recovered the money advanced, but had lost all they had gained by digging the canals and £368 besides. What were the prices of shares in the two concerns originally?
7. A, B, and C were three architects. A and B built four warehouses with flat roofs, each a large one, and each a small one, the linear width of the two large ones being the same, and also that of the two small ones. A built his as long and as bigh as they were wide, but B made the length and height of his large one equal to the width of his small one, and the length and height of his small one equal to the width of his large one, in such a manner that the difference between the solid content of those built by A and those built by B was 73728 cubic feet. C also built a warehouse upon a square plat of ground which was equal to the difference between the ground-plats occupied by those which A built, and found that it would have just stood on 2688 square feet, if he had added eight times as many square feet to the groundplat as there were linear feet in its width. How many feet wide were the several buildings erected by A, B, and C ?
QUESTIONS IN ALGEBRA.
common multiple of (). 6), (),
1. Investigate the rule for the expansion of a binomial; and shew that when the power is expressed by the integer (m), the number of terms is (m + 1).
2. Find the least common multiple of each being a fraction in its lowest terms: and also the greatest common measure of the two quantities (a) and (mb), when (m) is an integer. 3. Extract the cube root of a3
216 4. Solve the following equations:
2.x + 5 2 2.
= 2. X-2
I + y = 7. 2. x2 + x2 = 12,
22 + x2 = 24.
y* = 14.
x2 + y2 X xy = 78.
x4 + x2y + y* = 133.
y? t yz = 21.
22 + z = 24. 6. Solve the following equations :
2 1. 3x
3. x .C + 4 + 2x .X + 4 = 2 - 2 + 4. 7. A has £400 due to him from B, and £520 at the end of twelve years; at what time ought the two sums to be paid together, so that neither person may sustain loss; simple interest being allowed at the rate of £5 per cent. ?
8. Find the number of years for which an annuity of £10 may he purchased, by the present payment of £245 simple interest being calculated at £4. per cent.
9. A travels from C, one mile the first day, five the second, nine the third, &c. ; B sets out from the same place 4ļ days after A, and travels twice as many miles as A does in the last day, except four. In how many days will B overtake A ?
10. A's income of £400 is paid in equal parts at the end of 3, 6, 9, 12 months; and his expenses, which are as the numbers 1, 2, 3, 4, are paid at the same time; and the remaining sums being lent at an interest of £5 per cent. amount at the end of the year to £113 15s. Find the amount of his annual expenses. 11. AC
B D A, C....B, are (n+1) stones placed a yard from each other, and D is another assumed station :--Two persons, M and N, set out from A; M to carry the stones separately to A, and N to carry them to D. Find the distance B D, so that N nay travel twice as many yards as M.
12. Find the number of permutations of the letters in the word COLLEGE. 13. If the digits of a circulating decimal be a, b, c, its equiva
10'a + 10b + c lent fraction is,
999 14. 210 trees are to be planted in rows on a triangular piece of ground containing 9600 square yards; a tree being planted at each angle, and the extreme trees of every row parallel to a side being situated on the two remaining sides, the number of trees in succeeding rows being 2, 3, 4, &c.; find the distance between contiguous trees in the directions of the three sides, supposing them to be as the numbers 3, 4, 5.
1. Express in general any decimal of n places; a, b, c, &c. being the digits.
2. Prove the rule for finding the greatest common measure.
y + 18 = 5y.