Hence x+72=3241. A.'s share, Sum of all =1000l. the proof. 4. It is required to divide 1000l. between two persons, so that their shares of it shall be in the proportion of 7 to 9. Let x the first person's share, Then will 1000-x second person's share, And x 1000-x:: 7 : 9, by the question, Or 9x+7x=7000, or x= 7000 16 =4377. 10s. = 1st share, and 1000-x=1000-4371. 10s. = 5621. 10s. = 2d share. 5. i ne paving or a square court with stones; at 2s. a yard, will cost as much as the enclosing it with pallisades, at 58. a yard; required the side of the square? Let x= length of the side of the square sought, Or 2x=20, and x=10, the length of the side required. 6. Out of a cask of wine, which had leaked away a third part, 21 gallons were afterwards drawn, and the cask being then guaged, appeared to be half full; how much did it hold? Let x = the number of gallons the cask is supposed to have held, 1 Then it would have leaked awayx gallons, 3 Whence there had been taken out of it, altogether, 3 2 That is 63+x=x, or 126+2x=3x, Consequently 3x-2x=126, or x=126, the number of gallons required. 7. What fraction is that, to the numerator of which if 1 be added, its value will be denominator, its value will be 1. 3' 14 but if 1 be added to the Hence 3x+3=y, and 4x=y+1, or x= Therefore 3(+1)+3=y, or 3y+3+12=4y, y+1 4' by the question. y+1 4 That is y=15, and x= 3+1 15+1 16 = 4 Whence the fraction that was to be found, is 15 8. A market woman bought in a certain number of eggs at 2 a penny, and as many others at 3 a penny, and having sold them out again, altogether, at the rate of 5 for 2d., found she had lost 4d.; how many eggs had she? Let x= the number of eggs of each ort, 1 Then will x the price of the first sort, 1 x= the price of the second sort, And x= But 5 2 2x (the whole number of eggs) : 4x 5' the price of both sorts, when mixed 5 together at the rate of 5 for 2d., That is 15x+10x-24x=120, or x=120, the number of eggs of each sort, as required. 9. If can perform a piece of work in 10 days, and B in 13; in what time will they finish it, if they are both set about it together? Let the time sought be denoted by x, That is 13x+10x=130, or 23x=130, 130 15 Whence x= =5 days, the time required. 23 23 10. If one agent A, alone, can produce an effect e, in the time a, and another agent B, alone in the time b; in what time will both of them together produce the same effect? Let the time sought be denoted by x, Then a : e :: x:: ex α. part of the effect produced by A, ex And be: x part of the effect produced by B, b x e (the whole effect) by the question, Or-+ =1 by dividing each side by e, α b 11. How much rye at 4s. 6d. a bushel, must be mixed with 50 bushels of wheat, at 6s. a bushel, so that the mixture may be worth 5s. a bushel ? Let x= the number of bushels required, Then 9x is the price of the rye in sixpences, And 600 the price of the wheat in ditto, Also (50+x)X 10 the price of the mixture in ditto, Whence 9x+600-500+10r, by the question, Or, by transposition, 10x-9x=600-500, Consequently x=100 the number of bushels required. 12. A labourer engaged to serve for 40 days, on condition that for every day he worked he should receive 20d., but for every day he was absent he should forfeit 3d., now at the end of the time, he had to receive 17. 11s. 8d.; how many days did he work, and how many was he idle? Let the number of days that he worked be denoted by x, Then will 40x be the number of days he was idle, Also 20x the sum earned, and (40—x)×8. Or 320-8x the sum forfeited, Whence 20x-(320—8x) = 380d.(= 11. 11s. 8d.), by the question, That is 20x320+8x=380, 28x=380+320=700, Or Consequently x- 25, the number of days he 28 worked, and 40-x-40-25-15, the number of days he was idle. QUESTIONS FOR PRACTICE. 1. It is required to divide a line, of 15 inches in length, into two such parts, that one may be three fourths of the other. Ans. 84 and 62 2. My purse and money together are worth 20s, and the money is worth 7 times as much as the purse, how much is there in it? Ans. 17s. 6d. 3. A shepherd, being asked how many sheep he had in his flock, said, if I had as many more, half as many more, and 7 sheep and a half, I should have just 500; how many had he? Ans. 197. 4. A post is one fourth of its length in the mud, one third in the water, and 10 feet above the water; what is is its whole length? Ans. 24 feet 1 4 1 5. After paying away of my money, and then of the 5 remainder, I had 72 guineas left; what had I at first? Ans. 120 guineas 6. It is required to divide 300 between A, B, and c, so that ▲ may have twice as much as в, and c as much as A and B together. Ans. A 100l., в 50l., c 150l. 7. A person, at the time he was married, was 3 times as old as his wife; but after they had lived together 15 years, he was only twice as old; what were their ages on their wedding day? Ans. Bride's age 15, bridegroom's 45 8. What number is that from which, if 5 be subtracted, two thirds of the remainder will be 40? Ans. 65 9. At a certain election, 1296 persons voted, and the successful candidate had a majority of 120; how many voted for each? Ans. 708 for one, and 588 for the other |