III. 1. Two men, A and B, hired a pasture together for $55, and A was to pay $13 more than B. What did each pay? Suppose B paid x dollars; A was to pay 13 dollars more; therefore he paid ≈ +13. These put together must make the whole 55 dollars. 2x+13=55 It appears that 2 x is not so much as 55 by 13, therefore taking 13 from 55, Dividing by 2, B's share is $21, and A's, being 13 more, is $34, x+13=21+ 13 = 34 = A's share. Proof. 34+21= 55 the whole sum. 2. A man bought a horse and chaise for $300; the horse cost $28 more than the chaise. What was the price of each? 3. A man bequeathed his estate of $12000 to his son and daughter; the son was to have $2350 more than the daughter. What was the share of each ? 4. A father who has three sons, leaves them 16000 crowns. The will specifies that the eldest shall have 2000 crowns more than the second, and that the second shall have 1000 more than the youngest. What is the share of each? Let x denote the number of crowns in the share of the youngest, then x + 1000 will denote the share of the second, and +1000+2000 will denote the share of the eldest. These added together must make the whole sum. x+x+1000+x+1000 + 2000 = 16000 Putting together the x's and the numbers, 3x+4000 = 16000 It appears that 3 x is not so much as 16000 by 4000, therefore subtracting 4000 from 16000, 3 x 3 x Dividing by 3, 16000-4000 x= 4000 share of the youngest. The share of the youngest is 4000 crowns; add to this 1000, it makes 5000, the share of the second, x+1000 5000 share of the second. Add 2000 more, it makes 7000, the share of the eldest, x+1000+2000 = 7000 = share of the eldest. Proof. The several shares added make 16000 crowns which is the whole estate. 5. A draper bought three pieces of cloth, which together measured 159 yards; the second piece was 15 yards longer than the first, and the third was 24 yards longer than the second. What was the length of each? 6. A gentleman bequeathed an estate of $65000 to his wife, two sons, and three daughters. The wife was to have $2000 less than the elder son, and $3000 more than the younger son; and the portion of each of the daughters was $3500 less than that of the younger son. Required the share of each. The 1st example may be performed differently. Let x denote the number of dollars paid by A ; B paid $13 less, thereforex-13 will represent the number of dollars paid by B. These added together must make the whole. Putting the x's together, Ꮖ +x- 1355 2x 1355 It appears that 2x is more than 55 by 13, therefore add 13 to 55 to make 2 x, 55+ 13 2 x Dividing by 2, This gives A's share $34, from which subtract $13, and it gives B's share $21, as before, X- -13=21= B's share. In the same manner perform the 2d and 3d. The 4th may be solved in a similar manner. Let the elder son's share be represented by x. The second son's share, being $2000 less, will be x 2000. The younger son's share, being $1000 less still, will be x-2000-1000. These added together must make the whole sum. x+x-2000 + x -2000-1000 = 16000 Putting the x's together and the numbers together, 3x-5000 = 16000. It appears that 3 x is more than 16000 by 5000, therefore add 5000 to 16000, The elder son's share is $7000, as before. The others may be easily found from this. Again, let x denote the second son's share. The elder son's, -being $2000 more, will be x + 2000. The younger son's, being $1000 less, will be x-1000. These added together must make the whole. x + 2000+ x + x· 1000 16000 Putting the x's together and the numbers together, 3x+1000 3x 16000 15000 x= 5000 The second son's share is $5000, as before. From this the rest are easily found. Perform the 5th and 6th in a similar way. 7. At a certain election 943 men voted, and the candidate chosen had a majority of 65. How many voted for each? 8. A person employed 4 workmen; to the first of whom he gave 2 shillings more than to the second; to the second 3 shillings more than to the third; and to the third 4 more than to the fourth. Their wages amounted to 32 shillings. What did each receive? 9. A cask, which held 146 gallons, was filled with a mixture of brandy, wine, and water. In it there were 15 gallons of wine more than there were of brandy, and as much water as both wine and brandy. What quantity was there of each ? Observe, that after the question is put into equation, the purpose is to make x stand alone in one member of the equation, equal to a known quantity in the other member, then the value of x is found. In the preceding examples in this Art. x has been found only in the first member, but connected with known quantities by the signs + and In the solution of these equations the first thing was to unite all the x's into one term, and all the known quantities into another. Then, if the number which stood on the same side with x, had the sign + before it, that number was subtracted from the other member of the equation; but if it had the sign-before it, it was added to the other member. Then the second member was divided by the coefficient of x, and the answer was obtained. 10. A and B began to trade with equal stocks. In the first year A gained a sum equal to twice his stock and £27 over; Now the B gained a sum equal to his stock and £153 over. amount of both their gains was equal to 5 times the stock of either. What was the stock ? Let x denote the stock. Then A's gain was 2x + 27, and B's was x+153. These added together must make 5 times the stock, that is, 5 x. 5x=2x+27 + x + 153 Uniting the x's in 2d member, and the numbers, 5x=3x+180 Subtracting 3 x from both sides, 2 x 180 x= 90 11. A young man being asked his age, answered that if the age of his father, which was 44 years, were added to twice his own, the sum would be four times his own age. What was his age? 12. A man méeting some beggars, gave each of them 4 pence, and had 16 pence left; if he had given them 6 pence apiece, he would have wanted 12 pence more for that purpose. How many beggars were there, and how much mohad he? ney 13. A man has six sons, each of whom is 4 years older than his next younger brother; and the eldest is three times as old as the youngest. Required their ages. 14. Three persons, A, B, and C, make a joint contribution, which in the whole amounts to £76, of which A contributes a certain sum, B contributes as much as A and £10 more, and C as much as A and B both. Required their several contributions. 15. A boy, being sent to market to buy a certain quantity of meat, found that if he bought beef, which was 4 pence per pound, he would lay out all the money he was entrusted with; but if he bought mutton, which was 3 pence per pound, he would have 2 shillings left. How much meat was he sent for? 16. A man lying at the point of death left all his estate to his three sons, to be divided as follows: to A he gave one half of the whole wanting $500; to B one third; and to C the rest, which was $100 less than the share of B. What was the whole estate, and what was each son's share? Let a represent the whole estate. A's share will be B's share Ꮖ 500 100 3 These together will be equal to the whole estate, which was represented by x. The whole estate is $3600; the shares are $1300, $1200, and $1100, respectively. 17. A father intends by his will, that his three sons shall share his property in the following manner; the eldest is to receive 1000 crowns less than half the whole fortune; the second is to receive 800 crowns less than of the whole; and the third is to receive 600 crowns less than of the whole. Required the amount of the whole fortune, and the share of each. 18. A father leaves four sons, who share his property in the following manner; the first takes 3000 livres less than one half the fortune; the second, 1000 livres less than one third of the whole; the third, exactly one fourth; and the fourth takes 600 livres more than one fifth of the whole. What was the whole fortune, and what did each receive? |