which was $100 less than the share of B. What was the whole estate, and what was each son's share? Let x represent the whole estate. A's share will be 500 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? 19. In a mixture of copper, tin, and lead; 16 lb. less than one half of the whole was copper; 12 lb. less than one third of the whole was tin, and 4 lb. more than one fourth of the whole was lead. What quantity of each was there in the mixture? ; 20. A general having lost a battle, found that he had only 3600 men more than one half of his army left, fit for action 600 more than one eighth of them being wounded, and the rest, which amounted to one fifth of the whole army, either slain or taken prisoners. Of how many men did his army consist before the battle? 21. Seven eighths of a certain number exceeds four fifths of it by 6. What is that number? 22. A and B talking of their ages, A says to B, one third of my age exceeds its fourth by 5 years. What was his age? 23. A sum of money is to be divided between two persons, A and B, so that as often as A takes £9, B takes £4. Now it happens that A receives £15 more than B. What is the share of each? 24. In a mixture of wine and cider, 25 gallons more than half the whole was wine, and 5 gallons less than one third of the whole was cider. How many gallons were there of each? IV. 1. A man having some calves and some sheep, and being asked how many he had of each sort, answered, that he had 20 more sheep than calves, and that three times the number of sheep was equal to seven times the number of calves. How many were there of each ? Let x denote the number of calves. Then +20 will denote the number of sheep. 7 times the number of calves is 7 x; 3 times the number of sheep is 3x+60; for it is evident that to take 3 times x + 20, it is necessary to multiply both terms by 3. By the conditions these must be equal, = x= 15 number of calves. x+20=35= number of sheep. Ans. 15 calves, and 35 sheep. IV. x+18 Equations. 25 2. Two men talking of their ages, the first says, your age is 18 years more than mine, and twice your age is equal to three times mine. Required the age of each. 3. Three men, A, B, and C, make a joint contribution, which in the whole amounts to £276. A contributes a certain sum, B twice as much as A and £12 more, and C three times as much as B and £12 more. Required their several con tributions. 4. A man bought 7 oxen and 11 cows for $591. oxen he gave $15 apiece more than for the cows. did he give apiece for each? Let x denote the price of a cow. Then the price of an ox will be x + 15. For the How much 11 cows at x dollars apiece will come to 11 x dollars. If one ox cost x + 15 dollars, 7 oxen will cost 7 times x + 15, which is 7x + 105. The price of the oxen and of the cows added together will make $591, the whole price. 5. A man bought 20 pears and 7 oranges for 95 cents. For the oranges he gave 2 cents apiece more than for the pears. What did he give apiece for each ? 6. A man bought 20 oranges and 25 lemons for $1.95. For the oranges he gave 3 cents apiece more than for the lemons. What did he give apiece for each? 7. Two persons engage at play, A has 76 guineas, and B 52, before they begin. After a certain number of games lost and won between them, A rises with three times as many guineas as B. How many guineas did A win of B? Let x denote the number of guineas that A won of B. 52 -- A has now three times as many as B, that is, 3 times 52 — x, which is 156 3 x. It is evident that both 52 and x must be multiplied by 3, because 52 is a number too large by x, therefore 3 times 52 will be too large by 3 x. Ans. 20 guineas Proof. If A won 20 guineas of B, A will have 96 and B 32 3 times 32 are 96. This equation is rather more difficult to solve than any of the preceding. In the first place I subtract 76 from both members, so as to remove it from the first member. Then to get 3x out of the second member, which is there subtracted, I add 3x to both members; then the x's are all in the first member, and the known numbers in the other. N., B. Any term which has the sign+, either expressed or understood, may be removed from one member to the other by giving it the sign —; for this is the same as subtracting it from both sides. Thus x +3 10; x is not so much as 10 by 3, we therefore say x = 10 3. Again, 5 x = 18+ 3 x. 5 x is more than 18 by 3x, therefore we may say 5 x 18. -- Now -3x= Any term which has the sign before it may be removed from one member to the other by giving it the sign +. This is equivalent to adding the number to both sides. Thus 5x 3= 17. In this it appears that 5x is more than 17 by 3; therefore we say 5 x = 173. Again, 5 x = 32 - 3 x. Here it appears that 5 x is not so much as 32 by 3x; therefore we say 5x+3x=32. This is called transposition. Hence it appears that any term may be transposed from one member to the other, care being taken to change the sign. In the last example, 76 was transposed from the first member to the second, and the sign changed from + to; and 3 x as transposed from the second member to the first, and the sign changed from-to+. This has been done in many of the preceding examples. When a number, consisting of two or more terms, is to be multiplied, all the terms must be multiplied, and their signs preserved. In the last example, 52-x, multiplied by 3, gave a product 156 3 r. 8. A person bought two casks of wine, one of which held exactly three times as much as the other. From each he drew 4 gallons, and then there were four times as many gallons remaining in the larger as in the smaller. How many gallons were there in each at first? Let x denote the number of gallons in the less at first. And the greater The greater is now 4 times as large as the less; 4 times x - 4 is 4 x 16. x= 12 less. 3 x 36= greater. Ans. Less 12 gallons, greater 36 gallons. Proof. 36 is three times 12 according to the conditions. Take 4 from each, then one contains 32 and the other 8. 4 times 8. 32 is 9. A man when he was married was three times as old as his wife; after they had lived together 15 years, he was but twice as old. How old was each when they were married? 10. A farmer has two flocks of sheep, each containing the same number. From one of these he sells 39, and from the other 93; and finds just twice as many remaining in the one as in the other. How many did each flock originally contain? 11. A courier, who travels 60 miles per day, had been despatched 5 days, when a second was sent to overtake him; in order to which, he must go 75 miles per day; in what time will he overtake the former? 12. A and B engaged in trade, A with £240, and B with £96. A lost twice as much as B; and upon settling their accounts it appeared that A had three times as much remaining as B. How much did each lose? Let x denote B's loss, then 96 x will denote what he had remaining. 2 x will denote A's loss, and 240 — 2 x what he had remaining, &c. 13. Two persons began to play with equal sums of money; the first lost 14 shillings, and the other won 14 shillings, and then the second had twice as many shillings as the first. What sum had each at first ? |